BUILT-IN MATHEMATICA SYMBOL

LinearSolveFunction

represents a function for providing solutions to a matrix equation.

MORE INFORMATION

LinearSolveFunction[...] is generated by LinearSolve[m].

LinearSolveFunction works like Function.

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.

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

:

A 2×2 matrix:

In[1]:=

Create a function for solving the matrix equation

:

In[2]:=

Out[2]=

Find the solution

of

for

:

In[3]:=

Out[3]=

Find the solution

of

for a matrix

:

In[4]:=

Out[4]=

Scope (3)

Do computations using exact arithmetic:

Do computations using machine-number arithmetic:

Do computations using 24-digit 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

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:

Possible Issues (1)

Matrix factorizations are not found for rectangular matrices:

A solution is found: