f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zhbgst (f08usc)

## 1  Purpose

nag_zhbgst (f08usc) reduces a complex Hermitian-definite generalized eigenproblem $Az=\lambda Bz$ to the standard form $Cy=\lambda y$, where $A$ and $B$ are band matrices, $A$ is a complex Hermitian matrix, and $B$ has been factorized by nag_zpbstf (f08utc).

## 2  Specification

 #include #include
 void nag_zhbgst (Nag_OrderType order, Nag_VectType vect, Nag_UploType uplo, Integer n, Integer ka, Integer kb, Complex ab[], Integer pdab, const Complex bb[], Integer pdbb, Complex x[], Integer pdx, NagError *fail)

## 3  Description

To reduce the complex Hermitian-definite generalized eigenproblem $Az=\lambda Bz$ to the standard form $Cy=\lambda y$, where $A$, $B$ and $C$ are banded, nag_zhbgst (f08usc) must be preceded by a call to nag_zpbstf (f08utc) which computes the split Cholesky factorization of the positive definite matrix $B$: $B={S}^{\mathrm{H}}S$. The split Cholesky factorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.
This function overwrites $A$ with $C={X}^{\mathrm{H}}AX$, where $X={S}^{-1}Q$ and $Q$ is a unitary matrix chosen (implicitly) to preserve the bandwidth of $A$. The function also has an option to allow the accumulation of $X$, and then, if $z$ is an eigenvector of $C$, $Xz$ is an eigenvector of the original system.

## 4  References

Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
Kaufman L (1984) Banded eigenvalue solvers on vector machines ACM Trans. Math. Software 10 73–86

## 5  Arguments

1:    $\mathbf{order}$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 2.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{vect}$Nag_VectTypeInput
On entry: indicates whether $X$ is to be returned.
${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$
$X$ is not returned.
${\mathbf{vect}}=\mathrm{Nag_FormX}$
$X$ is returned.
Constraint: ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$ or $\mathrm{Nag_FormX}$.
3:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{ka}$IntegerInput
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the number of superdiagonals, ${k}_{a}$, of the matrix $A$.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the number of subdiagonals, ${k}_{a}$, of the matrix $A$.
Constraint: ${\mathbf{ka}}\ge 0$.
6:    $\mathbf{kb}$IntegerInput
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: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$.
7:    $\mathbf{ab}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$.
On entry: the upper or lower triangle of the $n$ by $n$ Hermitian 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 ${\mathbf{ab}}\left[{k}_{a}+i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{a}\right),\dots ,j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[i-j+\left(j-1\right)×{\mathbf{pdab}}\right]$, for $j=1,\dots ,n$ and $i=j,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{a}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{a}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ab}}\left[{k}_{a}+j-i+\left(i-1\right)×{\mathbf{pdab}}\right]$, for $i=1,\dots ,n$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{a}\right),\dots ,i$.
On exit: the upper or lower triangle of ab is overwritten by the corresponding upper or lower triangle of $C$ as specified by uplo.
8:    $\mathbf{pdab}$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: ${\mathbf{pdab}}\ge {\mathbf{ka}}+1$.
9:    $\mathbf{bb}\left[\mathit{dim}\right]$const ComplexInput
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 banded split Cholesky factor of $B$ as specified by uplo, n and kb and returned by nag_zpbstf (f08utc).
10:  $\mathbf{pdbb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array bb.
Constraint: ${\mathbf{pdbb}}\ge {\mathbf{kb}}+1$.
11:  $\mathbf{x}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{n}}\right)$ when ${\mathbf{vect}}=\mathrm{Nag_FormX}$;
• $1$ when ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$.
The $\left(i,j\right)$th element of the matrix $X$ is stored in
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the $n$ by $n$ matrix $X={S}^{-1}Q$, if ${\mathbf{vect}}=\mathrm{Nag_FormX}$.
If ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$, x is not referenced.
12:  $\mathbf{pdx}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{vect}}=\mathrm{Nag_FormX}$, ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$, ${\mathbf{pdx}}\ge 1$.
13:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ENUM_INT_2
On entry, ${\mathbf{vect}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{vect}}=\mathrm{Nag_FormX}$, ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$, ${\mathbf{pdx}}\ge 1$.
NE_INT
On entry, ${\mathbf{ka}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ka}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}>0$.
On entry, ${\mathbf{pdbb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdbb}}>0$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}>0$.
NE_INT_2
On entry, ${\mathbf{ka}}=〈\mathit{\text{value}}〉$ and ${\mathbf{kb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$.
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ka}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{ka}}+1$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7  Accuracy

Forming the reduced matrix $C$ is a stable procedure. However it involves implicit multiplication by ${B}^{-1}$. When nag_zhbgst (f08usc) is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if $B$ is ill-conditioned with respect to inversion.

## 8  Parallelism and Performance

nag_zhbgst (f08usc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of real floating-point operations is approximately $20{n}^{2}{k}_{B}$, when ${\mathbf{vect}}=\mathrm{Nag_DoNotForm}$, assuming $n\gg {k}_{A},{k}_{B}$; there are an additional $5{n}^{3}\left({k}_{B}/{k}_{A}\right)$ operations when ${\mathbf{vect}}=\mathrm{Nag_FormX}$.
The real analogue of this function is nag_dsbgst (f08uec).

## 10  Example

This example computes all the eigenvalues of $Az=\lambda Bz$, where
 $A = -1.13+0.00i 1.94-2.10i -1.40+0.25i 0.00+0.00i 1.94+2.10i -1.91+0.00i -0.82-0.89i -0.67+0.34i -1.40-0.25i -0.82+0.89i -1.87+0.00i -1.10-0.16i 0.00+0.00i -0.67-0.34i -1.10+0.16i 0.50+0.00i$
and
 $B = 9.89+0.00i 1.08-1.73i 0.00+0.00i 0.00+0.00i 1.08+1.73i 1.69+0.00i -0.04+0.29i 0.00+0.00i 0.00+0.00i -0.04-0.29i 2.65+0.00i -0.33+2.24i 0.00+0.00i 0.00+0.00i -0.33-2.24i 2.17+0.00i .$
Here $A$ is Hermitian, $B$ is Hermitian positive definite, and $A$ and $B$ are treated as band matrices. $B$ must first be factorized by nag_zpbstf (f08utc). The program calls nag_zhbgst (f08usc) to reduce the problem to the standard form $Cy=\lambda y$, then nag_zhbtrd (f08hsc) to reduce $C$ to tridiagonal form, and nag_dsterf (f08jfc) to compute the eigenvalues.

### 10.1  Program Text

Program Text (f08usce.c)

### 10.2  Program Data

Program Data (f08usce.d)

### 10.3  Program Results

Program Results (f08usce.r)