f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dsbgvd (f08ucc)

## 1  Purpose

nag_dsbgvd (f08ucc) computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form
 $Az=λBz ,$
where $A$ and $B$ are symmetric and banded, and $B$ is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

## 2  Specification

 #include #include
 void nag_dsbgvd (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Integer ka, Integer kb, double ab[], Integer pdab, double bb[], Integer pdbb, double w[], double z[], Integer pdz, NagError *fail)

## 3  Description

The generalized symmetric-definite band problem
 $Az = λ Bz$
is first reduced to a standard band symmetric problem
 $Cx = λx ,$
where $C$ is a symmetric band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The symmetric eigenvalue problem is then solved for the eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that the matrix of eigenvectors, $Z$, satisfies
 $ZT A Z = Λ and ZT B Z = I ,$
where $\Lambda$ is the diagonal matrix whose diagonal elements are the eigenvalues.

## 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
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:     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 $\mathrm{Nag_Lower}$.
4:     nIntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     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: ${\mathbf{ka}}\ge 0$.
6:     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: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$.
7:     ab[$\mathit{dim}$]doubleInput/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 $n$ symmetric 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.
8:     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: ${\mathbf{pdab}}\ge {\mathbf{ka}}+1$.
9:     bb[$\mathit{dim}$]doubleInput/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 $n$ symmetric 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 $B={S}^{\mathrm{T}}S$, as returned by nag_dpbstf (f08ufc).
10:   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: ${\mathbf{pdbb}}\ge {\mathbf{kb}}+1$.
11:   w[n]doubleOutput
On exit: the eigenvalues in ascending order.
12:   z[$\mathit{dim}$]doubleOutput
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 ${Z}^{\mathrm{T}}BZ=I$.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, z is not referenced.
13:   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 ${\mathbf{pdz}}\ge 1$.
14:   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; $〈\mathit{\text{value}}〉$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
NE_ENUM_INT_2
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_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, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$.
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_dpbstf (f08ufc) 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, assuming that $n\gg {k}_{a}$, is approximately proportional to ${n}^{2}{k}_{a}$ otherwise.
The complex analogue of this function is nag_zhbgvd (f08uqc).

## 9  Example

This example finds all the eigenvalues of the generalized band symmetric eigenproblem $Az=\lambda Bz$, where
 $A = 0.24 0.39 0.42 0.00 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 0.00 0.63 0.48 -0.03 and B = 2.07 0.95 0.00 0.00 0.95 1.69 -0.29 0.00 0.00 -0.29 0.65 -0.33 0.00 0.00 -0.33 1.17 .$

### 9.1  Program Text

Program Text (f08ucce.c)

### 9.2  Program Data

Program Data (f08ucce.d)

### 9.3  Program Results

Program Results (f08ucce.r)