void	nag_dtbrfs (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer n, Integer kd, Integer nrhs, const double ab[], Integer pdab, const double b[], Integer pdb, const double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3

Description

nag_dtbrfs (f07vhc) 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 function handles each right-hand side vector (stored as a column of the matrix

B

) independently, so we describe the function of nag_dtbrfs (f07vhc) in terms of a single right-hand side

b

and solution

x

Given a computed solution

x

, the function 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 function 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

Arguments

1: $order$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $uplo$ – Nag_UploTypeInput

On entry: specifies whether

A

is upper or lower triangular.

$uplo = Nag_Upper$: $A$ is upper triangular.
$uplo = Nag_Lower$: $A$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $trans$ – Nag_TransTypeInput

On entry: indicates the form of the equations.

$trans = Nag_NoTrans$: The equations are of the form $A X = B$ .
$trans = Nag_Trans$ or $Nag_ConjTrans$: The equations are of the form $A^{T} X = B$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

4: $diag$ – Nag_DiagTypeInput

On entry: indicates whether

A

is a nonunit or unit triangular matrix.

$diag = Nag_NonUnitDiag$: $A$ is a nonunit triangular matrix.
$diag = Nag_UnitDiag$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

diag = Nag_NonUnitDiag

Nag_UnitDiag

5: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: $kd$ – IntegerInput

On entry:

k_{d}

, the number of superdiagonals of the matrix

A

uplo = Nag_Upper

, or the number of subdiagonals if

uplo = Nag_Lower

Constraint:

kd \geq 0

7: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

8: $ab [\dim]$ – const doubleInput

Note: the dimension, dim, of the array ab must be at least

\max (1, pdab \times n)

On entry: the

n

n

triangular band matrix

A

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of

A_{i j}

, depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [k_{d} + i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k_{d}), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k_{d})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k_{d})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [k_{d} + j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k_{d}), \dots, i$ .

diag = Nag_UnitDiag

, the diagonal elements of

AB

are assumed to be

1

, and are not referenced.

9: $pdab$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array ab.

Constraint:

pdab \geq kd + 1

10: $b [\dim]$ – const doubleInput

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

r

right-hand side matrix

B

11: $pdb$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

12: $x [\dim]$ – const doubleInput

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

X

is stored in

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

r

solution matrix

X

, as returned by nag_dtbtrs (f07vec).

13: $pdx$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdx \geq \max (1, nrhs)$ .

14: $ferr [nrhs]$ – doubleOutput

On exit:

ferr [j - 1]

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

15: $berr [nrhs]$ – doubleOutput

On exit:

berr [j - 1]

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

16: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $kd = 〈value〉$ .
Constraint: $kd \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $nrhs = 〈value〉$ .
Constraint: $nrhs \geq 0$ .

On entry, $pdab = 〈value〉$ .
Constraint: $pdab > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .

On entry, $pdx = 〈value〉$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $pdab = 〈value〉$ and $kd = 〈value〉$ .
Constraint: $pdab \geq kd + 1$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $nrhs = 〈value〉$ .
Constraint: $pdb \geq \max (1, nrhs)$ .

On entry, $pdx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdx \geq \max (1, n)$ .

On entry, $pdx = 〈value〉$ and $nrhs = 〈value〉$ .
Constraint: $pdx \geq \max (1, nrhs)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

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

nag_dtbrfs (f07vhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dtbrfs (f07vhc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

A call to nag_dtbrfs (f07vhc), 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 function is nag_ztbrfs (f07vvc).

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 Function Document

nag_dtbrfs (f07vhc)

▸▿ 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