mrdivide, /

Solve systems of linear equations xA = B for x

Syntax

x = B/A

x = mrdivide(B,A)

Description

x = B/A solves the system of linear equations x*A = B for x. The matrices A and B must contain the same number of columns. MATLAB^® displays a warning message if A is badly scaled or nearly singular, but performs the calculation regardless.

• If A is a scalar, then B/A is equivalent to B./A.

• If A is a square n-by-n matrix and B is a matrix with n columns, then x = B/A is a solution to the equation x*A = B, if it exists.

• If A is a rectangular m-by-n matrix with m ~= n, and B is a matrix with n columns, then x = B/A returns a least-squares solution of the system of equations x*A = B.

x = mrdivide(B,A) is an alternative way to execute x = B/A, but is rarely used. It enables operator overloading for classes.

Examples

System of Equations

Solve a system of equations that has a unique solution, x*A = B.

A = [1 1 3; 2 0 4; -1 6 -1];
B = [2 19 8];
x = B/A

x = 1×3

    1.0000    2.0000    3.0000

Least-Squares on an Underdetermined System

Solve an underdetermined system, x*C = D.

C = [1 0; 2 0; 1 0];
D = [1 2];
x = D/C

Warning: Rank deficient, rank = 1, tol =  1.332268e-15.

x = 1×3

         0    0.5000         0

MATLAB® issues a warning but proceeds with calculation.

Verify that x is not an exact solution.

x*C-D

ans = 1×2

     0    -2

Input Arguments

A, B — Operands

vectors | full matrices | sparse matrices

Operands, specified as vectors, full matrices, or sparse matrices. A and B must have the same number of columns.

• If A or B has an integer data type, the other input must be scalar. Operands with an integer data type cannot be complex.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | char
Complex Number Support: Yes

Output Arguments

x — Solution

vector | full matrix | sparse matrix

Solution, returned as a vector, full matrix, or sparse matrix. If A is an m-by-n matrix and B is a p-by-n matrix, then x is a p-by-m matrix.

x is sparse only if both A and B are sparse matrices.

Tips

• The operators / and \ are related to each other by the equation B/A = (A'\B')'.

• If A is a square matrix, then B/A is roughly equal to B*inv(A), but MATLAB processes B/A differently and more robustly.

• Use decomposition objects to efficiently solve a linear system multiple times with different right-hand sides. decomposition objects 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.

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 syntax Z = X/Y, the Y operand must be a scalar.

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

C/C++ Code Generation

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

Usage notes and limitations:

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

• If the code generator determines that A is a scalar at compile time, then B/A is equivalent to B./A.

  Otherwise, even if the run-time value of A is a scalar, the code generator treats A as a matrix and requires the number of columns in A and B to be equal. So, if B has more than one column, the code generator produces an error.

GPU Code Generation

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

Usage notes and limitations:

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

• If the code generator determines that A is a scalar at compile time, then B/A is equivalent to B./A.

  Otherwise, even if the run-time value of A is a scalar, the code generator treats A as a matrix and requires the number of columns in A and B to be equal. So, if B has more than one column, the code generator produces an error.

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™.

The mrdivide function supports GPU array input with these usage notes and limitations:

• If B is rectangular, then it must also be nonsparse.

• The MATLAB mrdivide function prints a warning if B is badly scaled, nearly singular, or rank deficient. The gpuArray mrdivide 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™.

Usage notes and limitations:

• If B is rectangular, then it must also be nonsparse.

• The MATLAB mrdivide function prints a warning if B is badly scaled, nearly singular, or rank deficient. The distributed array mrdivide is unable to check for this condition. Take action to avoid this condition.

• If B is an M-by-N matrix with N > M, for distributed arrays, mrdivide computes a solution that minimizes norm(X). The result is the same as the result of PINV(B)*A.

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

Version History

Introduced before R2006a