# NAG CL Interfacef08utc (zpbstf)

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## 1Purpose

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

## 2Specification

 #include
 void f08utc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kb, Complex bb[], Integer pdbb, NagError *fail)
The function may be called by the names: f08utc, nag_lapackeig_zpbstf or nag_zpbstf.

## 3Description

f08utc computes a split Cholesky factorization of a complex Hermitian positive definite band matrix $B$. It is designed to be used in conjunction with 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.

## 5Arguments

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: 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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $B$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{kb}$Integer Input
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: $\mathbf{bb}\left[\mathit{dim}\right]$Complex Input/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 $n×n$ 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: $\mathbf{pdbb}$Integer Input
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 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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.
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 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 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 $B$.

## 7Accuracy

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

## 8Parallelism and Performance

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