# LinearSolveFunction

LinearSolveFunction[dimensions,data]

represents a function for providing solutions to a matrix equation.

# 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)

Matrix factorizations are not found for rectangular matrices:

A solution is found:

Wolfram Research (2003), LinearSolveFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/LinearSolveFunction.html.

#### Text

Wolfram Research (2003), LinearSolveFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/LinearSolveFunction.html.

#### CMS

Wolfram Language. 2003. "LinearSolveFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LinearSolveFunction.html.

#### APA

Wolfram Language. (2003). LinearSolveFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LinearSolveFunction.html

#### BibTeX

@misc{reference.wolfram_2024_linearsolvefunction, author="Wolfram Research", title="{LinearSolveFunction}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/LinearSolveFunction.html}", note=[Accessed: 20-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_linearsolvefunction, organization={Wolfram Research}, title={LinearSolveFunction}, year={2003}, url={https://reference.wolfram.com/language/ref/LinearSolveFunction.html}, note=[Accessed: 20-May-2024 ]}