1. Root Finding UC Berkeley Fall 2004, E77 http://jagger.me.berkeley.edu/~pack/e77 Copyright 2005, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. http://jagger.me.berkeley.edu/~pack/e77http://creativecommons.org/licenses/by-sa/2.0/

2. Solving Algebraic Equations From linearity, it is easy to solve the equation or even the system of equations But what about an equation like

3. Solving Algebraic Equations In general, given a continuous function f, how do you find the (any, all) solutions to Graph of f(x) versus x x f(x) “ roots ” of f

4. Iterative Methods Iterative methods start with a “guess”, labeled x 0 and through “easy” calculations, generate a sequence x 1, x 2, x 3, … Goal is that sequence satisfies and convergence to a limit, x* x* is a solution

5. Rate of Convergence Suppose {x n } is a sequence, converging to a limit x*. Let e n denote the difference x*-x n If there is a constant C, and positive exponent  such that for large n then the sequence {x n } converges with order  to x*. –“linear” convergence is  =1 The sequence 3, 2.5, 2.25, 2.125, 2.0625, … converges linearly to 2 (here C = ½) The sequence 3, 2.1, 2.01, 2.001, 2.0001,… converges linearly to 2 (here C=0.1) –“quadratic” convergence is  =2 The sequence converges quadratically to 2 (here C=1)

6. Stopping Criteria Suppose {x n } is a sequence, converging to a limit x*. The limit x* has the property f(x*)=0. Let tol be a positive number Absolute (in x) |x n – x*| < tol Relative (in x) |x n – x*| < tol × |x*| Absolute in f |f(x n )| < tol But we don ’ t know x* “ Absolute ” |x n – x n-1 | < tol “ Relative ” |x n – x n-1 | < tol × |x n | Sometimes (in bisection, for example) we can bound this, even without knowing x*.

7. Bisection Method, basic idea Suppose f is continuous, There must be a root x* between x L and x R. Let x M be the midpoint. |x M -x*|≤0.5|x R -x L | Based on sign of replace either x L or x R with x M, and repeat. Intermediate value theorem

8. Simple pseudo-code: Bisection % Start with xL, xR, fL = f(xL); fR = f(xR); while StoppingCriteriaNotMet xM = 0.5*(xL+xR); yM = f(xM); if fL*fM<0 % replace R with M xR = xM; else xL = xM; end

9. Examples: Bisection

10. Newton’s Method: motivation graph of f(x) Graph of h(x), the straight-line approximation to f at x 0 functions are equal at x 0 slope of h (everywhere) equals the slope of f at x 0 graph of h is straight line

11. Newton’s Method: motivation graph of f(x) Graph of h(x), the straight-line approximation to f at x 0 Approximation to f, near x 0 Approximate solving f(x)=0 by solving (easier) h(x)=0

12. Newton’s Method: iteration graph of f(x) Graph of h(x), the straight-line approximation to f at x 0 Algorithm: Repeat, using x* app as the “ initial ” point. General form of the iteration

13. Newton’s Method: iteration General form of the iteration Facts about Newton ’ s method to solve f(x) = 0 At each step, need to evaluate the function and its derivative at a single point Not guaranteed to converge. It might “ cycle ”, and it might diverge. If it does converge, the convergence is quadratic. More complicated than bisection, can work better. Generalizes to N equations in N unknowns

14. Function handles and feval Several Matlab functions solve your problem by repeatedly calling functions that you supply –finding zeros of a function ( fzero ) –Integrating a function ( trapz, quad ) –Integrating a differential equation ( ode45 ) –minimizing a function ( fminbnd, fminsearch ) For example, in calling fzero, you must pass a reference- to-the-function (called a function handle) that you want fzero to find the zeros of. Use the @ operator to get the reference. >> shan = @sin; >> class(shan) >> feval(shan,linspace(0,pi,5)) New data type: add to list with double, cell, char, struct. Any function (built-in, mfile, etc)

15. Function handles and feval General syntax for feval is [out1,out2,…] = feval(FuncH,arg1,arg2,…) Output argument list function handle Input argument list