NAG Library Function Document

nag_zpbstf (f08utc)

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:     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:     n – IntegerInput

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

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

4:     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:     bb[$\mathit{dim}$] – ComplexInput/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 $n$ Hermitian 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:     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:     fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

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.

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  Further Comments

9  Example