NAG FL Interfacef08utf (zpbstf)

1Purpose

f08utf computes a split Cholesky factorization of a complex Hermitian positive definite band matrix.

2Specification

Fortran Interface
 Subroutine f08utf ( uplo, n, kb, bb, ldbb, info)
 Integer, Intent (In) :: n, kb, ldbb Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (Inout) :: bb(ldbb,*) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f08utf_ (const char *uplo, const Integer *n, const Integer *kb, Complex bb[], const Integer *ldbb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f08utf, nagf_lapackeig_zpbstf or its LAPACK name zpbstf.

3Description

f08utf computes a split Cholesky factorization of a complex Hermitian positive definite band matrix $B$. It is designed to be used in conjunction with f08usf.
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: $\mathbf{uplo}$Character(1) Input
On entry: indicates whether the upper or lower triangular part of $B$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $B$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $B$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $B$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{kb}$Integer Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the number of superdiagonals, ${k}_{b}$, of the matrix $B$.
If ${\mathbf{uplo}}=\text{'L'}$, the number of subdiagonals, ${k}_{b}$, of the matrix $B$.
Constraint: ${\mathbf{kb}}\ge 0$.
4: $\mathbf{bb}\left({\mathbf{ldbb}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array bb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ Hermitian positive definite band matrix $B$.
The matrix is stored in rows $1$ to ${k}_{b}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $B$ within the band must be stored with element ${B}_{ij}$ in ${\mathbf{bb}}\left({k}_{b}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{b}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $B$ within the band must be stored with element ${B}_{ij}$ in ${\mathbf{bb}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{b}\right)\text{.}$
On exit: $B$ is overwritten by the elements of its split Cholesky factor $S$.
5: $\mathbf{ldbb}$Integer Input
On entry: the first dimension of the array bb as declared in the (sub)program from which f08utf is called.
Constraint: ${\mathbf{ldbb}}\ge {\mathbf{kb}}+1$.
6: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
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
 $E≤ck+1εSHS,$
$c\left(k+1\right)$ is a modest linear function of $k+1$, and $\epsilon$ is the machine precision. It follows that $\left|{e}_{ij}\right|\le c\left(k+1\right)\epsilon \sqrt{\left({b}_{ii}{b}_{jj}\right)}$.

8Parallelism and Performance

f08utf 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.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations is approximately $4n{\left(k+1\right)}^{2}$, assuming $n\gg k$.
A call to f08utf may be followed by a call to f08usf to solve the generalized eigenproblem $Az=\lambda Bz$, where $A$ and $B$ are banded and $B$ is positive definite.