NAG CL Interface
f07hrc (zpbtrf)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{order}$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface 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_UploType Input

On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.

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

The upper triangular part of $A$ is stored and $A$ is factorized as $U^{\mathrm{H}}U$, where $U$ is upper triangular.

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

The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{H}}$, where $L$ is lower triangular.

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

3: $\mathbf{n}$ – Integer Input

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

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

4: $\mathbf{kd}$ – Integer Input

On entry: $k_d$, the number of superdiagonals or subdiagonals of the matrix $A$.

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

5: $\mathbf{ab}\left[\mathit{dim}\right]$ – Complex Input/Output

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

On entry: the $n\times n$ Hermitian positive definite band matrix $A$.

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

if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,

$A_{ij}$ is stored in $\mathbf{ab}\left[k_d+i-j+(j-1)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=\max (1,j-k_d),\dots ,j$;

if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,

$A_{ij}$ is stored in $\mathbf{ab}\left[i-j+(j-1)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\min (n,j+k_d)$;

if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,

$A_{ij}$ is stored in $\mathbf{ab}\left[j-i+(i-1)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\min (n,i+k_d)$;

if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,

$A_{ij}$ is stored in $\mathbf{ab}\left[k_d+j-i+(i-1)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=\max (1,i-k_d),\dots ,i$.

On exit: the upper or lower triangle of $A$ is overwritten by the Cholesky factor $U$ or $L$ as specified by uplo, using the same storage format as described above.

6: $\mathbf{pdab}$ – Integer Input

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

Constraint: $\mathbf{pdab}\ge \mathbf{kd}+1$.

7: $\mathbf{fail}$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

NE_BAD_PARAM

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.

NE_INT

On entry, $\mathbf{kd}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{kd}\ge 0$.

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_2

On entry, $\mathbf{pdab}=⟨\mathit{\text{value}}⟩$ and $\mathbf{kd}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{pdab}\ge \mathbf{kd}+1$.

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 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

NE_POS_DEF

The leading minor of order $⟨\mathit{\text{value}}⟩$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$. There is no function specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling f07brc or as a full Hermitian matrix, by calling f07mrc.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results