单边Jacobi求解SVD和双边Jacobi求解SVD方法的Matlab区别
时间: 2024-01-10 10:04:41 浏览: 56
单边Jacobi求解SVD和双边Jacobi求解SVD方法都是用于求解SVD分解的算法,它们的区别如下:
1. 单边Jacobi求解SVD方法只对矩阵的一个方向进行迭代,而双边Jacobi求解SVD方法对矩阵的两个方向进行迭代,因此双边Jacobi方法比单边Jacobi方法更准确。
2. 单边Jacobi求解SVD方法只适用于对对称矩阵进行SVD分解,而双边Jacobi求解SVD方法适用于对任意矩阵进行SVD分解。
3. 单边Jacobi求解SVD方法的计算速度比双边Jacobi求解SVD方法快,但是精度相对较低。
在Matlab中,单边Jacobi求解SVD方法可以使用svds函数实现,而双边Jacobi求解SVD方法可以使用svd函数实现。另外,Matlab还提供了其他的SVD分解函数,如eigs函数和eig函数等。
相关问题
单边Jacobi求解SVD和双边Jacobi求解SVD方法的Matlab代码
单边Jacobi求解SVD方法的Matlab代码:
```matlab
function [U,S,V] = svd_jacobi(A)
% SVD-Jacobi
% U*S*V' = A
% A: m*n matrix
% U: m*m matrix
% S: m*n matrix
% V: n*n matrix
% Jacobi algorithm, EISPACK subroutine TRED2, TQLI
% Reference: J.H.Wilkinson, C.Reinsch, Handbook for Automatic Computation, Vol.II, Linear Algebra, Springer, 1971.
[m,n] = size(A);
if m < n
A = A';
[m,n] = size(A);
end
U = eye(m);
S = A';
V = eye(n);
maxiter = 30;
eps = 1e-15;
for iter = 1 : maxiter
for p = 1 : n - 1
for q = p + 1 : n
if abs(S(p,q)) > eps
theta = (S(q,q) - S(p,p)) / (2 * S(p,q));
t = sign(theta) / (abs(theta) + sqrt(1 + theta^2));
c = 1 / sqrt(1 + t^2);
s = c * t;
G = eye(n);
G(p,p) = c;
G(q,q) = c;
G(p,q) = -s;
G(q,p) = s;
S = G' * S * G;
V = V * G;
end
end
end
for p = 1 : m - 1
for q = p + 1 : m
if abs(S(p,q)) > eps
theta = (S(q,q) - S(p,p)) / (2 * S(p,q));
t = sign(theta) / (abs(theta) + sqrt(1 + theta^2));
c = 1 / sqrt(1 + t^2);
s = c * t;
G = eye(m);
G(p,p) = c;
G(q,q) = c;
G(p,q) = -s;
G(q,p) = s;
S = G' * S * G;
U = U * G;
end
end
end
if max(max(triu(abs(S),1)))) < eps
break;
end
end
if m > n
U = U(:,1:n);
S = S(1:n,:);
else
V = V(:,1:m);
S = S(:,1:m);
end
end
```
双边Jacobi求解SVD方法的Matlab代码:
```matlab
function [U,S,V] = svd_jacobi2(A)
% SVD-Jacobi
% U*S*V' = A
% A: m*n matrix
% U: m*m matrix
% S: m*n matrix
% V: n*n matrix
% Jacobi algorithm, EISPACK subroutine TRED2, TQLI
% Reference: J.H.Wilkinson, C.Reinsch, Handbook for Automatic Computation, Vol.II, Linear Algebra, Springer, 1971.
[m,n] = size(A);
if m < n
A = A';
[m,n] = size(A);
end
U = eye(m);
S = A';
V = eye(n);
maxiter = 30;
eps = 1e-15;
for iter = 1 : maxiter
for p = 1 : n - 1
for q = p + 1 : n
if abs(S(p,q)) > eps
theta = (S(q,q) - S(p,p)) / (2 * S(p,q));
t = sign(theta) / (abs(theta) + sqrt(1 + theta^2));
c = 1 / sqrt(1 + t^2);
s = c * t;
G = eye(n);
G(p,p) = c;
G(q,q) = c;
G(p,q) = -s;
G(q,p) = s;
S = G' * S * G;
V = V * G;
end
end
end
for p = 1 : m - 1
for q = p + 1 : m
if abs(S(p,q)) > eps
theta = (S(q,q) - S(p,p)) / (2 * S(p,q));
t = sign(theta) / (abs(theta) + sqrt(1 + theta^2));
c = 1 / sqrt(1 + t^2);
s = c * t;
G = eye(m);
G(p,p) = c;
G(q,q) = c;
G(p,q) = -s;
G(q,p) = s;
S = G' * S * G;
U = U * G;
end
end
end
if max(max(triu(abs(S),1)))) < eps
break;
end
end
if m > n
U = U(:,1:n);
S = S(1:n,:);
else
V = V(:,1:m);
S = S(:,1:m);
end
end
```
单边Jacobi求解svd的Matlab代码和讲解
单边Jacobi求解SVD(Singular Value Decomposition,奇异值分解)是一种基于迭代的数值方法,用于计算矩阵的奇异值和奇异向量。该方法包括对矩阵进行旋转操作,使其奇异值逐渐趋近于对角线上的元素,从而实现对矩阵的分解。
以下是Matlab代码实现单边Jacobi求解SVD的过程:
```matlab
function [U, S, V] = svd_jacobi(A, tol)
% 输入:矩阵A,迭代收敛容差tol
% 输出:矩阵A的奇异值分解结果U、S、V
[m,n] = size(A);
maxiter = 500; % 最大迭代次数
iter = 0;
converged = false;
U = eye(m);
V = eye(n);
% 迭代求解
while ~converged && iter < maxiter
converged = true;
% 对A的每一对非对角线元素进行旋转
for p = 1:n-1
for q = p+1:n
G = A(:,p) - A(:,q);
if norm(G) < tol % 如果G的模长小于tol,则不需要旋转
continue;
end
Jpq = [1, 0, 0; 0, cos(theta), -sin(theta); 0, sin(theta), cos(theta)];
Jpq([p,q], [p,q]) = [cos(theta), -sin(theta); sin(theta), cos(theta)];
A = A * Jpq;
V = V * Jpq;
converged = false;
end
end
% 对A的每一对非对角线元素进行旋转
for p = 1:m-1
for q = p+1:m
G = A(p,:) - A(q,:);
if norm(G) < tol % 如果G的模长小于tol,则不需要旋转
continue;
end
Jpq = [1, 0, 0; 0, cos(theta), -sin(theta); 0, sin(theta), cos(theta)];
Jpq([p,q], [p,q]) = [cos(theta), -sin(theta); sin(theta), cos(theta)];
A = Jpq * A;
U = U * Jpq';
converged = false;
end
end
iter = iter + 1;
end
S = diag(diag(A)); % 取出A的对角线元素,即奇异值
end
```
该代码主要实现了对矩阵A的每一对非对角线元素进行旋转的过程,其中使用了一个收敛容差tol和一个最大迭代次数maxiter,以确保迭代可以收敛并避免无限循环。在每次迭代中,如果G的模长小于tol,则不需要旋转,否则就将矩阵A和V或U分别与旋转矩阵Jpq相乘,以实现旋转操作。
需要注意的是,由于单边Jacobi求解SVD是一种迭代方法,因此其收敛速度可能会受到矩阵A的特征值分布和初始矩阵的选择等因素的影响。因此,我们需要根据实际情况选择合适的迭代次数和收敛容差,以确保求解结果的准确性和效率。