2.5.3. Linear System Solvers

  • sparse matrix/eigenvalue problem solvers live in scipy.sparse.linalg

  • the submodules:

    • dsolve: direct factorization methods for solving linear systems

    • isolve: iterative methods for solving linear systems

    • eigen: sparse eigenvalue problem solvers

  • all solvers are accessible from:

    >>> import scipy as sp

    >>> sp.sparse.linalg.__all__

    ['ArpackError', 'ArpackNoConvergence', ..., 'use_solver']

2.5.3.1. Sparse Direct Solvers

  • default solver: SuperLU

    • included in SciPy

    • real and complex systems

    • both single and double precision

  • optional: umfpack

    • real and complex systems

    • double precision only

    • recommended for performance

    • wrappers now live in scikits.umfpack

    • check-out the new scikits.suitesparse by Nathaniel Smith

Examples

  • import the whole module, and see its docstring:

    >>> help(sp.sparse.linalg.spsolve)

    Help on function spsolve in module scipy.sparse.linalg._dsolve.linsolve:

    ...

  • both superlu and umfpack can be used (if the latter is installed) as follows:

    • prepare a linear system:

      >>> import numpy as np

      >>> mtx = sp.sparse.spdiags([[1, 2, 3, 4, 5], [6, 5, 8, 9, 10]], [0, 1], 5, 5, "csc")

      >>> mtx.toarray()

      array([[ 1,  5,  0,  0,  0],

             [ 0,  2,  8,  0,  0],

             [ 0,  0,  3,  9,  0],

             [ 0,  0,  0,  4, 10],

             [ 0,  0,  0,  0,  5]])

      >>> rhs = np.array([1, 2, 3, 4, 5], dtype=np.float32)

    • solve as single precision real:

      >>> mtx1 = mtx.astype(np.float32)

      >>> x = sp.sparse.linalg.spsolve(mtx1, rhs, use_umfpack=False)

      >>> print(x)

      [106.   -21.     5.5   -1.5    1. ]

      >>> print("Error: %s" % (mtx1 * x - rhs))

      Error:  [0.  0.  0.  0.  0.]

    • solve as double precision real:

      >>> mtx2 = mtx.astype(np.float64)

      >>> x = sp.sparse.linalg.spsolve(mtx2, rhs, use_umfpack=True)

      >>> print(x)

      [106.   -21.     5.5   -1.5    1. ]

      >>> print("Error: %s" % (mtx2 * x - rhs))

      Error:  [0.  0.  0.  0.  0.]

    • solve as single precision complex:

      >>> mtx1 = mtx.astype(np.complex64)

      >>> x = sp.sparse.linalg.spsolve(mtx1, rhs, use_umfpack=False)

      >>> print(x)

      [106. +0.j  -21. +0.j    5.5+0.j   -1.5+0.j    1. +0.j]

      >>> print("Error: %s" % (mtx1 * x - rhs))

      Error:  [0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j]

    • solve as double precision complex:

      >>> mtx2 = mtx.astype(np.complex128)

      >>> x = sp.sparse.linalg.spsolve(mtx2, rhs, use_umfpack=True)

      >>> print(x)

      [106. +0.j  -21. +0.j    5.5+0.j   -1.5+0.j    1. +0.j]

      >>> print("Error: %s" % (mtx2 * x - rhs))

      Error:  [0.+0.j  0.+0.j  0.+0.j  0.+0.j  0.+0.j]

"""

Solve a linear system

=======================



Construct a 1000x1000 lil_array and add some values to it, convert it

to CSR format and solve A x = b for x:and solve a linear system with a

direct solver.

