带wilkinson位移的隐式对称QR算法matlab

以下是带Wilkinson位移的隐式对称QR算法的MATLAB代码：

function [Q, R] = wilkinsonQR(A)
% Input:
%   A: n x n matrix to be factorized
% Output:
%   Q: n x n orthogonal matrix
%   R: n x n upper triangular matrix such that A = QR
% 
% The algorithm uses Wilkinson's shift strategy to improve accuracy.

n = size(A, 1);
Q = eye(n);
for k = 1:n-1
    % Compute the Wilkinson shift
    mu = wilkinsonShift(A(k:k+1, k:k+1));
    
    % Update the diagonal block of A using the shift
    A(k:k+1, k:k+1) = A(k:k+1, k:k+1) - mu*eye(2);
    
    % Compute the Givens rotation to zero out the subdiagonal element
    [c, s] = givensRotation(A(k+1, k), A(k, k));
    
    % Apply the Givens rotation to the diagonal block of A and the right-hand side
    A(k:k+1, k:n) = [c s; -s c]'*A(k:k+1, k:n);
    Q(:, k:k+1) = Q(:, k:k+1)*[c s; -s c];
end
R = A;

其中，`wilkinsonShift`函数用于计算Wilkinson位移，`givensRotation`函数用于计算Givens旋转矩阵。这两个函数的代码如下：

function mu = wilkinsonShift(A)
% Compute the Wilkinson shift for a 2 x 2 matrix A

a = A(1, 1);
b = A(1, 2);
c = A(2, 1);
d = A(2, 2);

tau = (a - d)/2;
if tau >= 0
    mu = d - b^2/(tau + sign(tau)*sqrt(tau^2 + b^2));
else
    mu = d - b^2/(-tau + sign(tau)*sqrt(tau^2 + b^2));
end

function [c, s] = givensRotation(a, b)
% Compute the Givens rotation matrix G such that [c s; -s c]*[a; b] = [r; 0]

if b == 0
    c = 1;
    s = 0;
elseif abs(b) > abs(a)
    t = -a/b;
    s = 1/sqrt(1 + t^2);
    c = s*t;
else
    t = -b/a;
    c = 1/sqrt(1 + t^2);
    s = c*t;
end

这些函数的实现参考了《Numerical Linear Algebra》一书中的代码。