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IMSL_LUSOL

# IMSL_LUSOL

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 Lâ1PA = U.

The matrix U is upper-triangular, while Lâ1A â¡ Pn â 1 Ln â 2Pn â 2...L0 P0 A â¡ U. The factors Pi and Li are defined by the partial pivoting. Each Pi is an interchange of row i with row j â¥ 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 â 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 âloop unrolling and jammingâ 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 = Lâ1b and x = Uâ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/Îµ (where Îµ 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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