NAG Library Function Document

nag_zgbequ (f07btc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{order}$ – 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 $\mathbf{order}=\mathrm{Nag\_RowMajor}$. 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: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:    $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of rows of the matrix $A$.

Constraint: $\mathbf{m}\ge 0$.

3:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of columns of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

4:    $\mathbf{kl}$ – IntegerInput

On entry: $k_l$, the number of subdiagonals of the matrix $A$.

Constraint: $\mathbf{kl}\ge 0$.

5:    $\mathbf{ku}$ – IntegerInput

On entry: $k_u$, the number of superdiagonals of the matrix $A$.

Constraint: $\mathbf{ku}\ge 0$.

6:    $\mathbf{ab}\left[\mathit{dim}\right]$ – const ComplexInput

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

• $\max \left(1,\mathbf{pdab}\times \mathbf{n}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{m}\times \mathbf{pdab}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: the $m$ by $n$ 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 $A_{ij}$, for row $i=1,\dots ,m$ and column $j=\max \left(1,i-k_l\right),\dots ,\min \left(n,i+k_u\right)$, depends on the order argument as follows:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $A_{ij}$ is stored as $\mathbf{ab}\left[\left(j-1\right)\times \mathbf{pdab}+\mathbf{ku}+i-j\right]$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $A_{ij}$ is stored as $\mathbf{ab}\left[\left(i-1\right)\times \mathbf{pdab}+\mathbf{kl}+j-i\right]$.

See Section 9 in nag_zgbsv (f07bnc) for further details.

7:    $\mathbf{pdab}$ – 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: $\mathbf{pdab}\ge \mathbf{kl}+\mathbf{ku}+1$.

8:    $\mathbf{r}\left[\mathbf{m}\right]$ – doubleOutput

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR or $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_MAT_COL_ZERO, r contains the row scale factors, the diagonal elements of $D_R$. The elements of r will be positive.

9:    $\mathbf{c}\left[\mathbf{n}\right]$ – doubleOutput

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR, c contains the column scale factors, the diagonal elements of $D_C$. The elements of c will be positive.

10:  $\mathbf{rowcnd}$ – double *Output

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR or $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_MAT_COL_ZERO, rowcnd contains the ratio of the smallest value of $\mathbf{r}\left[i-1\right]$ to the largest value of $\mathbf{r}\left[i-1\right]$. If $\mathbf{rowcnd}\ge 0.1$ and amax is neither too large nor too small, it is not worth scaling by $D_R$.

11:  $\mathbf{colcnd}$ – double *Output

On exit: if $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_NOERROR, colcnd contains the ratio of the smallest value of $\mathbf{c}\left[i-1\right]$ to the largest value of $\mathbf{c}\left[i-1\right]$.

If $\mathbf{colcnd}\ge 0.1$, it is not worth scaling by $D_C$.

12:  $\mathbf{amax}$ – double *Output

On exit: $\max \left|a_{ij}\right|$. If amax is very close to overflow or underflow, the matrix $A$ should be scaled.

13:  $\mathbf{fail}$ – 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 $〈\mathit{\text{value}}〉$ had an illegal value.

NE_INT

On entry, $\mathbf{kl}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{kl}\ge 0$.

On entry, $\mathbf{ku}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ku}\ge 0$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 0$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

On entry, $\mathbf{pdab}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdab}>0$.

NE_INT_3

On entry, $\mathbf{pdab}=〈\mathit{\text{value}}〉$, $\mathbf{kl}=〈\mathit{\text{value}}〉$ and $\mathbf{ku}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdab}\ge \mathbf{kl}+\mathbf{ku}+1$.

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_MAT_COL_ZERO

Column $〈\mathit{\text{value}}〉$ of $A$ is exactly zero.

NE_MAT_ROW_ZERO

Row $〈\mathit{\text{value}}〉$ of $A$ is exactly zero.

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

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f07btce.c)

10.2

Program Data

Program Data (f07btce.d)

10.3

Program Results

Program Results (f07btce.r)