Documentation

mldivide, \

Solve systems of linear equations Ax = B for x

Syntax

Description

example

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

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

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

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

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

Examples

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System of Equations

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

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

    1.0000
    1.0000
    1.0000

Linear System with Singular Matrix

Solve a linear system of equations A*x = b involving 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 =
1.306145e-17. 

x =

    1.5000
    2.5000
   -0.5000
    0.5000

When rcond is between 0 and eps, 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, although rcond is small.

When rcond is equal to 0, the singular warning appears.

A = [1 0; 0 0];
b = [1; 1];
x = A\b
Warning: Matrix is singular to working precision. 

x =

     1
   Inf

In this case, division by zero leads to computations with Inf and/or NaN, making the computed result unreliable.

Least-Squares Solution of Underdetermined System

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

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

         0
    4.0000
    0.6667

Linear System with Sparse Matrix

Solve a simple system of linear equations using sparse matrices.

Consider the matrix equation A*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 — Coefficient matrixvector | full matrix | sparse matrix

Coefficient matrix, specified as a vector, full matrix, or sparse matrix. If A has m rows, then B must have m rows.

Data Types: single | double
Complex Number Support: Yes

B — Right-hand sidevector | full matrix | sparse matrix

Right-hand side, specified as a vector, full matrix, or sparse matrix. If B has m rows, then A must have m rows.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

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x — Solutionvector | 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 an m-by-p matrix, then x is an n-by-p matrix, including the case when p==1.

If A has full storage, x is also full. If A is sparse, then x has the same storage as B.

More About

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Tips

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

  • If the rank of A is less than the number of columns in A, then x = A\B is not necessarily the minimum norm solution. The more computationally expensive x = pinv(A)*B computes the minimum norm least-squares solution.

  • For full singular inputs, you can compute the least-squares solution using the function linsolve.

Algorithms

The versatility of mldivide in solving linear systems stems from its ability 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 between full (also called "dense") and sparse input arrays.

Algorithm for Full Inputs

The flow chart below shows the algorithm path when inputs A and B are full.

Algorithm for Sparse Inputs

If A is full and B is sparse then mldivide converts B to a full matrix and uses the full algorithm path (above) to compute a solution with full storage. If A is sparse, the storage of the solution x is the same as that of B and mldivide follows the algorithm path for sparse inputs, shown below.

See Also

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