LinearSolve

LinearSolve[m,b]

finds an x that solves the matrix equation m.x==b.

LinearSolve[m]

generates a LinearSolveFunction[] that can be applied repeatedly to different b.

Details and Options

  • LinearSolve works on both numerical and symbolic matrices, as well as SparseArray objects.
  • The argument b can be either a vector or a matrix. »
  • The matrix m can be square or rectangular. »
  • LinearSolve[m] and LinearSolveFunction[] provide an efficient way to solve the same approximate numerical linear system many times.
  • LinearSolve[m,b] is equivalent to LinearSolve[m][b].
  • For underdetermined systems, LinearSolve will return one of the possible solutions; Solve will return a general solution. »
  • LinearSolve has the following options and settings:
  • MethodAutomaticmethod to use
    Modulus0whether to solve modulo n
    ZeroTestAutomatictest to determine when expressions are zero
  • The ZeroTest option only applies to exact and symbolic matrices.
  • With Method->Automatic, the method is automatically selected depending upon input.
  • Explicit Method settings for exact and symbolic matrices include:
  • "CofactorExpansion"Laplace cofactor expansion
    "DivisionFreeRowReduction"Bareiss method of division-free row reduction
    "OneStepRowReduction"standard row reduction
  • Explicit Method settings for approximate numeric matrices include:
  • "Banded"banded matrix solver
    "Cholesky"Cholesky method for positive definite Hermitian matrices
    "Krylov"iterative Krylov sparse solver
    "Multifrontal"direct sparse LU decomposition
    "Pardiso"parallel direct sparse solver

Examples

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Basic Examples  (2)

With no righthand side, a LinearSolveFunction is returned:

Scope  (6)

Find the solution to with exact arithmetic:

Find the solution using numerical methods with machine precision:

Find the solution using 20digit arbitrary precision:

Find a solution for a singular matrix:

A solution is not always available:

Find a solution for a rectangular matrix:

Solve for when is a matrix:

When no righthand side for is given, a LinearSolveFunction is returned:

This contains data to solve the problem quickly for a few values of :

Sparse methods are used for sparse matrices:

Options  (7)

Method  (6)

"Banded"  (1)

Solve using a banded matrix method:

Check a relative error of the computed solution:

"Cholesky"  (1)

Solve using the Cholesky decomposition:

Check a relative error of the computed solution:

"Krylov"  (2)

The following suboptions can be specified for the method "Krylov":

  • "BasisSize"the size of the Krylov basis (GMRES only)
    "MaxIterations"the maximum number of iterations
    "Method"methods to be used
    "Preconditioner"which preconditioner to apply
    "PreconditionerSide"how to apply a preconditioner ("Left" or "Right")
    "ResidualNormFunction"A norm function that computes a norm of the residual of the solution
    "StartingVector"the initial vector to start iterations
    "Tolerance"the tolerance used to terminate iterations
  • Possible settings for "Method" include:

  • "BiCGSTAB"iterative method for arbitrary square matrices
    "ConjugateGradient"iterative method for Hermitian positive definite matrices
    "GMRES"iterative method for arbitrary square matrices
  • Possible settings for "Preconditioner" include:

  • "ILU0"a preconditioner based on an incomplete LU factorization of the original matrix without fill-in
    "ILUT"a variant of ILU0 with fill-in
    "ILUTP"a variant of ILUT with column permutation
  • Possible suboptions for "Preconditioner" include:

  • "FillIn"upper bound on the number of additional nonzero elements in a row introduced by the ILUT preconditioner
    "PermutationTolerance"when to permute columns
    "Tolerance"drop tolerance (any element of magnitude smaller than this tolerance is treated as zero)
  • Solve using a Krylov method:

    Check a relative error of the computed solution:

    "Multifrontal"  (1)

    Solve using a direct multifrontal method:

    Check a relative error of the computed solution:

    "Pardiso"  (1)

    Solve using a direct multifrontal method:

    Check a relative error of the computed solution:

    Modulus  (1)

    Find the solution x to m.x==b modulo 47:

    Verify the solution:

    Applications  (2)

    Newton's method for finding a root of a multivariate function:

    Approximately solve the boundary value problem :

    Show the error compared with the exact solution:

    Properties & Relations  (4)

    m is a 3×3 matrix:

    A system of linear equations:

    The solution computed by Solve:

    The solution computed by LinearSolve:

    Verify that they are the same:

    If is nonsingular, the solution of is the inverse of when is the identity matrix:

    In this case there is no solution to :

    Use LeastSquares to minimize :

    Compare to general minimization:

    There are multiple solutions to if is singular:

    LinearSolve returns only one solution:

    Use NullSpace to get the complete spanning set of solutions:

    Possible Issues  (2)

    Solution found for an underdetermined system is not unique:

    All solutions are found by Solve:

    With ill-conditioned matrices, numerical solutions may not be sufficiently accurate:

    The solution is more accurate if sufficiently high precision is used:

    Neat Examples  (3)

    Solve 100,000 equations using a direct method:

    Solve a million equations using an iterative method:

    Check a relative error of the solution:

    Solve the same system of equations using a banded matrix method:

    Check a relative error of the solution:

    Introduced in 1988
     (1.0)
     |
    Updated in 1996
     (3.0)
    2003
     (5.0)
    2014
     (10.0)