void	nag_dgbrfs (Nag_OrderType order, Nag_TransType trans, Integer n, Integer kl, Integer ku, Integer nrhs, const double ab[], Integer pdab, const double afb[], Integer pdafb, const Integer ipiv[], const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3

Description

nag_dgbrfs (f07bhc) returns the backward errors and estimated bounds on the forward errors for the solution of a real 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_dgbrfs (f07bhc) 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: $trans$ – Nag_TransTypeInput

On entry: indicates the form of the linear equations for which

X

is the computed solution.

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

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

3: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $kl$ – IntegerInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

5: $ku$ – IntegerInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

6: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

7: $ab [\dim]$ – const doubleInput

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

\max (1, pdab \times n)

On entry: the original

n

n

band matrix

A

as supplied to nag_dgbtrf (f07bdc) but with reduced requirements since the matrix is not factorized.

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

A_{i j}

, for row

i = 1, \dots, n

and column

j = \max (1, i - k_{l}), \dots, \min (n, i + k_{u})

, depends on the order argument as follows:

if $order = Nag_ColMajor$ , $A_{i j}$ is stored as $ab [(j - 1) \times pdab + ku + i - j]$ ;
if $order = Nag_RowMajor$ , $A_{i j}$ is stored as $ab [(i - 1) \times pdab + kl + j - i]$ .

See Section 9 in nag_dgbsv (f07bac) for further details.

8: $pdab$ – 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 kl + ku + 1

9: $afb [\dim]$ – const doubleInput

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

\max (1, pdafb \times n)

On entry: the

L U

factorization of

A

, as returned by nag_dgbtrf (f07bdc).

10: $pdafb$ – IntegerInput

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

Constraint:

pdafb \geq 2 \times kl + ku + 1

11: $ipiv [\dim]$ – const IntegerInput

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

\max (1, n)

On entry: the pivot indices, as returned by nag_dgbtrf (f07bdc).

12: $b [\dim]$ – 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

13: $pdb$ – 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 $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

14: $x [\dim]$ – doubleInput/Output

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_dgbtrs (f07bec).

On exit: the improved solution matrix

X

15: $pdx$ – 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 $order = Nag_RowMajor$ , $pdx \geq \max (1, nrhs)$ .

16: $ferr [nrhs]$ – 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

17: $berr [nrhs]$ – 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

18: $fail$ – 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, $kl = 〈value〉$ .
Constraint: $kl \geq 0$ .

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

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, $pdafb = 〈value〉$ .
Constraint: $pdafb > 0$ .

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

On entry, $pdx = 〈value〉$ .
Constraint: $pdx > 0$ .
NE_INT_2: 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_INT_3: On entry, $pdab = 〈value〉$ , $kl = 〈value〉$ and $ku = 〈value〉$ .
Constraint: $pdab \geq kl + ku + 1$ .

On entry, $pdafb = 〈value〉$ , $kl = 〈value〉$ and $ku = 〈value〉$ .
Constraint: $pdafb \geq 2 \times kl + ku + 1$ .
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_dgbrfs (f07bhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dgbrfs (f07bhc) 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

For each right-hand side, computation of the backward error involves a minimum of

4 n (k_{l} + k_{u})

floating-point operations. Each step of iterative refinement involves an additional

2 n (4 k_{l} + 3 k_{u})

operations. This assumes

n ≫ k_{l}

and

n ≫ k_{u}

. At most five steps of iterative refinement are performed, but usually only one or two steps are required.

Estimating the forward error 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 (2 k_{l} + k_{u})

operations.

The complex analogue of this function is nag_zgbrfs (f07bvc).

10

Example

This example solves the system of equations

A X = B

using iterative refinement and to compute the forward and backward error bounds, where

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0.00 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0.00 & 2.56 & 2.46 & 4.07 \\ 0.00 & 0.00 & - 4.78 & - 3.82 \end{matrix}) and B = (\begin{matrix} 4.42 & - 36.01 \\ 27.13 & - 31.67 \\ - 6.14 & - 1.16 \\ 10.50 & - 25.82 \end{matrix}) .

Here

A

is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_dgbtrf (f07bdc).

NAG Library Function Document

nag_dgbrfs (f07bhc)

▸▿ 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