数值分析幂法与反幂法-matlab程序

1、数值分析幂法与反幂法matlab程序随机产生一对称矩阵，对不同的原点位移和初值(至少取3个 ) 分别使用幂法求计算矩阵的主特征值及主特征向量，用反幂法求计算矩阵的按模最小特征值及特征向量。要求1）比较不同的原点位移和初值说明收敛性2）给出迭代结果，生成DOC文件。3）程序清单，生成M文件。解答： A=rand(5) % 随机产生 5*5矩阵求随机矩阵A =0.70940.16260.58530.69910.14930.75470.11900.22380.89090.25750.27600.49840.75130.95930.84070.67970.95970.25510.54720.25430

2、.65510.34040.50600.13860.8143 B=A+A %A矩阵和 A 的转置相加，得到随机对称矩阵 BB =1.41870.91730.86131.37880.80440.91730.23800.72221.85060.59790.86130.72221.50251.21441.34671.37881.85061.21441.09440.39290.80440.59791.34670.39291.62861.41870.91730.86131.37880.80440.91730.23800.72221.85060.5979B=0.86130.72221.50251.21441

3、.34671.37881.85061.21441.09440.39290.80440.59791.34670.39291.6286编写幂法、反幂法程序：functionm,u,index,k=pow(A,u,ep,it_max)% 求矩阵最大特征值的幂法，其中% A为矩阵；% ep为精度要求，缺省为 1e-5 ；% it_max 为最大迭代次数，缺省为 100；% m为绝对值最大的特征值；% u为对应最大特征值的特征向量；% index ，当 index=1 时，迭代成功，当 index=0 时，迭代失败 if nargin4it_max=100; endif nargin3 ep=1e-5;

4、end n=length(A); index=0; k=0; m1=0; m0=0.01;% 修改移位参数，原点移位法加速收敛，为 0时，即为幂法I=eye(n) T=A-m0*Iwhile k=it_max v=T*u; vmax,i=max(abs(v); m=v(i);u=v/m;if abs(m-m1)ep;index=1;break ;endm=m+m0;m1=m;k=k+1;endfunctionm,u,index,k=pow_inv(A,u， ep,it_max)% 求矩阵最大特征值的反幂法，其中% A为矩阵；% ep为精度要求，缺省为 1e-5 ；% it_max 为最大迭代次

5、数，缺省为 100；% m为绝对值最大的特征值；% u为对应最大特征值的特征向量；% index ，当 index=1 时，迭代成功，当 index=0 时，迭代失败 if nargin4it_max=100; endif nargin3 ep=1e-5;end n=length(A);index=0;k=0;m1=0;m0=0;% 修改移位参数，原点移位法加速收敛，为 0时，即为反幂法I=eye(n); T=A-m0*I; invT=inv(T); while k=it_maxv=invT*u;vmax,i=max(abs(v);m=v(i);u=v/m;ifabs(m-m1)epindex

6、=1;break ;endm1=m;k=k+1;endm=1/m;m=m+m0;修改输入的 m0的值，所得结果：幂法：u(0 )m0muindexk105.27100.92230.76850.96411.00000.7941 T11010.55.27100.92230.76850.96411.00000.7941 T11310.7941 T10.35.27100.92230.76850.96411.000011110.15.27100.92230.76850.96411.00000.7941 T110105.27100.92230.76850.96411.00000.7941 T1920.00

7、0015.27100.92230.76850.96411.00000.7941 T12930.7941 T40.0000055.27100.92230.76850.96411.00001950.0000015.27100.92230.76850.96411.00000.7941 T197805.27100.92230.76850.96411.00000.7941 T110450.000025.27100.92230.76850.96411.00000.7941 T010001360.7941 T890.000015.27100.92230.76850.96411.0000110150.0000

8、015.27100.92230.76850.96411.00000.7941 T110反幂法：(0 )um0muindexk10-0.003910.00005-0.0039110.00001-0.003910.000005-0.003910-0.0039230.00001-0.0039450.77250.81991.00000.40500.80780.77250.81991.00000.40500.80780.77250.81991.00000.40500.80780.77250.81991.00000.40500.80780.77250.81991.00000.40500.80780.77250.81991.00000.40500.8078T15T15T15T15T15T1578453689150.000005-0.00390.000001-0.00390-0.00390.00005-0.00390.00001-0.00390.000001-0.00390.77250.81991.00000.40500.80780.77250.81991.00000.40500.80780.