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Numerical Analysis - Advanced Topics in Root Finding -

Numerical Analysis - Advanced Topics in Root Finding -. Hanyang University Jong-Il Park. Summary: Root Finding. Error analysis of N-R method. Taylor series:. Newton-Raphson method:. --- (1). At the true solution x r :. --- (2). (1)  (2) :. Let. Quadratic convergence!.

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Numerical Analysis - Advanced Topics in Root Finding -

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  1. Numerical Analysis- Advanced Topics in Root Finding - Hanyang University Jong-Il Park

  2. Summary: Root Finding

  3. Error analysis of N-R method Taylor series: Newton-Raphson method: --- (1) At the true solution xr : --- (2) (1)  (2) : Let Quadratic convergence!

  4. Error Analysis of Secant Method • Convergence [Jeeves, 1958] • More efficient than N-R method if the calculation of f’(x) is complex • Modified secant method

  5. Multiple roots • Bracketing methods cannot cope with multiple roots • Open methods can find multiple roots • But the speed is slow in many cases • f’(x)0 will cause a problem(divide byzero) f(x) will always reach zero before f’(x) [Ralston and Rabinowitz, 1978]  if a zero check for f(x) is incorporated into the program, the computation can be terminated before f’(x) reaches zero • Alternative way using u(x)=f(x)/f’(x)

  6. Polynomial evaluation • Bad method • Worst method • Best method C code

  7. Polynomial differentiation or

  8. N-th derivatives

  9. Polynomial deflation • Multiplication by (x-a) • Synthetic division by (x-a)

  10. Bairstow’s method • Deflation method • Find r and s such that b0=b1=0 • Efficient routine using synthetic division exists • Good initial guess of r, s is important • Complex roots can be evaluated • Suitable for root polishing • In Numerical Recipes in C void qroot();  Read Sect.7.5.

  11. Laguerre method • Deflation method • Algorithm derivation • In Numerical Recipes in C: zroots() calls laguer();

  12. Application of Root Finding:Electric circuit design(1/2) • Problem: Find the proper R to dissipate energy to 1% at a specified rate(t=0.05s), given L=5H, C=10-4 F. • Solution:

  13. Application of Root Finding:Electric circuit design(2/2) • Reasonable initial range for R: • 0< R < 400 • To achieve r.e. of 10-4 % • Bisection method: 21 iterations • Other methods: ? • To achieve r.e. of 10-6 % • Bisection method: ? iterations • Other methods: ? • [Homework]

  14. Homework #4 [Due: 10/17] • Find the root of f(R)=0 and the number of iterations when the r.e.=10-4 and 10-6 respectively. • Solve the problems: 8-31, 32 • Explain the concept of “pointer to function” and describe how you use it in your homework #3.

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