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The IMSL_LUFAC procedure computes the LU factorization of a real or complex matrix.

Any of several related computations can be performed by using keywords. These extra tasks include computing the LU factorization of AT, computing an estimate of the L1 condition number, and returning L or U separately.

The IMSL_LUFAC procedure 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 i ≥ j. Thus, Pi is defined by that value of j. Every Li = mieiT 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 array containing the decomposition information.

The factorization efficiency is based on a technique of “loop unrolling and jamming” due to Dr. Leonard J. Harding of the University of Michigan, Ann Arbor, Michigan. When 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_LUFAC procedure fails if U, the upper triangular part of the factorization, has a zero diagonal element.

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