非線性方程數(shù)值方法

1、非線性方程數(shù)值方法1、 固定點迭代：求解方程 ) (x g x ? 的近似值，初始值為 0 x ，迭代式為 ) ( 1 n n x g x ? ?function [k,x1,err,x]=fixpt(g,x0,tol,maxl) %Input-g is the iteration function input as a string 'g' % -x0 is the initial guess for the

2、fixed point % -tol is the tolerance % -maxl is the maximum number of iterations %Output-k is the number of iteration that were carried out % -x1 is the approximation to the fixed point % -err is the e

3、rror in the approximation % -x contains the sequence {xn} x(1)=x0; for k=2:maxlx(k)=feval(g,x(k-1));err=abs(x(k)-x(k-1));relerr=err/(abs(x(k))+eps);x1=x(k);if(err0,break,end maxl=1+round((log(b-a)-log(delta))/log(2)

4、) for k=1:maxlc=(a+b)/2;yc=feval(f,c);if yc==0a=c;b=c;elseif yb*yc>0b=c;yb=yc;elsea=c;ya=yc;endif b-a<delta,break,end end c=(a+b)/2;err=abs(b-a);yc=feval(f,c); 例： f=inline('x*sin(x)-1') 5、Steffensen 加速

5、法：給定初始近似值 0 p ，快速尋找固定點方程 ) (x g x ? 的解，假設(shè) ) (x g 和) ( ' x g 是連續(xù)的，而且 1 ) ( ' ? x g ，通常的固定點迭代緩慢（線性）收斂到 p 。function [P,Q]=steff(f,df,p0,delta ,epsilon,maxl) %Input -f is the object function input as a string 'f

6、' % -df is the derivative of f input as a string 'df' % -p0 is the initial approximation to a zero of f % -delta is the tolerance for p0 % -epsilon is the tolerance for the function va

7、lues y % -maxl is the maximum number of iterations %Output -P is the Steffensen approximation to the zero % -Q is the matrix containing the Steffensen sequence %Initialize the matrix R R=zeros(maxl,3); R(1,1)

8、=p0; for k=1:maxlfor j=2:3%Denominator in Newton-Raphson method is calculated nrodenom=feval(df,R(k,j-1));%Calculate Newton-Raphson approximationsif nrodenom==0'division by zero in Newton-Raphson method'breakelse

9、R(k,j)=R(k,j-1)-feval(f,R(k,j-1))/nrodenom;end%Denominator in Aitken's Acceleration process calculatedaadenom=R(k,3)-2*R(k,2)+R(k,1);%Calculate Aitken's Acceleration approximationsif aadenom==0'division by ze

10、ro in Aitken''s Acceleration'breakelseR(k+1,1)=R(k,1)-(R(k,2)-R(k,1))^2/aadenom;end end%End program if division by zero occurredif(nrodenom==0)|(aadenom==0)breakend%Stopping criteria are evluated err=abs(R(k,

11、1)-R(k+1,1));relerr=err/(abs(R(k+1,1))+delta);y=feval(f,R(k+1,1));if(err<delta)|(relerr<delta)|(y<epsilon)%P and the matrix Q are determined P=R(k+1,1);Q=R(1:k+1,:);breakend end end 例： f=inline('x^2-5')