Next: Solving eigenvalue problems Up: Linear Algebra in Previous: LAPACK

## Solving linear systems

A good method (when A is dense and data uncertainty is present or possible) for solving Ax = b is to factorize A into PA = LU and then solve the two systems Ly = Pb and Ux = y (write Ax = b so PA x = Pb so LUx = Pb and put Ux = y, whence Ly = Pb). Each of these triangular systems is very easy to solve, by forward elimination and back substitution, respectively.

Finding is almost never a good idea, because it is more expensive than factorization (indeed you have to essentially factorize to find , in the usual case). Solving the two triangular systems is just as cheap (for a computer) as the matrix multiplication B b, where B is the inverse of A. If you want to know itself, (there are statistical interpretations, for example) then by all means, do that, but don't find merely to solve Ax = b.

We will talk about solving overdetermined systems Ax = b when there are more equations than unknowns, later.

The LAPACK routines for solving linear systems (there are lots) all allow you to compute the condition number of the system, as well (or at least a good approximation of it). The fundamental inequality is that if Ax = b and , then

where is the condition number of the matrix A. Note that it depends on the matrix A and not on the method used to solve Ax = b. In terms of the singular value decomposition, we have that is the ratio of the largest singular value to the smallest; this is infinite if the matrix is singular, and is large if the matrix is `nearly' singular. In fact, is the 2-norm matrix distance to the nearest rank k-1 matrix. See Golub and Van Loan for details.

Next: Solving eigenvalue problems Up: Linear Algebra in Previous: LAPACK

Robert Corless
Wed Jan 31 11:33:59 EST 1996