The routine may be called by the names f11jbf or nagf_sparse_real_symm_precon_ichol_solve.
f11jbf solves a system of linear equations
involving the preconditioning matrix , corresponding to an incomplete Cholesky decomposition of a sparse symmetric matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction), as generated by f11jaf.
In the above decomposition is a lower triangular sparse matrix with unit diagonal, is a diagonal matrix and is a permutation matrix. and are supplied to f11jbf through the matrix
which is a lower triangular n by n sparse matrix, stored in SCS format, as returned by f11jaf. The permutation matrix is returned from f11jaf via the array ipiv.
It is envisaged that a common use of f11jbf will be to carry out the preconditioning step required in the application of f11gef to sparse symmetric linear systems. f11jbf is used for this purpose by the Black Box routine f11jcf.
f11jbf may also be used in combination with f11jaf to solve a sparse symmetric positive definite system of linear equations directly (see Section 9.4 in f11jaf).
1: – IntegerInput
On entry: , the order of the matrix . This must be the same value as was supplied in the preceding call to f11jaf.
2: – Real (Kind=nag_wp) arrayInput
On entry: the values returned in the array a by a previous call to f11jaf.
3: – IntegerInput
On entry: the dimension of the arrays a, irow and icol as declared in the (sub)program from which f11jbf is called. This must be the same value returned by the preceding call to f11jaf.
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
It is expected that a common use of f11jbf will be to carry out the preconditioning step required in the application of f11gef to sparse symmetric linear systems. In this situation f11jbf is likely to be called many times with the same matrix . In the interests of both reliability and efficiency, you are recommended to set for the first of such calls, and to set for all subsequent calls.
This example reads in a symmetric positive definite sparse matrix and a vector . It then calls f11jaf, with and , to compute the complete Cholesky decomposition of :
Then it calls f11jbf to solve the system
It then repeats the exercise for the same matrix permuted with the bandwidth-reducing Reverse Cuthill–McKee permutation, calculated with f11yef.