NAG Library Function Document

nag_dgbequ (f07bfc)

1  Purpose

2  Specification

3  Description

4  References

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 $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or Nag_ColMajor.

2:     m – IntegerInput

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

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

3:     n – IntegerInput

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

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

4:     kl – IntegerInput

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

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

5:     ku – IntegerInput

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

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

6:     ab[$\mathit{dim}$] – const doubleInput

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 8 in nag_dgbsv (f07bac) for further details.

7:     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:     r[m] – 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:     c[n] – 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:   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:   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:   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:   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 $〈\mathit{\text{value}}〉$ 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.

NE_MAT_COL_ZERO

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

NE_MAT_ROW_ZERO

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

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f07bfce.c)

9.2  Program Data

Program Data (f07bfce.d)

9.3  Program Results

Program Results (f07bfce.r)