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The IMSL_LUSOL function solves a general system of real or complex linear equations Ax = b.

The IMSL_LUSOL function solves a system of linear algebraic equations with a real or complex coefficient matrix A. Any of several related computations can be performed by using keywords. These extra tasks include solving AHx = b or computing the solution of Ax = b given the LU factorization of A. The function first computes the LU factorization of A with partial pivoting such that L–1PA = U.

The matrix U is upper-triangular, while L–1A ≡ Pn – 1 Ln – 2Pn – 2...L0 P0 A ≡ U. The factors Pi and Li are defined by the partial pivoting. Each Pi is an interchange of row i with row j ≥ i. Thus, Pi is defined by that value of j. Every Li = miei T is an elementary elimination matrix. The vector mi is zero in entries 0, ... , i – 1. This vector is stored as column i in the strictly lower-triangular part of the working matrix containing the decomposition information.

The factorization efficiency is based on a technique of “loop unrolling and jamming” by Dr. Leonard J. Harding of the University of Michigan, Ann Arbor, Michigan. The solution of the linear system is then found by solving two simpler systems, y = L–1b and x = U–1y. When the solution to the linear system or the inverse of the matrix is sought, an estimate of the L1 condition number of A is computed using the same algorithm as in Dongarra et al. (1979). If the estimated condition number is greater than 1/ε (where ε is the machine precision), a warning message is issued. This indicates that very small changes in A may produce large changes in the solution x. The IMSL_LUSOL function fails if U, the upper-triangular part of the factorization, has a zero diagonal element.

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