Computes the LU factorization of a real general matrix and estimates its L1 condition number.

Routine LFCRG performs an LU factorization of a real general coefficient matrix. It also estimates the condition number of the matrix. The underlying code is based on either LINPACK , LAPACK, or ScaLAPACK code depending upon which supporting libraries are used during linking. For a detailed explanation see Using ScaLAPACK, LAPACK, LINPACK, and EISPACK in the Introduction section of this manual. The LU factorization is done using scaled partial pivoting. Scaled partial pivoting differs from partial pivoting in that the pivoting strategy is the same as if each row were scaled to have the same ∞-norm. Otherwise, partial pivoting is used.

The L1 condition number of the matrix A is defined to be κ(A) = ∥A∥1∥A-1∥1. Since it is expensive to compute ∥A-1∥1, the condition number is only estimated. The estimation algorithm is the same as used by LINPACK and is described in a paper by Cline et al. (1979).

If the estimated condition number is greater than 1/ɛ (where ɛ is machine precision), a warning error is issued. This indicates that very small changes in A can cause very large changes in the solution x. Iterative refinement can sometimes find the solution to such a system.

LFCRG fails if U, the upper triangular part of the factorization, has a zero diagonal element. This can occur only if A either is singular or is very close to a singular matrix.

The LU factors are returned in a form that is compatible with routines LFIRG, LFSRG and LFDRG. To solve systems of equations with multiple right-hand-side vectors, use LFCRG followed by either LFIRG or LFSRG called once for each right-hand side. The routine LFDRG can be called to compute the determinant of the coefficient matrix after LFCRG has performed the factorization.

Let F be the matrix FACT and let p be the vector IPVT. The triangular matrix U is stored in the upper triangle of F. The strict lower triangle of F contains the information needed to reconstruct L using

where Pk is the identity matrix with rows k and pk interchanged and Lk is the identity with Fik for
i = k + 1, …, N inserted below the diagonal. The strict lower half of F can also be thought of as containing the negative of the multipliers. LFCRG is based on the LINPACK routine SGECO; see Dongarra et al. (1979). SGECO uses unscaled partial pivoting.

The arguments which differ from the standard version of this routine are:

All other arguments are global and are the same as described for the standard version of the routine. In the argument descriptions above, MXLDA and MXCOL can be obtained through a call to SCALAPACK_GETDIM (see Utilities) after a call to SCALAPACK_SETUP (see Utilities) has been made. See the ScaLAPACK Example below.

The inverse of a 3 × 3 matrix is computed. LFCRG is called to factor the matrix and to check for singularity or ill-conditioning. LFIRG is called to determine the columns of the inverse.

The inverse of the same 3 × 3 matrix is computed as a distributed example. LFCRG is called to factor the matrix and to check for singularity or ill-conditioning. LFIRG is called to determine the columns of the inverse. SCALAPACK_MAP and SCALAPACK_UNMAP are IMSL utility routines (see Chapter 11, “Utilities”) used to map and unmap arrays to and from the processor grid. They are used here for brevity. DESCINIT is a ScaLAPACK tools routine which initializes the descriptors for the local arrays.