## 18.2 LAPACK

LAPACK is a collection of Fortran subprograms for advanced linear algebra problems like solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems. LAPACK replaces the older packages LINPACK and EISPACK. LAPACK subroutines were written to exploit BLAS as much as possible.

### Routines

Probably the most widely used LAPACK routines are the ones that solve systems of linear equations:

• xGESV - Solve AX=B for a general matrix A
• xPOSV - Solve AX=B for a symmetric positive definite matrix A
• xGBSV - Solve AX=B for a general banded matrix A

There are many more routines for other special types of matrices.

The source code and executables for some computers are available from Netlib at http://www.netlib.org/lapack. The complete LAPACK User's Guide is also on the Web.

### Documentation

Just like the BLAS routines, the LAPACK routines are virtually self-explanatory. Details of the input and output parameters for any given subroutine are contained in the header section of the file. For example, here is the header section of the LAPACK subroutine SGESV:

```      SUBROUTINE SGESV( N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
*  -- LAPACK driver routine (version 2.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     March 31, 1993
*
*     .. Scalar Arguments ..
INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
INTEGER            IPIV( * )
REAL               A( LDA, * ), B( LDB, * )
*     ..
*
*  Purpose
*  =======
*
*  SGESV computes the solution to a real system of linear equations
*     A * X = B,
*  where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
*
*  The LU decomposition with partial pivoting and row interchanges is
*  used to factor A as
*     A = P * L * U,
*  where P is a permutation matrix, L is unit lower triangular, and U is
*  upper triangular.  The factored form of A is then used to solve the
*  system of equations A * X = B.
*
*  Arguments
*  =========
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  A       (input/output) REAL array, dimension (LDA,N)
*          On entry, the N-by-N coefficient matrix A.
*          On exit, the factors L and U from the factorization
*          A = P*L*U; the unit diagonal elements of L are not stored.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (output) INTEGER array, dimension (N)
*          The pivot indices that define the permutation matrix P;
*          row i of the matrix was interchanged with row IPIV(i).
*
*  B       (input/output) REAL array, dimension (LDB,NRHS)
*          On entry, the N-by-NRHS matrix of right hand side matrix B.
*          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, U(i,i) is exactly zero.  The factorization
*                has been completed, but the factor U is exactly
*                singular, so the solution could not be computed.
*
*  =====================================================================
```