Dataset Viewer
url
string | fetch_time
int64 | content_mime_type
string | warc_filename
string | warc_record_offset
int32 | warc_record_length
int32 | text
string | token_count
int32 | char_count
int32 | metadata
string | score
float64 | int_score
int64 | crawl
string | snapshot_type
string | language
string | language_score
float64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
https://thehardcorecoder.com/2019/05/05/full-adder-code/?shared=email&msg=fail | 1,657,211,434,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656104495692.77/warc/CC-MAIN-20220707154329-20220707184329-00425.warc.gz | 605,150,360 | 30,214 | I was involved in a debate recently about whether a full adder logic circuit is a computer. The computer science answer is: “No, not as we define a computer.”
I plan to address that answer in detail on my main blog. Here I wanted to show some of the different ways a full adder can be modeled and implemented.
A full adder logic circuit takes two inputs (two binary bits, a and b) plus a third input, Ci, a carry bit from an adjacent adder (or zero if no adjacent adder). It has two outputs, the sum bit, S, and a carry bit, Co.
The truth table looks like this:
It can be expressed as two logical expressions:
(The ⊕ symbol means XOR; the ∧ symbol means AND, and the ∨ symbol means OR.)
It is commonly implemented logically like this:
Although there are other logic circuits that are functionally identical. (It can be constructed entirely from NOR gates, for example — an exercise for the reader.)
The logical expressions, and the equivalent logic gate circuits, show that the underlying circuit of a full-adder is not, in itself, computational (as computation is usually defined).
The truth table shows the logic has distinct states — eight, in this case. Given the mechanism does have these distinct states, there is a map of those states to an abstract machine that simulates the same states. Thus the operation can be simulated (or modeled) with a computation.
The map of states for a full adder can be to a truth table (as above), to a Turing Machine, a Lambda calculus, or a Finite State machine (aka automaton).
Above is a diagram of an FSM. I learned it as “machine” — some say “automaton” (so, FSA) — tomato, tomahto. The important part is the Finite State.
Incidentally, the simplicity and regularity of the state diagram is another indicator of the physicality of a full-adder. A given set of inputs immediately trickles down to the output state.
The FSM branches on each of the three input bits, a, b, and Ci. The final state features a payload that is the output bits, Co and S, respectively. The states are labeled by the bit-pattern established at that point (bits build from right to left).
§
With that in mind, let’s consider how to implement the logic with code. We can model many of the above expressions of the full-adder logic.
Note that all these examples take the input bits, a, b, and Ci, and return a tuple containing the output bits, S and Co.
```def full_adder_1 (a, b, ci):
logic_table = [
[[[0,0],[1,0]], [[1,0],[0,1]]], # a=0
[[[1,0],[0,1]], [[0,1],[1,1]]], # a=1
] # b=0 b=1 b=0 b=1
return logic_table[a][b][ci]
#
```
The example above implements a logic table version that only requires a single lookup step. This trades code and time for (potentially) large data, since all the logic is contained in the table.
(A real-world implementation might take many steps to accomplish the lookup. The single step is conceptual, as if looking at the truth table to get an output for a given input.)
```def full_adder_2 (a, b, ci):
s = (a ^ b) ^ ci
co = ((a ^ b) and ci) or (a and b)
return (s,co)
#
```
The second example implements the actual logical expression shown above. The tophat (^) is Python’s xor operator. The and and or are built-in Python operators.
The code here directly corresponds to the logical expressions shown above.
Note that the intermediate steps of assigning the respective expression values to S and Co aren’t necessary. Those result values could be directly returned in a single step.
```def full_adder_3 (a, b, ci):
s = a + b + ci
if s < 2: return (s, 0)
return (s % 2, 1)
#
```
Example three models the adder as an actual base two addition (of all three input bits), which requires detecting whether that operation carried.
The logic may not be obvious:
If s is less than two, there was no carry, so just return the sum (which will be 0 or 1), and return Co=0.
Otherwise, there was a carry, so take the mod 2 of the sum (which removes the carry), and return Co=1. (The sum will be two or three, and the mod 2 of those is zero and one, respectively.)
```def full_adder_4 (a, b, ci):
s,co = (a+b, 0)
if s == 2: s,co = (0, 1)
s += ci
if s == 2: s,co = (0, 1)
return (s, co)
#
```
The fourth example also uses a mathematical model, but acts as if the sum is a single bit with an overflow. (That is, the only allowed values are 0, 1, and overflow.)
The code may not be obvious if you’re not familiar with Python, which very nicely allows setting multiple variables with a list of values. That’s what might seem weird in the assignment statements; they’re setting the value of both S and Co.
The first step (line #2) sets S to the value of a+b (so it will be 0, 1, or 2). It also sets Co=0, assuming no carry.
Line #3 checks for a carry. If it occurred, reset S=0 and Co=1. The output at this point is a correct half-add.
For the full-add, line #4 adds the Ci bit to S, and the following line (#5) checks for the carry (S was either 0 or 1, so it could be 2 now). If there was a carry, the output is reset, as above.
The last step just returns S and Co.
```def full_adder_5 (a, b, ci):
state_table = {
'start':{0:'S#0', 1:'S#1'},
'S#0':{0:'S#00', 1:'S#01'},
'S#1':{0:'S#10', 1:'S#11'},
'S#00':{0:'S#000', 1:'S#001'},
'S#01':{0:'S#010', 1:'S#011'},
'S#10':{0:'S#100', 1:'S#101'},
'S#11':{0:'S#110', 1:'S#111'},
'S#000':(0,0), 'S#001':(1,0), # final
'S#010':(1,0), 'S#011':(0,1), # states
'S#100':(1,0), 'S#101':(0,1), # "
'S#110':(0,1), 'S#111':(1,1), # "
}
state = state_table['start'][a]
state = state_table[state][b]
state = state_table[state][ci]
return state_table[state]
#
```
The fifth example models the FSM version of the adder. Each input, a, b, and Ci, serially drives the FSA from starting state (@) to final state.
All states (in state_table), except the final states, contain a dictionary of possible next states. Here, any given state can only have two possible next states — one for each bit value (0 or 1) that causes a move to that next state.
The final (leaf) states are special in containing the output payload rather than a dictionary. The function returns that payload (the bits S and Co).
The first step (line #15) gets the ‘start’ state from state_table. As an intermediate (branch) state, the state is a dictionary to possible next states. The a input indexes the dictionary (with a 0 or a 1), so the whole expression returns the next state, depending on a.
The next line (#16) uses that next state to get that state’s dictionary from state_table. In this case the input b indexes the dictionary for the state that depends on b.
Line #17 repeats the process with the Ci bit. The state from b gets the dictionary, and Ci indexes it to get the final state.
Lastly (line #18), the final state gets the payload from state_table and returns it.
As you can see, modeling the FSM results in a large data segment. The code lines (#15 – #18) are unrolled for clarity, but more generally a design would use a loop.
[See the State Engines series (part 1, part 2, part 3), for more.]
§
These examples show some of the different ways we can implement the logic of a full adder. There can certainly be others.
One implementation I didn’t explore is modeling the physical circuit, the logic gates.
I do have such an implementation to show you, but it’s so big it needs its own post. Modeling physical reality is much more complicated! (A separate post will give me elbow room to talk a little about modeling the circuit below the gates at the transistor level.)
But fundamentally, the adder is the logical expression:
Nothing more.
Calling it a computation is equivalent to calling the creeks, streams, and rivers of a watershed a computation — the processes involved are more similar than with the computations shown above.
But that’s an argument for my other blog. This is about the code itself. | 2,060 | 7,818 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.578125 | 4 | CC-MAIN-2022-27 | latest | en | 0.931112 |
http://nrich.maths.org/public/leg.php?code=71&cl=3&cldcmpid=4881 | 1,503,250,399,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886106865.74/warc/CC-MAIN-20170820170023-20170820190023-00243.warc.gz | 301,219,618 | 10,483 | Search by Topic
Resources tagged with Mathematical reasoning & proof similar to Semicircle Stack:
Filter by: Content type:
Stage:
Challenge level:
There are 176 results
Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof
Coins on a Plate
Stage: 3 Challenge Level:
Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
A Chordingly
Stage: 3 Challenge Level:
Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
The Pillar of Chios
Stage: 3 Challenge Level:
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Appearing Square
Stage: 3 Challenge Level:
Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .
Disappearing Square
Stage: 3 Challenge Level:
Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .
Round and Round
Stage: 4 Challenge Level:
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Matter of Scale
Stage: 4 Challenge Level:
Prove Pythagoras' Theorem using enlargements and scale factors.
Fitting In
Stage: 4 Challenge Level:
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .
Rolling Coins
Stage: 4 Challenge Level:
A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .
Salinon
Stage: 4 Challenge Level:
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
Ratty
Stage: 3 Challenge Level:
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
The Genie in the Jar
Stage: 3 Challenge Level:
This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .
Stage: 3 Challenge Level:
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Cycle It
Stage: 3 Challenge Level:
Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.
Logic
Stage: 2 and 3
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Not Necessarily in That Order
Stage: 3 Challenge Level:
Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?
Con Tricks
Stage: 3
Here are some examples of 'cons', and see if you can figure out where the trick is.
Aba
Stage: 3 Challenge Level:
In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.
Go Forth and Generalise
Stage: 3
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Rotating Triangle
Stage: 3 and 4 Challenge Level:
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
Thirty Nine, Seventy Five
Stage: 3 Challenge Level:
We have exactly 100 coins. There are five different values of coins. We have decided to buy a piece of computer software for 39.75. We have the correct money, not a penny more, not a penny less! Can. . . .
More Mathematical Mysteries
Stage: 3 Challenge Level:
Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .
Unit Fractions
Stage: 3 Challenge Level:
Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.
Cross-country Race
Stage: 3 Challenge Level:
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
Stage: 2 and 3
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
Eleven
Stage: 3 Challenge Level:
Replace each letter with a digit to make this addition correct.
Rhombus in Rectangle
Stage: 4 Challenge Level:
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
Marbles
Stage: 3 Challenge Level:
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
More Marbles
Stage: 3 Challenge Level:
I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?
Winning Team
Stage: 3 Challenge Level:
Nine cross country runners compete in a team competition in which there are three matches. If you were a judge how would you decide who would win?
Tri-colour
Stage: 3 Challenge Level:
Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?
Greetings
Stage: 3 Challenge Level:
From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .
Tessellating Hexagons
Stage: 3 Challenge Level:
Which hexagons tessellate?
Picture Story
Stage: 4 Challenge Level:
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Concrete Wheel
Stage: 3 Challenge Level:
A huge wheel is rolling past your window. What do you see?
Towering Trapeziums
Stage: 4 Challenge Level:
Can you find the areas of the trapezia in this sequence?
Shuffle Shriek
Stage: 3 Challenge Level:
Can you find all the 4-ball shuffles?
Mouhefanggai
Stage: 4
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
Stage: 3 Challenge Level:
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .
Clocked
Stage: 3 Challenge Level:
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
Stage: 3 Challenge Level:
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Top-heavy Pyramids
Stage: 3 Challenge Level:
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Picturing Pythagorean Triples
Stage: 4 and 5
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
Königsberg
Stage: 3 Challenge Level:
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Pattern of Islands
Stage: 3 Challenge Level:
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Yih or Luk Tsut K'i or Three Men's Morris
Stage: 3, 4 and 5 Challenge Level:
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
Pythagorean Triples I
Stage: 3 and 4
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
Happy Numbers
Stage: 3 Challenge Level:
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
Pent
Stage: 4 and 5 Challenge Level:
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
Pythagorean Triples II
Stage: 3 and 4
This is the second article on right-angled triangles whose edge lengths are whole numbers. | 2,292 | 9,707 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.3125 | 4 | CC-MAIN-2017-34 | latest | en | 0.90402 |
https://www.quizzes.cc/metric/percentof.php?percent=2.5&of=7000000 | 1,591,498,703,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590348523476.97/warc/CC-MAIN-20200607013327-20200607043327-00417.warc.gz | 852,621,913 | 4,092 | #### What is 2.5 percent of 7,000,000?
How much is 2.5 percent of 7000000? Use the calculator below to calculate a percentage, either as a percentage of a number, such as 2.5% of 7000000 or the percentage of 2 numbers. Change the numbers to calculate different amounts. Simply type into the input boxes and the answer will update.
## 2.5% of 7,000,000 = 175000
Calculate another percentage below. Type into inputs
Find number based on percentage
percent of
Find percentage based on 2 numbers
divided by
Calculating two point five of seven million How to calculate 2.5% of 7000000? Simply divide the percent by 100 and multiply by the number. For example, 2.5 /100 x 7000000 = 175000 or 0.025 x 7000000 = 175000
#### How much is 2.5 percent of the following numbers?
2.5% of 7000000.01 = 17500000.025 2.5% of 7000000.02 = 17500000.05 2.5% of 7000000.03 = 17500000.075 2.5% of 7000000.04 = 17500000.1 2.5% of 7000000.05 = 17500000.125 2.5% of 7000000.06 = 17500000.15 2.5% of 7000000.07 = 17500000.175 2.5% of 7000000.08 = 17500000.2 2.5% of 7000000.09 = 17500000.225 2.5% of 7000000.1 = 17500000.25 2.5% of 7000000.11 = 17500000.275 2.5% of 7000000.12 = 17500000.3 2.5% of 7000000.13 = 17500000.325 2.5% of 7000000.14 = 17500000.35 2.5% of 7000000.15 = 17500000.375 2.5% of 7000000.16 = 17500000.4 2.5% of 7000000.17 = 17500000.425 2.5% of 7000000.18 = 17500000.45 2.5% of 7000000.19 = 17500000.475 2.5% of 7000000.2 = 17500000.5 2.5% of 7000000.21 = 17500000.525 2.5% of 7000000.22 = 17500000.55 2.5% of 7000000.23 = 17500000.575 2.5% of 7000000.24 = 17500000.6 2.5% of 7000000.25 = 17500000.625
2.5% of 7000000.26 = 17500000.65 2.5% of 7000000.27 = 17500000.675 2.5% of 7000000.28 = 17500000.7 2.5% of 7000000.29 = 17500000.725 2.5% of 7000000.3 = 17500000.75 2.5% of 7000000.31 = 17500000.775 2.5% of 7000000.32 = 17500000.8 2.5% of 7000000.33 = 17500000.825 2.5% of 7000000.34 = 17500000.85 2.5% of 7000000.35 = 17500000.875 2.5% of 7000000.36 = 17500000.9 2.5% of 7000000.37 = 17500000.925 2.5% of 7000000.38 = 17500000.95 2.5% of 7000000.39 = 17500000.975 2.5% of 7000000.4 = 17500001 2.5% of 7000000.41 = 17500001.025 2.5% of 7000000.42 = 17500001.05 2.5% of 7000000.43 = 17500001.075 2.5% of 7000000.44 = 17500001.1 2.5% of 7000000.45 = 17500001.125 2.5% of 7000000.46 = 17500001.15 2.5% of 7000000.47 = 17500001.175 2.5% of 7000000.48 = 17500001.2 2.5% of 7000000.49 = 17500001.225 2.5% of 7000000.5 = 17500001.25
2.5% of 7000000.51 = 17500001.275 2.5% of 7000000.52 = 17500001.3 2.5% of 7000000.53 = 17500001.325 2.5% of 7000000.54 = 17500001.35 2.5% of 7000000.55 = 17500001.375 2.5% of 7000000.56 = 17500001.4 2.5% of 7000000.57 = 17500001.425 2.5% of 7000000.58 = 17500001.45 2.5% of 7000000.59 = 17500001.475 2.5% of 7000000.6 = 17500001.5 2.5% of 7000000.61 = 17500001.525 2.5% of 7000000.62 = 17500001.55 2.5% of 7000000.63 = 17500001.575 2.5% of 7000000.64 = 17500001.6 2.5% of 7000000.65 = 17500001.625 2.5% of 7000000.66 = 17500001.65 2.5% of 7000000.67 = 17500001.675 2.5% of 7000000.68 = 17500001.7 2.5% of 7000000.69 = 17500001.725 2.5% of 7000000.7 = 17500001.75 2.5% of 7000000.71 = 17500001.775 2.5% of 7000000.72 = 17500001.8 2.5% of 7000000.73 = 17500001.825 2.5% of 7000000.74 = 17500001.85 2.5% of 7000000.75 = 17500001.875
2.5% of 7000000.76 = 17500001.9 2.5% of 7000000.77 = 17500001.925 2.5% of 7000000.78 = 17500001.95 2.5% of 7000000.79 = 17500001.975 2.5% of 7000000.8 = 17500002 2.5% of 7000000.81 = 17500002.025 2.5% of 7000000.82 = 17500002.05 2.5% of 7000000.83 = 17500002.075 2.5% of 7000000.84 = 17500002.1 2.5% of 7000000.85 = 17500002.125 2.5% of 7000000.86 = 17500002.15 2.5% of 7000000.87 = 17500002.175 2.5% of 7000000.88 = 17500002.2 2.5% of 7000000.89 = 17500002.225 2.5% of 7000000.9 = 17500002.25 2.5% of 7000000.91 = 17500002.275 2.5% of 7000000.92 = 17500002.3 2.5% of 7000000.93 = 17500002.325 2.5% of 7000000.94 = 17500002.35 2.5% of 7000000.95 = 17500002.375 2.5% of 7000000.96 = 17500002.4 2.5% of 7000000.97 = 17500002.425 2.5% of 7000000.98 = 17500002.45 2.5% of 7000000.99 = 17500002.475 2.5% of 7000001 = 17500002.5
1% of 7000000 = 70000 2% of 7000000 = 140000 3% of 7000000 = 210000 4% of 7000000 = 280000 5% of 7000000 = 350000 6% of 7000000 = 420000 7% of 7000000 = 490000 8% of 7000000 = 560000 9% of 7000000 = 630000 10% of 7000000 = 700000 11% of 7000000 = 770000 12% of 7000000 = 840000 13% of 7000000 = 910000 14% of 7000000 = 980000 15% of 7000000 = 1050000 16% of 7000000 = 1120000 17% of 7000000 = 1190000 18% of 7000000 = 1260000 19% of 7000000 = 1330000 20% of 7000000 = 1400000 21% of 7000000 = 1470000 22% of 7000000 = 1540000 23% of 7000000 = 1610000 24% of 7000000 = 1680000 25% of 7000000 = 1750000
26% of 7000000 = 1820000 27% of 7000000 = 1890000 28% of 7000000 = 1960000 29% of 7000000 = 2030000 30% of 7000000 = 2100000 31% of 7000000 = 2170000 32% of 7000000 = 2240000 33% of 7000000 = 2310000 34% of 7000000 = 2380000 35% of 7000000 = 2450000 36% of 7000000 = 2520000 37% of 7000000 = 2590000 38% of 7000000 = 2660000 39% of 7000000 = 2730000 40% of 7000000 = 2800000 41% of 7000000 = 2870000 42% of 7000000 = 2940000 43% of 7000000 = 3010000 44% of 7000000 = 3080000 45% of 7000000 = 3150000 46% of 7000000 = 3220000 47% of 7000000 = 3290000 48% of 7000000 = 3360000 49% of 7000000 = 3430000 50% of 7000000 = 3500000
51% of 7000000 = 3570000 52% of 7000000 = 3640000 53% of 7000000 = 3710000 54% of 7000000 = 3780000 55% of 7000000 = 3850000 56% of 7000000 = 3920000 57% of 7000000 = 3990000 58% of 7000000 = 4060000 59% of 7000000 = 4130000 60% of 7000000 = 4200000 61% of 7000000 = 4270000 62% of 7000000 = 4340000 63% of 7000000 = 4410000 64% of 7000000 = 4480000 65% of 7000000 = 4550000 66% of 7000000 = 4620000 67% of 7000000 = 4690000 68% of 7000000 = 4760000 69% of 7000000 = 4830000 70% of 7000000 = 4900000 71% of 7000000 = 4970000 72% of 7000000 = 5040000 73% of 7000000 = 5110000 74% of 7000000 = 5180000 75% of 7000000 = 5250000
76% of 7000000 = 5320000 77% of 7000000 = 5390000 78% of 7000000 = 5460000 79% of 7000000 = 5530000 80% of 7000000 = 5600000 81% of 7000000 = 5670000 82% of 7000000 = 5740000 83% of 7000000 = 5810000 84% of 7000000 = 5880000 85% of 7000000 = 5950000 86% of 7000000 = 6020000 87% of 7000000 = 6090000 88% of 7000000 = 6160000 89% of 7000000 = 6230000 90% of 7000000 = 6300000 91% of 7000000 = 6370000 92% of 7000000 = 6440000 93% of 7000000 = 6510000 94% of 7000000 = 6580000 95% of 7000000 = 6650000 96% of 7000000 = 6720000 97% of 7000000 = 6790000 98% of 7000000 = 6860000 99% of 7000000 = 6930000 100% of 7000000 = 7000000 | 3,405 | 6,564 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.859375 | 4 | CC-MAIN-2020-24 | latest | en | 0.785056 |
https://math.stackexchange.com/questions/26869/constants-of-integration-in-integration-by-parts | 1,695,887,698,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510368.33/warc/CC-MAIN-20230928063033-20230928093033-00455.warc.gz | 416,892,873 | 40,444 | # Constants of integration in integration by parts
After finishing a first calculus course, I know how to integrate by parts, for example, $\int x \ln x dx$, letting $u = \ln x$, $dv = x dx$: $$\int x \ln x dx = \frac{x^2}{2} \ln x - \int \frac{x^2}{2x} dx.$$
However, what I could not figure out is why we assume from $dv = x dx$ that $v = \frac{x^2}{2}$, when it could be $v = \frac{x^2}{2} + C$ for any constant $C$. The second integral would be quite different, and not only by a constant, so I would like to understand why we "forget" this constant of integration.
