NAG C Library Function Document

nag_zgbtrf (f07brc)

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 within the band of the matrix $A$.

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

5:    $\mathbf{ku}$ – IntegerInput

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

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

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

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 $n$ matrix $A$.

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 $\mathbf{ab}\left[\left(j-1\right)\times \mathbf{pdab}+\mathbf{kl}+\mathbf{ku}+i-j\right]$;
• 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.

On exit: ab is overwritten by details of the factorization.

The elements, $u_{ij}$, of the upper triangular band factor $U$ with $k_l+k_u$ super-diagonals, and the multipliers, $l_{ij}$, used to form the lower triangular factor $L$ are stored. The elements $u_{ij}$, for $i=1,\dots ,m$ and $j=i,\dots ,\min \left(n,i+k_l+k_u\right)$, and $l_{ij}$, for $i=1,\dots ,m$ and $j=\max \left(1,i-k_l\right),\dots ,i$, are stored where $A_{ij}$ is stored on entry.

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: $\mathbf{pdab}\ge 2\times \mathbf{kl}+\mathbf{ku}+1$.

8:    $\mathbf{ipiv}\left[\min \left(\mathbf{m},\mathbf{n}\right)\right]$ – IntegerOutput

On exit: the pivot indices that define the permutation matrix. At the $\mathit{i}$th step, if $\mathbf{ipiv}\left[\mathit{i}-1\right]>\mathit{i}$ then row $\mathit{i}$ of the matrix $A$ was interchanged with row $\mathbf{ipiv}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\min \left(m,n\right)$. $\mathbf{ipiv}\left[i-1\right]\le i$ indicates that, at the $i$th step, a row interchange was not required.

9:    $\mathbf{fail}$ – 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: $\mathbf{pdab}\ge 2\times \mathbf{kl}+\mathbf{ku}+1$.

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_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.

NE_SINGULAR

Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f07brce.c)

10.2

Program Data

Program Data (f07brce.d)

10.3

Program Results

Program Results (f07brce.r)