pymoto.solvers.SolverDenseQR

class pymoto.solvers.SolverDenseQR(A=None)

Solver for dense (square) matrices using a QR decomposition

__init__(A=None)

Initialize the solver

Parameters:

A (matrix, optional) – Optionally provide a matrix, which is used in :method:`update` right away.

Methods

__init__([A])

Initialize the solver

residual(A, x, b[, trans])

Calculates the (relative) residual of the linear system of equations

solve(rhs[, x0, trans])

Solves the linear system of equations using the QR factorization.

update(A)

Factorize the matrix as \(\mathbf{A}=\mathbf{Q}\mathbf{R}\), where \(\mathbf{Q}\) is orthogonal (\(\mathbf{Q}^\text{H}=\mathbf{Q}^{-1}\)) and \(\mathbf{R}\) is upper triangular.

Attributes

defined

update(A)

Factorize the matrix as \(\mathbf{A}=\mathbf{Q}\mathbf{R}\), where \(\mathbf{Q}\) is orthogonal (\(\mathbf{Q}^\text{H}=\mathbf{Q}^{-1}\)) and \(\mathbf{R}\) is upper triangular.

solve(rhs, x0=None, trans='N')

Solves the linear system of equations using the QR factorization.

trans

Equation

Solution of \(x\)

N

\(A x = b\)

\(R^{-1} Q^H b\)

T

\(A^T x = b\)

\(Q^* R^{-T} b\)

H

\(A^H x = b\)

\(Q R^{-H} b\)

The right-hand-side \(\mathbf{b}\) can be of size (N) or (N, K), where N is the size of matrix \(\mathbf{A}\) and K is the number of right-hand sides.

defined = True
static residual(A, x, b, trans='N')

Calculates the (relative) residual of the linear system of equations

The residual is calculated as \(r = \frac{\left| \mathbf{A} \mathbf{x} - \mathbf{b} \right|}{\left| \mathbf{b} \right|}\)

Parameters:

  • A – The matrix

  • x – Solution vector

  • b – Right-hand side

  • trans (optional) – Matrix tranformation (N is normal, T is transposed, H is hermitian transposed)

Returns:

Residual value