continuate.krylov

Krylov subspace methods

These methods are based on the Arnoldi process

\[AV_n = V_{n+1} H_{n+1},\]

where \(V_n\) denotes the basis of Krylov subspace, and \(H_n\) denotes the projected matrix with Hessenberg form.

continuate.krylov.arnoldi_common(A, r, krylov_tol=1e-09, **cfg)

Support generator for Arnoldi process

Parameters:

A : scipy.sparse.linalg.LinearOperator

* operator is needed.

r : np.array

The base of Krylov subspace \(K = \left<r, Ar, A^2r, ...\right>\)

Yields:

V : np.array (2d)

With shape \((N, n)\) (n starts with 2)

h : np.array (1d)

The last column of \(H_n\) with shape (n, )

continuate.krylov.default_options = {'krylov_maxiter': 100, 'krylov_tol': 1e-09}

default values of options

You can get these values through continuate.get_default_options()

Parameters:

krylov_tol : float

Tolerrance of Krylov iteration

krylov_maxiter : float

Max iteration number of Krylov iteration

continuate.krylov.gmres(A, b, x0=None, krylov_tol=1e-09, krylov_maxiter=100, **cfg)

Solve linear equations \(Ax=b\) by GMRES

Parameters:

A : scipy.sparse.linalg.LinearOperator

* operator is needed.

b : np.array

inhomogeneous term

x0 : np.array

Initial guess of linear problem

Returns:

x : np.array

The solution

Examples

>>> from numpy.random import random
>>> from scipy.sparse.linalg import aslinearoperator
>>> A = aslinearoperator(random((5, 5)))
>>> x = random(5)
>>> b = A*x
>>> ans = gmres(A, b)
>>> np.allclose(ans, x)
True

continuate.krylov.gmres_factorize(A, r, krylov_tol=1e-09, krylov_maxiter=100, **cfg)

Execute a factorization \(AV = VQR\) to solve minimization problem \(|Ax-b| = |VQ(Ry-g)| = |Ry-g|\).

Parameters:

A : scipy.sparse.linalg.LinearOperator

* operator is needed.

b : np.array

inhomogeneous term

x0 : np.array

Initial guess of linear problem

Returns:

V : np.array (2d)

The basis of Krylov subspace with shape \((N, n)\)

R : np.array (2d)

Projected, and QR-decomposed matrix \(AV = VQR\)

g : np.array (1d)

Q-rotated vector of \(b\)

Q : [np.array(2x2)]

Givens rotation matrix

Examples

>>> from numpy.random import random
>>> from scipy.sparse.linalg import aslinearoperator

>>> A = aslinearoperator(random((5, 5)))
>>> b = random(5)
>>> V, R, g, Q = gmres_factorize(A, b)
>>> V.shape, R.shape, g.shape
((5, 5), (5, 5), (5,))

>>> A = aslinearoperator(np.identity(5))
>>> V, R, g, Q = gmres_factorize(A, b)
>>> V.shape, R.shape, g.shape
((5, 1), (1, 1), (1,))