126_MATH 685 CSI 700 OR 682 Lecture Notes

Example: bisection iteration... Convergence of FPI... Convergence of Newton’s method... Inverse quadratic interpolation... Safeguarded methods Rapidly convergent methods for solving non

MATH 685/ CSI 700/ OR 682

Lecture Notes

Lecture 8 Nonlinear equations

Example

Existence/uniqueness

Examples in 1d

Example of a system in 2d

Multiplicity

Sensitivity and conditioning

Sensitivity and conditioning

Sensitivity and conditioning

Convergence rate

Convergence rate

Bisection method

Example: bisection iteration

Bisection method

Fixed-point iterations

Examples

Example: fixed point problems

Examples: FPI

Example: FPI

Convergence of FPI

Newton’s method

Newton’s method

Newton’s method

Convergence of Newton’s method

Newton’s method

Secant method

Secant method

Example

Higher-degree interpolation

Inverse interpolation

Inverse quadratic interpolation

Example

Linear fractional interpolation

Example

Safeguarded methods

 Rapidly convergent methods for solving nonlinear equations may not converge unless started close to solution, but safe methods are slow

 Hybrid methods combine features of both types of methods to achieve both speed and reliability

 Use rapidly convergent method, but maintain bracket around

solution

 If next approximate solution given by fast method falls outside bracketing interval, perform one iteration of safe method, such as bisection

 Fast method can then be tried again on smaller interval with

greater chance of success

 Ultimately, convergence rate of fast method should prevail

 Hybrid approach seldom does worse than safe method, and

usually does much better

 Popular combination is bisection and inverse quadratic

interpolation, for which no derivatives required

Zeros of polynomials

Systems of nonlinear

equations

Solving systems of nonlinear equations is much more difficult than scalar case because:

existence and number of solutions or good starting guess is much more complex

convergence to desired solution or to bracket

solution to produce absolutely safe method

dimension of problem

Fixed-point iteration (FPI)

Newton’s method

Example

Example

Convergence of Newton’s method

Cost of Newton’s method

Secant updating methods

Broyden’s method

Broyden’s method

Example

Example

Example

Example (cont)

Robust Newton-like methods

Trust-region methods