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LinearSolveFunction
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LinearSolveFunction

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LinearSolveFunction[dimensions,data]

represents a function for providing solutions to a matrix equation.

Details

Examples

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Basic Examples  (1)Summary of the most common use cases

A 2×2 matrix:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-czpt04

Create a function for solving the matrix equation :

In[2]:=2
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https://wolfram.com/xid/0n4sns32cuq-e10f1r
Out[2]=2

Find the solution of for :

In[3]:=3
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https://wolfram.com/xid/0n4sns32cuq-k49yl
Out[3]=3

Find the solution of for a matrix :

In[4]:=4
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https://wolfram.com/xid/0n4sns32cuq-b55sfv
Out[4]=4

Scope  (3)Survey of the scope of standard use cases

Do computations using exact arithmetic:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-ji50ru
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0n4sns32cuq-c2gspm
Out[2]=2

Do computations using machine‐number arithmetic:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-fakfpt
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0n4sns32cuq-i7v4fs
Out[2]=2

Do computations using 24‐digit arithmetic:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-efuy1p
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0n4sns32cuq-bo9q6n
Out[2]=2

Generalizations & Extensions  (1)Generalized and extended use cases

Get a factorization for a large sparse matrix:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-bx8gj9
Out[2]=2

Show the solution of :

In[3]:=3
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https://wolfram.com/xid/0n4sns32cuq-e22h4e
Out[5]=5

Applications  (2)Sample problems that can be solved with this function

Find the eigenvector associated with the smallest eigenvalue:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-if535v
Out[3]=3

Approximate value of the eigenvalue:

In[4]:=4
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https://wolfram.com/xid/0n4sns32cuq-pisuv3
Out[4]=4

Expressions for a finite difference approximation to the Laplacian in 2 dimensions:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-eps6v6

A matrix approximating the Laplacian operator:

In[2]:=2
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https://wolfram.com/xid/0n4sns32cuq-hw92do
Out[2]=2

A linear solve function for the matrix:

In[3]:=3
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https://wolfram.com/xid/0n4sns32cuq-efcalb
Out[3]=3

An approximate solution for the Poisson equation on the unit rectangle:

In[4]:=4
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https://wolfram.com/xid/0n4sns32cuq-h3nd5j
Out[4]=4

An approximate solution for the Poisson equation on the unit rectangle:

In[5]:=5
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https://wolfram.com/xid/0n4sns32cuq-cea55w
Out[5]=5

Properties & Relations  (2)Properties of the function, and connections to other functions

A LinearSolveFunction is effectively like a matrix inverse. A large sparse matrix:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-ppnh13
Out[1]=1

Time to compute and memory used by the linear solve function:

In[2]:=2
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https://wolfram.com/xid/0n4sns32cuq-g2n29
Out[2]=2

Time to compute and memory used by the inverse matrix:

In[3]:=3
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https://wolfram.com/xid/0n4sns32cuq-c0degb
Out[3]=3

Comparison of f and the inverse matrix for a random vector b:

In[4]:=4
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https://wolfram.com/xid/0n4sns32cuq-dy3l4
Out[5]=5

Create a LinearSolveFunction for a simple matrix:

In[1]:=1
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https://wolfram.com/xid/0n4sns32cuq-hvqrm
Out[1]=1

Compute the inverse matrix from f:

In[2]:=2
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https://wolfram.com/xid/0n4sns32cuq-lmh30y
Out[2]=2

Compute the inverse matrix using Inverse:

In[3]:=3
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https://wolfram.com/xid/0n4sns32cuq-hx4pmo
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

Matrix factorizations are not found for rectangular matrices:

In[1]:=1
✖
https://wolfram.com/xid/0n4sns32cuq-k3a05z
Out[1]=1

A solution is found:

In[2]:=2
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https://wolfram.com/xid/0n4sns32cuq-j3z2mg
Out[2]=2
Wolfram Research (2003), LinearSolveFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/LinearSolveFunction.html.
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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.

✖
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.

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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

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

BibTeX

@misc{reference.wolfram_2025_linearsolvefunction, author="Wolfram Research", title="{LinearSolveFunction}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/LinearSolveFunction.html}", note=[Accessed: 04-April-2025 ]}

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@misc{reference.wolfram_2025_linearsolvefunction, author="Wolfram Research", title="{LinearSolveFunction}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/LinearSolveFunction.html}", note=[Accessed: 04-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_linearsolvefunction, organization={Wolfram Research}, title={LinearSolveFunction}, year={2003}, url={https://reference.wolfram.com/language/ref/LinearSolveFunction.html}, note=[Accessed: 04-April-2025 ]}

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@online{reference.wolfram_2025_linearsolvefunction, organization={Wolfram Research}, title={LinearSolveFunction}, year={2003}, url={https://reference.wolfram.com/language/ref/LinearSolveFunction.html}, note=[Accessed: 04-April-2025 ]}