NAG Library Routine Document

F07VVF (ZTBRFS)

INTEGER	N, KD, NRHS, LDAB, LDB, LDX, INFO
REAL (KIND=nag_wp)	FERR(NRHS), BERR(NRHS), RWORK(N)
COMPLEX (KIND=nag_wp)	AB(LDAB,), B(LDB,), X(LDX,), WORK(2N)
CHARACTER(1)	UPLO, TRANS, DIAG

The routine may be called by its LAPACK name ztbrfs.

3 Description

F07VVF (ZTBRFS) returns the backward errors and estimated bounds on the forward errors for the solution of a complex triangular band system of linear equations with multiple right-hand sides

A X = B

A^{T} X = B

A^{H} 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 F07VVF (ZTBRFS) 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 $A X = B$ .
$TRANS ='T'$: The equations are of the form $A^{T} X = B$ .
$TRANS ='C'$: The equations are of the form $A^{H} X = B$ .

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, $*$ ) – COMPLEX (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 F07VVF (ZTBRFS) is called.

Constraint:

LDAB \geq KD + 1

9: B(LDB, $*$ ) – COMPLEX (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 F07VVF (ZTBRFS) is called.

Constraint:

LDB \geq \max (1, N)

11: X(LDX, $*$ ) – COMPLEX (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 F07VSF (ZTBTRS).

12: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07VVF (ZTBRFS) 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( $2 \times N$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace

16: RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace

17: INFO – INTEGEROutput

On exit:

INFO = 0

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

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th parameter 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 Further Comments

A call to F07VVF (ZTBRFS), for each right-hand side, involves solving a number of systems of linear equations of the form

A x = b

A^{H} x = b

; the number is usually

5

and never more than

11

. Each solution involves approximately

8 n k

real floating point operations (assuming

n ≫ k

The real analogue of this routine is F07VHF (DTBRFS).

9 Example

This example solves the system of equations

A X = B

and to compute forward and backward error bounds, where

A = (\begin{array}{r} - 1.94 + 4.43 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ - 3.39 + 3.44 i & 4.12 - 4.27 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 1.62 + 3.68 i & - 1.84 + 5.53 i & 0.43 - 2.66 i & 0.00 + 0.00 i \\ 0.00 + 0.00 i & - 2.77 - 1.93 i & 1.74 - 0.04 i & 0.44 + 0.10 i \end{array})

and

B = (\begin{array}{r} - 8.86 - 3.88 i & - 24.09 - 5.27 i \\ - 15.57 - 23.41 i & - 57.97 + 8.14 i \\ - 7.63 + 22.78 i & 19.09 - 29.51 i \\ - 14.74 - 2.40 i & 19.17 + 21.33 i \end{array}) .

NAG Library Routine DocumentF07VVF (ZTBRFS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07VVF (ZTBRFS)