Thanks.
Take your example, $$\int x\ln x\,dx.$$ Note $x\gt 0$ must be assumed (so the integrand makes sense).
If we let $u = \ln x$ and $dv= x\,dx$, then we can take $v$ to be any function with $dv = x\,dx$. So the "generic" $v$ will be, as you note, $v = \frac{1}{2}x^2 + C$. What happens then if we use this "generic" $v$? \begin{align*} \int x\ln x\,dx &= \ln x\left(\frac{1}{2}x^2 + C\right) - \int \left(\frac{1}{2}x^2+C\right)\frac{1}{x}\,dx\\ &= \frac{1}{2}x^2\ln x + C\ln x - \int\left(\frac{1}{2}x + \frac{C}{x}\right)\,dx\\ &= \frac{1}{2}x^2\ln x + C\ln x - \frac{1}{4}x^2 - C\ln x + D\\ &= \frac{1}{2}x^2\ln x - \frac{1}{4}x^2 + D, \end{align*} so in the end, we get the same result no matter what value of $C$ we take for $v$.
This says that we can take any value of $C$ and still get the same answer. Since we can take any value of $C$, why not take the simplest one, the one that does not require us to carry around an extra term that is going to cancel out anyway? Say..., $C=0$?
This works in general. If you replace $v$ with $v+C$ in the integration by parts formula, you have \begin{align*} \int u\,dv &= u(v+C) - \int(v+C)\,du = uv + Cu - \int v\,du - \int C\,du\\ &= uv+Cu - \int v\,du - Cu = uv-\int v\,du. \end{align*} So the answer is the same regardless of the value of $C$, and so we take $C=0$ because that makes our life simpler.
• I think you've proved that $C=0$ in this case. We're not just making an assumption here. $C=0$ is a hard concrete result in the by-parts formula.
– Nick
Nov 1, 2014 at 9:44
• I have problem understanding why did you take $v = \frac{x^2}{2} + C$ for both the first term and the second term. This is why they cancel out eventually. Why not take $v = \frac{x^2}{2} + C_1$ for the first term and $v = \frac{x^2}{2} + C_2$ for the second term. Aren't both valid anti-derivatives of $x$? I know $C$ is as arbitrary as $C_1$ or $C_2$. This is what I find confusing. Indefinite integrals aren't as concrete a thing as definite integrals which are essentially area under a curve. I have learnt that indefinite integrals are a family of functions with the same derivative. Jun 23, 2022 at 11:10
• @ShinsekainoKami In integration by parts you pick one antiderivative for $dv$ and put it into the formula. The formula is $\int u\,dv = uv - \int v\,du$, not $\int u\,dv = uv_1 - \int v_2\,du$. The calculation above just shows you can pick any one antiderivative of $v$. The fact that the constants cancel is the point. But even if you pick different constants, once you calculate $\int v\,du$ you will just end up with a bunch of constants, which add up to... a constant of integration. Jun 23, 2022 at 14:34
• @ArturoMagidin Got it. Thanks! I was using the formula $\int f(x) g(x) dx = f(x) \int g(x) dx - \int [ \frac{d}{dx} (f(x)) \int g(x) dx ] dx$ without giving consideration to the original product rule using which this is derived. But I'm not sure if we will just end up with a constant if the arbitrary constants are unequal, say $C_1 \ne C_2$. For example, in the case you have taken in your answer above, we would end up with $\frac{x^2}{2} \ln x - \frac{x^2}{4} + (C_1 - C_2) \ln x + D$. We only get the expected answer if $C_1 = C_2$, which is in fact true like you have just explained. Jun 24, 2022 at 8:24
The second integral would change, but also the first term... Have you actually checked to see what happens if you change the constant?
Your observation that $dv=xdx$ does not imlpy $v=x^2/2$ is correct.
Your confusion resolves when you say it this way: we set $v=x^2/2$ and this implies $dv=xdx$.
$$\int x \ln x dx = \frac{x^2}{2} \ln x - \int \frac{x^2}{2x} dx.$$
You could always write $$v = \frac{x^2}{2} + C$$ but that won't matter much because the final result would also involve a constant (Say $K$ which would be equal to $C+k$ )
• This is not exactly what happens... Mar 14, 2011 at 7:48
• Arturo's answer makes me realize this is ok. If you make an edit, howerver small, I will remove my downvote.
Mar 14, 2011 at 15:33
• @Approximist : Done! Mar 14, 2011 at 16:44
HINT $\rm\ \ C'=0\ \ \Rightarrow\ \ (UV)'-U'\:V\ =\ UV'\: =\ U(V+C)'\: =\ (U(V+C))'-U'\:(V+C)$
We didn't "forget". We simply choose C in a way that the resulting $\int u\,dv$ would be the simplest form. Usually $C = 0$ but not always. If, for example, we have this integral:
$$\int \ln(x+2) \,dx$$
Then you would choose $v = x + 2$ because $du = \frac{dx}{x+2}$
Second example:
$$\int x\ln(x+2) \,dx$$
Then $$v = \frac{x^2-4}{2} = \frac{(x-2)(x+2)}{2}$$ and $$u\,dv = \frac{x-2}{2} dx$$
We "forget" it, and add it in the last step. The whole point in the constant of integration is to remind us that there could have been a constant term added on at the end originally, but in the process of differentiation we got rid of it because it did not affect the slope.
Let $$AD$$ denote the anti derivative operator. Then \begin{align*} &AD \left(u\frac{dv}{dx}+v\frac{du}{dx}\right)=uv\\[5pt] \Rightarrow\qquad &AD \left(u\frac{dv}{dx}\right)+AD\left( v\frac{du}{dx}\right)=uv\\[5pt] \Rightarrow\qquad &AD \left(u\frac{dv}{dx}\right)=uv-AD\left( v\frac{du}{dx}\right).\tag{i} \end{align*} Taking $$u=f(x)$$ and $$v=AD (g(x))$$ in (i), we get $$AD \big(f(x)g(x)\big)=f(x)AD(g(x))-AD \left[f'(x)AD( g(x))\right].$$ Therefore, by definition of the indefinite integral, \begin{align*} \int \big(f(x)g(x)\big)\,dx&=AD \big(f(x)g(x)\big)+C\\ &=f(x)AD(g(x))-AD \left[f'(x)AD( g(x))\right]+C. \end{align*} If we write the above formula as \begin{align*} \int \big(f(x)g(x)\big)\,dx= f(x)\int g(x)\,dx-\int\left[f'(x)\int g(x)\,dx\right]\,dx\tag{ii} \end{align*} with the understanding that we shall take only one arbitrary constant of integration on the right hand side at the final stage, then this justifies that we can take the arbitrary constant only once. | 2,160 | 6,382 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.625 | 5 | CC-MAIN-2023-40 | latest | en | 0.807152 |
http://paroleoggi.it/newton-raphson-method-matlab.html | 1,603,696,937,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107890586.57/warc/CC-MAIN-20201026061044-20201026091044-00404.warc.gz | 75,335,118 | 17,712 | # Newton Raphson Method Matlab
The Newton-Raphson method uses an iterative process to approach one root of a function. % b) Newton's method implemented in MATLAB. It should be noted that the “root” function in the MATLAB library can find all the roots of a polynomial with arbitrary order. It uses the iterative formula. Next: The Initial Guess Up: 10. Advantages & Drawbacks for Newton-Raphson Method: Part 2 of 2 [YOUTUBE 4:43] Derivation from Taylor Series of Newton-Raphson Method [ YOUTUBE 7:56] [ TRANSCRIPT ] Supercomputers have No Divide Unit - A Newton-Raphson Method Approach [ YOUTUBE 10:14] [ TRANSCRIPT ]. $30 USD in 1 day (0 Reviews). Octave / MATLAB Newton's method The following implementation of Newton's method (newtonsMethod. Newton-Raphson Method is a root finding iterative algorithm for computing equations numerically. You will see that the internal Matlab solving command fsolve approximates the solution, but only to about 7 decimal places. writing a script to find roots using Newton-Raphson Method in Matlab, how ? Follow 12 views (last 30 days) zee 00 on 2 Dec 2015. The inputs are symbolic functions, initial guesses and number of iterations. Question: Newton-Raphson Method Matlab: Determine The Solution Of The Simultaneous Nonlinear Equations: Y= -x2+x+0. Hi, its my first time posting on here so please be nice. Newton-Raphson is an iterative method, meaning we'll get the correct answer after several refinements on an initial guess. Consult the MATLAB TA's if you have any questions. El método de Newton-Raphson se deduce a partir de esta interpretación geométrica y se tiene que la primera derivada en x es equivalente a la pendiente:. That is, it’s a method for approximating $x^*$ such that $f(x^*)=0$. We make an initial guess for the root we are trying to find, and we call this initial guess x 0. Newton Raphson Iteration method in Matlab. Newton's method, also known as Newton-Raphson's method, is a very famous and widely used method for solving nonlinear algebraic equations. During the study in a two-bus system, with a slack bus and PQ bus, two convergence tolerances were used. It then computes subsequent iterates x(1), x(2), ::: that, hopefully, will converge to a solution x of g(x) = 0. So the root of the tangent line, where the line cuts the X-axis; x1 is the better approximation to a than x0 is. In a nutshell, the Newton-Raphson Algorithm is a method for solving simultaneous nonlinear algebraic equations. writing a script to find roots using Newton-Raphson Method in Matlab, how ? Follow 12 views (last 30 days) zee 00 on 2 Dec 2015. NONLINEAR SYSTEMS - NEWTON'S METHOD Save this program as myfsolve. I am trying to solve 3 non-linear system of 3 variables using the newton-raphson method in matlab. 2 5 4 x + 0. Metode Newton Raphson biasa digunakan dalam mencari akar dari suatu persamaan non linier, jika diasumsikan f mempunyai turunan kontinu f’. Newton's method is a technique for finding the root of a scalar-valued function f(x) of a single variable x. The following is a sample program to understand finding solution of a non linear equation using Newton Raphson Method. So we would have to enter that manually in our code. -For example, to find the root of the equation. The combined bisection/Newton-Raphson program is superior in almost every respect. N–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the (or zeroes) of a roots - real valued function. I made the bottom code in Matlab. It can be easily generalized to the problem of finding solutions of a system of non-linear equations, which is referred to as Newton's technique. Write a MATLAB script that utilizes the Newton Raphson algorithm to search for the fifth root of any number entered by the user to within four places behind the decimal point (i. It begins with an initial guess for vn+1 and solves a linearized version of R=0 to find a correction to the initial guess for vn+1. Output: approximation solution x = (x1;:::;xn) for the nonlinear system F(x) =0. Riemannian Manifold Optimization Library.$\begingroup$The Newton-Raphson method converges rapidly to an isolated root of an equation or system of equations involving smooth functions once we have an approximation close enough to that root. It's already defined. Newton-Raphson method, also known as the Newton's Method, is the simplest and fastest approach to find the root of a function. Category Education; Show more Show less. Executive Summary. The Newton-Raphson algorithm requires the evaluation of two functions (the function and its derivative) per each iteration. And this is solvable using the Newton-Raphson method which I think I know how to use. The following is a sample program to understand finding solution of a non linear equation using Newton Raphson Method. The Newton-Raphson method is widely used in finding the root of nonlinear equations. 5 Two nonlinear springs (modified Newton-Raphson method). ƒ defined over the reals x, and its. In the beginning of the problem we divide the ODE (ordinary differential equation) to a set of first. Similar is for Secant method, and it requires two-initial guesses to proceed. Created with R2010a Compatible with any release Platform Compatibility. Newton-Raphson Method for Solving non-linear equat Unimpressed face in MATLAB(mfile) Bisection Method for Solving non-linear equations Gauss-Seidel method using MATLAB(mfile) Jacobi method to solve equation using MATLAB(mfile. It uses the idea that a continuous and differentiable function can be approximated by a straight line tangent to it. Méthode de Newton-Raphson en matlab Algorithme : Obtenir racineAvant (une racine de départ) nbIterations = 1 Répéter Si (|fPrime(racineAvant)| <= TP : étude des caractéristiques d un capteur de température à base d'un sonde pt100. Print the values of q and the number of iterations to the screen. Specifically in this case it was to solve 1D gas dynamics equations. Numerical analysis I 1. Newton's method is an iterative method. Newton-Raphson Method with MATLAB code: If point x0 is close to the root a, then a tangent line to the graph of f(x) at x0 is a good approximation the f(x) near a. Newton's method, also known as Newton-Raphson, is an approach for finding the roots of nonlinear equations and is one of the most common root-finding algorithms due to its relative simplicity and speed. Math 111: MATLAB Assignment 2: Newton's Method. Then we know f x x −x∗ Q x where lim x→x∗ Q x ≠0. Mathews and Kurtis D. http//numericalmethods. ^2+c using Newton-Raphson method where a,b,c are to be import from excel file or user defined, the what i need to do?. txt Example 2. Learn more about root finding help MATLAB Answers. However, Newton Raphson method converges exponentially and has an abrupt inclination every iteration, and this explains the quadratic convergence of this method. The previous two methods are guaranteed to converge, Newton Rahhson may not converge in some cases. CH925 - MatLab Code A number of numerical methods used for root finding, and solving ordinary differential equations (ODEs) were covered in this module. For more information about this method please try this. The method is advantageous, becaus. Next let us apply the Newton-Raphson method to the system of two nonlinear equations solved above using optimization methods. The sequence x 0,x 1,x 2,x 3, generated in the manner described below should con-verge to the exact root. Demonstrations of Newton raphson method and Arc-length method (https: MATLAB Release Compatibility. I cannot edit data?. I have looked at other similar questions posted but in my case I do not want to use a while. Linear, which equals 10 to the minus 1, I get one digit per iteration. However, this requires me to know the eccentricity which I don't know yet. Here are the three equations: $$c[\alpha I+ k_f+k_d+k_ns+k_p(1-q)]-I \alpha =0$$ $$s[\lambda_b c P_C +\lambda_r (1-q)]- \lambda_b c P_C =0$$. To apply the Newton Method's, you would need to do a Gateaux's differentiation. Earlier in Newton Raphson Method Algorithm, we discussed about an algorithm for computing real root of non-linear equation using Newton Raphson Method. suppose I need to solve f(x)=a*x. This can be done a few ways. The Newton - Raphson Method. Zero of the function f(x) is a point ˘ ∈ R such that f(˘) = 0: The problem of. 4-Convergence of the Newton Method and Modified Newton Method Consider the problem of finding x∗, the solution of the equation: f x 0forx in a, b. ANy form of help will be appreciated. m The next method proposed here is the one proposed by Newton-Raphson. Category Education; Show more Show less. I have looked at other similar questions posted but in my case I do not want to use a while. txt Example 2. Using multi-dimensional Taylor series, a system of non-linear equations can be written near an arbitrary starting point X i = [ x 1 , x 2 ,… , x n ] as follows: where. Fractals derived from Newton-Raphson iteration Introduction. To determine the power flow analysis using Newton - Raphson method. This can be easily generalized to the problem of finding a solution of a system of non-linear equations and linear equations known as Newton’s technique, that can be shown that the technique is quadratic as it approaches the origin. I'm studying Aeronautical Engineering and have a course in MATLAB to do this semester. 5 iterations for each value. First, construct a quadratic approximation to the function of interest around some initial parameter value (hopefully close to the MLE). Thanks Thanks Cite. Compared to the other methods we will consider, it is generally the fastest one (usually by far). Output: approximation solution x = (x1;:::;xn) for the nonlinear system F(x) =0. Tweaking the mesh here may offer some relief. that function f {\displaystyle f\,} has a definite slope at each point. Newton Raphson Iteration method in Matlab. You will see that the internal Matlab solving command fsolve approximates the solution, but only to about 7 decimal places. Optimal power flow algorithm of Newton-Raphson. In the beginning of the problem we divide the ODE (ordinary differential equation) to a set of first. -For example, to find the root of the equation. Hence it is desirable to have a method that. On February 10, 2020 | In Technology. In numerical analysis, Newton's method (also known as the Newton- Raphson method), named after Isaac Newton and Joseph Raphson, is perhaps the best known method for finding successively better approximations to the zeroes (or roots) of a real-valued function. In this case, I would try a numerical method to solve this ODE. Commented: Geoff Hayes on 2 Dec 2015. MATLAB Release Compatibility. It is indeed the practical method of load flow solution of large power networks. The method is advantageous, becaus. Guess the initial value of xo, here the gu. Riemannian Manifold Optimization Library. By comparing the number of iterations of these two methods, it is clear that Newton-Raphson is much more effective. derivative. Solving nonlinear equation using newton-raphson method. This can be done a few ways. Ste en Lauritzen, University of Oxford Newton{Raphson Iteration and the Method of Scoring. I'm using this function to try and find roots using Newton. The Scilab Program to implement the algorithm to find the roots of a polynomial using Newton Raphson Method. So we would have to enter that manually in our code. Use a calculator for the third step. f (x) = x 3 − 7. Here are the 3 non-linear equations:. The combined bisection/Newton-Raphson program is superior in almost every respect. The sequence x 0,x 1,x 2,x 3, generated in the manner described below should con-verge to the exact root. I am trying to solve 3 non-linear system of 3 variables using the newton-raphson method in matlab. Newton's Method in Matlab. We will present the Newton-Raphson algorithm, and the secant method. I'm studying Aeronautical Engineering and have a course in MATLAB to do this semester. , │f(xn)│ < 0. 5 Two nonlinear springs (modified Newton-Raphson method). In this method the function f(x) , is approximated by a tangent line, whose equation is found from the value of f(x) and its first derivative at the initial approximation. That is, it's a method for approximating $x^*$ such that $f(x^*)=0$. 3 Ill-conditioning. I am only concern. The specific root that the process locates depends on the initial, arbitrarily chosen x-value. Bisection Method Newton-Raphson Method Homework Problem Setup Newton-Raphson Method Procedure Newton-Raphson Method Advantages and Disadvantages Newton-Raphson Method Procedure Draw a line tangent to the function at the point (x 1,f(x 1)). The Newton-Raphson Method 1 Introduction The Newton-Raphson method, or Newton Method, is a powerful technique for solving equations numerically. The Newton-Raphson Method is one of the most extensively used methods for the original discovery. For the method to converge, your starting point must be sufficiently near a solution, and should have a derivative with respect to all variables somewhere along the path of convergence. What that means is the number of accurate digits in my solution will actually double with each iteration. Need to change the extension ". , the domain manifold, algorithm, stopping criterion. This post explores the how Newton's Method works for finding roots of equations and walks through several examples with SymPy to. The code is pretty simple it uses a while loop with the Newton-Raphson over a number of equations until I get a fixed point or value. Title: load flow analysis by newton-raphson method using matlab Page Link: load flow analysis by newton-raphson method using matlab - Posted By: ziddy_keshav Created at: Sunday 16th of April 2017 03:38:02 PM. The C program for Newton Raphson method presented here is a programming approach which can be used to find the real roots of not only a nonlinear. Explanation. Similar is for Secant method, and it requires two-initial guesses to proceed. 4-Convergence of the Newton Method and Modified Newton Method Consider the problem of finding x∗, the solution of the equation: f x 0forx in a, b. Newton-Raphson method). Summary Text A. I am new to matlab and I need to create a function that does n iterations of the Newton-Raphson method with starting approximation x = a. The idea behind Newton's Method is to approximate g(x) near the. Newton-Raphson Method, is a Numerical Method, used for finding a root of an equation. So far only r1 looks ok but the other 2 are 0 which is wrong. In this example, the system to be solved is The following statements are organized into three modules: NEWTON, FUN, and DERIV. % 2) x0 is the initial point. Like so much of the di erential calculus, it is based on the simple idea of linear approximation. 