nag_dsbgvd (f08ucc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dsbgvd (f08ucc)

+ Contents

    1  Purpose
    7  Accuracy

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 <nag.h>
#include <nagf08.h>
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 Λ  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 order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobNag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
job=Nag_EigVals
Only eigenvalues are computed.
job=Nag_DoBoth
Eigenvalues and eigenvectors are computed.
Constraint: job=Nag_EigVals or Nag_DoBoth.
3:     uploNag_UploTypeInput
On entry: if uplo=Nag_Upper, the upper triangles of A and B are stored.
If uplo=Nag_Lower, the lower triangles of A and B are stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     nIntegerInput
On entry: n, the order of the matrices A and B.
Constraint: n0.
5:     kaIntegerInput
On entry: if uplo=Nag_Upper, the number of superdiagonals, ka, of the matrix A.
If uplo=Nag_Lower, the number of subdiagonals, ka, of the matrix A.
Constraint: ka0.
6:     kbIntegerInput
On entry: if uplo=Nag_Upper, the number of superdiagonals, kb, of the matrix B.
If uplo=Nag_Lower, the number of subdiagonals, kb, of the matrix B.
Constraint: kakb0.
7:     ab[dim]doubleInput/Output
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
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 Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[ka+i-j+j-1×pdab], for j=1,,n and i=max1,j-ka,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+ka;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+ka;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[ka+j-i+i-1×pdab], for i=1,,n and j=max1,i-ka,,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: pdabka+1.
9:     bb[dim]doubleInput/Output
Note: the dimension, dim, of the array bb must be at least max1,pdbb×n.
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 Bij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Bij is stored in bb[kb+i-j+j-1×pdbb], for j=1,,n and i=max1,j-kb,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Bij is stored in bb[i-j+j-1×pdbb], for j=1,,n and i=j,,minn,j+kb;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Bij is stored in bb[j-i+i-1×pdbb], for i=1,,n and j=i,,minn,i+kb;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Bij is stored in bb[kb+j-i+i-1×pdbb], for i=1,,n and j=max1,i-kb,,i.
On exit: the factor S from the split Cholesky factorization B=STS, 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: pdbbkb+1.
11:   w[n]doubleOutput
On exit: the eigenvalues in ascending order.
12:   z[dim]doubleOutput
Note: the dimension, dim, of the array z must be at least
  • max1,pdz×n when job=Nag_DoBoth;
  • 1 otherwise.
The i,jth element of the matrix Z is stored in
  • z[j-1×pdz+i-1] when order=Nag_ColMajor;
  • z[i-1×pdz+j-1] when order=Nag_RowMajor.
On exit: if job=Nag_DoBoth, z contains the matrix Z of eigenvectors, with the ith column of Z holding the eigenvector associated with w[i-1]. The eigenvectors are normalized so that ZTBZ=I.
If job=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 job=Nag_DoBoth, pdz max1,n ;
  • otherwise pdz1.
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.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
The algorithm failed to converge; value off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
NE_ENUM_INT_2
On entry, job=value, pdz=value and n=value.
Constraint: if job=Nag_DoBoth, pdz max1,n ;
otherwise pdz1.
NE_INT
On entry, ka=value.
Constraint: ka0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdbb=value.
Constraint: pdbb>0.
On entry, pdz=value.
Constraint: pdz>0.
NE_INT_2
On entry, ka=value and kb=value.
Constraint: kakb0.
On entry, pdab=value and ka=value.
Constraint: pdabka+1.
On entry, pdbb=value and kb=value.
Constraint: pdbbkb+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 fail.errnum=n+value, for 1valuen, then nag_dpbstf (f08ufc) returned fail.errnum=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.

8  Further Comments

The total number of floating point operations is proportional to n3  if job=Nag_DoBoth and, assuming that nka , is approximately proportional to n2 ka  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 = λ 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)


nag_dsbgvd (f08ucc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012