NAG Library Function Document

nag_dpbrfs (f07hhc)

+− 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

nag_dpbrfs (f07hhc) returns error bounds for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides,

A X = B

. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

2 Specification

#include <nag.h>

#include <nagf07.h>

void	nag_dpbrfs (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, const double ab[], Integer pdab, const double afb[], Integer pdafb, const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)

3 Description

nag_dpbrfs (f07hhc) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides

A 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_dpbrfs (f07hhc) 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 3.2.1.3 in the Essential Introduction 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 the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$uplo = Nag_Upper$: The upper triangular part of $A$ is stored and $A$ is factorized as $U^{T} U$ , where $U$ is upper triangular.
$uplo = Nag_Lower$: The lower triangular part of $A$ is stored and $A$ is factorized as $L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: kd – IntegerInput

On entry:

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

5: nrhs – IntegerInput

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

6: 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

original symmetric positive definite band matrix

A

as supplied to nag_dpbtrf (f07hdc).

7: pdab – IntegerInput

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

Constraint:

pdab \geq kd + 1

8: afb[ $\dim$ ] – const doubleInput

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

\max (1, pdafb \times n)

On entry: the Cholesky factor of

A

, as returned by nag_dpbtrf (f07hdc).

9: 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 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[ $\dim$ ] – 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_dpbtrs (f07hec).

On exit: the improved solution matrix

X

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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, $pdafb = ⟨value⟩$ .
Constraint: $pdafb > 0$ .
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, $pdafb = ⟨value⟩$ and $kd = ⟨value⟩$ .
Constraint: $pdafb \geq kd + 1$ .
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.

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_dpbrfs (f07hhc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dpbrfs (f07hhc) 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 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

8 n k

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

12 n k

operations. This assumes

n ≫ k

. 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

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

4 n k

operations.

The complex analogue of this function is nag_zpbrfs (f07hvc).

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} 5.49 & 2.68 & 0.00 & 0.00 \\ 2.68 & 5.63 & - 2.39 & 0.00 \\ 0.00 & - 2.39 & 2.60 & - 2.22 \\ 0.00 & 0.00 & - 2.22 & 5.17 \end{matrix}) and B = (\begin{matrix} 22.09 & 5.10 \\ 9.31 & 30.81 \\ - 5.24 & - 25.82 \\ 11.83 & 22.90 \end{matrix}) .

Here

A

is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_dpbtrf (f07hdc).

NAG Library Function Documentnag_dpbrfs (f07hhc)

+− 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

NAG Library Function Document

nag_dpbrfs (f07hhc)