pymoto.solvers.SOR

class pymoto.solvers.SOR(A=None, w=1.0)

Successive over-relaxation preconditioner The matrix \(A = L + D + U\) is split into a lower triangular, diagonal, and upper triangular part. \(M = \left(\frac{D}{\omega} + L\right) \frac{\omega D^{-1}}{2-\omega} \left(\frac{D}{\omega} + U\right)\)

__init__(A=None, w=1.0)

Initialize the SOR preconditioner

Parameters:

A (optional) – The matrix w (optional): Weight factor \(0 < \omega < 2\)

Methods

__init__([A, w])

Initialize the SOR preconditioner

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 \(\mathbf{A} \mathbf{x} = \mathbf{b}\)

update(A)

Updates with a new matrix of the same structure

Attributes

defined

update(A)

Updates with a new matrix of the same structure

Parameters:

A (matrix) – The new matrix of size (N, N)

Returns:

self

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

Solves the linear system of equations \(\mathbf{A} \mathbf{x} = \mathbf{b}\)

Parameters:

  • rhs – Right hand side \(\mathbf{b}\) of shape (N) or (N, K) for multiple right-hand-sides

  • x0 (optional) – Initial guess for the solution

  • trans (optional) – Option to transpose matrix ‘N’: A @ x == rhs (default) Normal matrix ‘T’: A^T @ x == rhs Transposed matrix ‘H’: A^H @ x == rhs Hermitian transposed matrix (conjugate transposed)

Returns:

Solution vector \(\mathbf{x}\) of same shape as \(\mathbf{b}\)

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