mldivide,\

Solve systems of linear equationsAx = Bforx

Syntax

x = A\B

x = mldivide(A,B)

Description

x=A\Bsolves the system of linear equationsA*x = B. The matricesAandBmust have the same number of rows. MATLAB^®displays a warning message ifAis badly scaled or nearly singular, but performs the calculation regardless.

IfAis a scalar, thenA\B相当于A.\B.
IfAis a squaren-by-nmatrix andBis a matrix withnrows, thenx = A\Bis a solution to the equationA*x = B, if it exists.
IfAis a rectangularm-by-nmatrix withm ~= n, andBis a matrix withmrows, thenA\Breturns a least-squares solution to the system of equationsA*x= B.

x= mldivide(A,B)is an alternative way to executex=A\B, but is rarely used. It enables operator overloading for classes.

Examples

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System of Equations

Open Live Script

Solve a simple system of linear equations,A*x = B.

A = magic(3); B = [15; 15; 15]; x = A\B

x =3×11.0000 1.0000 1.0000

Linear System with Singular Matrix

Open Live Script

Solve a linear system of equationsA*x = binvolving a singular matrix,A.

A = magic(4); b = [34; 34; 34; 34]; x = A\b

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 4.625929e-18.

x =4×10.9804 0.9412 1.0588 1.0196

Whenrcondis between0andeps, MATLAB® issues a nearly singular warning, but proceeds with the calculation. When working with ill-conditioned matrices, an unreliable solution can result even though the residual(b-A*x)is relatively small. In this particular example, the norm of the residual is zero, and an exact solution is obtained, althoughrcondis small.

Whenrcondis equal to0, the singular warning appears.

A = [1 0; 0 0]; b = [1; 1]; x = A\b

Warning: Matrix is singular to working precision.

x =2×11 Inf

In this case, division by zero leads to computations withInfand/orNaN, making the computed result unreliable.

Least-Squares Solution of Underdetermined System

Open Live Script

Solve a system of linear equations,A*x = b.

A = [1 2 0; 0 4 3]; b = [8; 18]; x = A\b

x =3×10 4.0000 0.6667

Linear System with Sparse Matrix

Open Live Script

Solve a simple system of linear equations using sparse matrices.

Consider the matrix equationA*x = B.

A = sparse([0 2 0 1 0; 4 -1 -1 0 0; 0 0 0 3 -6; -2 0 0 0 2; 0 0 4 2 0]); B = sparse([8; -1; -18; 8; 20]); x = A\B

x = (1,1) 1.0000 (2,1) 2.0000 (3,1) 3.0000 (4,1) 4.0000 (5,1) 5.0000

Input Arguments

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`A`,`B`—Operands
vectors|full matrices|sparse matrices

Operands, specified as vectors, full matrices, or sparse matrices.AandBmust have the same number of rows.

IfAorBhas an integer data type, the other input must be scalar. Operands with an integer data type cannot be complex.

Output Arguments

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`x`— Solution
vector | full matrix | sparse matrix

Solution, returned as a vector, full matrix, or sparse matrix. IfAis anm-by-nmatrix andBis anm-by-pmatrix, thenxis ann-by-pmatrix, including the case whenp==1.

IfAhas full storage,xis also full. IfAis sparse, thenxhas the same storage asB.

Tips

The operators/and\are related to each other by the equationB/A = (A'\B')'.
IfAis a square matrix, thenA\Bis roughly equal toinv(A)*B, but MATLAB processesA\Bdifferently and more robustly.
If the rank ofAis less than the number of columns inA, thenx = A\Bis not necessarily the minimum norm solution. You can compute the minimum norm least-squares solution usingx =lsqminnorm(A,B)orx =pinv(A)*B.
Usedecompositionobjects to efficiently solve a linear system multiple times with different right-hand sides.decompositionobjects are well-suited to solving problems that require repeated solutions, since the decomposition of the coefficient matrix does not need to be performed multiple times.

