NAG Library Function Document

nag_dpbstf (f08ufc)

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:    $\mathbf{uplo}$ – Nag_UploTypeInput

On entry: indicates whether the upper or lower triangular part of $B$ is stored.

$\mathbf{uplo}=\mathrm{Nag\_Upper}$

The upper triangular part of $B$ is stored.

$\mathbf{uplo}=\mathrm{Nag\_Lower}$

The lower triangular part of $B$ is stored.

Constraint: $\mathbf{uplo}=\mathrm{Nag\_Upper}$ or $\mathrm{Nag\_Lower}$.

3:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $B$.

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

4:    $\mathbf{kb}$ – IntegerInput

On entry: if $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the number of superdiagonals, $k_b$, of the matrix $B$.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the number of subdiagonals, $k_b$, of the matrix $B$.

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

5:    $\mathbf{bb}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, of the array bb must be at least $\max \left(1,\mathbf{pdbb}\times \mathbf{n}\right)$.

On entry: the $n$ by $n$ symmetric positive definite band matrix $B$.

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of $B_{ij}$, depends on the order and uplo arguments as follows:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $B_{ij}$ is stored in $\mathbf{bb}\left[k_b+i-j+\left(j-1\right)\times \mathbf{pdbb}\right]$, for $j=1,\dots ,n$ and $i=\max \left(1,j-k_b\right),\dots ,j$;
• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $B_{ij}$ is stored in $\mathbf{bb}\left[i-j+\left(j-1\right)\times \mathbf{pdbb}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\min \left(n,j+k_b\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $B_{ij}$ is stored in $\mathbf{bb}\left[j-i+\left(i-1\right)\times \mathbf{pdbb}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\min \left(n,i+k_b\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $B_{ij}$ is stored in $\mathbf{bb}\left[k_b+j-i+\left(i-1\right)\times \mathbf{pdbb}\right]$, for $i=1,\dots ,n$ and $j=\max \left(1,i-k_b\right),\dots ,i$.

On exit: $B$ is overwritten by the elements of its split Cholesky factor $S$.

6:    $\mathbf{pdbb}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix $B$ in the array bb.

Constraint: $\mathbf{pdbb}\ge \mathbf{kb}+1$.

7:    $\mathbf{fail}$ – 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, $\mathbf{kb}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{kb}\ge 0$.

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

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

NE_INT_2

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

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_POS_DEF

The factorization could not be completed, because the updated element $b\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$ would be the square root of a negative number. Hence $B$ is not positive definite. This may indicate an error in forming the matrix $B$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example