Computes the Cholesky factorization of a sparse Hermitian positive definite matrix A by an OpenMP parallelized supernodal algorithm and solves the sparse Hermitian positive definite system of linear equations Ax = b.

The type d_complex functions are imsl_z_sparse_cholesky_smp and imsl_z_free_numeric_factor.

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

The function imsl_c_sparse_cholesky_smp solves a system of linear algebraic equations having a sparse Hermitian positive definite coefficient matrix A. In this function’s default usage, a symbolic factorization of a permutation of the coefficient matrix is computed first. Then a numerical factorization exploiting OpenMP parallelism is performed. The solution of the linear system is then found using the numeric factor.

The symbolic factorization step of the computation consists of determining a minimum degree ordering and then setting up a sparse supernodal data structure for the Cholesky factor, L. This step only requires the “pattern” of the sparse coefficient matrix, i.e., the locations of the nonzero elements but not any of the elements themselves. Thus, the val field in the Imsl_c_sparse_elem structure is ignored. If an application generates different sparse Hermitian positive definite coefficient matrices that all have the same sparsity pattern, then by using IMSL_RETURN_SYMBOLIC_FACTOR and IMSL_SUPPLY_SYMBOLIC_FACTOR, the symbolic factorization needs only be computed once.

Given the sparse data structure for the Cholesky factor L, as supplied by the symbolic factor, the numeric factorization produces the entries in L so that

Here P is the permutation matrix determined by the minimum degree ordering.

The numerical factorization is an implementation of a parallel supernodal algorithm that combines a left-looking and a right-looking column computation scheme. This algorithm is described in detail in Rauber et al. (1999).

If an application requires that several linear systems be solved where the coefficient matrix is the same but the right-hand sides change, the options IMSL_RETURN_NUMERIC_FACTOR and IMSL_SUPPLY_NUMERIC_FACTOR can be used to precompute the Cholesky factor. Then the IMSL_SOLVE_ONLY option can be used to efficiently solve all subsequent systems.

Given the numeric factorization, the solution x is obtained by the following calculations:

The permutation information, P, is carried in the numeric factor structure Imsl_c_numeric_factor.

As a simple example of default use, consider the following Hermitian positive definite matrix

We construct the solution xT = (1 + i, 2 + 2i, 3 + 3i) to the system Ax = b by setting b:=Ax = (‑2 + 2i, 5 +15i, 36 + 28i)T. The number of nonzeros in the lower triangle is nz = 5. The solution is computed and printed.

This example shows how a symbolic factor can be re-used. Consider matrix A, a Hermitian positive definite matrix with value 6 on the diagonal and value -1- i on its lower codiagonal and the lower band at distance 50 from the diagonal. At first, the system Ax = b is solved and the symbolic factorization of A is returned. Then, the system Cy = d with C = A+4*I, I the identity matrix, is solved using the symbolic factorization just computed. This is possible because A and C have the same nonzero structure and therefore also the same symbolic factorization. The solution errors are printed.

In this example, set A = E(2500, 50). First solve the system Ax = b1 and return the numeric factorization resulting from that call. Then solve the system Ax = b2 using the numeric factorization just computed. The ratio of execution times is printed. Be aware that timing results are highly machine dependent.