Algorithms

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The versatility ofmldivide在解决线性系统源于能力to take advantage of symmetries in the problem by dispatching to an appropriate solver. This approach aims to minimize computation time. The first distinction the function makes is betweenfull(also called “dense”) andsparseinput arrays.

Algorithm for Full Inputs

The flow chart below shows the algorithm path when inputsAandBarefull.

Algorithm for Sparse Inputs

IfAis full andBis sparse thenmldivideconvertsBto a full matrix and uses the full algorithm path (above) to compute a solution with full storage. IfAis sparse, the storage of the solutionxis the same as that ofBandmldividefollows the algorithm path forsparseinputs, shown below.

Extended Capabilities

Tall Arrays
Calculate with arrays that have more rows than fit in memory.

This function supports tall arrays with the limitation:

For the syntaxZ = X\Y, the arrayXmust be a scalar or a tall matrix with the same number of rows asY.

For more information, seeTall Arrays for Out-of-Memory Data.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

For sparse matrix inputs, the language standard must be C99 or later.

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

Usage notes and limitations:

For sparse matrix inputs, the language standard must be C99 or later.

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, seeRun 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:

IfAis rectangular, then it must also be nonsparse.
The MATLABmldividefunction prints a warning ifAis badly scaled, nearly singular, or rank deficient. ThegpuArraymldivideis unable to check for this condition. Take action to avoid this condition.
64-bit integers are not supported.

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

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

Usage notes and limitations:

The MATLABmldividefunction prints a warning ifAis badly scaled, nearly singular, or rank deficient. The distributed arraymldivideis unable to check for this condition. Take action to avoid this condition.

For more information, seeRun MATLAB Functions with Distributed Arrays(Parallel Computing Toolbox).

Version History

Introduced before R2006a

expand all

R2022a: LDL factorization no longer used for full matrices

Behavior changed in R2022a

The LDL factorization is no longer used for full matrices that are Hermitian indefinite. Instead, the LU factorization is used for these matrices.

mldivide,\

Syntax

Description

Examples

System of Equations

Linear System with Singular Matrix

Least-Squares Solution of Underdetermined System

Linear System with Sparse Matrix

Input Arguments

`A`,`B`—Operands
vectors|full matrices|sparse matrices

Output Arguments

`x`— Solution
vector | full matrix | sparse matrix

Tips

Algorithms

Algorithm for Full Inputs

Algorithm for Sparse Inputs

Extended Capabilities

Tall Arrays
Calculate with arrays that have more rows than fit in memory.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

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

Thread-Based Environment
Run code in the background using MATLAB®`backgroundPool`or accelerate code with Parallel Computing Toolbox™`ThreadPool`.

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

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

Version History

R2022a: LDL factorization no longer used for full matrices

See Also

Topics

mldivide,\

Syntax

Description

Examples

System of Equations

Linear System with Singular Matrix

Least-Squares Solution of Underdetermined System

Linear System with Sparse Matrix

Input Arguments

A,B—Operandsvectors|full matrices|sparse matrices

Output Arguments

x— Solutionvector | full matrix | sparse matrix

Tips

Algorithms

Algorithm for Full Inputs

Algorithm for Sparse Inputs

Extended Capabilities

Tall ArraysCalculate with arrays that have more rows than fit in memory.

C/C++ Code GenerationGenerate C and C++ code using MATLAB® Coder™.

GPU Code GenerationGenerate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Thread-Based EnvironmentRun code in the background using MATLAB®backgroundPoolor accelerate code with Parallel Computing Toolbox™ThreadPool.

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

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

Version History

R2022a: LDL factorization no longer used for full matrices

See Also

Topics

`A`,`B`—Operands
vectors|full matrices|sparse matrices

`x`— Solution
vector | full matrix | sparse matrix

Tall Arrays
Calculate with arrays that have more rows than fit in memory.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

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

Thread-Based Environment
Run code in the background using MATLAB®`backgroundPool`or accelerate code with Parallel Computing Toolbox™`ThreadPool`.

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

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