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第六节 极小化方法

第六节 极小化方法. 一、与线性方程组等价的变分问题. 二、最速下降法. 三、共轭斜量法 ( 共轭梯度法 ). 四、预条件共轭斜量法. 一、与线性方程组等价的变分问题. 二、最速下降法. clear;x=-18:0.5:18; y=x'; X=ones(size(y))*x; Y=y*ones(size(x)); Z=0.5*(X.^2+6*Y.^2+4*X.*Y)-(4*X+10*Y); meshc(Z); colormap(hot) xlabel('x'),ylabel('y'),zlabel('z').

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第六节 极小化方法

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  1. 第六节 极小化方法 一、与线性方程组等价的变分问题 二、最速下降法 三、共轭斜量法(共轭梯度法) 四、预条件共轭斜量法

  2. 一、与线性方程组等价的变分问题

  3. 二、最速下降法

  4. clear;x=-18:0.5:18; y=x'; X=ones(size(y))*x; Y=y*ones(size(x)); Z=0.5*(X.^2+6*Y.^2+4*X.*Y)-(4*X+10*Y); meshc(Z); colormap(hot) xlabel('x'),ylabel('y'),zlabel('z')

  5. 三、共轭斜量法 (CG) (共轭梯度法)

  6. 公式化简

  7. 公式化简

  8. 四、预条件共轭斜量法(PCG)

  9. 预条件共轭斜量法 实际计算，可通过变换，转化成用原方程组的量来计算。

  10. 预优矩阵的选取 下面介绍几种选取预优矩阵的方案:

  11. MATLAB调用格式: r=cholinc(sa,tol) 或 r=cholinc(sa,’0’) 其中系数矩阵a必须用稀疏形式 sa. Cho.m a=[3,-1,0,2;-1,3,-1,0;0,-1,3,-1;2,0,-1,3]; sa=sparse(a); r=cholinc(sa,'0') full(r)

  12. 预条件共轭斜量法MATLAB的三种调用格式: 1.不用预优矩阵的共轭斜量法 x=pcg(a,b,tol,kmax) 2.用预优矩阵的共轭斜量法 (1)x=pcg(a,b,tol,kmax,m) (2) r=chol(m) x=pcg(a,b,tol,kmax,r’,r,x0) 3.未给定预优矩阵的共轭斜量法 r=cholinc(sa,’0’) x=pcg(a,b,tol,kmax,r’,r,x0)

  13. 二版习题P150----12 三版习题P140----28，29，30

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