This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# LinearSolveFunction

 LinearSolveFunctionrepresents a function for providing solutions to a matrix equation.
• LinearSolveFunction[...][b] finds the solution to the matrix equation for the specific vector or matrix .
• In standard output format, only the dimensions element of a LinearSolveFunction object is printed explicitly. The remaining elements are indicated by .
• dimensions specifies the dimensions of the matrix m from which the LinearSolveFunction was constructed.
A 2×2 matrix:
Create a function for solving the matrix equation :
Find the solution of for :
Find the solution of for a matrix :
A 2×2 matrix:
Create a function for solving the matrix equation :
 Out[2]=
Find the solution of for :
 Out[3]=
Find the solution of for a matrix :
 Out[4]=
 Scope   (3)
Do computations using exact arithmetic:
Do computations using machine-number arithmetic:
Do computations using 24-digit arithmetic:
Get a factorization for a large sparse matrix:
Show the solution of :
 Applications   (2)
Find the eigenvector associated with the smallest eigenvalue:
Approximate value of the eigenvalue:
Expressions for a finite difference approximation to the Laplacian in 2 dimensions:
A matrix approximating the Laplacian operator:
A linear solve function for the matrix:
An approximate solution for the Poisson equation on the unit rectangle:
An approximate solution for the Poisson equation on the unit rectangle:
A LinearSolveFunction is effectively like a matrix inverse. A large sparse matrix:
Time to compute and memory used by the linear solve function:
Time to compute and memory used by the inverse matrix:
Comparison of and the inverse matrix for a random vector :
Create a LinearSolveFunction for a simple matrix:
Compute the inverse matrix from :
Compute the inverse matrix using Inverse: