Solution of Nonlinear Equations: Lecture (II)

1
Chapter 2

• Solution of Nonlinear Equations Lecture (II)

2
Outline

• Numerical methods
• (2) Newton-Raphson (or simply Newtons) method

3
Newton-Paphson method

• Newtons method algorithm
• x1 The initial guess for the root of f(x) 0.
• x2 The next approximation to the root. The point
  of intersection of the tangent to the curve at x1
  with the x axis gives x2.
• The iterative procedure stops when meeting a
  convergence criterion
• f(xi) ? ?, xi xi-1 ? ?, or (xi-xi-1)/xi
  ? ?.

4
Derivation of the Newtons method

• Taylors series expansion of the function f(x)
  about an arbitrary point x1
• Considering only the first two terms in the
  expansion
• f(x) ? f(x1) (x x1) f (x1)
• Set f(x) ? f(x1) (x x1) f (x1) 0, and
  solve for the root
• To further improve the root, replace x2 with x1
  to obtain x3, and so on.

5
Notes on Newtons method

• Newtons method requires the derivative of the
  function, f df/dx some may be quite
  complicated.
• f(x) may not be available in explicit form, in
  which case numerical differentiation techniques
  are required.
• Newtons method converges very fast in most
  cases. However, it may not converge (see examples
  on the left).

6
Example Newtons method

• Example 2.8 Find the root of the equation
• using the Newton-Raphson method with starting
  point x1 0.0, and the convergence criterion,
  f(xi) ? ? with ? 10-5. Note The derivative
  of tan-1(u) is given by

7
Plot function f(x)

• Lets first plot the function f(x) from x 0 to
  1 to gain some insight on the behavior of the
  function.

Root
8
Flowchart
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Implement Newtons method by hand

• Show work step by step.
• Also, summarize the results in the Table below.

i xi f(xi) Is f(xi) ? ?? f(xi) (answer if the previous column is No
1
2
3
4

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Implement Newtons method write an M-file

• For the Example given on slide 6, write an
  M-file to compute the root of the equation using
  Newton-Raphson method.
• Follow the flowchart provided previously.
• Save the M-file as myNewton-Raphson.m.
• A copy of the M-file will be handed out later.