finds an x that solves the matrix equation m.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. »
- LinearSolve has the following options and settings:
Method Automatic method to use Modulus 0 whether to solve modulo n ZeroTest Automatic test 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
Examplesopen allclose all
Basic Examples (2)
With no right‐hand side, a LinearSolveFunction is returned:
When no right‐hand side for is given, a LinearSolveFunction is returned:
|"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|
|"BiCGSTAB"||iterative method for arbitrary square matrices|
|"ConjugateGradient"||iterative method for Hermitian positive definite matrices|
|"GMRES"||iterative method for arbitrary square matrices|
|"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|
|"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)|
Properties & Relations (4)
The solution computed by Solve:
The solution computed by LinearSolve:
Use LeastSquares to minimize :
LinearSolve returns only one solution:
Use NullSpace to get the complete spanning set of solutions:
Possible Issues (2)
All solutions are found by Solve: