Integer, Intent (In)	::	n, kd, nrhs, ldab, ldb
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	ab(ldab,*)
Real (Kind=nag_wp), Intent (Inout)	::	b(ldb,*)
Character (1), Intent (In)	::	uplo, trans, diag

C Header Interface

#include nagmk26.h

void	f07vef_ ( const char uplo, const char trans, const char diag, const Integer n, const Integer kd, const Integer nrhs, const double ab[], const Integer ldab, double b[], const Integer ldb, Integer *info, const Charlen length_uplo, const Charlen length_trans, const Charlen length_diag)

The routine may be called by its LAPACK name dtbtrs.

3

Description

f07vef (dtbtrs) solves a real triangular band system of linear equations

A X = B

A^{T} X = B

4

References

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

Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

5

Arguments

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 f07vef (dtbtrs) is called.

Constraint:

ldab \geq kd + 1

9: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/Output

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

On exit: the

n

r

solution matrix

X

10: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f07vef (dtbtrs) is called.

Constraint:

ldb \geq \max (1, n)

11: $info$ – 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.

$info > 0$: Element $〈value〉$ of the diagonal is exactly zero. $A$ is singular and the solution has not been computed.

7

Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

|E| \leq c (k) ε |A|,

c (k)

is a modest linear function of

k

, and

ε

is the machine precision.

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (k) cond (A, x) ε,   provided c (k) cond (A, x) ε < 1,

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty}

Note that

cond (A, x) \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

;

cond (A, x)

can be much smaller than

cond (A)

and it is also possible for

cond (A^{T})

to be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling f07vhf (dtbrfs), and an estimate for

κ_{\infty} (A)

can be obtained by calling f07vgf (dtbcon) with

norm ='I'

8

Parallelism and Performance

f07vef (dtbtrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07vef (dtbtrs) 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

The total number of floating-point operations is approximately

2 n k r

k ≪ n

The complex analogue of this routine is f07vsf (ztbtrs).

10

Example

This example solves the system of equations

A X = B

, 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}) .

Here

A

is treated as a lower triangular band matrix with one subdiagonal.

NAG Library Routine Document

f07vef (dtbtrs)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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