SVD

template<typename T, DataLocation L, template<typename, typename> typename S>
internal::SVD<T, L> svd(const Matrix<T, L, S> &A, SVDMethod method = kSVDDefault)

Computes the (complete) Singular Value Decomposition (SVD) of a matrix \( \mathbf{A} \): \( \mathbf{A} = \mathbf{USV}^T\), where \( \mathbf{U} \) and \( \mathbf{V} \) are unitary matrices and \( \mathbf{S} \) contains the singular values of \( \mathbf{A} \).

Usage:

Matrix<double> A;
Vector<double> S;
Matrix<double> U;
Matrix<double> VT;
S = svd(A);                             // Calculates the singular values of A
tie(U, S, VT) = svd(A);                 // Calculates the singular values S as well as U and VT

Warning

The routine returns \( \mathbf{V}^T \) instead of \( \mathbf{V} \).

Parameters:

• A – [in] input matrix

• method – [in] method used for calculating the SVD. See SVDMethod for all options.

Returns:

An object representing the factorization.

template<typename T, DataLocation L, template<typename, typename> typename S>
internal::SVD<T, L> compact_svd(const Matrix<T, L, S> &A, SVDMethod method = kSVDDefault)

Computes the reduced Singular Value Decomposition (SVD) of a matrix \( \mathbf{A} \): \( \mathbf{A} = \mathbf{USV}^T\), where \( \mathbf{U} \) and \( \mathbf{V} \) are unitary matrices, and \( \mathbf{S} \) contains the singular values of \( \mathbf{A} \). Only the first min(A.rows(), A.cols()) rows/columns of \( \mathbf{U} \) and \( \mathbf{V} \) are computed.

Usage:

Matrix<double> A;
Vector<double> S;
Matrix<double> U;
Matrix<double> VT;
S = compact_svd(A);                             // Calculates the singular values of A
tie(U, S, VT) = compact_svd(A);                 // Calculates the singular values S as well as U and VT

Warning

The routine returns \( \mathbf{V}^T \) instead of \( \mathbf{V} \).

Parameters:

• A – [in] input matrix

• method – [in] method used for calculating the SVD. See SVDMethod for all options.

Returns:

An object representing the factorization.