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

Examples

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Basic Examples  (2)

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With no righthand side, a LinearSolveFunction is returned:

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Scope  (6)

Options  (6)

Applications  (2)

Properties & Relations  (4)

Possible Issues  (2)

Neat Examples  (3)

See Also

Inverse  Solve  NullSpace  CoefficientArrays  CholeskyDecomposition  PseudoInverse  LeastSquares  RowReduce  LinearSolveFunction  MatrixPower

Tutorials

Introduced in 1988
(1.0)
| Updated in 2014
(10.0)