The routine may be called by the names f01abf or nagf_matop_real_symm_posdef_inv.
To compute the inverse of a real symmetric positive definite matrix , f01abf first computes a Cholesky factorization of as , where is lower triangular. An approximation to is found by computing and then the product . The residual matrix is calculated using additional precision, and a correction to is found by solving . is replaced by , and this iterative refinement of the inverse is repeated until full machine accuracy has been obtained.
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
1: – Real (Kind=nag_wp) arrayInput/Output
On entry: the upper triangle of the positive definite symmetric matrix . The elements of the array below the diagonal need not be set.
On exit: the lower triangle of the inverse matrix is stored in the elements of the array below the diagonal, in rows to ; is stored in for . The upper triangle of the original matrix is unchanged.
2: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f01abf is called.
3: – IntegerInput
On entry: , the order of the matrix .
4: – Real (Kind=nag_wp) arrayOutput
On exit: the lower triangle of the inverse matrix , with stored in , for .
5: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f01abf is called.
6: – Real (Kind=nag_wp) arrayWorkspace
7: – IntegerInput/Output
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).
Errors or warnings detected by the routine:
The matrix is not positive definite, possibly due to rounding errors.
The refinement process failed to converge. The matrix is ill-conditioned.
On entry, and .
On entry, and .
On entry, .
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
f01abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01abf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The time taken by f01abf is approximately proportional to .