linsolve (MATLAB Functions)

Solve a linear system of equations

Syntax

X = linsolve(A,B)
X = linsolve(A,B,opts)

Description

X = linsolve(A,B) solves the linear system A*X = B using LU factorization with partial pivoting when A is square and QR factorization with column pivoting otherwise. The number of columns of A must equal the number of rows of B. If A is m-by-n and B is n-by-k, then X is m-by-k. linsolve returns a warning if A is square and ill conditioned or if it is not square and rank deficient.

[X, R] = linsolve(A,B) suppresses these warnings and returns R, which is the reciprocal of the condition number of A if A is square, or the rank of A if A is not square.

X = linsolve(A,B,opts) solves the linear system A*X = B or A'*X = B, using the solver that is most appropriate given the properties of the matrix A, which you specify in opts. For example, if A is upper triangular, you can set opts.UT = true to make linsolve use a solver designed for upper triangular matrices. If A has the properties in opts, linsolve is faster than mldivide, because linsolve does not perform any tests to verify that A has the specified properties.

Caution If A does not have the properties that you specify in opts, linsolve returns incorrect results and does not return an error message. If you are not sure whether A has the specified properties, use mldivide instead.

The TRANSA field of the opts structure specifies the form of the linear system you want to solve:

If you set opts.TRANSA = false, linsolve(A,B,opts) solves A*X = B.
If you set opts.TRANSA = true, linsolve(A,B,opts) solves A'*X = B.

The following table lists all the field of opts and their corresponding matrix properties. The values of the fields of opts must be logical and the default value for all fields is false.

Field Name
Matrix Property

LT
Lower triangular

UT
Upper triangular

UHESS
Upper Hessenberg

SYM
Real symmetric or complex Hermitian

POSDEF
Positive definite

RECT
General rectangular

TRANSA
Conjugate transpose -- specifies whether the function solves A*X = B or A'*X = B

Field Name	Matrix Property
`LT`	Lower triangular
`UT`	Upper triangular
`UHESS`	Upper Hessenberg
`SYM`	Real symmetric or complex Hermitian
`POSDEF`	Positive definite
`RECT`	General rectangular
TRANSA	Conjugate transpose -- specifies whether the function solves `AX = B` or `A'X = B`

The following table lists all combinations of field values in opts that are valid for linsolve. A true/false entry indicates that linsolve accepts either true or false.

LT
UT
UHESS
SYM
POSDEF
RECT
TRANSA

true
false
false
false
false
true/false
true/false

false
true
false
false
false
true/false
true/false

false
false
true
false
false
false
true/false

false
false
false
true
true
false
true/false

false
false
false
false
false
true/false
true/false

LT	UT	UHESS	SYM	POSDEF	RECT	TRANSA
`true`	`false`	`false`	`false`	`false`	`true`/`false`	`true`/`false`
`false`	`true`	`false`	`false`	`false`	`true`/`false`	`true`/`false`
`false`	`false`	`true`	`false`	`false`	`false`	`true`/`false`
`false`	`false`	`false`	`true`	`true`	`false`	`true`/`false`
`false`	`false`	`false`	`false`	`false`	`true`/`false`	`true`/`false`

Example

The following code solves the system A'x = b for an upper triangular matrix A using both mldivide and linsolve.

A = triu(rand(5,3)); x = [1 1 1 0 0]'; b = A'*x;
y1 = (A')\b         
opts.UT = true; opts.TRANSA = true;
y2 = linsolve(A,b,opts)

y1 =

    1.0000
    1.0000
    1.0000
         0
         0


y2 =

    1.0000
    1.0000
    1.0000
         0
         0

Note If you are working with matrices having different properties, it is useful to create an options structure for each type of matrix, such as opts_sym. This way you do not need to change the fields whenever you solve a system with a different type of matrix A.

See Also

mldivide

linkprop linspace