文档介绍：Introduction to puting -- A Matrix Vector Approach Using Matlab Written by Charles Loan
陈文斌
复旦大学
Chapter 8 Nonlinear Equations and Optimization
Finding Roots
Minimizing a function of one variable
Minimizing multivariate functions
Solving systems of nonlinear equations
We consider several types of nonlinear problems. They differ in whether or not the solution sought is a vector or a scalar and whether or not the goal is to produce a root or a minimizer of some given function. The presentation anized around a family of “orbit” problems. Suppose the vector-valued functions
Specify the location at time t of a pair of s that go around the Sun. Assume that the orbits are elliptical and that the sun is situated at (0,0).
Question 1. At what times are the s and the sun collinear? If f(t) is the the sine of the angle between P1 and P2, then this problem is equivalent to finding a zero of f(t). We focus on the bisection and Newton methods, and the Matlab zero-finder:
fzero
Question 2. How close do the two s get for a period of time? If f(t) is distance from P1 to P2, then this is a single-variable minimization problem. We develop the method of golden section search and discuss the Matlab minmizer fmin.
Question 3. How close do the two orbits get? The method of steepest descent and Matlab multivariable minimizer fmins are designed to solve problems of this variety.
Question 4. Where (if at all) do the two orbits intersect? This is an example of a multivariable root-finding problem:
The Newton framework for systems of nonlinear equations is discussed. Related topics include the use of finite differences to approximate the Jacobian and the Gauss-Newton method for the nonlinear least squares problem.
Finding Roots
quintic polynomial
Algorithms in this area are iterative and proceed by producing a sequence of numbers that converge to a root of interest.
Where do we start the iteration?
Does the iteration converge and how fast?
How do we know when to terminate