Question

我试图在MATLAB中编写Demmel - Kahan（＆＃34;计算精确奇异值分解＆＃34;）的SVD算法。

我写了几乎所有的代码，但我不了解这个算法的一些步骤。

算法所说的步骤＆＃34;专门处理，从iLower计算2x2子矩阵的SVD到iUpper + 1＆＃34; ...我是否必须将零偏移QR对角化应用于2x2子矩阵？

然后当它说我必须应用标准移位算法时，它没有解释如何使用来自双对角矩阵的对角线和上对角矢量来做到这一点。

但是，我不认为问题只出现在这些步骤中，因为有时候我的代码会运行，结果就像matlab中的svd（）内置函数一样。

我将这篇文章与Demmel - Kahan的算法以及我在这里写的代码联系起来，也许有人可以帮助我。

谢谢！ Demmel - Kahan High Accurancy Singular Value Decomposition

dir=0;
eps = 1e-010; 
tol = 100*eps; 
maxit = 3*(n^2); 
fudge=min(m,n); 
s=diag(B);
e=diag(B,1);
lambda=abs(s);

for j=n-1:-1:1
    lambdas(j)=abs(s(j))*(lambda(j+1)/(lambda(j+1)+abs(e(j))));
end
mu(1)=abs(s(1));
for j = 1:n-1
    mu(j+1) = abs(s(j+1))*(mu(j)/(mu(j)+abs(e(j))));
end
lambdaMin=min(lambdas);
muMin=min(mu);
sigmaLower = min(lambdaMin,muMin);
s1=max(abs(s));
e1=max(abs(e));
sigmaUpper = max(s1,e1);
thresh = max((tol*sigmaLower),(maxit*realmin));
for iter=1:maxit
    for i=1:n-1
        if abs(e(i))<=thresh
            iUpper=i;
        else iUpper=n;
        end
    end
    if iUpper==1    
        break;
    end

    for i=n-1:-1:1
        if abs(e(i))<=thresh
            iPrime=i;
        else iPrime=0;
        end
    end
    iLower=iPrime+1;
        tmp=B(iLower:iUpper,iLower:iUpper);
    if iUpper==iLower+1
                [s,e]=ZeroShiftQR(s,e,iLower,iUpper,1);
                e(iLower)=0;
        B(iLower:iUpper,iLower:iUpper)=diag(s)+diag(e,1);
        continue;
    end
        if B(iLower:iUpper,iLower:iUpper)~=tmp
        s=diag(B);
        e=diag(B,1);
        if abs(s(iLower))<=abs(s(iUpper))
            dir=1;
        else dir=2;
        end
    end
% Convergence criteria
if dir==1 % Convergence criteria 1b,1a,2a
    if abs(e(iUpper-1)/lambdas(iUpper))<=tol % 1b to e(iUpper-1)
        e(iUpper-1)=0;
    end
    for j=1:n-1 % Criteria 1a
        if abs(e(j)/mu(j))<=tol
            e(j)=0;
        end
    end
    % Criteria 2a to e(iUpper-1)
    tmp=mu(1)/sqrt(n-1);
    for i=2:n-1
        if mu(i)/sqrt(n-1)<tmp
            tmp=mu(i)/sqrt(n-1);
        end
    end
    if e(iUpper-1)*e(iUpper-1)<=0.5*tol*(tmp*tmp-abs(s(iUpper))*abs(s(iUpper)))
        e(iUpper-1)=0;
    end
else % Convergence criteria 1a,1b,2b
    if abs(e(iLower)/mu(iLower))<=tol % Criteria 1a to e(iLower)
        e(iLower)=0;
    end
    for j=n-1:-1:1 % Criteria 1b
        if abs(e(j)/lambdas(j))<=tol
            e(j)=0;
        end
    end
    % Criteria 2b 
    tmp=lambdas(2)/sqrt(n-1);
    for i=3:n-1
        if lambdas(i)/sqrt(n-1)<tmp
           tmp=lambdas(i)/sqrt(n-1);
        end
    end
    if e(iLower)*e(iLower)<=0.5*tol*(tmp*tmp-abs(s(iLower))*abs(s(iLower)))
        e(iLower)=0;
    end
end
% Compute the shift
if (fudge*tol*sigmaLower/sigmaUpper)<=eps
    shift=0;
else if dir==1
        w=s(iUpper);
        C=diag(s)+diag(e,1);
        C=C*C';
        C=C(end-1:end,end-1:end);
        [lambda_1,lambda_2] = eig2(C); % Calculate the eigenvalues of C
        if abs(C(2,2)-lambda_1)<abs(C(2,2)-lambda_2)
            shift=lambda_1;
        else shift=lambda_2;
        end
    else w=s(iLower);
         C=diag(s)+diag(e,1);
         C=C*C';
         C=C(1:2,1:2);
         [lambda_1,lambda_2] = eig2(C);
         if abs(C(2,2)-lambda_1)<abs(C(2,2)-lambda_2)
            shift=lambda_1;
         else shift=lambda_2;
         end
    end
    if (shift/w)^2<=eps
        shift=0;
    end
end
% QR iterations
if shift==0
    if dir==1
        [s,e]=ZeroShiftQR(s,e,iLower,iUpper,1); % Zero-shift QR iterations with direction down
        if e(iUpper-1)<=thresh
            e(iUpper-1)=0;
        end
    else [s,e]=ZeroShiftQR(s,e,iLower,iUpper,2); % Zero-shift QR iterations with direction up
        if e(iLower)<=thresh
            e(iLower)=0;
        end
    end
else if dir==1
        [s,e]=ZeroShiftQR(s,e,iLower,iUpper,1); % This is where it should be applied the shifted QR dir. down
        if e(iUpper-1)<=thresh
            e(iUpper-1)=0;
        end
    else  [s,e]=ZeroShiftQR(s,e,iLower,iUpper,2); % This is where it should be applied the shifted QR dir. up
        if e(iLower)<=thresh
            e(iLower)=0;
        end
    end
end     
end
% Sort singular values
s=sort(abs(s),'descend')

SVD＆＃34;高精度＆＃34; Demmel - Kahan算法

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