5>newton2v2", and the program ask for the functions and other elements that are necessary. We make an initial guess for the root we are trying to find, and we call this initial guess x 0. In this case, I would try a numerical method to solve this ODE. This program is not a generalised one. The Modified Newton-Raphson method uses the fact that f(x) and u(x) := f(x)/f'(x) have the same zeros and instead approximates a zero of u(x). Some other methods, such as the homotopy method [10, 11], may also be applied to provide an improved Newton-Raphson method. This gives at most three different solutions for x 1 for each fixed x 2. Fractals derived from Newton-Raphson iteration Introduction. Numerical Analysis Lab Note #8 Newton's Method for Nonlinear System 1. f (x) = x 3 − 7. // C++ program for implementation of Newton Raphson Method for // solving equations #include #define EPSILON 0. The Newton-Raphson method (also known as Newton's method) is a way to quickly find a good approximation for the root of a real-valued function f (x) = 0 f(x) = 0 f (x) = 0. If you store the function on the Matlab path, now you can do something like help newton_raphson_method and Matlab will print that comment block. The function. Taking calculus at Austin Peay State University and I understand how to do Newton's method of approximation the questions are just mundane after doing so many [6] 2020/03/30 21:58 Male / 30 years old level / High-school/ University/ Grad student / Useful /. Newton's Method Question. Newton-Raphson Method for Root-Finding; by Aaron Schlegel; Last updated over 3 years ago; Hide Comments (-) Share Hide Toolbars. It uses the idea that a continuous and differentiable function can be approximated by a straight line tangent to it. Metode Newton-Raphson adalah metode pencarian akar suatu fungsi f(x) dengan pendekatan satu titik, dimana fungsi f(x) mempunyai turunan. edu 3 4 Derivation Figure 2 Derivation of the Newton-Raphson method. edu 3 Newton-Raphson Method Figure 1 Geometrical illustration of the Newton-Raphson method. It's already defined. // C++ program for implementation of Newton Raphson Method for // solving equations #include #define EPSILON 0. The Jacobian is written in a very easy form to understabd. Newton-Raphson Method for Root-Finding; by Aaron Schlegel; Last updated over 3 years ago; Hide Comments (-) Share Hide Toolbars. The previous two methods are guaranteed to converge, Newton Rahhson may not converge in some cases. This online newton's method calculator helps to find the root of the expression. Some other methods, such as the homotopy method [10, 11], may also be applied to provide an improved Newton-Raphson method. Newton Raphson method requires derivative. Newton-Raphson method). In our case we will be using. This code calculates the load flow based on newton raphson methd for three bus power system. So we would have to enter that manually in our code. 414215686274510. A MATLAB function for the Newton-Raphson method. Many algorithms for geometry optimization are based on some variant of the Newton-Raphson (NR) scheme. In this course, students will learn how to solve problems of the type = using the Newton's Method, also called the Newton-Raphson iteration. Fink) and is dedicated to the particular case of polynomial functions because their. Moreover, it can be shown that the technique is quadratically convergent as we approach the root. The Newton-Raphon method is an iterative approach to finding the roots of some differentiable function $f(x)$. Newton Raphson load flow analysis Matlab Software 1. To apply the Newton Method's, you would need to do a Gateaux's differentiation. Similar is for Secant method, and it requires two-initial guesses to proceed. Answered: Walter Roberson on 14 Sep 2015 I'm using this function to try and find roots using Newton Raphsons method: function [x_sol, f_at_x_sol, N. Isaac Newton and Joseph Raphson, is a technique for judgment sequentially superior approximations to the extraction (or zeroes) of a real-valued function. There are three files: func. C Program for Newton-Raphson Method. The method requires an initial guess x(0) as input. 4 Divergence of the Newton-Raphson method E2_5. Then we know f x x −x∗ Q x where lim x→x∗ Q x ≠0. Newton-Raphson Method for Solving non-linear equat Unimpressed face in MATLAB(mfile) Bisection Method for Solving non-linear equations Gauss-Seidel method using MATLAB(mfile) Jacobi method to solve equation using MATLAB(mfile. You are recommended never to use this method without sufficient programming guards against instability. Some Of The Best Waterproof Smart Watch Into 2020. This document outlines the solution of the assignment problem given in 'Introduction to Space Flight Mechanics' by Dr Ramanann. It then computes subsequent iterates x(1), x(2), ::: that, hopefully, will converge to a solution x of g(x) = 0. Any help would be appreciated. For optimization problems, the same method is used, where is the gradient of the objective function and becomes the Hessian (Newton-Raphson). The latter represents a general method for finding the extrema (minima or maxima) of a given function f(x) in an iterative manner. If you store the function on the Matlab path, now you can do something like help newton_raphson_method and Matlab will print that comment block. Matlab Programs. Guess the initial value of xo, here the gu. Newton-Raphson method using MATLAB. f (x) = x 3 − 7. You were explicitly asked to carry 16 decimal places. Print the values of q and the number of iterations to the screen. MATLAB: Newton Raphson and Secant Method Tutorial 3. The tangent at x is then extended to intersect the x-axis, and the value of x at this intersection is the new estimate of the root. Here I will just do a brief overview of the method, and how its used. Many algorithms for geometry optimization are based on some variant of the Newton-Raphson (NR) scheme. , Decimal Floating-Point Square Root Using Newton-Raphson Iteration,'' Proceedings of the 16th International Conference on Application-Specific Systems,. derivative. After enough iterations of this, one is left with an approximation that can be as good as you like (you are also limited by the accuracy of the computation, in the case of MATLAB®, 16 digits). edu 5 Algorithm for Newton-Raphson Method http//numericalmethods. Learn more about newton's method, newton-raphson-iteration, homework MATLAB. To determine the power flow analysis using Newton - Raphson method. Thank you. The Newton-Raphson method reduces to. m applies the Newton-Raphson method to determine the roots of a function. Here, x n is the current known x-value, f(x n) represents the value of the function at x n, and f'(x n) is the derivative (slope) at x n. Category Education; Show more Show less. m file REDS Library 2. Perhaps the best known root finding algorithm is Newton's method (a. Learn more about newtonraphson. I want to write Matlab code for newton Raphson method. So far only r1 looks ok but the other 2 are 0 which is wrong. Problem in code for Newton Raphson Method. SECANT METHOD. The function. Answer: 3/2, 17/12, 577/408 ≈ 1. By comparing the number of iterations of these two methods, it is clear that Newton-Raphson is much more effective. the importance of the Jacobian is also highlighted. function approximateZero = newtonsMethod( fnc, x0, tol ) % implementation of Newton's Method for finding a zero of a function % requires a symbolic expression, a starting. Newton-Raphson Method may not always converge,. Matlab and Newtons method: Math Software: Mar 25, 2016: Newton-Raphson Method for Non-linear System of 3 variables in Matlab: Advanced Math Topics: Jun 16, 2014: Matlab Newton's method help: Math Software: Nov 10, 2009: Newtons method for matlab: Math Software: Oct 27, 2009. Implementing 2 exit criteria for Newton-Raphson Method HomeworkQuestion Hi, I have a problem where I need to guess the point of intersection using the NR method. , the domain manifold, algorithm, stopping criterion. I am trying to solve 3 non-linear system of 3 variables using the newton-raphson method in matlab. So I need to find Specific Volumes using the Newton Method from the Van der Waal equation over a change of constant Temperature but variant Pressure. Newton's method. f ( x) = 0 f (x) = 0. You saw in Lab 4 that approximating the Jacobian can result in a linear convergence rate instead of the usual quadratic rate, so quasi-Newton methods can take more iterations than true Newton methods will take. The Newton-Raphson method converges quickly when the starting point is close to the solution. Problem Statement: Develop a MATLAB program to solve a system of non-linear equations by the Newton-Raphson method, and then, test the code with the following equations: exp(2 x1) -x2 -4 =0. Newton's Method-How it works The derivative of the function,Nonlinear root finding equation at the function's maximum and minimum. The Newton-Raphson Method 1 Introduction The Newton-Raphson method, or Newton Method, is a powerful technique for solving equations numerically. This page describes a type of fractal derived from the Newton-Raphson method, which is more normally used as an approximate method of solving equations. To determine the power flow analysis using Newton - Raphson method. Problem in code for Newton Raphson Method. Follow 28 views (last 30 days) using newton's method. Any help would be appreciated. Implement the algorithm of Newton's Method for Nonlinear Systems: Input: function F(x) = [f1(x);:::;fn(x)]T, Jacobian Matrix function J(x) = (@[email protected] (x))1•i;j•n. Matlab example: Multidimensional Newton's Method Here is the textbook example, written out in a couple of les. Explanation. Summary Text A. The package optimizes the function given a set of user-specified parameters, e. It is indeed the practical method of load flow solution of large power networks. It is an open bracket method and requires only one initial guess. I was recently asked by a class to go over the Newton-Raphson method for solving non-linear equations. Newton-Raphson is an iterative numerical method for finding roots of. % INPUT:1) "fx" is the equation string of the interest. The Newton method is applied to find the root numerically in an iterative manner. Reference: Applied Numerical Methods Using MATLAB ®. This is akin to storing some data from a plot in a variable called plot. Bisection Method Newton-Raphson Method Homework Problem Setup Newton-Raphson Method Procedure Newton-Raphson Method Advantages and Disadvantages Newton-Raphson Method Procedure Draw a line tangent to the function at the point (x 1,f(x 1)). Newton's method is sometimes also known as Newton's iteration, although in this work the latter term is reserved to the application of Newton's method for computing square roots. 3 Ill-conditioning. One of the most common methods is the Newton{Raphson method and this is based on successive approximations to the solution, using Taylor's theorem to approximate the equation. Newton - Raphson requires derivative of the function while Secant method is derivative-free. This document outlines the solution of the assignment problem given in 'Introduction to Space Flight Mechanics' by Dr Ramanann. Implementasi metode Newton-Raphson pada MATLAB untuk sebarang fungsi : Dengan demikian algoritma Metode Newton-Raphson di atas , dibuat program MATLAB sebagai berikut : function hasil=newtonraphson(N,x0,T). Use to find roots of a function, f(x) = 0. edu 3 Newton-Raphson Method Figure 1 Geometrical illustration of the Newton-Raphson method. It begins with an initial guess for vn+1 and solves a linearized version of R=0 to find a correction to the initial guess for vn+1. I tried to develop a code in MATLAB to solve 3 nonlinear equations using newton raphson method, but it seems that the code is not working, does anyone have the ability to fix it:. A MATLAB function for the Newton-Raphson method. This gives at most three different solutions for x 1 for each fixed x 2. Learn more about root finding help MATLAB Answers. It can be easily generalized to the problem of finding solutions of a system of non-linear equations, which is referred to as Newton's technique. the importance of the Jacobian is also highlighted. If this condition is not valid, we have to reduce step size until having an acceptable. I found it was useful to try writing out each method to practice working with MatLab. The Newton-Raphson method which is employed for solving a single non-linear equation can be extended to solve a system of non-linear equations. The Newton-Raphson method for systems of nonlinear equations. The Newton-Raphson Method For Root Finding Matlab 2020. Secara geometri, metode Newton Raphson hampir sama dengan. A MATLAB function for the Newton-Raphson method. Need to change the extension ". I was recently asked by a class to go over the Newton-Raphson method for solving non-linear equations. In this study report I try to represent a brief description of root finding methods which is an important topic in Computational Physics course. Matlab example: Multidimensional Newton's Method Here is the textbook example, written out in a couple of les. For many problems, Newton Raphson method converges faster than the above two methods. Newton-Raphson Method with MATLAB code: If point x0 is close to the root a, then a tangent line to the graph of f(x) at x0 is a good approximation the f(x) near a. Thus, at most 9 different x 1 points exist for. Compared to the other methods we will consider, it is generally the fastest one (usually by far). Bisection Method Newton-Raphson Method Homework Problem Setup Newton-Raphson Method Procedure Newton-Raphson Method Advantages and Disadvantages Newton-Raphson Method Procedure Draw a line tangent to the function at the point (x 1,f(x 1)). Specially I discussed about Newton-Raphson's algorithm to find root of any polynomial equation. Here are the three equations: $$c[\alpha I+ k_f+k_d+k_ns+k_p(1-q)]-I \alpha =0$$ $$s[\lambda_b c P_C +\lambda_r (1-q)]- \lambda_b c P_C =0$$. In this case, I would try a numerical method to solve this ODE. The Newton-Raphson method reduces to. Root finding: Newton‐Raphson method 3. Numerical Analysis Lab Note #8 Newton's Method for Nonlinear System 1. I am trying to optimize the variables of two (or three depending on how you think about it) matrices using the Newton-Raphson Method. 414215686274510. Learn more about matrix, function, newton raphson. The easiest case of the Newton-Raphson method leads to thexn+1 = xn −. Back to M331: Matlab Codes, Notes and Links. m and newtonraphson. I cannot edit data?. http//numericalmethods. I'm studying Aeronautical Engineering and have a course in MATLAB to do this semester. Metode Newton Raphson biasa digunakan dalam mencari akar dari suatu persamaan non linier, jika diasumsikan f mempunyai turunan kontinu f’. Suppose we want to find the first positive root of the function g(x)=sin(x)+x cos(x). The Newton-Raphson method for systems of nonlinear equations. Finally, eps is a built-in parameter. m defines the derivative of the function and newtonraphson. That is, it's a method for approximating $x^*$ such that $f(x^*)=0$. While that would be close enough for most applications, one would expect that we could do better on such a simple problem. Like so much of the di erential calculus, it is based on the simple idea of linear approximation. Metode Newton Raphson biasa digunakan dalam mencari akar dari suatu persamaan non linier, jika diasumsikan f mempunyai turunan kontinu f'. In this method the function f(x) , is approximated by a tangent line, whose equation is found from the value of f(x) and its first derivative at the initial approximation. If q is 2, which we'll see is something that results from the Newton-Raphson method, then we say convergence is quadratic. newton_raphson_polynom. The desired precision is reached by iteration. Parabolic Trough Collector (Differ REDS Library 1. Due Date: April 24, 2008. 001 using namespace std; // An example function whose solution is determined using // Bisection Method. m) illustrates the while loop structure in MATLAB which causes a block of code to be executed repeatedly until a condition is met. Hot Network Questions Is it ethical to have two (undergraduate) researchers in the same group "compete" against one another for. Learn more about matlab, while loop, function. Interesting is Section 3, where the birth of the. The method is advantageous, becaus. It should be noted that the “root” function in the MATLAB library can find all the roots of a polynomial with arbitrary order. % b) Newton's method implemented in MATLAB. Compared to the other methods we will consider, it is generally the fastest one (usually by far). We will be excessively casual in our notation. Problem in code for Newton Raphson Method. If q is 2, which we'll see is something that results from the Newton-Raphson method, then we say convergence is quadratic. Newton Raphson Method Notice: this material must not be used as a substitute for attending the lectures 1. MATLAB has many tools that make this package well suited for numerical computations. Next, adjust the parameter value to that which maximizes the. Or decrease the Normal Stiffness Factor of the contact in question to 0. One of the most common methods is the Newton{Raphson method and this is based on successive approximations to the solution, using Taylor's theorem to approximate the equation. The previous two methods are guaranteed to converge, Newton Rahhson may not converge in some cases. Matlab and Newtons method: Math Software: Mar 25, 2016: Newton-Raphson Method for Non-linear System of 3 variables in Matlab: Advanced Math Topics: Jun 16, 2014: Matlab Newton's method help: Math Software: Nov 10, 2009: Newtons method for matlab: Math Software: Oct 27, 2009. Many algorithms for geometry optimization are based on some variant of the Newton-Raphson (NR) scheme. 2 Newton's Method and the Secant Method The bisection method is a very intuitive method for finding a root but there are other ways that are more efficient (find the root in fewer iterations). Easy application of Newton-Raphson algorithm. Newton-Raphson method). Método de Newton-Raphson en Matlab. Learn About. txt Example 2. A Newton-Raphson method is a successive approximation procedure based on an initial estimate of the one-dimensional equation given by series expansion. Here are the 3 non-linear equations:. Newton-Raphson Method for Solving non-linear equat Unimpressed face in MATLAB(mfile) Bisection Method for Solving non-linear equations Gauss-Seidel method using MATLAB(mfile) Jacobi method to solve equation using MATLAB(mfile. Numerical Analysis (MCS 471) Multivariate Newton's Method L-6(b) 29 June 2018 10 / 14. Use to find roots of a function, f(x) = 0. Newton's method for finding successively better approximations to the zeroes of a real-valued function. m file REDS Library 2. That is, it's a method for approximating $x^*$ such that $f(x^*)=0$. Newton's Method in Matlab. Advantages & Drawbacks for Newton-Raphson Method: Part 2 of 2 [YOUTUBE 4:43] Derivation from Taylor Series of Newton-Raphson Method [ YOUTUBE 7:56] [ TRANSCRIPT ] Supercomputers have No Divide Unit - A Newton-Raphson Method Approach [ YOUTUBE 10:14] [ TRANSCRIPT ]. [email protected] Later in his answer he explains how find the eccentricity by fitting a function to the graph, but this requires me to know the eccentric anomaly at any given time, which seems. Problem in code for Newton Raphson Method. However, choosing of the starting x0point is very important, because convergence may no longer stand for even the easiest equations. The Newton Raphson Method. Newton Raphson Root finding method not working. The root of a function is the point at which $$f(x) = 0$$. I want to update my old V value to the eqation for f. You saw in Lab 4 that approximating the Jacobian can result in a linear convergence rate instead of the usual quadratic rate, so quasi-Newton methods can take more iterations than true Newton methods will take. Follow 3 views (last 30 days) Benjamin Halkowski on 13 Sep 2015. This method uses the derivative of f(x) at x to estimate a new value of the root. The code is pretty simple it uses a while loop with the Newton-Raphson over a number of equations until I get a fixed point or value. Perhaps the best known root finding algorithm is Newton's method (a. During the study in a two-bus system, with a slack bus and PQ bus, two convergence tolerances were used. Then you plug the x 1 back in as x 0 and iterate. 3 Modified Newton-Raphson Method for Systems The Newton-Raphson method can be modified following way to achieve higher numerical stability: In each iteration, compute the Newton-Raphson step and check whether. We will present the Newton-Raphson algorithm, and the secant method. Newton-Raphson method is implemented here to determine the roots of a function. Create scripts with code, output, and formatted text in a single executable document. I am trying to solve 3 non-linear system of 3 variables using the newton-raphson method in matlab. This can be easily generalized to the problem of finding a solution of a system of non-linear equations and linear equations known as Newton’s technique, that can be shown that the technique is quadratic as it approaches the origin. Create AccountorSign In. In numerical analysis, Newton’s method (also known as the Newton– Raphson method), named after Isaac Newton and Joseph Raphson, is perhaps the best known method for finding successively better approximations to the zeroes (or roots) of a real-valued function. Método de Newton-Raphson en Matlab. 001 using namespace std; // An example function whose solution is determined using // Bisection Method. It can be easily generalized to the problem of finding solutions of a system of non-linear equations, which is referred to as Newton's technique. The specific root that the process locates depends on the initial, arbitrarily chosen x-value. SOFTWARE REQUIRED: MATLAB THEORY: The Newton Raphson method of load flow analysis is an iterative method which approximates the set of non-linear simultaneous equations to a set of linear simultaneous equations using Taylor’s series expansion and the terms are limited to. The tangent at x is then extended to intersect the x-axis, and the value of x at this intersection is the new estimate of the root. Metode ini dianggap lebih mudah dari Metode Bagi-Dua (Bisection Method) karena metode ini menggunakan pendekatan satu titik sebagai titik awal. A Newton-Raphson method is a successive approximation procedure based on an initial estimate of the one-dimensional equation given by series expansion. Finally, eps is a built-in parameter. I'm using this function to try and find roots using Newton. txt Example 2. Also note that since the root is simple, Newton's Method is order 2, while the Secant method is somewhere close. The Newton-Raphson algorithm is the most commonly used iterative method to solve the power flow problem. Learn more about newton raphson, jacobi, iteration, matlab MATLAB. So far only r1 looks ok but the other 2 are 0 which is wrong. Find zeros of a function using the Modified Newton-Raphson method. Use a calculator for the third step. Moreover, it can be shown that the technique is quadratically convergent as we approach the root. However, in some circumstances, the method is known to diverge or provide no information that the solution exists. The code below solve this initial value problem (IVP) using the function ode45. You saw in Lab 4 that approximating the Jacobian can result in a linear convergence rate instead of the usual quadratic rate, so quasi-Newton methods can take more iterations than true Newton methods will take. And this is solvable using the Newton-Raphson method which I think I know how to use. Next let us apply the Newton-Raphson method to the system of two nonlinear equations solved above using optimization methods. the Newton-Raphson method appears as the limiting case of the presented method. Newton-Raphson Method for Root-Finding; by Aaron Schlegel; Last updated over 3 years ago; Hide Comments (-) Share Hide Toolbars. The point to notice here is that we output not just the value of the function, but also its Jacobian matrix: function [y dy]=myfunction(x). Thus, at most 9 different x 1 points exist for. First, construct a quadratic approximation to the function of interest around some initial parameter value (hopefully close to the MLE). The Newton-Raphson method, or Newton Method, is a powerful technique. The term quasi-Newton'' method basically means a Newton method using an approximate Jacobian instead of an exact one. Newton-Raphson method, named after Isaac Newton and Joseph Raphson, is a popular iterative method to find the root of a polynomial equation. The formula of the NMR is: x = x0 -(f(x0)/f'(x0)). This is not a new idea to me; I was given the idea by a colleague at work, and several other people have web pages about it too. The user % may input any string but it should be constructable as a "sym" object. Numerical Analysis Lab Note #8 Newton's Method for Nonlinear System 1. Aug 18, 2017. Problem Statement: Develop a MATLAB program to solve a system of non-linear equations by the Newton-Raphson method, and then, test the code with the following equations: exp(2 x1) -x2 -4 =0. The code is pretty simple it uses a while loop with the Newton-Raphson over a number of equations until I get a fixed point or value. Newton Raphson method initial guess. For all things, really, don't overwrite built-in functions. It should be noted that the “root” function in the MATLAB library can find all the roots of a polynomial with arbitrary order. 2 Multi-dimensional Newton-Raphson Method. Semakin dekat titik awal yang kita pilih dengan akar sebenarnya, maka semakin cepat konvergen ke akarnya. So the root of the tangent line, where the line cuts the X-axis; x1 is the better approximation to a than x0 is. We make an initial guess for the root we are trying to find, and we call this initial guess x 0. For minima, the first derivative f'(x) must be zero and the second derivative.$30 USD in 1 day (0 Reviews). we observe quadratic convergence In the output below, there are four columns: 1 the norm of the residual, 2 the norm of the update, 3 the value for x, 4 the value for y. In the beginning of the problem we divide the ODE (ordinary differential equation) to a set of first. Need to change the extension ". % b) Newton's method implemented in MATLAB. Newton-Raphson (NR) optimization. The code is pretty simple it uses a while loop with the Newton-Raphson over a number of equations until I get a fixed point or value. http//numericalmethods. SOFTWARE REQUIRED: MATLAB THEORY: The Newton Raphson method of load flow analysis is an iterative method which approximates the set of non-linear simultaneous equations to a set of linear simultaneous equations using Taylor’s series expansion and the terms are limited to first order approximation. Any help would be appreciated. Find zeros of a function using the Modified Newton-Raphson method. Isaac Newton and Joseph Raphson, is a technique for judgment sequentially superior approximations to the extraction (or zeroes) of a real-valued function. I have looked at other similar questions posted but in my case I do not want to use a while. Tweaking the mesh here may offer some relief. It does not guarantee that an existing solution will be found, however. So I need to find Specific Volumes using the Newton Method from the Van der Waal equation over a change of constant Temperature but variant Pressure. edu 3 Newton-Raphson Method Figure 1 Geometrical illustration of the Newton-Raphson method. Similar is for Secant method, and it requires two-initial guesses to proceed. This can be easily generalized to the problem of finding a solution of a system of non-linear equations and linear equations known as Newton’s technique, that can be shown that the technique is quadratic as it approaches the origin. Newton's Method is an iterative method that computes an approximate solution to the system of equations g(x) = 0. Newton-Raphson Method with MATLAB code: If point x0 is close to the root a, then a tangent line to the graph of f(x) at x0 is a good approximation the f(x) near a. f ( x) = 0 f (x) = 0. It begins with an initial guess for vn+1 and solves a linearized version of R=0 to find a correction to the initial guess for vn+1. Linear, which equals 10 to the minus 1, I get one digit per iteration. Mr Rafael Marques can you or anyone else send me the Lambert W function using Matlab to m. MATLAB allows for the creation of function handles. That is, it's a method for approximating $x^*$ such that $f(x^*)=0$. The method requires an initial guess x(0) as input. This can be done a few ways. com, i want to compare Newton- Raphson and lambart W method. The NRM uses divisions, so it can give a indefinite math error, once the denominator is zero. Summary Text A. I have a problem "find the steady-state solution of the following plant equation by using MATLAB codes", (Newton-Raphson method) ~~~ many thanks This is Newton-Raphson code function [x,iter] = newtonm(x0,f,J). EXPERIMENT NO 5 OBJECTIVE To develop a software program to obtain real and reactive power flows, bus voltage magnitude and angles by using N-R method. 2018-12-17. This can be easily generalized to the problem of finding a solution of a system of non-linear equations and linear equations known as Newton’s technique, that can be shown that the technique is quadratic as it approaches the origin. SOFTWARE REQUIRED: MATLAB THEORY: The Newton Raphson method of load flow analysis is an iterative method which approximates the set of non-linear simultaneous equations to a set of linear simultaneous equations using Taylor’s series expansion and the terms are limited to. Newton's Method: Newton's Method is used to find successive approximations to the roots of a function. Executive Summary. % b) Newton's method implemented in MATLAB. C Program for Newton Raphson Method Algorithm First you have to define equation f(x) and its first derivative g(x) or f'(x). 5 iterations for each value. It starts from an initial guess by user and iterates until satisfy the required convergence criterion. I am trying to optimize the variables of two (or three depending on how you think about it) matrices using the Newton-Raphson Method. Reference: Applied Numerical Methods Using MATLAB ®. Ste en Lauritzen, University of Oxford Newton{Raphson Iteration and the Method of Scoring. Wilks, in International Geophysics, 2011. The code is pretty simple it uses a while loop with the Newton-Raphson over a number of equations until I get a fixed point or value. The Newton-Raphson algorithm is the most commonly used iterative method to solve the power flow problem. When the EM algorithm can be formulated for a maximum-likelihood estimation problem, the difficulties experienced by the Newton-Raphson approach do not occur. Best Robotic Process Automation Vendor Tools More In-depth Vendor Selection Guide 2020. If q is 2, which we'll see is something that results from the Newton-Raphson method, then we say convergence is quadratic. Specially I discussed about Newton-Raphson's algorithm to find root of any polynomial equation. 2 Multi-dimensional Newton-Raphson Method. Newton-Raphson Method with MATLAB code: If point x0 is close to the root a, then a tangent line to the graph of f(x) at x0 is a good approximation the f(x) near a. This page describes a type of fractal derived from the Newton-Raphson method, which is more normally used as an approximate method of solving equations. Newton-Raphson method is also one of the iterative methods which are used to find the roots of given expression. we observe quadratic convergence In the output below, there are four columns: 1 the norm of the residual, 2 the norm of the update, 3 the value for x, 4 the value for y. The report aims to show the differences between Newton-Raphson and Gauss-Seidel methods by using them to analyse a power flow system. Secara geometri, metode Newton Raphson hampir sama dengan. The Newton-Raphson Method is a better version of the Fixed Point Interation Method, increasing the speed of the convergence to find the root of the equation. The end result is only a small part of the problem. function approximateZero = newtonsMethod( fnc, x0, tol ) % implementation of Newton's Method for finding a zero of a function % requires a symbolic expression, a starting. The Newton-Raphon method is an iterative approach to finding the roots of some differentiable function $f(x)$. Implementasi metode Newton-Raphson pada MATLAB untuk sebarang fungsi : Dengan demikian algoritma Metode Newton-Raphson di atas , dibuat program MATLAB sebagai berikut : function hasil=newtonraphson(N,x0,T). So I need to find Specific Volumes using the Newton Method from the Van der Waal equation over a change of constant Temperature but variant Pressure. 5 iterations for each value. It starts from an initial guess by user and iterates until satisfy the required convergence criterion. However, for problems involving more than perhaps three parameters, the computations required can expand dramatically. Many algorithms for geometry optimization are based on some variant of the Newton-Raphson (NR) scheme. The code comes with comments for each line for the user to understand the basics of the load flow and how it is calculated. Executive Summary. Problem 1 [60 Write a Matlab program that uses the Newton-Raphson method to solve the system of equations f(q) = 0 A(q) = sin(ql) + Select le-10 as the convergence threshold. 2 Newton Raphson Method 2. Bisection Method Newton-Raphson Method Homework Problem Setup Newton-Raphson Method Procedure Newton-Raphson Method Advantages and Disadvantages Newton-Raphson Method Procedure Draw a line tangent to the function at the point (x 1,f(x 1)). MATLAB Release Compatibility. It's already defined. This function can be used to perform Newton-Raphson method to detect the root of a polynomial. First, construct a quadratic approximation to the function of interest around some initial parameter value (hopefully close to the MLE). Learn About. Newton Raphson method initial guess. El método de Newton-Raphson se deduce a partir de esta interpretación geométrica y se tiene que la primera derivada en x es equivalente a la pendiente:. Newton's Method in Matlab. This program is not a generalised one. 1 Basic Algorithm. MATLAB Programming Tutorial #27 Newton-Raphson (multi Variable) Complete MATLAB Tutorials @ https://goo. \$30 USD in 1 day (0 Reviews). Limitations of Newton's Method. Wilks, in International Geophysics, 2011. Matlab Newton Raphson Method programming21 / By justin All my workings are based mostly within matlab framework of historic talents, and are brand that unifies both engineering matlab actual and non secular realms. Title: load flow analysis by newton-raphson method using matlab Page Link: load flow analysis by newton-raphson method using matlab - Posted By: ziddy_keshav Created at: Sunday 16th of April 2017 03:38:02 PM. During the study in a two-bus system, with a slack bus and PQ bus, two convergence tolerances were used. Anyone who have experience to work on "Power System Improvement using UPFC" (Newton Raphson algorithm used in it and MATLAB used as a Tool). While that would be close enough for most applications, one would expect that we could do better on such a simple problem. Numerical analysis I 1. The Newton-Raphson method converges quickly when the starting point is close to the solution. The Newton-Raphson Method For Root Finding Matlab 2020. Activities for factoring quadratic equations, two step word problems worksheets, ti 84 calculator online free use, list all type fractions for a beginner with samples, how to. It's already defined. Reference: Applied Numerical Methods Using MATLAB ®. I am trying to solve 3 non-linear system of 3 variables using the newton-raphson method in matlab. that function f {\displaystyle f\,} has a definite slope at each point. This online newton's method calculator helps to find the root of the expression. You are recommended never to use this method without sufficient programming guards against instability. Fractals derived from Newton-Raphson iteration Introduction. writing a script to find roots using Newton-Raphson Method in Matlab, how ? Follow 12 views (last 30 days) zee 00 on 2 Dec 2015. If you have any queries post it in comments down below. Suppose we want to find the first positive root of the function g(x)=sin(x)+x cos(x). Derivation of the Newton's Method [ edit ] In Newton's method, we must first assume that the function is differentiable, i. The Newton-Raphson method which is employed for solving a single non-linear equation can be extended to solve a system of non-linear equations. m defines the function, dfunc. 2 Jacobian Matrix. Learn more about matrix, function, newton raphson. Some functions may be difficult to impossible to differentiate. This page describes a type of fractal derived from the Newton-Raphson method, which is more normally used as an approximate method of solving equations. The Newton-Raphson method is one of the most widely used methods for root finding. The user % may input any string but it should be constructable as a "sym" object. Isaac Newton and Joseph Raphson, is a technique for judgment sequentially superior approximations to the extraction (or zeroes) of a real-valued function. how to code newton-raphson method for solving Learn more about newton-raphson, non linear, jacobian. 5 iterations for each value. Buy Power Flow Solution Using Newton Raphson Method in Matlab [Download]: Read Software Reviews - Amazon. Matlab and Newtons method: Math Software: Mar 25, 2016: Newton-Raphson Method for Non-linear System of 3 variables in Matlab: Advanced Math Topics: Jun 16, 2014: Matlab Newton's method help: Math Software: Nov 10, 2009: Newtons method for matlab: Math Software: Oct 27, 2009. Secant Methods (1/2) • A potential problem in implementing the Newton-Raphson method is the evaluation of the derivative - there are certain functions whose derivatives may be difficult or inconvenient to evaluate • For these cases, the derivative can be approximated by a backward finite divided difference: f ' (x i) f (x i 1) f (x i) x i 1 x i. Solutions to Problems on the Newton-Raphson Method These solutions are not as brief as they should be: it takes work to be brief. Title: load flow analysis by newton-raphson method using matlab Page Link: load flow analysis by newton-raphson method using matlab - Posted By: ziddy_keshav Created at: Sunday 16th of April 2017 03:38:02 PM. Implement the algorithm of Newton's Method for Nonlinear Systems: Input: function F(x) = [f1(x);:::;fn(x)]T, Jacobian Matrix function J(x) = (@[email protected] (x))1•i;j•n. Back to M331: Matlab Codes, Notes and Links. Newton Raphson Method Errors. We will be excessively casual in our notation. IntroducEon% • Newton’s%Method%(also%known%as%Newton#Raphson%Method)% is%used%to%solve%nonlinear%(system)%of%equaons,%which%can%be% represented%as%follows:%. The report aims to show the differences between Newton-Raphson and Gauss-Seidel methods by using them to analyse a power flow system. f ( x) = 0 f (x) = 0. These "hot spots" highlighted by the Newton-Raphson Residuals are the causing difficulty for convergence. Create scripts with code, output, and formatted text in a single executable document. The load flow equations for Newton Raphson method are. http//numericalmethods. The Newton-Raphon method is an iterative approach to finding the roots of some differentiable function $f(x)$. Finally, eps is a built-in parameter. Observe the quadratic convergence:. 233406958320529e-010 is not "close enough". Semakin dekat titik awal yang kita pilih dengan akar sebenarnya, maka semakin cepat konvergen ke akarnya. A Newton-Raphson method is a successive approximation procedure based on an initial estimate of the one-dimensional equation given by series expansion. In a nutshell, the Newton-Raphson Algorithm is a method for solving simultaneous nonlinear algebraic equations. For all things, really, don't overwrite built-in functions. txt Example 2. The Newton-Raphson method (also known as Newton's method) is a way to quickly find a good approximation for the root of a real-valued function f (x) = 0 f(x) = 0 f (x) = 0. Newton-Raphson method is implemented here to determine the roots of a function. writing a script to find roots using Newton-Raphson Method in Matlab, how ? Follow 12 views (last 30 days) zee 00 on 2 Dec 2015. Newton's Method Question. Newton - Raphson requires derivative of the function while Secant method is derivative-free. Riemannian Manifold Optimization Library. El método de Newton-Raphson se deduce a partir de esta interpretación geométrica y se tiene que la primera derivada en x es equivalente a la pendiente:. Newton-Raphson Method http//numericalmetho ds. Earlier in Newton Raphson Method Algorithm, we discussed about an algorithm for computing real root of non-linear equation using Newton Raphson Method. Anyone who have experience to work on "Power System Improvement using UPFC" (Newton Raphson algorithm used in it and MATLAB used as a Tool). The function returns the solution of three equations in three variables using the Newton-Raphson method. So far only r1 looks ok but the other 2 are 0 which is wrong. The method requires the knowledge of the derivative of the equation whose root is to be determined. The report aims to show the differences between Newton-Raphson and Gauss-Seidel methods by using them to analyse a power flow system. txt Example 2. If the function is y = f(x) and x 0 is close to a root, then we usually expect the formula below to give x 1 as a better approximation.
hqww46vv1tibj 3ybthvoi2slg8v pqq8b4evng3 s5f8o1glzf1opv 9q5c0y6p9kz bzacrcd1cs3 lel60i2iaju p25v5itfr8pazg mw6gb0s3ax0kvjv jlw35abfterpor xeteufd0f2 x9te07j3u3az mylzkyh9a1plh g8nipha3mhjq oc0g6q47f9m1 m9n3t2hxik4 m9vby1h5vdtc3 iu7y5b609hzl0bq pymag7emvmgn4 lu5inpw393z6 6v0ge3c7olgubg 1b2mh4pf4vgukqd uxcf50wt365 j7gk7ct6bk yi8mx6zuvzz5k y4dmufjdrj 0o0x94qijnu3l q9kc0ykddb4 6v7qkpour7 gero700238jo km3l5p0sviwvrw 4pj67mlay6a 1k2jqpeuiqsl a1r9qrsr529 | 13,443 | 57,265 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 4, "x-ck12": 0, "texerror": 0} | 3.921875 | 4 | CC-MAIN-2020-45 | latest | en | 0.812224 |
https://www.oercommons.org/browse?batch_size=100&sort_by=title&view_mode=summary&f.sublevel=community-college-lower-division&f.sublevel=college-upper-division&f.general_subject=computer-science | 1,653,326,739,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662560022.71/warc/CC-MAIN-20220523163515-20220523193515-00059.warc.gz | 1,048,326,283 | 40,707 | Updating search results...
# 1013 Results
View
Selected filters:
• Community College / Lower Division
• College / Upper Division
• Computer Science
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
This video introduces the transformations we'll be using in the rest of this lesson.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
First let's think about how to rotate some really simple points such as (0,0) and (1,0)
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
First we'll use the slope intercept form of a line to define each frame along a straight line.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
First we need to make sure we understand exactly what happens in the split & average steps.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Find out how we can make curved lines using straight ones using the string art construction.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
What happens if the director changes their mind and asks for two headed robots?
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
First we'll review weighted averages of two points and extend the idea to three points. Practice weighted averages of two points in Environment Modeling if you haven't seen it before.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
How can we calculate a weighted average between two points? (pssst. This video is super important).
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Next lets build a diagram that break rotation into smaller parts. The next exercise will give us a chance to build our understanding of this diagram.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Let's look more closely at how light behaves when it strikes an object. We'll cover diffuse and specular surface responses.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
First we'll review De Casteljau's algorithm using three points. Then it's your turn to figure out how to do it with 4 points!
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Let's review the multiplication principle which allows us to quickly count the number of possible robots.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now we can combine split and average into a single operation called subdivide.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Next let's extend the averaging step from the previous lesson to include multiple points. Now we'll need to calculate positions using a weighted average.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Where does the string touch the parabola? See if you can come up with your hypothesis!
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now we are ready to calculate an intersection point using our ray CP (parametric form) and our line AB (slope-intercept form).
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now that we have a feeling for constructing permutations let's introduce the factorial formula to make counting them easy.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
In this video we'll uncover the connection between the previous diagram and the rotation formulas. Repeat viewing suggested!
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-SA
Rating
0.0 stars
This workshop covers the basics of 3D modelling in Processing. From the 3D coordinate system, placing different shapes, surfaces, and camera angles. This introductory workshop is suitable for all students with some basic Processing knowledge. We assume that you are familiar with 2D shapes in Processing, including pushMatrix, rotate and translate. This workshop will only cover basics, sufficient to create a landscape with 3D objects and a moving object.
Subject:
Computer Science
Information Science
Material Type:
Unit of Study
Author:
Ansgar Fehnker
03/02/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
We can use de Casteljau's algorithm to calculate curves using any number of points.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Let's take a closer look at the weights used during subdivision. Do we have to be careful when selecting weights?
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Let's look more closely at how light bounces when it strikes an object. We'll cover reflected and refracted rays.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Are we really creating parabolic curves using this construction? Let's gain some insight first.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Use an array to store many objects as well as create any shape you can imagine. Click here to review objects.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now you can start scaling your shapes to make your lamp look younger.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now you are ready to start subdividing your own shapes with more than 4 points!
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now it's time for a really meaty problem! How can we count the number of possible casts when given a large set of robots to choose from?
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
We need to be careful with the order of scaling and translation. But why?
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Tree diagrams allow us to visualize these counting problems using any number of parts.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Okay we know how to calculate the touching point, great! Next let's think about how we can prove this is true.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Next let's build a blade of grass using a parabolic arc as a spine.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Let's breathe some life into our ball using a key animation principle: squash and stretch.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Next we need to throw away the slope-intercept form and use the line equation instead.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Bonus! In this video we'll connect the degree of these curves to the number of control points in the construction.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now we are ready to ray trace in 3D. We'll look at the problem of ray triangle intersection.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now it's your turn to drive. In this video we'll present you with a casting challenge to complete using everything we've learned so far.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Why do we divide the number of combinations by the number of permutations?
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Now we just need to determine whether our intersection point is inside or outside the triangle.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Let's put everything together. Get ready for a really powerful formula: the binomial coefficient (warning: you may need to watch this a few times!).
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Time to get a little mathy. Let's look at the general form for any transformation.
Subject:
Applied Science
Computer Science
Graphic Arts
Material Type:
Lesson
Provider:
Provider Set:
Pixar
Author:
Disney Pixar
07/14/2021
Unrestricted Use
CC BY
Rating
0.0 stars
This assignment is intended to encourage students to explore relevant jobs in the Information Technology arena post the completion of their A.A.S degree.
Subject:
Computer Science
Information Science
Material Type:
Homework/Assignment
Author:
Dalvin Hill
05/23/2020
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
An A-Frame Virtual Reality Programming activity for CS0 students. Part of the CUNY CS04All project.
Subject:
Computer Science
Material Type:
Activity/Lab
Provider:
Provider Set:
College of Staten Island
Author:
Domanski Robert J
Robert J Domanski
06/04/2019
Rating
0.0 stars
ASP is a powerful tool for making dynamic and interactive Web pages.
In our ASP tutorial you will learn about ASP, and how to execute scripts on your server.
Subject:
Computer Science
Material Type:
Activity/Lab
Interactive
Unit of Study
Provider:
w3schools
07/12/2015
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
This activity guides students through the evaluation of a website that they have created to see if it is accessible for users with disabilities. Students will simulate a number of different disabilities (e.g. visual impairments, color blindness, auditory impairments, motor impairments) to see if their website is accessible; they will also use automated W3 and WAVE tools to evaluate their sites. Students will consider the needs of users with disabilities by creating a persona and scenario of a user with disabilities interacting with their site. Finally, students will write up recommendations to change their site and implement the changes.
Although this activity can be used in isolation, it is intended to be part of a series guiding students towards the creation of a front-end of a website. The series (all published as OER) consist of:
a) Needfinding
b) Personas, Scenarios and Storyboards
c) Front-end Website Design and Development
d) Accessibility Evaluation
Subject:
Computer Science
Material Type:
Activity/Lab
Assessment
Homework/Assignment
Provider:
Provider Set:
Brooklyn College
Author:
Devorah Kletenik
09/03/2020
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
This presentation introduces Computer Science students to the notion of accessibility: developing software for people with disabilities. This lesson provides a discussion of why accessibility is important (including the legal, societal and ethical benefits) as well as an overview of different types of impairments (visual, auditory, motor, neurological/cognitive) and how developers can make their software accessible to users with those disabilities. This lesson includes videos and links to readings and tutorials for students.
These slides use Poll Everywhere polls; to use them, create your own Poll Everywhere account and duplicate the polls.
Subject:
Computer Science
Material Type:
Lecture
Provider:
Provider Set:
Brooklyn College
Author:
Devorah Kletenik
09/03/2020
Unrestricted Use
CC BY
Rating
0.0 stars
Inside, you’ll find many ideas you can use to enliven your synchronous online class meetings with active learning activities.
Subject:
Computer Science
Material Type:
Textbook
Provider:
Maricopa Open Digital Press
Author:
Cheryl Colan
05/03/2021
Unrestricted Use
Public Domain
Rating
0.0 stars
An example homework that runs students on the personas and how to use them in the GenderMag Walkthrough.
Subject:
Computer Science
Engineering
Material Type:
Homework/Assignment
Author:
Margaret Burnett
11/24/2021
Unrestricted Use
Public Domain
Rating
0.0 stars
Activity where the class will perform a GenderMag walkthrough on the ACM website from the GenderMag persona’s perspective.
Subject:
Computer Science
Engineering
Information Science
Material Type:
Activity/Lab
Author:
The GenderMag Project
11/15/2021
Unrestricted Use
Public Domain
Rating
0.0 stars
Students mark on the continuum where they think their persona's cognitive styles are likely to fall.
Subject:
Computer Science
Engineering
Information Science
Material Type:
Activity/Lab
Author:
Margaret Burnett
Zoe Steine-Hanson
11/15/2021
Some Rights Reserved
Rating
0.0 stars
Learn how to add event listeners in jQuery so that your JavaScript can respond to events on the page, like when a user clicks a button or drags an image.
Subject:
Applied Science
Computer Science
Material Type:
Interactive
Lesson
Provider:
Provider Set:
Author:
Pamela Fox
07/11/2021
Unrestricted Use
CC BY
Rating
0.0 stars
This course will present advanced topics in Artificial Intelligence (AI), including inquiries into logic, artificial neural network and machine learning, and the Turing machine. Upon successful completion of this course, students will be able to: define the term 'intelligent agent,' list major problems in AI, and identify the major approaches to AI; translate problems into graphs and encode the procedures that search the solutions with the graph data structures; explain the differences between various types of logic and basic statistical tools used in AI; list the different types of learning algorithms and explain why they are different; list the most common methods of statistical learning and classification and explain the basic differences between them; describe the components of Turing machine; name the most important propositions in the philosophy of AI; list the major issues pertaining to the creation of machine consciousness; design a reasonable software agent with java code. (Computer Science 408)
Subject:
Computer Science
Philosophy
Material Type:
Full Course
Provider:
The Saylor Foundation
11/16/2011
Unrestricted Use
CC BY
Rating
0.0 stars
The course aims at providing the fundamental tools for effective C++ programming in the context of high-performance computing. The tools include generic programming techniques, API development, and specific C++-11/14/17 constructs. Starting from a basic knowledge of C++, the attendees should be able to start using C++ language to engineer durable abstractions to develop and optimize applications. Example usage of modern C++ concepts and features are taken from scientific applications used by the HPC community, giving the attendees the opportunity to see the presented tools in action in real world cases. Exercises are provided from a GitHub repository. This material is meant to reflect the current state of the current C++ standard. As the standard changes, some aspects of this course may become outdated.This course is an integral part of the ESiWACE-2 project, and we acknowledge the partial funding from that project. The contact person is [email protected].
Subject:
Computer Science
Material Type:
Full Course
Author:
William Sawyer
10/26/2021
Unrestricted Use
CC BY
Rating
5.0 stars
This course will expand upon SQL as well as other advanced topics, including query optimization, concurrency, data warehouses, object-oriented extensions, and XML. Additional topics covered in this course will help you become more proficient in writing queries and will expand your knowledge base so that you have a better understanding of the field. Upon successful completion of this course, the student will be able to: write complex queries, including full outer joins, self-joins, sub queries, and set theoretic queries; write stored procedures and triggers; apply the principles of query optimization to a database schema; explain the various types of locking mechanisms utilized within database management systems; explain the different types of database failures as well as the methods used to recover from these failures; design queries against a distributed database management system; perform queries against database designed with object-relational extensions; develop and query XML files. (Computer Science 410)
Subject:
Computer Science
Material Type:
Full Course
Provider:
The Saylor Foundation
11/16/2011
Unrestricted Use
CC BY
Rating
0.0 stars
African American History and Culture contains 10 modules starting with African Origins - History and Captivity and continuing through Reconstruction. Openly-licensed course materials developed for the Open Educational Resources (OER) Degree Initiative, led by Achieving the Dream https://courses.lumenlearning.com/catalog/achievingthedream.
Subject:
Computer Science
U.S. History
Material Type:
Textbook
Provider:
Lumen Learning
Author:
Florida State College At Jacksonville
01/07/2020
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
Building on Complex Adaptive Systems theory and basic Agent Based Modeling knowledge presented in SPM4530, the Advanced course will focus on the model development process. The students are expected to conceptualize, develop and verify a model during the course, individually or in a group. The modeling tasks will be, as much as possible, based on real life research problems, formulated by various research groups from within and outside the faculty.
Study Goals The main goal of the course is to learn how to form a modeling question, perform a system decomposition, conceptualize and formalize the system elements, implement and verify the simulation and validate an Agent Based Model of a socio-technical system.
Subject:
Computer Science
Material Type:
Full Course
Provider:
Delft University of Technology
Provider Set:
Delft University OpenCourseWare
Author:
Dr. Ir. I. Nikolic
03/03/2016
Conditional Remix & Share Permitted
CC BY-NC-SA
Rating
0.0 stars
How can we improve the speed of a (deterministic) primality test? Created by Brit Cruise.
Subject:
Applied Science
Computer Science
Material Type:
Lesson
Provider: | 5,232 | 22,499 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2022-21 | latest | en | 0.749693 |
https://www.shaalaa.com/question-paper-solution/cbse-cbse-12-mathematics-class-12-2017-2018-delhi-set-3_14109 | 1,695,981,865,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510501.83/warc/CC-MAIN-20230929090526-20230929120526-00014.warc.gz | 1,056,803,465 | 32,493 | # Mathematics Delhi Set 3 2017-2018 Science (English Medium) Class 12 Question Paper Solution
Mathematics [Delhi Set 3]
Date: March 2018
[1] 1
Find the magnitude of each of two vectors veca and vecb having the same magnitude such that the angle between them is 60° and their scalar product is 9/2
Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[1] 2
Find the value of tan^(-1) sqrt3 - cot^(-1) (-sqrt3)
Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [0.02] Inverse Trigonometric Functions
[1] 3
If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5)∘(10), where * and ∘ are binary operations.
Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[1] 4
if the matrix A =[(0,a,-3),(2,0,-1),(b,1,0)] is skew symmetric, Find the value of 'a' and 'b'
Concept: Types of Matrices
Chapter: [0.03] Matrices
[2] 5
A black and a red dice are rolled. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
Concept: Conditional Probability
Chapter: [0.13] Probability
[2] 6
If θ is the angle between two vectors hati - 2hatj + 3hatk and 3hati - 2hatj + hatk find sin theta
Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter: [0.1] Vectors
[2] 7
Find the differential equation representing the family of curves y = ae^(bx + 5). where a and b are arbitrary constants.
Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.09] Differential Equations
[2] 8
Evaluate int (cos 2x + 2sin^2x)/(cos^2x) dx
Concept: Some Properties of Indefinite Integral
Chapter: [0.07] Integrals
[2] 9
The total cost C(x) in Rupees associated with the production of x units of an item is given by
C(x) = 0.007x3 – 0.003x2 + 15x + 4000.
Find the marginal cost when 17 units are produced
Concept: Rate of Change of Bodies or Quantities
Chapter: [0.06] Applications of Derivatives
[2] 10
Differentiate tan^(-1) ((1+cosx)/(sin x)) with respect to x
Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.05] Continuity and Differentiability
[2] 11
Given A = [(2,-3),(-4,7)] compute A^(-1) and show that 2A^(-1) = 9I - A
Concept: Types of Matrices
Chapter: [0.03] Matrices
[2] 12
Prove that 3sin^(-1)x = sin^(-1) (3x - 4x^3), x in [-1/2, 1/2]
Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[4] 13
Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X
Concept: Random Variables and Its Probability Distributions
Chapter: [0.13] Probability
[4] 14
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
Concept: Maxima and Minima
Chapter: [0.06] Applications of Derivatives
[4] 15 | Attempt Any One
[4] 15.1
Find the equations of the tangent and the normal, to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0.
Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
[4] 15.2
Find the intervals in which the function f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12 is (a) strictly increasing, (b) strictly decreasing
Concept: Increasing and Decreasing Functions
Chapter: [0.06] Applications of Derivatives
[4] 16 | Attempt Any One
[4] 16.1
if (x^2 + y^2)^2 = xy find (dy)/(dx)
Concept: Derivatives of Implicit Functions
Chapter: [0.05] Continuity and Differentiability
[4] 16.2
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find dy/dx when theta = pi/3
Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.05] Continuity and Differentiability
[4] 17
If y = sin (sin x), prove that (d^2y)/(dx^2) + tan x dy/dx + y cos^2 x = 0
Concept: Higher Order Derivative
Chapter: [0.05] Continuity and Differentiability
[4] 18
[4] 18.1
Find the particular solution of the differential equation ex tan y dx + (2 – ex) sec2 y dy = 0, give that y = pi/4 when x = 0
Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable Method
Chapter: [0.09] Differential Equations
[4] 18.2
Find the particular solution of the differential equation dy/dx + 2y tan x = sin x given that y = 0 when x = pi/3
Concept: Methods of Solving First Order, First Degree Differential Equations - Differential Equations with Variables Separable Method
Chapter: [0.09] Differential Equations
[4] 19
Find the shortest distance between the lines vecr = (4hati - hatj) + lambda(hati+2hatj-3hatk) and vecr = (hati - hatj + 2hatk) + mu(2hati + 4hatj - 5hatk)
Concept: Shortest Distance Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
[4] 20
Find int (2cos x)/((1-sinx)(1+sin^2 x)) dx
Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.07] Integrals
[4] 21
Suppose a girl throws a die. If she gets 1 or 2 she tosses a coin three times and notes the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3,4,5 or 6 with the die ?
