矩阵迭代法求解系统（矩阵）的前n阶固有频率和固有振型

博大广阔

%matrix iteration method
%this method assumes that the  frequencies are distrinv and well
%seprated that w1<w2<w3...
%base princile:according to the expansion theorem:
%X1=c1*u1+c2*u2+c3*u3...... =C*Uwhitch ui is eigenvector
%conbine with D*X=r*X
%we obtain: X(r+1)=D*Xr=[c1/w1^2*r,c2/w2^2r,c3/w3^2r....]*U
%since the natrual frequences are assumed to be w1<w2<w3...
%so we can neglect the second to end
%X(r+1)=(c1/w1^2*r )*u1
%so :w1^2=Xr/X(r+1);  终止条件Xr=X(r+1); X(r+1)就是对应的一阶阵型。because D*Xr=r*Xr

%为计算第R+1阶的特征向量和特征值，应用正交条件消去前R阶
%Ci=ui*M*X/ui*M*ui 当ui是第i个阵型的正则化后的阵型ui*M*ui=1；X是假设的阵型，Ci是第i个阵型对该假设阵型的贡献量

function [R,XX]=matrix_interation(K,M,number,accuracy)
  if nargin<3;warndlg('wrong inpiut');end
if nargin<4; accuracy=0.001;end

D=K*M;  
for n=1:1:number                    %number is the need solved digenvector
     Xold=ones(size(D,2),1);erro=10^4;
      while erro>=accuracy
   Xnew=D*Xold;
  r=Xnew(1);
   Xnew=Xnew/Xnew(1);
   erro=norm((Xnew-Xold),2);
   Xold=Xnew;
   end
Xs=Xold/sqrt(Xold.'*M*Xold);
D=D-r*Xs*Xs.'*M;                 %消去前面几项,更新D矩阵
%%%保存各个阵型和固有频率
R(n)=r;
XX(:,n)=Xs;
end
R=sqrt(1./R);
XX=XX;
end










%note:
%1)the actual number of interations necessary to find the value of w1 to
%within a desired degree of accuracy depends on how colsely the arbitrary
%trail vector X1 resembles the fundamental mode X1 and how well w1 and w2
%are separated
%2)has a district advandge:没有计算误差
%3）has a unsual case the trail X1 is exactly proportional to oneof Xi.

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