## SOLVE Function

**solves a system of linear equations**
**SOLVE(** *A*, *B***)**

The inputs to the SOLVE function are as follows:
*A*
- is an
*n* ×*n* nonsingular matrix.

*B*
- is an
*n* ×*p* matrix.

The SOLVE function solves the set of linear
equations **AX** = **B** for **X**.
**A** must be square and nonsingular.

**X** = SOLVE(**A,B**) is equivalent to using
the INV function as **X** = INV(**A**)***B**.
However, the SOLVE function is recommended over the INV
function because it is more efficient and more accurate.
An example follows:
x=solve(a,b);

The solution method used is discussed
in Forsythe, Malcolm, and Moler (1967).

The SOLVE function (as well as the DET and INV functions)
uses the following criterion to decide whether the input
matrix, **A** = [*a*_{ij}]_{i,j = 1, ... ,n}, is singular:

where *MACHEPS* is the relative machine precision.
All matrix elements less than or equal to *sing*
are now considered rounding errors of the largest
matrix elements, so they are taken to be zero.
For example, if a diagonal or triangular coefficient matrix has
a diagonal value less than or equal to *sing*, the matrix
is considered singular by the DET, INV, and SOLVE functions.

Previously, a much smaller singularity criterion was
used, which caused algebraic operations to be performed
on values that were essentially floating point error.
This occasionally yielded numerically unstable results.
The new criterion is much more conservative,
and it generates far fewer erroneous results.
In some cases, you may need to scale
the data to avoid singular matrices.
If you think the new criterion is too strong,

- try the GINV function to compute the generalized inverse
- examine the size of the singular
values returned by the SVD function.
The SVD function can be used to compute a generalized
inverse with a user-specified singularity criterion.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.