TransportMaps.LinAlg._linalg

Module Contents

Functions

square_root(A[, square_root_type])	Factorizes \(A\) and returns the square root it
inverse_square_root(A[, square_root_type])	Factorizes \(A\) and returns the square root of \(A^{-1}\)
matrix_inverse(A)
solve_linear_system(A, b[, transposed])	Solve the system \(Ax = b\)
solve_square_root_linear_system(A, b)	Solve the system \(AA^{\top}x = b\)
log_det(A)	Compute \(\log\det A\)

TransportMaps.LinAlg._linalg.square_root(A, square_root_type='sym')

Factorizes \(A\) and returns the square root it

Parameters:

A (ndarray [\(d,d\)]) – matrix

Kwargs:

square_root_type (str): type of square root.

For square_root_type=='sym', \(L=U\Lambda^{\frac{1}{2}}U^T\) where \(A = U\Lambda U^T\) is the eigenvalue decomposition of \(A\). For square_root_type=='tri' or square_root_type=='chol', \(L=C\) where \(A=CC^T\) is the Cholesky decomposition of \(A\). For square_root_type=='kl', \(L=U\Lambda^{\frac{1}{2}}\) where \(A = U\Lambda U^T\) is the eigenvalue decomposition of \(A\) (this corresponds to the Karuenen-Loeve expansion). The eigenvalues and eigenvectors are ordered with \(\lambda_i\geq\lambda_{i+1}\).

Returns:

\(L\) – square root

TransportMaps.LinAlg._linalg.inverse_square_root(A, square_root_type='sym')

Factorizes \(A\) and returns the square root of \(A^{-1}\)

Parameters:

A (ndarray [\(d,d\)]) – matrix

Kwargs:

square_root_type (str): type of square root.

For square_root_type=='sym', \(L=U\Lambda^{\frac{1}{2}}U^T\) where \(A = U\Lambda U^T\) is the eigenvalue decomposition of \(A\). For square_root_type=='tri' or square_root_type=='chol', \(L=C\) where \(A=CC^T\) is the Cholesky decomposition of \(A\). For square_root_type=='kl', \(L=U\Lambda^{\frac{1}{2}}\) where \(A = U\Lambda U^T\) is the eigenvalue decomposition of \(A\) (this corresponds to the Karuenen-Loeve expansion). The eigenvalues and eigenvectors are ordered with \(\lambda_i\geq\lambda_{i+1}\).

Returns:

\(L^{-1}\) – square root of the inverse

TransportMaps.LinAlg._linalg.matrix_inverse(A)

TransportMaps.LinAlg._linalg.solve_linear_system(A, b, transposed=False)

Solve the system \(Ax = b\)

It checks whether A has some good properties.

TransportMaps.LinAlg._linalg.solve_square_root_linear_system(A, b)

Solve the system \(AA^{\top}x = b\)

TransportMaps.LinAlg._linalg.log_det(A)

Compute \(\log\det A\)