NAG Library Routine Document

F07VHF (DTBRFS)

INTEGER	N, KD, NRHS, LDAB, LDB, LDX, IWORK(N), INFO
REAL (KIND=nag_wp)	AB(LDAB,), B(LDB,), X(LDX,), FERR(NRHS), BERR(NRHS), WORK(3N)
CHARACTER(1)	UPLO, TRANS, DIAG

The routine may be called by its LAPACK name dtbrfs.

3 Description

F07VHF (DTBRFS) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular band system of linear equations with multiple right-hand sides

A X = B

A^{T} X = B

. The routine handles each right-hand side vector (stored as a column of the matrix

B

) independently, so we describe the function of F07VHF (DTBRFS) in terms of a single right-hand side

b

and solution

x

Given a computed solution

x

, the routine computes the component-wise backward error

β

. This is the size of the smallest relative perturbation in each element of

A

and

b

such that

x

is the exact solution of a perturbed system

\begin{matrix} (A + δ A) x = b + δ b \\ |δ a_{i j}| \leq β |a_{i j}| and |δ b_{i}| \leq β |b_{i}| . \end{matrix}

Then the routine estimates a bound for the component-wise forward error in the computed solution, defined by:

\max_{i} |x_{i} - {\hat{x}}_{i}| / \max_{i} |x_{i}|

where

\hat{x}

is the true solution.

For details of the method, see the F07 Chapter Introduction.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: $UPLO$ – CHARACTER(1)Input

On entry: specifies whether

A

is upper or lower triangular.

$UPLO ='U'$: $A$ is upper triangular.
$UPLO ='L'$: $A$ is lower triangular.

Constraint:

UPLO ='U'

'L'

2: $TRANS$ – CHARACTER(1)Input

On entry: indicates the form of the equations.

$TRANS ='N'$: The equations are of the form $A X = B$ .
$TRANS ='T'$ or $'C'$: The equations are of the form $A^{T} X = B$ .

Constraint:

TRANS ='N'

'T'

'C'

3: $DIAG$ – CHARACTER(1)Input

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$DIAG ='N'$: $A$ is a nonunit triangular matrix.
$DIAG ='U'$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

DIAG ='N'

'U'

4: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

5: $KD$ – INTEGERInput

On entry:

k_{d}

, the number of superdiagonals of the matrix

A

UPLO ='U'

, or the number of subdiagonals if

UPLO ='L'

Constraint:

KD \geq 0

6: $NRHS$ – INTEGERInput

On entry:

r

, the number of right-hand sides.

Constraint:

NRHS \geq 0

7: $AB (LDAB *)$ – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array AB must be at least

\max (1, N)

On entry: the

n

n

triangular band matrix

A

The matrix is stored in rows

1

k_{d} + 1

, more precisely,

if $UPLO ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $AB (k_{d} + 1 + i - j, j) for \max (1, j - k_{d}) \leq i \leq j$ ;
if $UPLO ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $AB (1 + i - j, j) for j \leq i \leq \min (n, j + k_{d}) .$

DIAG ='U'

, the diagonal elements of

A

are assumed to be

1

, and are not referenced.

8: $LDAB$ – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07VHF (DTBRFS) is called.

Constraint:

LDAB \geq KD + 1

9: $B (LDB *)$ – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array B must be at least

\max (1, NRHS)

On entry: the

n

r

right-hand side matrix

B

10: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07VHF (DTBRFS) is called.

Constraint:

LDB \geq \max (1, N)

11: $X (LDX *)$ – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array X must be at least

\max (1, NRHS)

On entry: the

n

r

solution matrix

X

, as returned by F07VEF (DTBTRS).

12: $LDX$ – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07VHF (DTBRFS) is called.

Constraint:

LDX \geq \max (1, N)

13: $FERR (NRHS)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

FERR (j)

contains an estimated error bound for the

j

th solution vector, that is, the

j

th column of

X

, for

j = 1, 2, \dots, r

14: $BERR (NRHS)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

BERR (j)

contains the component-wise backward error bound

β

for the

j

th solution vector, that is, the

j

th column of

X

, for

j = 1, 2, \dots, r

15: $WORK (3 \times N)$ – REAL (KIND=nag_wp) arrayWorkspace

16: $IWORK (N)$ – INTEGER arrayWorkspace

17: $INFO$ – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$INFO < 0$: If $INFO = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

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.

8 Parallelism and Performance

F07VHF (DTBRFS) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F07VHF (DTBRFS) 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.

9 Further Comments

A call to F07VHF (DTBRFS), for each right-hand side, involves solving a number of systems of linear equations of the form

A x = b

A^{T} x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n k

floating-point operations (assuming

n ≫ k

The complex analogue of this routine is F07VVF (ZTBRFS).

10 Example

This example solves the system of equations

A X = B

and to compute forward and backward error bounds, where

A = (\begin{matrix} - 4.16 & 0.00 & 0.00 & 0.00 \\ - 2.25 & 4.78 & 0.00 & 0.00 \\ 0.00 & 5.86 & 6.32 & 0.00 \\ 0.00 & 0.00 & - 4.82 & 0.16 \end{matrix}) and B = (\begin{matrix} - 16.64 & - 4.16 \\ - 13.78 & - 16.59 \\ 13.10 & - 4.94 \\ - 14.14 & - 9.96 \end{matrix}) .

NAG Library Routine DocumentF07VHF (DTBRFS)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

F07VHF (DTBRFS)