f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zhbgvx (f08upc)

## 1  Purpose

nag_zhbgvx (f08upc) computes selected the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form
 $Az=λBz ,$
where $A$ and $B$ are Hermitian and banded, and $B$ is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

## 2  Specification

 #include #include
 void nag_zhbgvx (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, Integer ka, Integer kb, Complex ab[], Integer pdab, Complex bb[], Integer pdbb, Complex q[], Integer pdq, double vl, double vu, Integer il, Integer iu, double abstol, Integer *m, double w[], Complex z[], Integer pdz, Integer jfail[], NagError *fail)

## 3  Description

The generalized Hermitian-definite band problem
 $Az = λ Bz$
is first reduced to a standard band Hermitian problem
 $Cx = λx ,$
where $C$ is a Hermitian band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The Hermitian eigenvalue problem is then solved for the required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that
 $zH A z = λ and zH B z = 1 .$

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press

## 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 Nag_ColMajor.
2:     jobNag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
${\mathbf{job}}=\mathrm{Nag_EigVals}$
Only eigenvalues are computed.
${\mathbf{job}}=\mathrm{Nag_DoBoth}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$.
3:     rangeNag_RangeTypeInput
On entry: if ${\mathbf{range}}=\mathrm{Nag_AllValues}$, all eigenvalues will be found.
If ${\mathbf{range}}=\mathrm{Nag_Interval}$, all eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
If ${\mathbf{range}}=\mathrm{Nag_Indices}$, the ilth to iuth eigenvalues will be found.
Constraint: ${\mathbf{range}}=\mathrm{Nag_AllValues}$, $\mathrm{Nag_Interval}$ or $\mathrm{Nag_Indices}$.
4:     uploNag_UploTypeInput
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangles of $A$ and $B$ are stored.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangles of $A$ and $B$ are stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
5:     nIntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     kaIntegerInput
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: ${\mathbf{ka}}\ge 0$.
7:     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: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$.
8:     ab[$\mathit{dim}$]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 contents of ab are overwritten.
9:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge {\mathbf{ka}}+1$.
10:   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 upper or lower triangle of 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: the factor $S$ from the split Cholesky factorization $B={S}^{\mathrm{H}}S$, as returned by nag_zpbstf (f08utc).
11:   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: ${\mathbf{pdbb}}\ge {\mathbf{kb}}+1$.
12:   q[$\mathit{dim}$]ComplexOutput
Note: the dimension, dim, of the array q must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$ when ${\mathbf{job}}=\mathrm{Nag_DoBoth}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, the $n$ by $n$ matrix, $Q$ used in the reduction of the standard form, i.e., $Cx=\lambda x$, from symmetric banded to tridiagonal form.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, q is not referenced.
13:   pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
• if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdq}}\ge 1$.
14:   vldoubleInput
15:   vudoubleInput
On entry: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Indices}$, vl and vu are not referenced.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
16:   ilIntegerInput
17:   iuIntegerInput
On entry: if ${\mathbf{range}}=\mathrm{Nag_Indices}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Interval}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
• if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
18:   abstoldoubleInput
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to
 $abstol+ε maxa,b ,$
where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖T‖}_{1}$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $C$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this function returns with NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).
19:   mInteger *Output
On exit: the total number of eigenvalues found. $0\le {\mathbf{m}}\le {\mathbf{n}}$.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$, ${\mathbf{m}}={\mathbf{n}}$.
If ${\mathbf{range}}=\mathrm{Nag_Indices}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
20:   w[n]doubleOutput
On exit: the eigenvalues in ascending order.
21:   z[$\mathit{dim}$]ComplexOutput
Note: the dimension, dim, of the array z must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdz}}×{\mathbf{n}}\right)$ when ${\mathbf{job}}=\mathrm{Nag_DoBoth}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $Z$ is stored in
• ${\mathbf{z}}\left[\left(j-1\right)×{\mathbf{pdz}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{z}}\left[\left(i-1\right)×{\mathbf{pdz}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, z contains the matrix $Z$ of eigenvectors, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{w}}\left[i-1\right]$. The eigenvectors are normalized so that ${Z}^{\mathrm{H}}BZ=I$.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, z is not referenced.
22:   pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
• if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdz}}\ge 1$.
23:   jfail[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array jfail must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, then
• if NE_NOERROR, the first m elements of jfail are zero;
• if NE_CONVERGENCE, jfail contains the indices of the eigenvectors that failed to converge.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, jfail is not referenced.
24:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
The algorithm failed to converge; $〈\mathit{\text{value}}〉$ eigenvectors did not converge.
NE_ENUM_INT_2
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdq}}\ge 1$.
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdz}}\ge 1$.
NE_ENUM_INT_3
On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$, ${\mathbf{il}}=〈\mathit{\text{value}}〉$, ${\mathbf{iu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
NE_ENUM_REAL_2
On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$, ${\mathbf{vl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{vu}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
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, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdq}}>0$.
On entry, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdz}}>0$.
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.
NE_MAT_NOT_POS_DEF
If ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}={\mathbf{n}}+〈\mathit{\text{value}}〉$, for $1\le 〈\mathit{\text{value}}〉\le {\mathbf{n}}$, then nag_zpbstf (f08utc) returned ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}=〈\mathit{\text{value}}〉$: $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

## 7  Accuracy

If $B$ is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of $B$ differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of $B$ would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.

The total number of floating point operations is proportional to ${n}^{3}$ if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$ and ${\mathbf{range}}=\mathrm{Nag_AllValues}$, and assuming that $n\gg {k}_{a}$, is approximately proportional to ${n}^{2}{k}_{a}$ if ${\mathbf{job}}=\mathrm{Nag_EigVals}$. Otherwise the number of floating point operations depends upon the number of eigenvectors computed.
The real analogue of this function is nag_dsbgvx (f08ubc).

## 9  Example

This example finds the eigenvalues in the half-open interval $\left(0.0,2.0\right]$, and corresponding eigenvectors, of the generalized band Hermitian eigenproblem $Az=\lambda Bz$, where
 $A = -1.13i+0.00 1.94-2.10i -1.40+0.25i 0.00i+0.00 1.94+2.10i -1.91i+0.00 -0.82-0.89i -0.67+0.34i -1.40-0.25i -0.82+0.89i -1.87i+0.00 -1.10-0.16i 0.00i+0.00 -0.67-0.34i -1.10+0.16i 0.50i+0.00$
and
 $B = 9.89i+0.00 1.08-1.73i 0.00i+0.00 0.00i+0.00 1.08+1.73i 1.69i+0.00 -0.04+0.29i 0.00i+0.00 0.00i+0.00 -0.04-0.29i 2.65i+0.00 -0.33+2.24i 0.00i+0.00 0.00i+0.00 -0.33-2.24i 2.17i+0.00 .$

### 9.1  Program Text

Program Text (f08upce.c)

### 9.2  Program Data

Program Data (f08upce.d)

### 9.3  Program Results

Program Results (f08upce.r)