text: # Newton Raphson Method Text Book Notes: Nonlinear Equations
  
  ```Newton Method of Solving Nonlinear
  Equations
  In the Newton-Raphson method, the root is not bracketed. Only one initial
  guess of the root is needed to get the iterative process started to find the root of an
  equation. Hence, the method falls in the category of open methods.
  Newton-Raphson method is based on the principle that if the initial guess
  of the root of f(x)=0 is at xi, then if one draws the tangent to the curve at f(xi), the
  point xi+1 where the tangent crosses the x-axis is an improved estimate of the root
  (Figure 1).
  Using the definition of the slope of a function, at x  xi
  f (xi ) = tan θ
   i)==
  f (x
  f(xi )  0
  xi  xi 1
  which gives
  xi 1 = xi -
  f(xi )
  f'(x i )
  (1)
  Equation (1) is called the Newton-Raphson formula for solving nonlinear
  equations of the form f x   0 . So starting with an initial guess, xi, one can find
  the next guess, xi+1, by using equation (1). One can repeat this process until one
  finds the root within a desirable tolerance.
  Algorithm
  The steps of the Newton-Raphson method to find the root of an equation f(x) = 0
  are
  1. Evaluate f(x) symbolically
  2. Use an initial guess of the root, xi, to estimate the new value of the root
  xi+1 as
  xi 1 = xi -
  f(xi )
  f'(x i )
  3. Find the absolute relative approximate error, a as
  a =
  xi 1- xi
   100
  xi 1
  4. Compare the absolute relative approximate error, a
  with the pre-
  specified relative error tolerance, s . If a &gt; s , then go to step 2, else
  stop the algorithm. Also, check if the number of iterations has exceeded
  the maximum number of iterations.
  f(x)
  x f x 
  f(xi)
  i,
  i
  f(xi+1)
  
  xi+2
  xi+1
  xi
  X
  ```
explanation: The document is a set of textbook notes explaining the Newton-Raphson method for solving nonlinear equations. It provides a detailed description of the method, including the mathematical formula and an algorithm for its implementation. The text does not contain a specific question or a question and answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Newton Raphson Method Text Book Notes: Nonlinear Equations
  
  ```Newton Method of Solving Nonlinear
  Equations
  In the Newton-Raphson method, the root is not bracketed. Only one initial
  guess of the root is needed to get the iterative process started to find the root of an
  equation. Hence, the method falls in the category of open methods.
  Newton-Raphson method is based on the principle that if the initial guess
  of the root of f(x)=0 is at xi, then if one draws the tangent to the curve at f(xi), the
  point xi+1 where the tangent crosses the x-axis is an improved estimate of the root
  (Figure 1).
  Using the definition of the slope of a function, at x  xi
  f (xi ) = tan θ
   i)==
  f (x
  f(xi )  0
  xi  xi 1
  which gives
  xi 1 = xi -
  f(xi )
  f'(x i )
  (1)
  Equation (1) is called the Newton-Raphson formula for solving nonlinear
  equations of the form f x   0 . So starting with an initial guess, xi, one can find
  the next guess, xi+1, by using equation (1). One can repeat this process until one
  finds the root within a desirable tolerance.
  Algorithm
  The steps of the Newton-Raphson method to find the root of an equation f(x) = 0
  are
  1. Evaluate f(x) symbolically
  2. Use an initial guess of the root, xi, to estimate the new value of the root
  xi+1 as
  xi 1 = xi -
  f(xi )
  f'(x i )
  3. Find the absolute relative approximate error, a as
  a =
  xi 1- xi
   100
  xi 1
  4. Compare the absolute relative approximate error, a
  with the pre-
  specified relative error tolerance, s . If a &gt; s , then go to step 2, else
  stop the algorithm. Also, check if the number of iterations has exceeded
  the maximum number of iterations.
  f(x)
  x f x 
  f(xi)
  i,
  i
  f(xi+1)
  
  xi+2
  xi+1
  xi
  X
  ```
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a set of textbook notes explaining the Newton-Raphson method for solving nonlinear equations. It provides a detailed description of the method, including the mathematical formula and an algorithm for its implementation. The text does not contain a specific question or a question and answer.
  
  Label: no-question