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

3 Description

nag_zpbequ (f07htc) 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_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 2.3.1.3 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: 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: $n$ – 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 of the matrix

A

uplo = Nag_Upper

, or the number of subdiagonals if

uplo = Nag_Lower

Constraint:

kd \geq 0

5: $ab [\dim]$ – const ComplexInput

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 Hermitian 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: $pdab$ – 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

7: $s [n]$ – doubleOutput

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 2.7 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, $pdab = 〈value〉$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7 Accuracy

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

8 Parallelism and Performance

nag_zpbequ (f07htc) is not threaded in any implementation.

9 Further Comments

The real analogue of this function is nag_dpbequ (f07hfc).

10 Example

This example equilibrates the Hermitian positive definite matrix

A

given by

A = (\begin{array}{l} 9.39 & 1.08 - 1.73 i & 0 & 0 \\ 1.08 + 1.73 i & 1.69 & (- 0.04 + 0.29 i) \times 10^{10} & 0 \\ 0 & (- 0.04 - 0.29 i) \times 10^{10} & 2.65 \times 10^{20} & (- 0.33 + 2.24 i) \times 10^{10} \\ 0 & 0 & (- 0.33 - 2.24 i) \times 10^{10} & 2.17 \end{array}) .

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

NAG Library Function Documentnag_zpbequ (f07htc)

▸▿ 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_zpbequ (f07htc)