# LinearSolveFunction

LinearSolveFunction[dimensions,data]

represents a function for providing solutions to a matrix equation.

# Details • is generated by LinearSolve[m].
• LinearSolveFunction works like Function.
• LinearSolveFunction[][b] finds the solution to the matrix equation for the specific vector or matrix .
• dimensions specifies the dimensions of the matrix m from which the LinearSolveFunction was constructed.

# Examples

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## Basic Examples(1)

A 2×2 matrix:

Create a function for solving the matrix equation :

Find the solution of for :

Find the solution of for a matrix :

## Scope(3)

Do computations using exact arithmetic:

Do computations using machinenumber arithmetic:

Do computations using 24digit arithmetic:

## Generalizations & Extensions(1)

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:

## Properties & Relations(2)

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 f and the inverse matrix for a random vector b:

Create a LinearSolveFunction for a simple matrix:

Compute the inverse matrix from f:

Compute the inverse matrix using Inverse:

## Possible Issues(1) 