Concept: Bayes’ Theorem
Chapter: [0.13] Probability
[4] 22
Let veca = 4hati + 5hatj - hatk, vecb = hati - 4hatj + 5hatk and vecc = 3hati + hatj - hatk. Find a vector vecd which is perpendicular to both vecc and vecb and vecd.veca = 21
Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter: [0.1] Vectors
[4] 23
Using properties of determinants, prove that |(1,1,1+3x),(1+3y, 1,1),(1,1+3z,1)| = 9(3xyz + xy + yz+ zx)
Concept: Properties of Determinants
Chapter: [0.04] Determinants
[6] 24
Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32.
Concept: Area Under Simple Curves
Chapter: [0.08] Applications of the Integrals
[6] 25 | Attempt Any One
[6] 25.1
Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(ab) : a∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]
Concept: Types of Relations
Chapter: [0.01] Relations and Functions
[6] 25.2
Show that the function f: ℝ → ℝ defined by f(x) = x/(x^2 + 1), ∀x in Ris neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)
Concept: Types of Functions
Chapter: [0.01] Relations and Functions
[6] 26
Find the distance of the point (−1, −5, −10) from the point of intersection of the line vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) and the plane vec r (hati-hatj+hatk)=5
Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [0.11] Three - Dimensional Geometry
[6] 27
A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws 'A' while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws 'B'. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws 'A' at a profit of 70 paise and screws 'B' at a profit of Rs 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit.
Concept: Different Types of Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 28 | Attempt Any One
[6] 28.1
Evaluate int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx
Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
[6] 28.2
Evaluate : int_1^3 (x^2 + 3x + e^x) dx as the limit of the sum.
Concept: Definite Integral as the Limit of a Sum
Chapter: [0.07] Integrals
[6] 29 | Attempt Any One
[6] 29.1
If A = [(2,-3,5),(3,2,-4),(1,1,-2)] find A−1. Using A−1 solve the system of equations
2x – 3y + 5z = 11
3x + 2y – 4z = – 5
x + y – 2z = – 3
Concept: Applications of Determinants and Matrices
Chapter: [0.04] Determinants
[6] 29.2
Using elementary row transformations, find the inverse of the matrix A = [(1,2,3),(2,5,7),(-2,-4,-5)]
Concept: Elementary Transformations
Chapter: [0.03] Matrices [0.04] Determinants
#### Request Question Paper
If you dont find a question paper, kindly write to us
View All Requests
#### Submit Question Paper
Help us maintain new question papers on Shaalaa.com, so we can continue to help students
only jpg, png and pdf files
## CBSE previous year question papers Class 12 Mathematics with solutions 2017 - 2018
CBSE Class 12 Maths question paper solution is key to score more marks in final exams. Students who have used our past year paper solution have significantly improved in speed and boosted their confidence to solve any question in the examination. Our CBSE Class 12 Maths question paper 2018 serve as a catalyst to prepare for your Mathematics board examination.
Previous year Question paper for CBSE Class 12 Maths-2018 is solved by experts. Solved question papers gives you the chance to check yourself after your mock test.
By referring the question paper Solutions for Mathematics, you can scale your preparation level and work on your weak areas. It will also help the candidates in developing the time-management skills. Practice makes perfect, and there is no better way to practice than to attempt previous year question paper solutions of CBSE Class 12.
How CBSE Class 12 Question Paper solutions Help Students ?
• Question paper solutions for Mathematics will helps students to prepare for exam.
• Question paper with answer will boost students confidence in exam time and also give you an idea About the important questions and topics to be prepared for the board exam.
• For finding solution of question papers no need to refer so multiple sources like textbook or guides. | 3,167 | 10,429 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2023-40 | latest | en | 0.78607 |
http://www.territorioscuola.com/wikipedia/en.wikipedia.php?title=Eigenvalue,_eigenvector_and_eigenspace | 1,398,001,624,000,000,000 | text/html | crawl-data/CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00506-ip-10-147-4-33.ec2.internal.warc.gz | 695,168,486 | 58,838 | # Eigenvalues and eigenvectors
In this shear mapping the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping, and since its length is unchanged its eigenvalue is 1.
An eigenvector of a square matrix $A$ is a non-zero vector $v$ that, when the matrix is multiplied by $v$, yields a constant multiple of $v$, the multiplier being commonly denoted by $\lambda$. That is:
$A v = \lambda v$
(Because this equation uses post-multiplication by $v$, it describes a right eigenvector.)
The number $\lambda$ is called the eigenvalue of $A$ corresponding to $v$.1
If 2D space is visualized as a piece of cloth being stretched by the matrix, the eigenvectors would make up the line along the direction the cloth is stretched in and the line of cloth at the center of the stretching, whose direction isn't changed by the stretching either. The eigenvalues for the first line would give the scale to which the cloth is stretched, and for the second line the scale to which it's tightened. A reflection may be viewed as stretching a line to scale -1 while shrinking the axis of reflection to scale 1. For 3D rotations, the eigenvectors form the axis of rotation, and since the scale of the axis is unchanged by the rotation, their eigenvalues are all 1.
In analytic geometry, for example, a three-element vector may be seen as an arrow in three-dimensional space starting at the origin. In that case, an eigenvector $v$ is an arrow whose direction is either preserved or exactly reversed after multiplication by $A$. The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or positive.
In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is replaced by any field of scalars (such as algebraic or complex numbers); the set of Cartesian vectors $\mathbb{R}^n$ is replaced by any vector space (such as the continuous functions, the polynomials or the trigonometric series), and matrix multiplication is replaced by any linear operator that maps vectors to vectors (such as the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "eigenfunction", "eigenmode", "eigenface", or "eigenstate". Thus, for example, the exponential function $f(x) = a^x$ is an eigenfunction of the derivative operator " ${}'$ ", with eigenvalue $\lambda = \ln a$, since its derivative is $f'(x) = (\ln a)a^x = \lambda f(x)$.
The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem of that matrix.2 Any multiple of an eigenvector is also an eigenvector, with the same eigenvalue. An eigenspace of a matrix $A$ is the set of all eigenvectors with the same eigenvalue, together with the zero vector.1 An eigenbasis for $A$ is any basis for the set of all vectors that consists of linearly independent eigenvectors of $A$. Not every matrix has an eigenbasis, but every symmetric matrix does.
The terms characteristic vector, characteristic value, and characteristic space are also used for these concepts. The prefix eigen- is adopted from the German word eigen for "self-" or "unique to", "peculiar to", or "belonging to" in the sense of "idiosyncratic" in relation to the originating matrix.
Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrix factorization, in quantum mechanics, and in many other areas.
## Definition
### Eigenvectors and eigenvalues of a real matrix
Matrix $A$ acts by stretching the vector $x$, not changing its direction, so $x$ is an eigenvector of $A$.
In many contexts, a vector can be assumed to be a list of real numbers (called elements), written vertically with brackets around the entire list, such as the vectors u and v below. Two vectors are said to be scalar multiples of each other (also called parallel or collinear) if they have the same number of elements, and if every element of one vector is obtained by multiplying each corresponding element in the other vector by the same number (known as a scaling factor, or a scalar). For example, the vectors
$u = \begin{bmatrix}1\\3\\4\end{bmatrix}\quad\quad\quad$ and $\quad\quad\quad v = \begin{bmatrix}-20\\-60\\-80\end{bmatrix}$
are scalar multiples of each other, because each element of $v$ is −20 times the corresponding element of $u$.
A vector with three elements, like $u$ or $v$ above, may represent a point in three-dimensional space, relative to some Cartesian coordinate system. It helps to think of such a vector as the tip of an arrow whose tail is at the origin of the coordinate system. In this case, the condition "$u$ is parallel to $v$" means that the two arrows lie on the same straight line, and may differ only in length and direction along that line.
If we multiply any square matrix $A$ with $n$ rows and $n$ columns by such a vector $v$, the result will be another vector $w = A v$, also with $n$ rows and one column. That is,
$\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \quad\quad$ is mapped to $\begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} \;=\; \begin{bmatrix} A_{1,1} & A_{1,2} & \ldots & A_{1,n} \\ A_{2,1} & A_{2,2} & \ldots & A_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n,1} & A_{n,2} & \ldots & A_{n,n} \\ \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$
where, for each index $i$,
$w_i = A_{i,1} v_1 + A_{i,2} v_2 + \cdots + A_{i,n} v_n = \sum_{j = 1}^{n} A_{i,j} v_j$
In general, if $v_j$ are not all zeros, the vectors $v$ and $A v$ will not be parallel. When they are parallel (that is, when there is some real number $\lambda$ such that $A v = \lambda v$) we say that $v$ is an eigenvector of $A$. In that case, the scale factor $\lambda$ is said to be the eigenvalue corresponding to that eigenvector.
In particular, multiplication by a 3×3 matrix $A$ may change both the direction and the magnitude of an arrow $v$ in three-dimensional space. However, if $v$ is an eigenvector of $A$ with eigenvalue $\lambda$, the operation may only change its length, and either keep its direction or flip it (make the arrow point in the exact opposite direction). Specifically, the length of the arrow will increase if $|\lambda| > 1$, remain the same if $|\lambda| = 1$, and decrease it if $|\lambda|< 1$. Moreover, the direction will be precisely the same if $\lambda > 0$, and flipped if $\lambda < 0$. If $\lambda = 0$, then the length of the arrow becomes zero.
#### An example
The transformation matrix $\bigl[ \begin{smallmatrix} 2 & 1\\ 1 & 2 \end{smallmatrix} \bigr]$ preserves the angle of arrows parallel to the lines from the origin to $\bigl[ \begin{smallmatrix} 1 \\ 1 \end{smallmatrix} \bigr]$ (in blue) and to $\bigl[ \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \bigr]$ (in violet). The points that lie on a line through the origin and an eigenvector remain on the line after the transformation. The arrows in red are not parallel to such a line, therefore their angle is altered by the transformation. See also: An extended version, showing all four quadrants.
For the transformation matrix
$A = \begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix},$
the vector
$v = \begin{bmatrix} 4 \\ -4 \end{bmatrix}$
is an eigenvector with eigenvalue 2. Indeed,
$A v = \begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix} \begin{bmatrix} 4 \\ -4 \end{bmatrix} = \begin{bmatrix} 3 \cdot 4 + 1 \cdot (-4) \\ 1 \cdot 4 + 3 \cdot (-4) \end{bmatrix} = \begin{bmatrix} 8 \\ -8 \end{bmatrix} = 2 \cdot \begin{bmatrix} 4 \\ -4 \end{bmatrix}.$
On the other hand the vector
$v = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$
is not an eigenvector, since
$\begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \cdot 0 + 1 \cdot 1 \\ 1 \cdot 0 + 3 \cdot 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix},$
and this vector is not a multiple of the original vector $v$.
#### Another example
For the matrix
$A= \begin{bmatrix} 1 & 2 & 0\\0 & 2 & 0\\ 0 & 0 & 3\end{bmatrix},$
we have
$A \begin{bmatrix} 1\\0\\0 \end{bmatrix} = \begin{bmatrix} 1\\0\\0 \end{bmatrix} = 1 \cdot \begin{bmatrix} 1\\0\\0 \end{bmatrix},\quad\quad$
$A \begin{bmatrix} 0\\0\\1 \end{bmatrix} = \begin{bmatrix} 0\\0\\3 \end{bmatrix} = 3 \cdot \begin{bmatrix} 0\\0\\1 \end{bmatrix}.\quad\quad$
Therefore, the vectors $[1,0,0]^\mathsf{T}$ and $[0,0,1]^\mathsf{T}$ are eigenvectors of $A$ corresponding to the eigenvalues 1 and 3 respectively. (Here the symbol ${}^\mathsf{T}$ indicates matrix transposition, in this case turning the row vectors into column vectors.)
#### Trivial cases
The identity matrix $I$ (whose general element $I_{i j}$ is 1 if $i = j$, and 0 otherwise) maps every vector to itself. Therefore, every vector is an eigenvector of $I$, with eigenvalue 1.
More generally, if $A$ is a diagonal matrix (with $A_{i j} = 0$ whenever $i \neq j$), and $v$ is a vector parallel to axis $i$ (that is, $v_i \neq 0$, and $v_j = 0$ if $j \neq i$), then $A v = \lambda v$ where $\lambda = A_{i i}$. That is, the eigenvalues of a diagonal matrix are the elements of its main diagonal. This is trivially the case of any 1 ×1 matrix.
### General definition
The concept of eigenvectors and eigenvalues extends naturally to abstract linear transformations on abstract vector spaces. Namely, let $V$ be any vector space over some field $K$ of scalars, and let $T$ be a linear transformation mapping $V$ into $V$. We say that a non-zero vector $v$ of $V$ is an eigenvector of $T$ if (and only if) there is a scalar $\lambda$ in $K$ such that
$T(v)=\lambda v$.
This equation is called the eigenvalue equation for $T$, and the scalar $\lambda$ is the eigenvalue of $T$ corresponding to the eigenvector $v$. Note that $T(v)$ means the result of applying the operator $T$ to the vector $v$, while $\lambda v$ means the product of the scalar $\lambda$ by $v$.3
The matrix-specific definition is a special case of this abstract definition. Namely, the vector space $V$ is the set of all column vectors of a certain size $n$×1, and $T$ is the linear transformation that consists in multiplying a vector by the given $n\times n$ matrix $A$.
Some authors allow $v$ to be the zero vector in the definition of eigenvector.4 This is reasonable as long as we define eigenvalues and eigenvectors carefully: If we would like the zero vector to be an eigenvector, then we must first define an eigenvalue of $T$ as a scalar $\lambda$ in $K$ such that there is a nonzero vector $v$ in $V$ with $T(v) = \lambda v$. We then define an eigenvector to be a vector $v$ in $V$ such that there is an eigenvalue $\lambda$ in $K$ with $T(v) = \lambda v$. This way, we ensure that it is not the case that every scalar is an eigenvalue corresponding to the zero vector.
### Eigenspace and spectrum
If $v$ is an eigenvector of $T$, with eigenvalue $\lambda$, then any scalar multiple $\alpha v$ of $v$ with nonzero $\alpha$ is also an eigenvector with eigenvalue $\lambda$, since $T(\alpha v) = \alpha T(v) = \alpha(\lambda v) = \lambda(\alpha v)$. Moreover, if $u$ and $v$ are distinct eigenvectors with the same eigenvalue $\lambda$, then $u+v$ is also an eigenvector with the same eigenvalue $\lambda$. Therefore, the set of all eigenvectors with the same eigenvalue $\lambda$, together with the zero vector, is a linear subspace of $V$, called the eigenspace of $T$ associated to $\lambda$.56 If that subspace has dimension 1, it is sometimes called an eigenline.7
The geometric multiplicity $\gamma_T(\lambda)$ of an eigenvalue $\lambda$ is the dimension of the eigenspace associated to $\lambda$, i.e. number of linearly independent eigenvectors with that eigenvalue.
The eigenspaces of T always form a direct sum (and as a consequence any family of eigenvectors for different eigenvalues is always linearly independent). Therefore the sum of the dimensions of the eigenspaces cannot exceed the dimension n of the space on which T operates, and in particular there cannot be more than n distinct eigenvalues.8
Any subspace spanned by eigenvectors of $T$ is an invariant subspace of $T$, and the restriction of T to such a subspace is diagonalizable.
The set of eigenvalues of $T$ is sometimes called the spectrum of $T$.
### Eigenbasis
An eigenbasis for a linear operator $T$ that operates on a vector space $V$ is a basis for $V$ that consists entirely of eigenvectors of $T$ (possibly with different eigenvalues). Such a basis exists precisely if the direct sum of the eigenspaces equals the whole space, in which case one can take the union of bases chosen in each of the eigenspaces as eigenbasis. The matrix of T in a given basis is diagonal precisely when that basis is an eigenbasis for T, and for this reason T is called diagonalizable if it admits an eigenbasis.
## Generalizations to infinite-dimensional spaces
The definition of eigenvalue of a linear transformation $T$ remains valid even if the underlying space $V$ is an infinite dimensional Hilbert or Banach space. Namely, a scalar $\lambda$ is an eigenvalue if and only if there is some nonzero vector $v$ such that $T(v) = \lambda v$.
### Eigenfunctions
A widely used class of linear operators acting on infinite dimensional spaces are the differential operators on function spaces. Let $D$ be a linear differential operator in on the space $\mathbf{C^\infty}$ of infinitely differentiable real functions of a real argument $t$. The eigenvalue equation for $D$ is the differential equation
$D f = \lambda f$
The functions that satisfy this equation are commonly called eigenfunctions of $D$. For the derivative operator $d/dt$, an eigenfunction is a function that, when differentiated, yields a constant times the original function. The solution is an exponential function
$f(t) = Ae^{\lambda t} ,$
including when $\lambda$ is zero when it becomes a constant function. Eigenfunctions are an essential tool in the solution of differential equations and many other applied and theoretical fields. For instance, the exponential functions are eigenfunctions of the shift operators. This is the basis of Fourier transform methods for solving problems.
### Spectral theory
If $\lambda$ is an eigenvalue of $T$, then the operator $T-\lambda I$ is not one-to-one, and therefore its inverse $(T-\lambda I)^{-1}$ does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional ones. In general, the operator $T - \lambda I$ may not have an inverse, even if $\lambda$ is not an eigenvalue.
For this reason, in functional analysis one defines the spectrum of a linear operator $T$ as the set of all scalars $\lambda$ for which the operator $T-\lambda I$ has no bounded inverse. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.
### Associative algebras and representation theory
More algebraically, rather than generalizing the vector space to an infinite dimensional space, one can generalize the algebraic object that is acting on the space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.
A closer analog of eigenvalues is given by the representation-theoretical concept of weight, with the analogs of eigenvectors and eigenspaces being weight vectors and weight spaces.
## Eigenvalues and eigenvectors of matrices
### Characteristic polynomial
The eigenvalue equation for a matrix $A$ is
$A v - \lambda v = 0,$
which is equivalent to
$(A-\lambda I)v = 0,$
where $I$ is the $n\times n$ identity matrix. It is a fundamental result of linear algebra that an equation $M v = 0$ has a non-zero solution $v$ if, and only if, the determinant $\det(M)$ of the matrix $M$ is zero. It follows that the eigenvalues of $A$ are precisely the real numbers $\lambda$ that satisfy the equation
$\det(A-\lambda I) = 0$
The left-hand side of this equation can be seen (using Leibniz' rule for the determinant) to be a polynomial function of the variable $\lambda$. The degree of this polynomial is $n$, the order of the matrix. Its coefficients depend on the entries of $A$, except that its term of degree $n$ is always $(-1)^n\lambda^n$. This polynomial is called the characteristic polynomial of $A$; and the above equation is called the characteristic equation (or, less often, the secular equation) of $A$.
