F08HAF (DSBEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08HAF (DSBEV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08HAF (DSBEV) computes all the eigenvalues and, optionally, all the eigenvectors of a real n by n symmetric band matrix A of bandwidth 2kd+1 .

2  Specification

SUBROUTINE F08HAF ( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, INFO)
INTEGER  N, KD, LDAB, LDZ, INFO
REAL (KIND=nag_wp)  AB(LDAB,*), W(N), Z(LDZ,*), WORK(3*N-2)
CHARACTER(1)  JOBZ, UPLO
The routine may be called by its LAPACK name dsbev.

3  Description

The symmetric band matrix A is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the QR algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     JOBZ – CHARACTER(1)Input
On entry: indicates whether eigenvectors are computed.
JOBZ='N'
Only eigenvalues are computed.
JOBZ='V'
Eigenvalues and eigenvectors are computed.
Constraint: JOBZ='N' or 'V'.
2:     UPLO – CHARACTER(1)Input
On entry: if UPLO='U', the upper triangular part of A is stored.
If UPLO='L', the lower triangular part of A is stored.
Constraint: UPLO='U' or 'L'.
3:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
4:     KD – INTEGERInput
On entry: if UPLO='U', the number of superdiagonals, kd, of the matrix A.
If UPLO='L', the number of subdiagonals, kd, of the matrix A.
Constraint: KD0.
5:     ABLDAB* – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least max1,N.
On entry: the upper or lower triangle of the n by n symmetric band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if UPLO='U', the elements of the upper triangle of A within the band must be stored with element Aij in ABkd+1+i-jj​ for ​max1,j-kdij;
  • if UPLO='L', the elements of the lower triangle of A within the band must be stored with element Aij in AB1+i-jj​ for ​jiminn,j+kd.
On exit: AB is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix T are returned in AB using the same storage format as described above.
6:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F08HAF (DSBEV) is called.
Constraint: LDABKD+1.
7:     WN – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
8:     ZLDZ* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Z must be at least max1,N if JOBZ='V', and at least 1 otherwise.
On exit: if JOBZ='V', Z contains the orthonormal eigenvectors of the matrix A, with the ith column of Z holding the eigenvector associated with Wi.
If JOBZ='N', Z is not referenced.
9:     LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08HAF (DSBEV) is called.
Constraints:
  • if JOBZ='V', LDZ max1,N ;
  • otherwise LDZ1.
10:   WORK3×N-2 – REAL (KIND=nag_wp) arrayWorkspace
11:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0
If INFO=i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2 = Oε A2 ,  
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8  Parallelism and Performance

F08HAF (DSBEV) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08HAF (DSBEV) 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.

9  Further Comments

The total number of floating-point operations is proportional to n3 if JOBZ='V' and is proportional to kd n2  otherwise.
The complex analogue of this routine is F08HNF (ZHBEV).

10  Example

This example finds all the eigenvalues and eigenvectors of the symmetric band matrix
A = 1 2 3 0 0 2 2 3 4 0 3 3 3 4 5 0 4 4 4 5 0 0 5 5 5 ,  
together with approximate error bounds for the computed eigenvalues and eigenvectors.

10.1  Program Text

Program Text (f08hafe.f90)

10.2  Program Data

Program Data (f08hafe.d)

10.3  Program Results

Program Results (f08hafe.r)


F08HAF (DSBEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016