NAG Library Function Document

nag_zpbtrf (f07hrc)

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 $\mathrm{Nag\_ColMajor}$.

2:     uplo – Nag_UploTypeInput

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:     n – IntegerInput

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

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

4:     kd – IntegerInput

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

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

5:     ab[$\mathit{dim}$] – ComplexInput/Output

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

On entry: the $n$ by $n$ 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 $\mathbf{ab}\left[k_d+i-j+\left(j-1\right)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=\max \left(1,j-k_d\right),\dots ,j$;
• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[i-j+\left(j-1\right)\times \mathbf{pdab}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\min \left(n,j+k_d\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[j-i+\left(i-1\right)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\min \left(n,i+k_d\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ab}\left[k_d+j-i+\left(i-1\right)\times \mathbf{pdab}\right]$, for $i=1,\dots ,n$ and $j=\max \left(1,i-k_d\right),\dots ,i$.

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

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, $\mathbf{kd}=⟨\mathit{\text{value}}⟩$.
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.

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 nag_zgbtrf (f07brc) or as a full Hermitian matrix, by calling nag_zhetrf (f07mrc).

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f07hrce.c)

10.2  Program Data

Program Data (f07hrce.d)

10.3  Program Results

Program Results (f07hrce.r)