f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zpbstf (f08utc)

## 1  Purpose

nag_zpbstf (f08utc) computes a split Cholesky factorization of a complex Hermitian positive definite band matrix.

## 2  Specification

 #include #include
 void nag_zpbstf (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kb, Complex bb[], Integer pdbb, NagError *fail)

## 3  Description

nag_zpbstf (f08utc) computes a split Cholesky factorization of a complex Hermitian positive definite band matrix $B$. It is designed to be used in conjunction with nag_zhbgst (f08usc).
The factorization has the form $B={S}^{\mathrm{H}}S$, where $S$ is a band matrix of the same bandwidth as $B$ and the following structure: $S$ is upper triangular in the first $\left(n+k\right)/2$ rows, and transposed — hence, lower triangular — in the remaining rows. For example, if $n=9$ and $k=2$, then
 $S = s11 s12 s13 s22 s23 s24 s33 s34 s35 s44 s45 s55 s64 s65 s66 s75 s76 s77 s86 s87 s88 s97 s98 s99 .$

None.

## 5  Arguments

1:     orderNag_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:     uploNag_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:     nIntegerInput
On entry: $n$, the order of the matrix $B$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     kbIntegerInput
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdbb}}×{\mathbf{n}}\right)$.
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 ${\mathbf{bb}}\left[{k}_{b}+i-j+\left(j-1\right)×{\mathbf{pdbb}}\right]$, for $j=1,\dots ,n$ and $i=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{b}\right),\dots ,j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${B}_{ij}$ is stored in ${\mathbf{bb}}\left[i-j+\left(j-1\right)×{\mathbf{pdbb}}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{b}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${B}_{ij}$ is stored in ${\mathbf{bb}}\left[j-i+\left(i-1\right)×{\mathbf{pdbb}}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{b}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${B}_{ij}$ is stored in ${\mathbf{bb}}\left[{k}_{b}+j-i+\left(i-1\right)×{\mathbf{pdbb}}\right]$, for $i=1,\dots ,n$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{b}\right),\dots ,i$.
On exit: $B$ is overwritten by the elements of its split Cholesky factor $S$.
6:     pdbbIntegerInput
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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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

The computed factor $S$ is the exact factor of a perturbed matrix $\left(B+E\right)$, where
 $E≤ck+1εSHS,$
$c\left(k+1\right)$ is a modest linear function of $k+1$, and $\epsilon$ is the machine precision. It follows that $\left|{e}_{ij}\right|\le c\left(k+1\right)\epsilon \sqrt{\left({b}_{ii}{b}_{jj}\right)}$.

## 8  Parallelism and Performance

nag_zpbstf (f08utc) is not threaded by NAG in any implementation.
nag_zpbstf (f08utc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

The total number of floating-point operations is approximately $4n{\left(k+1\right)}^{2}$, assuming $n\gg k$.
A call to nag_zpbstf (f08utc) may be followed by a call to nag_zhbgst (f08usc) to solve the generalized eigenproblem $Az=\lambda Bz$, where $A$ and $B$ are banded and $B$ is positive definite.