MATLAB Functions

Matrix inverse

Y = inv(X)

Description

Y = inv(X)

returns the inverse of the square matrix X. A warning message is printed if X is badly scaled or nearly singular.

In practice, it is seldom necessary to form the explicit inverse of a matrix. A frequent misuse of inv arises when solving the system of linear equations
Ax = b. One way to solve this is with x = inv(A)*b. A better way, from both an execution time and numerical accuracy standpoint, is to use the matrix division operator x = A\b. This produces the solution using Gaussian elimination, without forming the inverse. See \ and / for further information.

Examples

Here is an example demonstrating the difference between solving a linear system by inverting the matrix with inv(A)*b and solving it directly with A\b. A matrix A of order 100 has been constructed so that its condition number, cond(A), is 1.e10, and its norm, norm(A), is 1. The exact solution x is a random vector of length 100 and the right-hand side is b = A*x. Thus the system of linear equations is badly conditioned, but consistent.

On a 20 MHz 386SX notebook computer, the statements

tic, y = inv(A)*b, toc
err = norm(y-x)
res = norm(A*y-b)

produce

elapsed_time = 
    9.6600
err =
    2.4321e-07
res =
    1.8500e-09

while the statements

tic, z = A\b, toc
err = norm(z-x)
res = norm(A*z-b)

produce

elapsed_time =
    3.9500
err =
    6.6161e-08
res =
    9.1103e-16

It takes almost two and one half times as long to compute the solution with
y = inv(A)*b as with z = A\b. Both produce computed solutions with about the same error, 1.e-7, reflecting the condition number of the matrix. But the size of the residuals, obtained by plugging the computed solution back into the original equations, differs by several orders of magnitude. The direct solution produces residuals on the order of the machine accuracy, even though the system is badly conditioned.

The behavior of this example is typical. Using A\b instead of inv(A)*b is two to three times as fast and produces residuals on the order of machine accuracy, relative to the magnitude of the data.

Algorithm

The inv command uses the subroutines ZGEDI and ZGEFA from LINPACK. For more information, see the LINPACK Users' Guide.

Diagnostics

From inv, if the matrix is singular,

Matrix is singular to working precision.

On machines with IEEE arithmetic, this is only a warning message. inv then returns a matrix with each element set to Inf. On machines without IEEE arithmetic, like the VAX, this is treated as an error.

If the inverse was found, but is not reliable, this message is displayed.

Warning: Matrix is close to singular or badly scaled.
    Results may be inaccurate. RCOND = xxx