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.sparse.linalg as spla
    
    >>> spla.__all__
    ['LinearOperator', 'Tester', 'arpack', 'aslinearoperator', 'bicg',
    'bicgstab', 'cg', 'cgs', 'csc_matrix', 'csr_matrix', 'dsolve',
    'eigen', 'eigen_symmetric', 'factorized', 'gmres', 'interface',
    'isolve', 'iterative', 'lgmres', 'linsolve', 'lobpcg', 'lsqr',
    'minres', 'np', 'qmr', 'speigs', 'spilu', 'splu', 'spsolve', 'svd',
    'test', 'umfpack', 'use_solver', 'utils', 'warnings']

2.5.3.1. Sparse Direct Solvers

  • default solver: SuperLU 4.0
    • 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

2.5.3.1.1. Examples

  • import the whole module, and see its docstring:

    >>> from scipy.sparse.linalg import dsolve
    
    >>> help(dsolve)
  • both superlu and umfpack can be used (if the latter is installed) as follows:

    • prepare a linear system:

      >>> import numpy as np
      
      >>> from scipy import sparse
      >>> mtx = sparse.spdiags([[1, 2, 3, 4, 5], [6, 5, 8, 9, 10]], [0, 1], 5, 5)
      >>> mtx.todense()
      matrix([[ 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 = dsolve.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 = dsolve.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 = dsolve.spsolve(mtx1, rhs, use_umfpack=False)
      >>> print(x)
      [ 106.0+0.j -21.0+0.j 5.5+0.j -1.5+0.j 1.0+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 = dsolve.spsolve(mtx2, rhs, use_umfpack=True)
      >>> print(x)
      [ 106.0+0.j -21.0+0.j 5.5+0.j -1.5+0.j 1.0+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_matrix 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.sparse as sps
from matplotlib import pyplot as plt
from scipy.sparse.linalg.dsolve import linsolve
rand = np.random.rand
mtx = sps.lil_matrix((1000, 1000), dtype=np.float64)
mtx[0, :100] = rand(100)
mtx[1, 100:200] = mtx[0, :100]
mtx.setdiag(rand(1000))
plt.clf()
plt.spy(mtx, marker='.', markersize=2)
plt.show()
mtx = mtx.tocsr()
rhs = rand(1000)
x = linsolve.spsolve(mtx, rhs)
print('rezidual: %r' % np.linalg.norm(mtx * x - rhs))

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)

2.5.3.2.1. Common Parameters

  • mandatory:

    A
    : {sparse matrix, dense 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.

    tol
    : float

    Relative tolerance to achieve before terminating.

    maxiter
    : integer

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

    M
    : {sparse matrix, dense 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.

    callback
    : function

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

2.5.3.2.2. LinearOperator Class

from scipy.sparse.linalg.interface import LinearOperator
  • 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
>>> from scipy.sparse.linalg import LinearOperator
>>> def mv(v):
... return np.array([2*v[0], 3*v[1]])
...
>>> A = LinearOperator((2, 2), matvec=mv)
>>> A
<2x2 LinearOperator with unspecified dtype>
>>> A.matvec(np.ones(2))
array([ 2., 3.])
>>> A * np.ones(2)
array([ 2., 3.])

2.5.3.2.3. A Few Notes on Preconditioning

  • problem specific

  • often hard to develop

  • if not sure, try ILU
    • available in dsolve as spilu()

2.5.3.3. Eigenvalue Problem Solvers

2.5.3.3.1. 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 scipy
    from scipy.sparse.linalg import lobpcg
    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 = scipy.rand(A.shape[0], K)
    # preconditioner based on ml
    M = ml.aspreconditioner()
    # compute eigenvalues and eigenvectors with LOBPCG
    W,V = lobpcg(A, X, M=M, tol=1e-8, largest=False)
    #plot the eigenvectors
    import pylab
    pylab.figure(figsize=(9,9))
    for i in range(K):
    pylab.subplot(3, 3, i+1)
    pylab.title('Eigenvector %d' % i)
    pylab.pcolor(V[:,i].reshape(N,N))
    pylab.axis('equal')
    pylab.axis('off')
    pylab.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