f07fhf returns error bounds for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides, . It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.
The routine may be called by the names f07fhf, nagf_lapacklin_dporfs or its LAPACK name dporfs.
3Description
f07fhf returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides . The routine handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of f07fhf in terms of a single right-hand side and solution .
Given a computed solution , the routine computes the component-wise backward error
. This is the size of the smallest relative perturbation in each element of and such that is the exact solution of a perturbed system
Then the routine estimates a bound for the component-wise forward error in the computed solution, defined by:
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – Character(1)Input
On entry: specifies whether the upper or lower triangular part of is stored and how is to be factorized.
The upper triangular part of is stored and is factorized as , where is upper triangular.
The lower triangular part of is stored and is factorized as , where is lower triangular.
Constraint:
or .
2: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of right-hand sides.
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a
must be at least
.
On entry: the original symmetric positive definite matrix as supplied to f07fdf.
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07fhf is called.
Constraint:
.
6: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array af
must be at least
.
On entry: the Cholesky factor of , as returned by f07fdf.
7: – IntegerInput
On entry: the first dimension of the array af as declared in the (sub)program from which f07fhf is called.
Constraint:
.
8: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b
must be at least
.
On entry: the right-hand side matrix .
9: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07fhf is called.
Constraint:
.
10: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array x
must be at least
.
On entry: the solution matrix , as returned by f07fef.
On exit: the improved solution matrix .
11: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07fhf is called.
Constraint:
.
12: – Real (Kind=nag_wp) arrayOutput
On exit: contains an estimated error bound for the th solution vector, that is, the th column of , for .
13: – Real (Kind=nag_wp) arrayOutput
On exit: contains the component-wise backward error bound for the th solution vector, that is, the th column of , for .
14: – Real (Kind=nag_wp) arrayWorkspace
15: – Integer arrayWorkspace
16: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7Accuracy
The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07fhf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07fhf 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.
9Further Comments
For each right-hand side, computation of the backward error involves a minimum of floating-point operations. Each step of iterative refinement involves an additional operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form ; the number is usually or and never more than . Each solution involves approximately operations.