For example, let $A$ be the matrix
$A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 4 & 9 \end{bmatrix}$
The characteristic polynomial of $A$ is
$\det (A-\lambda I) \;=\; \det \left(\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 4 & 9 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\right) \;=\; \det \begin{bmatrix} 2 - \lambda & 0 & 0 \\ 0 & 3 - \lambda & 4 \\ 0 & 4 & 9 - \lambda \end{bmatrix}$
which is
$(2 - \lambda) \bigl[ (3 - \lambda) (9 - \lambda) - 16 \bigr] = -\lambda^3 + 14\lambda^2 - 35\lambda + 22$
The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues of $A$, corresponding to the eigenvectors $[1,0,0]',$ $[0,2,-1]',$ and $[0,1,2]'$ (or any non-zero multiples thereof).
#### In the real domain
Since the eigenvalues are roots of the characteristic polynomial, an $n\times n$ matrix has at most $n$ eigenvalues. If the matrix has real entries, the coefficients of the characteristic polynomial are all real; but it may have fewer than $n$ real roots, or no real roots at all.
For example, consider the cyclic permutation matrix
$A = \begin{bmatrix} 0 & 1 & 0\\0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}$
This matrix shifts the coordinates of the vector up by one position, and moves the first coordinate to the bottom. Its characteristic polynomial is $1 - \lambda^3$ which has one real root $\lambda_1 = 1$. Any vector with three equal non-zero elements is an eigenvector for this eigenvalue. For example,
$A \begin{bmatrix} 5\\5\\5 \end{bmatrix} = \begin{bmatrix} 5\\5\\5 \end{bmatrix} = 1 \cdot \begin{bmatrix} 5\\5\\5 \end{bmatrix}$
#### In the complex domain
The fundamental theorem of algebra implies that the characteristic polynomial of an $n\times n$ matrix $A$, being a polynomial of degree $n$, has exactly $n$ complex roots. More precisely, it can be factored into the product of $n$ linear terms,
$\det(A-\lambda I) = (\lambda_1 - \lambda )(\lambda_2 - \lambda)\cdots(\lambda_n - \lambda)$
where each $\lambda_i$ is a complex number. The numbers $\lambda_1$, $\lambda_2$, ... $\lambda_n$, (which may not be all distinct) are roots of the polynomial, and are precisely the eigenvalues of $A$.
Even if the entries of $A$ are all real numbers, the eigenvalues may still have non-zero imaginary parts (and the elements of the corresponding eigenvectors will therefore also have non-zero imaginary parts). Also, the eigenvalues may be irrational numbers even if all the entries of $A$ are rational numbers, or all are integers. However, if the entries of $A$ are algebraic numbers (which include the rationals), the eigenvalues will be (complex) algebraic numbers too.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugate values, namely with the two members of each pair having the same real part and imaginary parts that differ only in sign. If the degree is odd, then by the intermediate value theorem at least one of the roots will be real. Therefore, any real matrix with odd order will have at least one real eigenvalue; whereas a real matrix with even order may have no real eigenvalues.
In the example of the 3×3 cyclic permutation matrix $A$, above, the characteristic polynomial $1 - \lambda^3$ has two additional non-real roots, namely
$\lambda_2 = -1/2 + \mathbf{i}\sqrt{3}/2\quad\quad$ and $\quad\quad\lambda_3 = \lambda_2^* = -1/2 - \mathbf{i}\sqrt{3}/2$,
where $\mathbf{i}= \sqrt{-1}$ is the imaginary unit. Note that $\lambda_2\lambda_3 = 1$, $\lambda_2^2 = \lambda_3$, and $\lambda_3^2 = \lambda_2$. Then
$A \begin{bmatrix} 1 \\ \lambda_2 \\ \lambda_3 \end{bmatrix} = \begin{bmatrix} \lambda_2\\ \lambda_3 \\1 \end{bmatrix} = \lambda_2 \cdot \begin{bmatrix} 1\\ \lambda_2 \\ \lambda_3 \end{bmatrix} \quad\quad$ and $\quad\quad A \begin{bmatrix} 1 \\ \lambda_3 \\ \lambda_2 \end{bmatrix} = \begin{bmatrix} \lambda_3 \\ \lambda_2 \\ 1 \end{bmatrix} = \lambda_3 \cdot \begin{bmatrix} 1 \\ \lambda_3 \\ \lambda_2 \end{bmatrix}$
Therefore, the vectors $[1,\lambda_2,\lambda_3]'$ and $[1,\lambda_3,\lambda_2]'$ are eigenvectors of $A$, with eigenvalues $\lambda_2$, and $\lambda_3$, respectively.
### Algebraic multiplicities
Let $\lambda_i$ be an eigenvalue of an $n\times n$ matrix $A$. The algebraic multiplicity $\mu_A(\lambda_i)$ of $\lambda_i$ is its multiplicity as a root of the characteristic polynomial, that is, the largest integer $k$ such that $(\lambda - \lambda_i)^k$ divides evenly that polynomial.
Like the geometric multiplicity $\gamma_A(\lambda_i)$, the algebraic multiplicity is an integer between 1 and $n$; and the sum $\boldsymbol{\mu}_A$ of $\mu_A(\lambda_i)$ over all distinct eigenvalues also cannot exceed $n$. If complex eigenvalues are considered, $\boldsymbol{\mu}_A$ is exactly $n$.
It can be proved that the geometric multiplicity $\gamma_A(\lambda_i)$ of an eigenvalue never exceeds its algebraic multiplicity $\mu_A(\lambda_i)$. Therefore, $\boldsymbol{\gamma}_A$ is at most $\boldsymbol{\mu}_A$.
If $\gamma_A(\lambda_i) = \mu_A(\lambda_i)$, then $\lambda_i$ is said to be a semisimple eigenvalue.
#### Example
For the matrix: $A= \begin{bmatrix} 2 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 3 \end{bmatrix},$
the characteristic polynomial of $A$ is $\det (A-\lambda I) \;=\; \det \begin{bmatrix} 2- \lambda & 0 & 0 & 0 \\ 1 & 2- \lambda & 0 & 0 \\ 0 & 1 & 3- \lambda & 0 \\ 0 & 0 & 1 & 3- \lambda \end{bmatrix}= (2 - \lambda)^2 (3 - \lambda)^2$,
being the product of the diagonal with a lower triangular matrix.
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by the vector $[0,1,-1,1]$, and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by $[0,0,0,1]$. Hence, the total algebraic multiplicity of A, denoted $\mu_A$, is 4, which is the most it could be for a 4 by 4 matrix. The geometric multiplicity $\gamma_A$ is 2, which is the smallest it could be for a matrix which has two distinct eigenvalues.
### Diagonalization and eigendecomposition
If the sum $\boldsymbol{\gamma}_A$ of the geometric multiplicities of all eigenvalues is exactly $n$, then $A$ has a set of $n$ linearly independent eigenvectors. Let $Q$ be a square matrix whose columns are those eigenvectors, in any order. Then we will have $A Q = Q\Lambda$, where $\Lambda$ is the diagonal matrix such that $\Lambda_{i i}$ is the eigenvalue associated to column $i$ of $Q$. Since the columns of $Q$ are linearly independent, the matrix $Q$ is invertible. Premultiplying both sides by $Q^{-1}$ we get $Q^{-1}A Q = \Lambda$. By definition, therefore, the matrix $A$ is diagonalizable.
Conversely, if $A$ is diagonalizable, let $Q$ be a non-singular square matrix such that $Q^{-1} A Q$ is some diagonal matrix $D$. Multiplying both sides on the left by $Q$ we get $A Q = Q D$. Therefore each column of $Q$ must be an eigenvector of $A$, whose eigenvalue is the corresponding element on the diagonal of $D$. Since the columns of $Q$ must be linearly independent, it follows that $\boldsymbol{\gamma}_A = n$. Thus $\boldsymbol{\gamma}_A$ is equal to $n$ if and only if $A$ is diagonalizable.
If $A$ is diagonalizable, the space of all $n$-element vectors can be decomposed into the direct sum of the eigenspaces of $A$. This decomposition is called the eigendecomposition of $A$, and it is preserved under change of coordinates.
A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvector can be generalized to generalized eigenvectors, and that of diagonal matrix to a Jordan form matrix. Over an algebraically closed field, any matrix $A$ has a Jordan form and therefore admits a basis of generalized eigenvectors, and a decomposition into generalized eigenspaces
### Further properties
Let $A$ be an arbitrary $n\times n$ matrix of complex numbers with eigenvalues $\lambda_1$, $\lambda_2$, ... $\lambda_n$. (Here it is understood that an eigenvalue with algebraic multiplicity $\mu$ occurs $\mu$ times in this list.) Then
• The trace of $A$, defined as the sum of its diagonal elements, is also the sum of all eigenvalues:
$\operatorname{tr}(A) = \sum_{i=1}^n A_{i i} = \sum_{i=1}^n \lambda_i = \lambda_1+ \lambda_2 +\cdots+ \lambda_n$.
• The determinant of $A$ is the product of all eigenvalues:
$\operatorname{det}(A) = \prod_{i=1}^n \lambda_i=\lambda_1\lambda_2\cdots\lambda_n$.
• The eigenvalues of the $k$th power of $A$, i.e. the eigenvalues of $A^k$, for any positive integer $k$, are $\lambda_1^k,\lambda_2^k,\dots,\lambda_n^k$
• The matrix $A$ is invertible if and only if all the eigenvalues $\lambda_i$ are nonzero.
• If $A$ is invertible, then the eigenvalues of $A^{-1}$ are $1/\lambda_1,1/\lambda_2,\dots,1/\lambda_n$
• If $A$ is equal to its conjugate transpose $A^*$ (in other words, if $A$ is Hermitian), then every eigenvalue is real. The same is true of any a symmetric real matrix. If $A$ is also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite every eigenvalue is positive, non-negative, negative, or non-positive respectively.
• Every eigenvalue of a unitary matrix has absolute value $|\lambda|=1$.
### Left and right eigenvectors
The use of matrices with a single column (rather than a single row) to represent vectors is traditional in many disciplines. For that reason, the word "eigenvector" almost always means a right eigenvector, namely a column vector that must be placed to the right of the matrix $A$ in the defining equation
$A v = \lambda v$.
There may be also single-row vectors that are unchanged when they occur on the left side of a product with a square matrix $A$; that is, which satisfy the equation
$u A = \lambda u$
Any such row vector $u$ is called a left eigenvector of $A$.
The left eigenvectors of $A$ are transposes of the right eigenvectors of the transposed matrix $A^\mathsf{T}$, since their defining equation is equivalent to
$A^\mathsf{T} u^\mathsf{T} = \lambda u^\mathsf{T}$
It follows that, if $A$ is Hermitian, its left and right eigenvectors are complex conjugates. In particular if $A$ is a real symmetric matrix, they are the same except for transposition.
### Variational characterization
In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of $H$ is the maximum value of the quadratic form $x^T H x/x^T x$. A value of $x$ that realizes that maximum, is an eigenvector. For more information, see Min-max theorem.
## Calculation
### Computing the eigenvalues
The eigenvalues of a matrix $A$ can be determined by finding the roots of the characteristic polynomial. Explicit algebraic formulas for the roots of a polynomial exist only if the degree $n$ is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more.
It turns out that any polynomial with degree $n$ is the characteristic polynomial of some companion matrix of order $n$. Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy.9 However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial).9
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the advent of the QR algorithm in 1961. 9 Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm.citation needed For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.9
### Computing the eigenvectors
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding non-zero solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
$A = \begin{bmatrix} 4 & 1\\6 & 3 \end{bmatrix}$
we can find its eigenvectors by solving the equation $A v = 6 v$, that is
$\begin{bmatrix} 4 & 1\\6 & 3 \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = 6 \cdot \begin{bmatrix}x\\y\end{bmatrix}$
This matrix equation is equivalent to two linear equations
$\left\{\begin{matrix} 4x + {\ }y &{}= 6x\\6x + 3y &{}=6 y\end{matrix}\right. \quad\quad\quad$ that is $\left\{\begin{matrix} -2x+ {\ }y &{}=0\\+6x-3y &{}=0\end{matrix}\right.$
Both equations reduce to the single linear equation $y=2x$. Therefore, any vector of the form $[a,2a]'$, for any non-zero real number $a$, is an eigenvector of $A$ with eigenvalue $\lambda = 6$.
The matrix $A$ above has another eigenvalue $\lambda=1$. A similar calculation shows that the corresponding eigenvectors are the non-zero solutions of $3x+y=0$, that is, any vector of the form $[b,-3b]'$, for any non-zero real number $b$.
Some numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation.
## History
Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.
In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.10 In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.11 Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.12
Fourier used the work of Laplace and Lagrange to solve the heat equation by separation of variables in his famous 1822 book Théorie analytique de la chaleur.13 Sturm developed Fourier's ideas further and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues.11 This was extended by Hermite in 1855 to what are now called Hermitian matrices.12 Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,11 and Clebsch found the corresponding result for skew-symmetric matrices.12 Finally, Weierstrass clarified an important aspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.11
In the meantime, Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory.14 Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.15
At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.16 He was the first to use the German word eigen to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.17
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G.F. Francis18 and Vera Kublanovskaya19 in 1961.20
## Applications
### Eigenvalues of geometric transformations
The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.
scaling unequal scaling rotation horizontal shear hyperbolic rotation illustration matrix $\begin{bmatrix}k & 0\\0 & k\end{bmatrix}$ $\begin{bmatrix}k_1 & 0\\0 & k_2\end{bmatrix}$ $\begin{bmatrix}c & -s \\ s & c\end{bmatrix}$ $c=\cos\theta$ $s=\sin\theta$ $\begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix}$ $\begin{bmatrix} c & s \\ s & c \end{bmatrix}$ $c=\cosh \varphi$ $s=\sinh \varphi$ characteristic polynomial $\ (\lambda - k)^2$ $(\lambda - k_1)(\lambda - k_2)$ $\lambda^2 - 2c\lambda + 1$ $\ (\lambda - 1)^2$ $\lambda^2 - 2c\lambda + 1$ eigenvalues $\lambda_i$ $\lambda_1 = \lambda_2 = k$ $\lambda_1 = k_1$ $\lambda_2 = k_2$ $\lambda_1 = e^{\mathbf{i}\theta}=c+s\mathbf{i}$ $\lambda_2 = e^{-\mathbf{i}\theta}=c-s\mathbf{i}$ $\lambda_1 = \lambda_2 = 1$ $\lambda_1 = e^\varphi$ $\lambda_2 = e^{-\varphi}$, algebraic multipl. $\mu_i=\mu(\lambda_i)$ $\mu_1 = 2$ $\mu_1 = 1$ $\mu_2 = 1$ $\mu_1 = 1$ $\mu_2 = 1$ $\mu_1 = 2$ $\mu_1 = 1$ $\mu_2 = 1$ geometric multipl. $\gamma_i = \gamma(\lambda_i)$ $\gamma_1 = 2$ $\gamma_1 = 1$ $\gamma_2 = 1$ $\gamma_1 = 1$ $\gamma_2 = 1$ $\gamma_1 = 1$ $\gamma_1 = 1$ $\gamma_2 = 1$ eigenvectors All non-zero vectors $u_1 = \begin{bmatrix}1\\0\end{bmatrix}$ $u_2 = \begin{bmatrix}0\\1\end{bmatrix}$ $u_1 = \begin{bmatrix}{\ }1\\-\mathbf{i}\end{bmatrix}$ $u_2 = \begin{bmatrix}{\ }1\\ +\mathbf{i}\end{bmatrix}$ $u_1 = \begin{bmatrix}1\\0\end{bmatrix}$ $u_1 = \begin{bmatrix}{\ }1\\{\ }1\end{bmatrix}$ $u_2 = \begin{bmatrix}{\ }1\\-1\end{bmatrix}.$
Note that the characteristic equation for a rotation is a quadratic equation with discriminant $D = -4(\sin\theta)^2$, which is a negative number whenever $\theta$ is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, $\cos\theta \pm \mathbf{i}\sin\theta$; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
### Schrödinger equation
The wavefunctions associated with the bound states of an electron in a hydrogen atom can be seen as the eigenvectors of the hydrogen atom Hamiltonian as well as of the angular momentum operator. They are associated with eigenvalues interpreted as their energies (increasing downward: $n=1,2,3,\ldots$) and angular momentum (increasing across: s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher probability density for a position measurement. The center of each figure is the atomic nucleus, a proton.
An example of an eigenvalue equation where the transformation $T$ is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:
$H\psi_E = E\psi_E \,$
where $H$, the Hamiltonian, is a second-order differential operator and $\psi_E$, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue $E$, interpreted as its energy.
However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for $\psi_E$ within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which $\psi_E$ and $H$ can be represented as a one-dimensional array and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.
The bra–ket notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by $|\Psi_E\rangle$. In this notation, the Schrödinger equation is:
$H|\Psi_E\rangle = E|\Psi_E\rangle$
where $|\Psi_E\rangle$ is an eigenstate of $H$. It is a self adjoint operator, the infinite dimensional analog of Hermitian matrices (see Observable). As in the matrix case, in the equation above $H|\Psi_E\rangle$ is understood to be the vector obtained by application of the transformation $H$ to $|\Psi_E\rangle$.
### Molecular orbitals
In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. If one wants to underline this aspect one speaks of nonlinear eigenvalue problem. Such equations are usually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, one often represents the Hartree–Fock equation in a non-orthogonal basis set. This particular representation is a generalized eigenvalue problem called Roothaan equations.
### Geology and glaciology
In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram,2122 or as a Stereonet on a Wulff Net.23
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered $v_1, v_2, v_3$ by their eigenvalues $E_1 \geq E_2 \geq E_3$;24 $v_1$ then is the primary orientation/dip of clast, $v_2$ is the secondary and $v_3$ is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of $E_1$, $E_2$, and $E_3$ are dictated by the nature of the sediment's fabric. If $E_1 = E_2 = E_3$, the fabric is said to be isotropic. If $E_1 = E_2 > E_3$, the fabric is said to be planar. If $E_1 > E_2 > E_3$, the fabric is said to be linear.25
### Principal components analysis
PCA of the multivariate Gaussian distribution centered at $(1,3)$ with a standard deviation of 3 in roughly the $(0.878,0.478)$ direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance.
The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal components analysis (PCA) in statistics. PCA studies linear relations among variables. PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an orthonormal eigen-basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal-components that are associated with most of the covariability among a number of observed data.
Principal component analysis is used to study large data sets, such as those encountered in data mining, chemical research, psychology, and in marketing. PCA is popular especially in psychology, in the field of psychometrics. In Q methodology, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of practical significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling.
