text: Click here to load reader
  
  # Simpson rule Composite Simpson rule
  
  Feb 22, 2016
  
  ## Documents
  
  niyati
  
  Simpson rule Composite Simpson rule. Simpson rule for numerical integration. y=sin(x) for x within [1.2 2]. a=1.2;b=2; x= linspace (a,b);plot(x,sin(x));hold on. Quadratic polynomial. Approximate sin using a quadratic polynomial . c=0.5*(a+b); x=[a b c]; y=sin(x); p=polyfit(x,y,2); - PowerPoint PPT Presentation
  
  2008, Applied Mathematics NDHU1Simpson ruleComposite Simpson rule
  
  Simpson rule for numerical integration
  
  2008, Applied Mathematics NDHU22008, Applied Mathematics NDHU3a=1.2;b=2;x=linspace(a,b);plot(x,sin(x));hold on
  
  y=sin(x) for x within [1.2 2]2008, Applied Mathematics NDHU4Quadratic polynomialc=0.5*(a+b);x=[a b c]; y=sin(x);p=polyfit(x,y,2);z=linspace(a,b);plot(z,polyval(p,z) ,'r')
  
  Approximate sin using a quadratic polynomial 2008, Applied Mathematics NDHU5a=1;b=3;x=linspace(a,b);plot(x,sin(x))hold on;c=0.5*(a+b);x=[a b c]; y=sin(x);p=polyfit(x,y,2);z=linspace(a,b);plot(z,polyval(p,z) ,'r')
  
  Approximate sin(x) within [1 3] by a quadratic polynomial 2008, Applied Mathematics NDHU6a=1;b=3;x=linspace(a,b);plot(x,sin(x))hold on;c=0.5*(a+b);x=[a b ]; y=sin(x);p=polyfit(x,y,1);z=linspace(a,b);plot(z,polyval(p,z) ,'r')
  
  Approximate sin(x) within [1 3] by a line
  
  2008, Applied Mathematics NDHU7
  
  Strategy I : Apply Trapezoid rule to calculate area under a lineStrategy II : Apply Simpson rule to calculate area under a second order polynomial
  
  Strategy II is more accurate and general than strategy I, since a line is a special case of quadratic polynomialSimpson ruleOriginal task: integration of f(x) within [a,b]c=(a+b)/2Numerical taskApproximate f(x) within [a,b] by a quadratic polynomial, p(x)Integration of p(x) within [a,b]2008, Applied Mathematics NDHU82008, Applied Mathematics NDHU9
  
  Blue: f(x)=sin(x) within [1,3] Red: a quadratic polynomial that pass (1,sin(1)), (2,sin(2)),(3,sin(3))2008, Applied Mathematics NDHU10
  
  The area under a quadratic polynomial is a sum of area I, II and III2008, Applied Mathematics NDHU11 The area under a quadratic polynomial is a sum of area I, II and III
  
  Partition [a,b] to three equal-size intervals, and use the high of the middle point c to produce three Trapezoids
  
  Use the composite Trapezoid rule to determine area I, II and III h/2*(f(a)+f(c) +f(c)+f(c)+ f(c)+f(b)) substitute h=(b-a)/3 and c=(a+b)/2 area I + II + III = (b-a)/6 *(f(a)+4*f((a+b)/2) +f(b))
  
  2008, Applied Mathematics NDHU12
  
  2008, Applied Mathematics NDHU13
  
  2008, Applied Mathematics NDHU14
  
  2008, Applied Mathematics NDHU15
  
  152008, Applied Mathematics NDHU16
  
  Shift Lagrange polynomialRescale Lagrange polynomial2008, Applied Mathematics NDHU17Shortcut to Simpson 1/3 rule
  
  Partition [a,b] to three equal intervalsDraw three trapezoids, Q(I), Q(II) and Q(III)Calculate the sum of areas of the three trapezoidsProof2008, Applied Mathematics NDHU18
  
  Composite Simpson rulePartition [a,b] into n intervalIntegrate each interval by Simpson ruleh=(b-a)/2n
  
  2008, Applied Mathematics NDHU19Apply Simpson sule to each interval2008, Applied Mathematics NDHU20
  
  2008, Applied Mathematics NDHU21Composite Simpson rule
  
  2008, Applied Mathematics NDHU22
  
  Set nSet fxh=(b-a)/(2*n)ans=0;for i=0:n-1 aa=a+2*i*h;cc=aa+h;bb=cc+h;ans=ans+h/3*(fx(aa)+4*fx(cc)+fx(bb))exit2008, Applied Mathematics NDHU23Composite Simpson rule
  
  h=(b-a)/2n2008, Applied Mathematics NDHU24Composite Simpson rule
  
  h=(b-a)/2nSimpson's Rule for Numerical Integration
  
  Welcome message from author
  This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
  Related Documents
  ##### Chain Rule – Differentiating Composite Functions.
  Category: Documents
  ##### SIMPSON STRONG-TIE Online Ordering .Online Ordering SIMPSON....
  Category: Documents
  ##### 1 §4.2 The Chain Rule (Pages 251~261) Composite Functions:....
  Category: Documents
  ##### First Rule of Composite Materials
  Category: Documents
  ##### Los simpson
  Category: Education
  ##### Bart Simpson
  Category: Documents
metadata: {"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}