# linsolve

Solve linear system of equations

## Description

example

X = linsolve(A,B) solves the linear system AX = B using one of these methods:

• When A is square, linsolve uses LU factorization with partial pivoting.

• For all other cases, linsolve uses QR factorization with column pivoting.

linsolve warns if A is ill conditioned (for square matrices) or rank deficient (for rectangular matrices).

example

X = linsolve(A,B,opts) uses an appropriate solver as determined by the options structure opts. The fields in opts are logical values describing properties of the matrix A. For example, if A is an upper triangular matrix, you can set opts.UT = true to make linsolve use a solver designed for upper triangular matrices. linsolve does not test to verify that A has the properties specified in opts.

example

[X,r] = linsolve(___) also returns r, which is the reciprocal of the condition number of A (for square matrices) or the rank of A (for rectangular matrices). You can use any of the input argument combinations in previous syntaxes. With this syntax, linsolve does not warn if A is ill conditioned or rank deficient.

## Examples

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Solve a linear system with both mldivide and linsolve to compare performance.

mldivide is the recommended way to solve most linear systems of equations in MATLAB®. However, the function performs several checks on the input matrix to determine whether it has any special properties. If you know about the properties of the coefficient matrix ahead of time, then you can use linsolve to avoid time-consuming checks for large matrices.

Create a 10000-by-10000 magic square matrix and extract the lower triangular portion. Set the LT field of the opts structure to true to indicate that A is a lower triangular matrix.

A = tril(magic(1e4));
opts.LT = true;

Create a vector of ones for the right-hand side of the linear equation $\mathrm{Ax}=\mathit{b}$. The number of rows in A and b must be equal.

b = ones(size(A,2),1);

Solve the linear system $\mathrm{Ax}=\mathit{b}$ using mldivide and time the calculation.

tic
x1 = A\b;
t1 = toc
t1 = 0.0524

Now, solve the system again using linsolve. Specify the options structure so that linsolve can select an appropriate solver for a lower triangular matrix.

tic
x2 = linsolve(A,b,opts);
t2 = toc
t2 = 0.0197

Compare the execution times to see how much faster linsolve is. As with any timing comparison, the results can vary between different computers and releases of MATLAB.

speedup = t1/t2
speedup = 2.6606

Solve a linear system using linsolve with two outputs to suppress matrix conditioning warnings.

Create a 20-by-20 Hilbert test matrix. This matrix is nearly singular, with the largest singular value being about 2e18 larger than the smallest.

A = hilb(20);

Solve a linear system involving A with linsolve. Since A is nearly singular, linsolve returns a warning.

b = ones(20,1);
x = linsolve(A,b);
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND =  5.628781e-20.

Now, solve the same linear system, but specify two outputs to linsolve. MATLAB® suppresses the warning, and the second output r contains the reciprocal condition number of A. You can use this syntax to handle ill-conditioned matrices with special cases in your code, without the code producing a warning.

[x,r] = linsolve(A,b)
x = 20×1
109 ×

-0.0000
0.0000
-0.0004
0.0071
-0.0592
0.2819
-0.7821
1.1830
-0.7030
-0.1061
⋮

r = 5.6288e-20

## Input Arguments

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Coefficient matrix. A appears in the system of linear equations on the left as AX = B. The number of rows in A must equal the number of rows in B.

A cannot be sparse. To solve a linear system involving a sparse matrix, use mldivide or decomposition instead.

Data Types: single | double
Complex Number Support: Yes

Input array, specified as a vector or matrix. B appears in the system of linear equations on the right as AX = B. If B is a matrix, then each column in the matrix represents a different vector for the right-hand side.

The number of rows in A must equal the number of rows in B.

Data Types: single | double
Complex Number Support: Yes

Coefficient matrix properties, specified as a structure. Use this structure to specify properties of A that linsolve uses to select an appropriate solver for the linear system. The fields in the structure contain true/false values to indicate whether A has each property. By default all fields in the structure are assumed to be false. This table lists the possible fields in opts and their corresponding matrix properties.

FieldMatrix Property

LT

Lower triangular (nonzero values appearing only on or below the main diagonal)

UT

Upper triangular (nonzero values appearing only on or above the main diagonal)

UHESS

Upper Hessenberg (all zero values below the first subdiagonal)

SYM

Real symmetric or complex Hermitian (matrix equal to its transpose)

POSDEF

Positive definite (all positive eigenvalues)

RECT

Rectangular matrix (different number of rows and columns)

TRANSA

Conjugate transpose — Specifies whether the function solves A*X = B (when opts.TRANSA = false) or the transposed problem A'*X = B (when opts.TRANSA = true)

Example: opts.UT = true specifies that A is upper triangular.

Example: opts.SYM = true, opts.POSDEF = true sets two fields to specify that A is symmetric and positive definite.

#### Valid Combinations

The rows of this table list all combinations of field values in opts that are valid for linsolve. Empty cells are the default value of false, and a true/false entry indicates that linsolve accepts either value.

LT

UT

UHESS

SYM

POSDEF

RECT

TRANSA

A is lower triangular

true

true/false

true/false

A is upper triangular

true

true/false

true/false

A is upper Hessenberg

true

true/false

A is symmetric

true

true/false

true/false

A is rectangular

true/false

true/false

#### Notes on Usage

• If A has the properties in opts, then linsolve is faster compared to mldivide, because linsolve invokes the appropriate solver immediately and does not perform any tests to verify that A has the specified properties.

• If A does not have the properties that you specify in opts, then linsolve returns incorrect results and does not always return an error message. Therefore, if you are unsure whether A has the specified properties, use mldivide or decomposition instead.

Data Types: struct

## Output Arguments

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Linear system solution, returned as a vector or matrix that satisfies AX = B (or ATX = B if opts.TRANSA = true). The size of X depends on whether opts.TRANSA = true:

• If A is m-by-n and B is m-by-k, then X is n-by-k and is the solution to AX = B.

• If opts.TRANSA = true, then A is m-by-n and B is n-by-k. In this case, X is m-by-k and is the solution to ATX = B.

Reciprocal condition number or rank, returned as a scalar.

• If A is a square matrix, then r is the reciprocal condition number of A.

• If A is a rectangular matrix, then r is the rank of A.

• If opts is specified, then r is the reciprocal of the condition number of A unless RECT is true and both LT and UT are false, in which case, r gives the rank of A.

## Tips

• The speed benefit of linsolve can vary depending on the matrix structure and the relative optimization of the underlying algorithms. In some cases (such as with small matrices) there might not be any speed-up compared to mldivide. The speed benefit with linsolve arises by avoiding costly checks on the properties of large matrices, or by choosing an algorithm that is better suited to the input than the choice that mldivide makes.

## Version History

Introduced before R2006a