Solves a real general system of linear equations Ax = b. Using optional arguments, any of several related computations can be performed. These extra tasks include computing the LU factorization of A using partial pivoting, computing the inverse matrix A-1, solving ATx = b, or computing the solution of Ax = b given the LU factorization of A.

The type double function is imsl_d_lin_sol_gen.

A pointer to the solution x of the linear system Ax = b. To release this space, use imsl_free. If no solution was computed, then NULL is returned.

The function imsl_f_lin_sol_gen solves a system of linear algebraic equations with a real coefficient matrix A. It first computes the LU factorization of A with partial pivoting such that L-1A = U. Let F be the matrix p_factor returned by optional argument IMSL_FACTOR. 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-1 using

The factors Pi and Li are defined by partial pivoting. Pi is the identity matrix with rows i and p_pvt[i‑1] interchanged. Li is the identity matrix with Fji , for j = i + 1, …, n, inserted below the diagonal in column i.

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. Additionally, the accuracy of the solution can be improved by iterative refinement. IMSL uses mixed precision iterative refinement in single precision and fixed precision iterative refinement in double precision. In double precision, the residuals b-Ax are computed with high accuracy using algorithms based on Ogita, Rump and Oishi (2005). 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 function imsl_f_lin_sol_gen fails if U, the upper triangular part of the factorization, has a zero diagonal element.

This example solves a system of three linear equations. This is the simplest use of the function. The equations follow below:

This example solves the transpose problem ATx = b and returns the LU factorization of A with partial pivoting. The same data as the initial example is used, except the solution x = A-Tb is returned in an array allocated in the main program. The L matrix is returned in implicit form.

Pi is the identity matrix with row i and row pvt[i-1] interchanged.

Li is the identity matrix with Fji, for j = i + 1, n, inserted below the diagonal in column i, where F is factor:

U is the upper triangle of factor:

This example computes the inverse of the 3 × 3 matrix A of the initial example and solves the same linear system. The matrix product C = A-1A is computed and printed. The function imsl_f_mat_mul_rect is used to compute C. The approximate result C = I is obtained.

This example computes the solution of two systems. Only the right-hand sides differ. The matrix and first right-hand side are given in the initial example. The second right-hand side is the vector c = [0.5, 0.3, 0.4]T. The factorization information is computed with the first solution and is used to compute the second solution. The factorization work done in the first step is avoided in computing the second solution.