"""



import numpy as np

import scipy as sp

import matplotlib.pyplot as plt



rng = np.random.default_rng(27446968)



mtx = sp.sparse.lil_array((1000, 1000), dtype=np.float64)

mtx[0, :100] = rng.random(100)

mtx[1, 100:200] = mtx[[0], :100]

mtx.setdiag(rng.random(1000))



plt.clf()

plt.spy(mtx, marker=".", markersize=2)

plt.show()



mtx = mtx.tocsr()

rhs = rng.random(1000)



x = sp.sparse.linalg.spsolve(mtx, rhs)



print(f"residual: {np.linalg.norm(mtx @ x - rhs)!r}")

2.5.3.2. Iterative Solvers

  • the isolve module contains the following solvers:

    • bicg (BIConjugate Gradient)

    • bicgstab (BIConjugate Gradient STABilized)

    • cg (Conjugate Gradient) - symmetric positive definite matrices only

    • cgs (Conjugate Gradient Squared)

    • gmres (Generalized Minimal RESidual)

    • minres (MINimum RESidual)

    • qmr (Quasi-Minimal Residual)

Common Parameters

  • mandatory:

    A{sparse array/matrix, dense array/matrix, LinearOperator}

    The N-by-N matrix of the linear system.

    b{array, matrix}

    Right hand side of the linear system. Has shape (N,) or (N,1).

  • optional:

    x0{array, matrix}

    Starting guess for the solution.

    tolfloat

    Relative tolerance to achieve before terminating.

    maxiterinteger

    Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.

    M{sparse array/matrix, dense array/matrix, LinearOperator}

    Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.

    callbackfunction

    User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

LinearOperator Class

  • common interface for performing matrix vector products

  • useful abstraction that enables using dense and sparse matrices within the solvers, as well as matrix-free solutions

  • has shape and matvec() (+ some optional parameters)

  • example:

>>> import numpy as np

>>> import scipy as sp

>>> def mv(v):

...     return np.array([2 * v[0], 3 * v[1]])

...

>>> A = sp.sparse.linalg.LinearOperator((2, 2), matvec=mv)

>>> A

<2x2 _CustomLinearOperator with dtype=float64>

>>> A.matvec(np.ones(2))

array([2.,  3.])

>>> A * np.ones(2)

array([2.,  3.])

A Few Notes on Preconditioning

  • problem specific

  • often hard to develop

  • if not sure, try ILU

2.5.3.3. Eigenvalue Problem Solvers

The eigen module

  • arpack * a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems

  • lobpcg (Locally Optimal Block Preconditioned Conjugate Gradient Method) * works very well in combination with PyAMG * example by Nathan Bell:

    """

    Compute eigenvectors and eigenvalues using a preconditioned eigensolver

    =======================================================================

    

    In this example Smoothed Aggregation (SA) is used to precondition

    the LOBPCG eigensolver on a two-dimensional Poisson problem with

    Dirichlet boundary conditions.

    """

    

    import numpy as np

    import scipy as sp

    import matplotlib.pyplot as plt

    

    from pyamg import smoothed_aggregation_solver

    from pyamg.gallery import poisson

    

    N = 100

    K = 9

    A = poisson((N, N), format="csr")

    

    # create the AMG hierarchy

    ml = smoothed_aggregation_solver(A)

    

    # initial approximation to the K eigenvectors

    X = np.random.random((A.shape[0], K))

    

    # preconditioner based on ml

    M = ml.aspreconditioner()

    

    # compute eigenvalues and eigenvectors with LOBPCG

    W, V = sp.sparse.linalg.lobpcg(A, X, M=M, tol=1e-8, largest=False)

    

    

    # plot the eigenvectors

    plt.figure(figsize=(9, 9))

    

    for i in range(K):

        plt.subplot(3, 3, i + 1)

        plt.title("Eigenvector %d" % i)

        plt.pcolor(V[:, i].reshape(N, N))

        plt.axis("equal")

        plt.axis("off")

    plt.show()

  • example by Nils Wagner:

  • output:

    $ python examples/lobpcg_sakurai.py

    Results by LOBPCG for n=2500

    

    [ 0.06250083  0.06250028  0.06250007]

    

    Exact eigenvalues

    

    [ 0.06250005  0.0625002   0.06250044]

    

    Elapsed time 7.01
../../_images/lobpcg_eigenvalues.png