linsolve

Solve linear system of equations

Syntax

X = linsolve(A,B)

X = linsolve(A,B,opts)

[X,r] = linsolve(___)

Description

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).

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.

[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

Solve Linear System

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 = 
0.1122

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 = 
0.0261

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 = 
4.2985

Suppress Matrix Condition Warnings

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

A — Coefficient matrix

matrix

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

B — Input array

vector | matrix

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

opts — Coefficient matrix properties

structure

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.

Field	Matrix 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

X — Linear system solution

vector | matrix

Linear system solution, returned as a vector or matrix that satisfies AX = B (or A^TX = 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 A^TX = B.

r — Reciprocal condition number or rank

scalar

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The opts structure must be a constant scalar. Code generation does not support arrays of options structures.

• Code generation only optimizes these cases:

  • UT

  • LT

  • UHESS = true (the TRANSA can be either true or false)

  • SYM = true and POSDEF = true

  Other options are equivalent to using mldivide.

• Code generation does not support sparse matrix inputs for this function.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Usage notes and limitations:

• The opts structure must be a constant scalar. Code generation does not support arrays of options structures.

• Code generation only optimizes these cases:

  • UT

  • LT

  • UHESS = true (the TRANSA can be either true or false)

  • SYM = true and POSDEF = true

  Other options are equivalent to using mldivide.

• Code generation does not support sparse matrix inputs for this function.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• The two-output syntax [x,r] = linsolve(___) is not supported.

• The MATLAB^® linsolve function prints a warning if A is badly scaled, nearly singular, or rank deficient. The gpuArray linsolve is unable to check for this condition. Take action to avoid this condition.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a