2 Solutions of Equations in One Variable-Numerical Analysis,Burden课件.pptVIP

Ch2 Solutions of Equations in One Variable;2.1 The Bisection Methods;Algorithm of Bisection;Convergence Theorem;INPUT a, b; TOL ; N (maximum number of iteration).OUTPUT approximation solution x or message of failure.Step 1 Set i=1;FA=f(a)Step 2 while i≤N do step 3-6 Step 3 set x=a+(b-a)/2; FX=f(x) Step 4 If FX=0 or (b-a)/2TOL then OUTPUT x; STOP. Step 5 Set i=i+1 Step 6 If FA·FX0 then set a=x ;FA=FX else set b=xStep 7 OUTPUT(‘method failed after N iteration,N=’,N)STOP;Other stopping rules can be applied in Step 4: |xN- xN-1|?; |f(xN )|?; |xN - xN-1|/ |xN|? etc. Without additional knowledge about f and x, the last rule is the best stopping criteria to apply because it comes closest to testing relative error. ;2.2 Fixed-Point Iteration (Example);Theorem 2.2 (two propositions) (existence 、uniqueness error estimation);(A priori estimation posteriori estimation! );Proof;Moreover,;Remarks 1. A priori estimation; posteriori estimation; 2. Example: 12+3x-2sinx=0; 3. The Th. gives sufficient conditions.; function [x_star,index,it]=iterate(phi,x0,ep,it_max) if nargin4 it_max=100; end if nargin3 ep=1e-5; end index=0;k=1; while k=it_max x=feval(phi,x0); if abs(x-x0)ep index=1; break; end x0=x; k=k+1 end x_star=x;it=k;;Example　Solve: Solution:　 　　x=solve(‘exp(-x)=sin(pi*x/2)’) (or solve(‘exp(-x)-sin(pi*x/2)’)) Return gives： 　　　x= 　　　　.44357353410429277965457309417668 (32-figure)　　　　　　　　;2.3 Newton’s Method---An important iteration;Geometry explanation; Matlab;;活澄佛迭缚蛮砖剑粥太筑违森枢蕊向茄橡日另畅奈泌诈栓米苟两玻薛短灸2 Solutions of Equations in One Variable-Numerical Analysis,Burden课件2 Solutions of Equations in One Variable-Numerical Analysis,Burden课件;Secant Method (2-step method);Geometry Describe ;Method of False Position(Regula Falsi);2.4 Convergence Order for Iterative Methods; The relation between convergent (local step) stable