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Solve linear system of equations
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Syntax
X = linsolve(A,B)
X = linsolve(A,B,opts)
[X,r] = linsolve(___)
Description
example
X = linsolve(A,B) solves the linear system AX = B using one of these methods:
When
Ais square,linsolveuses LU factorization with partial pivoting.For all other cases,
linsolveuses 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 Linear System
Open Live Script
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 . The number of rows in A and b must be equal.
b = ones(size(A,2),1);
Solve the linear system using mldivide and time the calculation.
ticx1 = A\b; t1 = toc
t1 = 0.0551
Now, solve the system again using linsolve. Specify the options structure so that linsolve can select an appropriate solver for a lower triangular matrix.
ticx2 = linsolve(A,b,opts);t2 = toc
t2 = 0.0226
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 = 2.4402
Suppress Matrix Condition Warnings
Open Live Script
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 = 20×1109 × -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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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 |
|---|---|
| Lower triangular (nonzero values appearing only on or below the main diagonal) |
| Upper triangular (nonzero values appearing only on or above the main diagonal) |
| Upper Hessenberg (all zero values below the first subdiagonal) |
| Real symmetric or complex Hermitian (matrix equal to its transpose) |
| Positive definite (all positive eigenvalues) |
| Rectangular matrix (different number of rows and columns) |
| Conjugate transpose — Specifies whether the function solves |
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.
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Notes on Usage
If
Ahas the properties inopts, thenlinsolveis faster compared tomldivide, becauselinsolveinvokes the appropriate solver immediately and does not perform any tests to verify thatAhas the specified properties.If
Adoes not have the properties that you specify inopts, thenlinsolvereturns incorrect results and does not always return an error message. Therefore, if you are unsure whetherAhas the specified properties, use mldivide or decomposition instead.
Data Types: struct
Output Arguments
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X — Linear system solution
vector | matrix
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
Aism-by-nandBism-by-k, thenXisn-by-kand is the solution to AX = B.If
opts.TRANSA = true, thenAism-by-nandBisn-by-k. In this case,Xism-by-kand is the solution to ATX = B.
r — Reciprocal condition number or rank
scalar
Reciprocal condition number or rank, returned as a scalar.
If
Ais a square matrix, thenris the reciprocal condition number ofA.If
Ais a rectangular matrix, thenris the rank ofA.If
optsis specified, thenris the reciprocal of the condition number ofAunlessRECTistrueand bothLTandUTarefalse, in which case,rgives the rank ofA.
Tips
The speed benefit of
linsolvecan 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 tomldivide. The speed benefit withlinsolvearises 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 thatmldividemakes.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
The
optsstructure must be a constant scalar. Code generation does not support arrays of options structures.Code generation only optimizes these cases:
UTLTUHESS=true(theTRANSAcan be eithertrueorfalse)SYM=trueandPOSDEF=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
optsstructure must be a constant scalar. Code generation does not support arrays of options structures.Code generation only optimizes these cases:
UTLTUHESS=true(theTRANSAcan be eithertrueorfalse)SYM=trueandPOSDEF=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®
linsolvefunction prints a warning ifAis badly scaled, nearly singular, or rank deficient. ThegpuArraylinsolveis 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
See Also
mldivide | decomposition | lsqminnorm
Topics
- Systems of Linear Equations
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