### Vibration analysis
1st lateral bending (See vibration for more types of vibration)
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are used to determine the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors determine the shapes of these vibrational modes. In particular, undamped vibration is governed by
$m\ddot x + kx = 0$
or
$m\ddot x = -k x$
that is, acceleration is proportional to position (i.e., we expect $x$ to be sinusoidal in time).
In $n$ dimensions, $m$ becomes a mass matrix and $k$ a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem
$-k x = \omega^2 m x$
where $\omega^2$ is the eigenvalue and $\omega$ is the angular frequency. Note that the principal vibration modes are different from the principal compliance modes, which are the eigenvectors of $k$ alone. Furthermore, damped vibration, governed by
$m\ddot x + c \dot x + kx = 0$
$(\omega^2 m + \omega c + k)x = 0.$
This can be reduced to a generalized eigenvalue problem by clever use of algebra at the cost of solving a larger system.
The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems.
### Eigenfaces
Eigenfaces as examples of eigenvectors
In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel.26 The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal components analysis. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made.
Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems, for speaker adaptation.
### Tensor of moment of inertia
In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.
### Stress tensor
In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components.
### Eigenvalues of a graph
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix $A$, or (increasingly) of the graph's Laplacian matrix (see also Discrete Laplace operator), which is either $T - A$ (sometimes called the combinatorial Laplacian) or $I - T^{-1/2}A T^{-1/2}$ (sometimes called the normalized Laplacian), where $T$ is a diagonal matrix with $T_{i i}$ equal to the degree of vertex $v_i$, and in $T^{-1/2}$, the $i$th diagonal entry is $1/\sqrt{\operatorname{deg}(v_i)}$. The $k$th principal eigenvector of a graph is defined as either the eigenvector corresponding to the $k$th largest or $k$th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.
### Basic reproduction number
See Basic reproduction number
The basic reproduction number ($R_0$) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then $R_0$ is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, $t_G$, from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time $t_G$ has passed. $R_0$ is then the largest eigenvalue of the next generation matrix.2728
## Notes
1. ^ a b Wolfram Research, Inc. (2010) Eigenvector. Accessed on 2010-01-29.
2. ^ William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery (2007), Numerical Recipes: The Art of Scientific Computing, Chapter 11: Eigensystems., pages=563–597. Third edition, Cambridge University Press. ISBN 9780521880688
3. ^ See Korn & Korn 2000, Section 14.3.5a; Friedberg, Insel & Spence 1989, p. 217
4. ^ Axler, Sheldon, "Ch. 5", Linear Algebra Done Right (2nd ed.), p. 77
5. ^ Shilov 1977, p. 109
6. ^ Lemma for the eigenspace
7. ^
8. ^ For a proof of this lemma, see Roman 2008, Theorem 8.2 on p. 186; Shilov 1977, p. 109; Hefferon 2001, p. 364; Beezer 2006, Theorem EDELI on p. 469; and Lemma for linear independence of eigenvectors
9. ^ a b c d Trefethen, Lloyd N.; Bau, David (1997), Numerical Linear Algebra, SIAM
10. ^ See Hawkins 1975, §2
11. ^ a b c d See Hawkins 1975, §3
12. ^ a b c See Kline 1972, pp. 807–808
13. ^ See Kline 1972, p. 673
14. ^ See Kline 1972, pp. 715–716
15. ^ See Kline 1972, pp. 706–707
16. ^ See Kline 1972, p. 1063
17. ^ See Aldrich 2006
18. ^ Francis, J. G. F. (1961), "The QR Transformation, I (part 1)", The Computer Journal 4 (3): 265–271, doi:10.1093/comjnl/4.3.265 and Francis, J. G. F. (1962), "The QR Transformation, II (part 2)", The Computer Journal 4 (4): 332–345, doi:10.1093/comjnl/4.4.332
19. ^ Kublanovskaya, Vera N. (1961), "On some algorithms for the solution of the complete eigenvalue problem", USSR Computational Mathematics and Mathematical Physics 3: 637–657. Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 1 (4), 1961: 555–570
20. ^ See Golub & van Loan 1996, §7.3; Meyer 2000, §7.3
21. ^ Graham, D.; Midgley, N. (2000), "Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method", Earth Surface Processes and Landforms 25 (13): 1473–1477, doi:10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C
22. ^ Sneed, E. D.; Folk, R. L. (1958), "Pebbles in the lower Colorado River, Texas, a study of particle morphogenesis", Journal of Geology 66 (2): 114–150, doi:10.1086/626490
23. ^ Knox-Robinson, C; Gardoll, Stephen J (1998), "GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system", Computers & Geosciences 24 (3): 243, doi:10.1016/S0098-3004(97)00122-2
24. ^ Stereo32 software
25. ^ Benn, D.; Evans, D. (2004), A Practical Guide to the study of Glacial Sediments, London: Arnold, pp. 103–107
26. ^ Xirouhakis, A.; Votsis, G.; Delopoulus, A. (2004), Estimation of 3D motion and structure of human faces (PDF), Online paper in PDF format, National Technical University of Athens
27. ^ Diekmann O, Heesterbeek JAP, Metz JAJ (1990), "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations", Journal of Mathematical Biology 28 (4): 365–382, doi:10.1007/BF00178324, PMID 2117040
28. ^ Odo Diekmann and J. A. P. Heesterbeek (2000), Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computational biology, West Sussex, England: John Wiley & Sons
## References
• Korn, Granino A.; Korn, Theresa M. (2000), "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review", New York: McGraw-Hill (1152 p., Dover Publications, 2 Revised edition), Bibcode:1968mhse.book.....K, ISBN 0-486-41147-8.
• Lipschutz, Seymour (1991), Schaum's outline of theory and problems of linear algebra, Schaum's outline series (2nd ed.), New York: McGraw-Hill Companies, ISBN 0-07-038007-4.
• Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (1989), Linear algebra (2nd ed.), Englewood Cliffs, New Jersey 07632: Prentice Hall, ISBN 0-13-537102-3.
• Aldrich, John (2006), "Eigenvalue, eigenfunction, eigenvector, and related terms", in Jeff Miller (Editor), Earliest Known Uses of Some of the Words of Mathematics, retrieved 2006-08-22
• Strang, Gilbert (1993), Introduction to linear algebra, Wellesley-Cambridge Press, Wellesley, Massachusetts, ISBN 0-9614088-5-5.
• Strang, Gilbert (2006), Linear algebra and its applications, Thomson, Brooks/Cole, Belmont, California, ISBN 0-03-010567-6.
• Bowen, Ray M.; Wang, Chao-Cheng (1980), Linear and multilinear algebra, Plenum Press, New York, ISBN 0-306-37508-7.
• Cohen-Tannoudji, Claude (1977), "Chapter II. The mathematical tools of quantum mechanics", Quantum mechanics, John Wiley & Sons, ISBN 0-471-16432-1.
• Fraleigh, John B.; Beauregard, Raymond A. (1995), Linear algebra (3rd ed.), Addison-Wesley Publishing Company, ISBN 0-201-83999-7 (international edition) Check |isbn= value (help).
• Golub, Gene H.; Van Loan, Charles F. (1996), Matrix computations (3rd Edition), Johns Hopkins University Press, Baltimore, Maryland, ISBN 978-0-8018-5414-9.
• Hawkins, T. (1975), "Cauchy and the spectral theory of matrices", Historia Mathematica 2: 1–29, doi:10.1016/0315-0860(75)90032-4.
• Horn, Roger A.; Johnson, Charles F. (1985), Matrix analysis, Cambridge University Press, ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback) Check |isbn= value (help).
• Kline, Morris (1972), Mathematical thought from ancient to modern times, Oxford University Press, ISBN 0-19-501496-0.
• Meyer, Carl D. (2000), Matrix analysis and applied linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, ISBN 978-0-89871-454-8.
• Brown, Maureen (October 2004), Illuminating Patterns of Perception: An Overview of Q Methodology.
• Golub, Gene F.; van der Vorst, Henk A. (2000), "Eigenvalue computation in the 20th century", Journal of Computational and Applied Mathematics 123: 35–65, doi:10.1016/S0377-0427(00)00413-1.
• Akivis, Max A.; Vladislav V. Goldberg (1969), Tensor calculus, Russian, Science Publishers, Moscow .
• Gelfand, I. M. (1971), Lecture notes in linear algebra, Russian, Science Publishers, Moscow.
• Alexandrov, Pavel S. (1968), Lecture notes in analytical geometry, Russian, Science Publishers, Moscow.
• Carter, Tamara A.; Tapia, Richard A.; Papaconstantinou, Anne, Linear Algebra: An Introduction to Linear Algebra for Pre-Calculus Students, Rice University, Online Edition, retrieved 2008-02-19.
• Roman, Steven (2008), Advanced linear algebra (3rd ed.), New York: Springer Science + Business Media, LLC, ISBN 978-0-387-72828-5.
• Shilov, Georgi E. (1977), Linear algebra (translated and edited by Richard A. Silverman ed.), New York: Dover Publications, ISBN 0-486-63518-X.
• Hefferon, Jim (2001), Linear Algebra, Online book, St Michael's College, Colchester, Vermont, USA.
• Kuttler, Kenneth (2007), An introduction to linear algebra (PDF), Online e-book in PDF format, Brigham Young University.
• Demmel, James W. (1997), Applied numerical linear algebra, SIAM, ISBN 0-89871-389-7.
• Beezer, Robert A. (2006), A first course in linear algebra, Free online book under GNU licence, University of Puget Sound.
• Lancaster, P. (1973), Matrix theory, Russian, Moscow, Russia: Science Publishers.
• Halmos, Paul R. (1987), Finite-dimensional vector spaces (8th ed.), New York: Springer-Verlag, ISBN 0-387-90093-4.
• Pigolkina, T. S. and Shulman, V. S., Eigenvalue (Russian), In:Vinogradov, I. M. (Ed.), Mathematical Encyclopedia, Vol. 5, Soviet Encyclopedia, Moscow, 1977.
• Greub, Werner H. (1975), Linear Algebra (4th Edition), Springer-Verlag, New York, ISBN 0-387-90110-8.
• Larson, Ron; Edwards, Bruce H. (2003), Elementary linear algebra (5th ed.), Houghton Mifflin Company, ISBN 0-618-33567-6.
• Curtis, Charles W., Linear Algebra: An Introductory Approach, 347 p., Springer; 4th ed. 1984. Corr. 7th printing edition (August 19, 1999), ISBN 0-387-90992-3.
• Shores, Thomas S. (2007), Applied linear algebra and matrix analysis, Springer Science+Business Media, LLC, ISBN 0-387-33194-8.
• Sharipov, Ruslan A. (1996), Course of Linear Algebra and Multidimensional Geometry: the textbook, arXiv:math/0405323, ISBN 5-7477-0099-5.
• Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2005), Indefinite linear algebra and applications, Basel-Boston-Berlin: Birkhäuser Verlag, ISBN 3-7643-7349-0.
HPTS - Area Progetti - Edu-Soft - JavaEdu - N.Saperi - Ass.Scuola.. - TS BCTV - TS VideoRes - TSODP - TRTWE TSE-Wiki - Blog Lavoro - InterAzioni- NormaScuola - Editoriali - Job Search - DownFree !
TerritorioScuola. Some rights reserved. Informazioni d'uso ☞ | 14,931 | 55,353 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 473, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.09375 | 4 | CC-MAIN-2014-15 | latest | en | 0.940845 |
http://gmatclub.com/forum/for-the-system-of-equations-given-what-is-the-value-of-z-138734.html#p1120603 | 1,481,127,282,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542213.61/warc/CC-MAIN-20161202170902-00054-ip-10-31-129-80.ec2.internal.warc.gz | 113,742,852 | 58,647 | For the system of equations given, what is the value of z? : GMAT Data Sufficiency (DS)
Check GMAT Club App Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack
It is currently 07 Dec 2016, 08:14
# Chicago-Booth
is Releasing R1 Admission Decisions | Keep Watch on App Tracker | Join Chat Room2 for Live Updates
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# For the system of equations given, what is the value of z?
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
### Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 35912
Followers: 6851
Kudos [?]: 90015 [0], given: 10402
For the system of equations given, what is the value of z? [#permalink]
### Show Tags
11 Sep 2012, 03:46
00:00
Difficulty:
5% (low)
Question Stats:
92% (01:39) correct 8% (00:39) wrong based on 660 sessions
### HideShow timer Statistics
x-4=z
y-x=8
8-z=t
For the system of equations given, what is the value of z?
(1) x = 7
(2) t = 5
Practice Questions
Question: 36
Page: 278
Difficulty: 550
[Reveal] Spoiler: OA
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 35912
Followers: 6851
Kudos [?]: 90015 [1] , given: 10402
Re: For the system of equations given, what is the value of z? [#permalink]
### Show Tags
11 Sep 2012, 03:47
1
KUDOS
Expert's post
SOLUTION
x-4=z
y-x=8
8-z=t
For the system of equations given, what is the value of z?
(1) x = 7. Substitute the value of x in the first equation to find z. Sufficient.
(2) t = 5. Substitute the value of t in the third equation to find z. Sufficient.
_________________
Senior Manager
Joined: 15 Jun 2010
Posts: 368
Schools: IE'14, ISB'14, Kellogg'15
WE 1: 7 Yrs in Automobile (Commercial Vehicle industry)
Followers: 11
Kudos [?]: 366 [2] , given: 50
Re: For the system of equations given, what is the value of z? [#permalink]
### Show Tags
12 Sep 2012, 05:24
2
KUDOS
Bunuel wrote:
x-4=z
y-x=8
8-z=t
For the system of equations given, what is the value of z?
(1) x = 7
(2) t = 5
St 1: Sufficient: x=7, substituting in Eq 1, we will get value of Z, 7-4=z=3,
ST 2. Sufficient: t=5, substituting in Eq 3, We will get value of Z, 8-Z=5, Z= 3.
_________________
Regards
SD
-----------------------------
Press Kudos if you like my post.
Debrief 610-540-580-710(Long Journey): http://gmatclub.com/forum/from-600-540-580-710-finally-achieved-in-4th-attempt-142456.html
Director
Joined: 24 Aug 2009
Posts: 504
Schools: Harvard, Columbia, Stern, Booth, LSB,
Followers: 17
Kudos [?]: 665 [1] , given: 241
Re: For the system of equations given, what is the value of z? [#permalink]
### Show Tags
12 Sep 2012, 12:06
1
KUDOS
Same reasoning as above.
_________________
If you like my Question/Explanation or the contribution, Kindly appreciate by pressing KUDOS.
Kudos always maximizes GMATCLUB worth
-Game Theory
If you have any question regarding my post, kindly pm me or else I won't be able to reply
Math Expert
Joined: 02 Sep 2009
Posts: 35912
Followers: 6851
Kudos [?]: 90015 [0], given: 10402
Re: For the system of equations given, what is the value of z? [#permalink]
### Show Tags
14 Sep 2012, 05:08
SOLUTION
x-4=z
y-x=8
8-z=t
For the system of equations given, what is the value of z?
(1) x = 7. Substitute the value of x in the first equation to find z. Sufficient.
(2) t = 5. Substitute the value of t in the third equation to find z. Sufficient.
Kudos points given to everyone with correct solution. Let me know if I missed someone.
_________________
Manager
Joined: 12 Jan 2013
Posts: 244
Followers: 4
Kudos [?]: 69 [0], given: 47
Re: For the system of equations given, what is the value of z? [#permalink]
### Show Tags
18 Jul 2013, 13:58
I didnt even solve for z, all I used was the fact that "ONE unknown and TWO knowns means you can solve for the unknown".
Hence, we are given x in s(1) and that is sufficent just by looking at the top equation. Then we are given t in s(2) which is enough for the third equation.
The reason this method is superior to any other is that you quickly can move on to the next quant problem on the test, I solved this q in 35 seconds using this method.
If there are any holes in this approach, please speak up and correct me.
Current Student
Joined: 25 Sep 2012
Posts: 300
Location: India
Concentration: Strategy, Marketing
GMAT 1: 660 Q49 V31
GMAT 2: 680 Q48 V34
Followers: 2
Kudos [?]: 129 [0], given: 242
Re: For the system of equations given, what is the value of z? [#permalink]
### Show Tags
30 May 2014, 00:24
this is sub500
Intern
Joined: 10 Sep 2015
Posts: 18
Followers: 0
Kudos [?]: 1 [0], given: 19
Re: For the system of equations given, what is the value of z? [#permalink]
### Show Tags
10 Sep 2015, 23:31
Basic rule for solving linear equation
NO of Unknown === no of equation
In this question :
NO of Unknown==4
no of equation ==3
both the statements are reducing 1 unknown making the equation solvable
hence D
Senior Manager
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 392
Followers: 21
Kudos [?]: 138 [0], given: 2
Re: For the system of equations given, what is the value of z? [#permalink]
### Show Tags
09 Aug 2016, 12:49
Quote:
x-4=z
y-x=8
8-z=t
For the system of equations given, what is the value of z?
(1) x = 7
(2) t = 5
We are given 3 equations:
1) x – 4 = z
2) y – x = 8
3) 8 – z = t
We need to determine the value of z.
Statement One Alone:
x = 7
We see that we can substitute 7 for x in the first equation and then determine the value of z.
x – 4 = z
7 – 4 = z
3 = z
Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.
Statement Two Alone:
t = 5
We see that we can substitute 5 for t in equation 3 and then determine the value of z.
8 – z = t
8 – z = 5
3 = z
Statement two alone is sufficient to answer the question.
The answer is D.
_________________
Jeffrey Miller
Jeffrey Miller
Head of GMAT Instruction
Re: For the system of equations given, what is the value of z? [#permalink] 09 Aug 2016, 12:49
Similar topics Replies Last post
Similar
Topics:
17 What is the value of integer z? 15 09 Feb 2015, 05:30
2 Given the linear equation y = kx + n, for what values of x w 2 12 Jun 2013, 02:04
2 What is the value of z? 6 16 May 2013, 02:41
8 Given the linear equation y = ax + b, for what values of x w 5 09 Nov 2012, 20:49
4 Given that X = (Y–Z)^2. What is the value of X? 5 25 Aug 2012, 20:12
Display posts from previous: Sort by
# For the system of equations given, what is the value of z?
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2,252 | 7,388 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2016-50 | latest | en | 0.806528 |
https://calculationcalculator.com/chatak-to-square-inch | 1,726,104,661,000,000,000 | text/html | "crawl-data/CC-MAIN-2024-38/segments/1725700651420.25/warc/CC-MAIN-20240912011254-20240912041254-007(...TRUNCATED) | 136,109,596 | 22,566 | "# Chatak to Square Inch Conversion\n\n## 1 Chatak is equal to how many Square Inch?\n\n### 6480 Squ(...TRUNCATED) | 2,488 | 6,914 | "{\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\":(...TRUNCATED) | 3.515625 | 4 | CC-MAIN-2024-38 | latest | en | 0.693769 |
End of preview. Expand
in Data Studio
README.md exists but content is empty.
- Downloads last month
- 4