void	f07bfc (Nag_OrderType order, Integer m, Integer n, Integer kl, Integer ku, const double ab[], Integer pdab, double r[], double c[], double rowcnd, double colcnd, double amax, NagError fail)

The function may be called by the names: f07bfc, nag_lapacklin_dgbequ or nag_dgbequ.

3 Description

f07bfc computes the diagonal scaling matrices. The diagonal scaling matrices are chosen to try to make the elements of largest absolute value in each row and column of the matrix

B

given by

B = D_{R} A D_{C}

have absolute value

1

. The diagonal elements of

D_{R}

and

D_{C}

are restricted to lie in the safe range

(δ, 1 / δ)

, where

δ

is the value returned by function X02AMC. Use of these scaling factors is not guaranteed to reduce the condition number of

A

but works well in practice.

4 References

None.

5 Arguments

1: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: $kl$ – Integer Input

On entry:

k_{l}

, the number of subdiagonals of the matrix

A

Constraint:

kl \geq 0

5: $ku$ – Integer Input

On entry:

k_{u}

, the number of superdiagonals of the matrix

A

Constraint:

ku \geq 0

6: $ab [\dim]$ – const double Input

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

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

On entry: the

m \times n

band matrix

A

whose scaling factors are to be computed.

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, m

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 f07bac for further details.

7: $pdab$ – Integer Input

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

8: $r [m]$ – double Output

On exit: if

fail . code =

NE_NOERROR or

fail . code =

NE_MAT_COL_ZERO, r contains the row scale factors, the diagonal elements of

D_{R}

. The elements of r will be positive.

9: $c [n]$ – double Output

On exit: if

fail . code =

NE_NOERROR, c contains the column scale factors, the diagonal elements of

D_{C}

. The elements of c will be positive.

10: $rowcnd$ – double * Output

On exit: if

fail . code =

NE_NOERROR or

fail . code =

NE_MAT_COL_ZERO, rowcnd contains the ratio of the smallest value of

r [i - 1]

to the largest value of

r [i - 1]

. If

rowcnd \geq 0.1

and amax is neither too large nor too small, it is not worth scaling by

D_{R}

11: $colcnd$ – double * Output

On exit: if

fail . code =

NE_NOERROR, colcnd contains the ratio of the smallest value of

c [i - 1]

to the largest value of

c [i - 1]

colcnd \geq 0.1

, it is not worth scaling by

D_{C}

12: $amax$ – double * Output

On exit:

\max | a_{i j} |

. If amax is very close to overflow or underflow, the matrix

A

should be scaled.

13: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdab = ⟨ value ⟩$ .
Constraint: $pdab > 0$ .
NE_INT_3: On entry, $pdab = ⟨ value ⟩$ , $kl = ⟨ value ⟩$ and $ku = ⟨ value ⟩$ .
Constraint: $pdab \geq kl + ku + 1$ .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_COL_ZERO: Column $⟨ value ⟩$ of $A$ is exactly zero.
NE_MAT_ROW_ZERO: Row $⟨ value ⟩$ of $A$ is exactly zero.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computed scale factors will be close to the exact scale factors.

8 Parallelism and Performance

f07bfc is not threaded in any implementation.

9 Further Comments

The complex analogue of this function is f07btc.

10 Example

This example equilibrates the band matrix

A

given by

A = (\begin{array}{l} - 0.23 & 2.54 & - 3.66 \times 10^{−10} & 0 \\ - 6.98 \times 10^{10} & 2.46 \times 10^{10} & - 2.73 & - 2.13 \times 10^{10} \\ 0 & 2.56 & 2.46 \times 10^{−10} & 4.07 \\ 0 & 0 & - 4.78 \times 10^{−10} & - 3.82 \end{array}) .

Details of the scaling factors, and the scaled matrix are output.

f07bf: FL CL CPP AD

NAG CL Interfacef07bfc (dgbequ)

▸▿ 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 CL Interface
f07bfc (dgbequ)