void	f07hfc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, const double ab[], Integer pdab, double s[], double scond, double amax, NagError *fail)

The function may be called by the names: f07hfc, nag_lapacklin_dpbequ or nag_dpbequ.

3 Description

f07hfc computes a diagonal scaling matrix

S

chosen so that

s_{j} = 1 / \sqrt{a_{j j}} .

This means that the matrix

B

given by

B = S A S,

has diagonal elements equal to unity. This in turn means that the condition number of

B

κ_{2} (B)

, is within a factor

n

of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).

4 References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

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: $uplo$ – Nag_UploType Input

On entry: indicates whether the upper or lower triangular part of

A

is stored in the array ab, as follows:

$uplo = Nag_Upper$: The upper triangle of $A$ is stored.
$uplo = Nag_Lower$: The lower triangle of $A$ is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $kd$ – Integer Input

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

5: $ab [\dim]$ – const double Input

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

\max (1, pdab \times n)

On entry: the upper or lower triangle of the symmetric positive definite 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 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 $j = \max (1, i - k_{d}), \dots, i$ .

Only the elements of the array ab corresponding to the diagonal elements of

A

are referenced. (Row

(k_{d} + 1)

of ab when

uplo = Nag_Upper

, row

1

of ab when

uplo = Nag_Lower

6: $pdab$ – 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 kd + 1

7: $s [n]$ – double Output

On exit: if

fail . code =

NE_NOERROR, s contains the diagonal elements of the scaling matrix

S

8: $scond$ – double * Output

On exit: if

fail . code =

NE_NOERROR, scond contains the ratio of the smallest value of s to the largest value of s. If

scond \geq 0.1

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

S

9: $amax$ – double * Output

On exit:

\max | a_{i j} |

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

A

should be scaled.

10: $fail$ – 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, $kd = ⟨ value ⟩$ .
Constraint: $kd \geq 0$ .

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

On entry, $pdab = ⟨ value ⟩$ .
Constraint: $pdab > 0$ .
NE_INT_2: On entry, $pdab = ⟨ value ⟩$ and $kd = ⟨ value ⟩$ .
Constraint: $pdab \geq kd + 1$ .
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_NOT_POS_DEF: The $⟨ value ⟩$ th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).
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

Background information to multithreading can be found in the Multithreading documentation.

f07hfc is not threaded in any implementation.

9 Further Comments

The complex analogue of this function is f07htc.

10 Example

This example equilibrates the symmetric positive definite matrix

A

given by

A = (\begin{array}{l} 5.49 & 2.68 \times 10^{10} & 0 & 0 \\ 2.68 \times 10^{10} & 5.63 \times 10^{20} & - 2.39 \times 10^{10} & 0 \\ 0 & - 2.39 \times 10^{10} & 2.60 & - 2.22 \\ 0 & 0 & - 2.22 & 5.17 \end{array}) .

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

f07hf: FL CL CPP AD PY MB

NAG CL Interfacef07hfc (dpbequ)

▸▿ 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
f07hfc (dpbequ)