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.

LinearSolve[a,b]

finds an x that solves the array equation a.x==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. »
  • For an n1×…×nk×m array a and an n1×…×nk×d1×…×dl array b, LinearSolve[a,b] gives an m×d1×…×dl array x, such that a.xb.
  • LinearSolve has the following options and settings:
  • Method Automaticmethod to use
    Modulus 0whether 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  (3)

Solve the matrix-vector equation with and :

Verify the solution:

Solve the matrix equation with and :

Verify the solution:

Solve a rectangular matrix equation:

Verify the solution:

Scope  (16)

Basic Uses  (9)

Solve at machine precision:

Solve a case where is a matrix:

Solve for a complex matrix:

Find a solution for an exact, rectangular matrix:

Compute a solution at arbitrary precision:

Solve for a symbolic matrix:

Solve the system when is a matrix:

Solve over a finite field:

Solve for CenteredInterval matrices:

Find random representatives mrep and brep of m and b:

Verify that sol contains LinearSolve[mrep,brep]:

Solve for when is a matrix of different dimensions:

When no right‐hand side for is given, a LinearSolveFunction is returned:

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

Special Matrices  (6)

Solve with sparse matrices:

As the result is typically not sparse, the result is returned as an ordinary list:

Sparse methods are used to efficiently solve sparse matrices:

Visualize the result:

Solve a system with structured matrices:

Use a different type of matrix structure:

An identity matrix always produces a trivial solution:

Solve a linear system whose coefficient matrix is a Hilbert matrix:

Solve a system whose coefficients are univariate polynomials of degree :

Arrays  (1)

Solve with a 2×3×6 array and a 2×3×4×5 array :

The result is a 6×4×5 array:

Verify the solution:

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 Pardiso:

    Check a relative error of the computed solution:

    Modulus  (1)

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

    Verify the solution:

    Applications  (11)

    Spans and Linear Independence  (3)

    The following three vectors are not linearly independent:

    The equation with a generic right-hand side does not have a solution:

    Equivalently, the equation with the identity matrix on the right-hand side has no solution:

    The following three vectors are linearly independent:

    The equation with a generic right-hand side has a solution:

    Equivalently, the equation with the identity matrix on the right-hand side has a solution:

    The solution is the inverse of :

    Determine if the following vectors are linearly independent or not:

    As does not have a solution for an arbitrary , they are not linearly independent:

    Equation Solving and Invertibility  (6)

    Solve the following system of equations:

    Rewrite the system in matrix form:

    Use LinearSolve to find a solution:

    Show that the solution is unique using NullSpace:

    Verify the result using SolveValues:

    Find all solutions of the following system of equations:

    First, write the coefficient matrix , variable vector and constant vector :

    Verify the rewrite:

    LinearSolve gives a particular solution:

    NullSpace gives a basis for solutions to the homogeneous equation :

    Define to be an arbitrary linear combination of the elements of :

    The general solution is the sum of and :

    Determine if the following matrix has an inverse:

    Since the system has no solution, does not have an inverse:

    Verify the result using Inverse:

    Determine if the following matrix has a nonzero determinant:

    Since the system has a solution, 's determinant must be nonzero:

    Confirm the result using Det:

    Find the inverse of the following matrix:

    To find the inverse, first solve the system :

    Verify the result using Inverse:

    Solve the system , with several different by means of computing a LinearSolveFunction:

    Perform the computation by inverting the matrix and multiplying by the inverse:

    The results are practically identical, even though LinearSolveFunction is multiple times faster:

    Calculus  (2)

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

    Compare with the answer found by FindRoot:

    Approximately solve the boundary value problem using discrete differences:

    Show the error compared with the exact solution:

    Properties & Relations  (9)

    For an invertible matrix , LinearSolve[m,b] gives the same result as SolveValues for the corresponding system of equations:

    Create the corresponding system of linear equations:

    Confirm that SolveValues gives the same result:

    LinearSolve always returns the trivial solution to the homogenous equation :

    Use NullSpace to get the complete spanning set of solutions if is singular:

    Compare with the result of SolveValues:

    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:

    If can be solved, LeastSquares is equivalent to LinearSolve:

    For a square matrix, LinearSolve[m,b] has a solution for a generic b iff Det[m]!=0:

    For a square matrix, LinearSolve[m,b] has a solution for a generic b iff m has full rank:

    For a square matrix, LinearSolve[m,b] has a solution for a generic b iff m has an inverse:

    For a square matrix, LinearSolve[m,b] has a solution for a generic b iff m has a trivial null space:

    Possible Issues  (3)

    Solution found for an underdetermined system is not unique:

    All solutions are found by Solve:

    LinearSolve gave the solution corresponding to :

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

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

    Some of the linear solvers available are not deterministic. Set up a system of equations:

    Create a solver function:

    The "Pardiso" solver is not deterministic:

    The Automatic solver method is deterministic:

    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:

    Wolfram Research (1988), LinearSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/LinearSolve.html (updated 2024).

    Text

    Wolfram Research (1988), LinearSolve, Wolfram Language function, https://reference.wolfram.com/language/ref/LinearSolve.html (updated 2024).

    CMS

    Wolfram Language. 1988. "LinearSolve." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/LinearSolve.html.

    APA

    Wolfram Language. (1988). LinearSolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LinearSolve.html

    BibTeX

    @misc{reference.wolfram_2024_linearsolve, author="Wolfram Research", title="{LinearSolve}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/LinearSolve.html}", note=[Accessed: 21-December-2024 ]}

    BibLaTeX

    @online{reference.wolfram_2024_linearsolve, organization={Wolfram Research}, title={LinearSolve}, year={2024}, url={https://reference.wolfram.com/language/ref/LinearSolve.html}, note=[Accessed: 21-December-2024 ]}