f07uhf returns error bounds for the solution of a real triangular system of linear equations with multiple right-hand sides, or , using packed storage.
The routine may be called by the names f07uhf, nagf_lapacklin_dtprfs or its LAPACK name dtprfs.
3Description
f07uhf returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular system of linear equations with multiple right-hand sides or , using packed storage. The routine handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of f07uhf 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 is upper or lower triangular.
is upper triangular.
is lower triangular.
Constraint:
or .
2: – Character(1)Input
On entry: indicates the form of the equations.
The equations are of the form .
or
The equations are of the form .
Constraint:
, or .
3: – Character(1)Input
On entry: indicates whether is a nonunit or unit triangular matrix.
is a nonunit triangular matrix.
is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be .
Constraint:
or .
4: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
5: – IntegerInput
On entry: , the number of right-hand sides.
Constraint:
.
6: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array ap
must be at least
.
On entry: the triangular matrix , packed by columns.
More precisely,
if , the upper triangle of must be stored with element in for ;
if , the lower triangle of must be stored with element in for .
If , the diagonal elements of are assumed to be , and are not referenced; the same storage scheme is used whether or ‘U’.
7: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b
must be at least
.
On entry: the right-hand side matrix .
8: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07uhf is called.
Constraint:
.
9: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array x
must be at least
.
On entry: the solution matrix , as returned by f07uef.
10: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07uhf is called.
Constraint:
.
11: – Real (Kind=nag_wp) arrayOutput
On exit: contains an estimated error bound for the th solution vector, that is, the th column of , for .
12: – 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 .
13: – Real (Kind=nag_wp) arrayWorkspace
14: – Integer arrayWorkspace
15: – 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.
f07uhf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07uhf 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
A call to f07uhf, for each right-hand side, involves solving a number of systems of linear equations of the form or ; the number is usually or and never more than . Each solution involves approximately floating-point operations.