F08HBF (DSBEVX) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08HBF (DSBEVX)

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

F08HBF (DSBEVX) computes selected eigenvalues and, optionally, eigenvectors of a real n by n symmetric band matrix A of bandwidth 2kd+1 . Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

2  Specification

SUBROUTINE F08HBF ( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, JFAIL, INFO)
INTEGER  N, KD, LDAB, LDQ, IL, IU, M, LDZ, IWORK(5*N), JFAIL(*), INFO
REAL (KIND=nag_wp)  AB(LDAB,*), Q(LDQ,*), VL, VU, ABSTOL, W(N), Z(LDZ,*), WORK(7*N)
CHARACTER(1)  JOBZ, RANGE, UPLO
The routine may be called by its LAPACK name dsbevx.

3  Description

The symmetric band matrix A is first reduced to tridiagonal form, using orthogonal similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

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
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

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:     RANGE – CHARACTER(1)Input
On entry: if RANGE='A', all eigenvalues will be found.
If RANGE='V', all eigenvalues in the half-open interval VL,VU will be found.
If RANGE='I', the ILth to IUth eigenvalues will be found.
Constraint: RANGE='A', 'V' or 'I'.
3:     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'.
4:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
5:     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.
6:     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.
7:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F08HBF (DSBEVX) is called.
Constraint: LDABKD+1.
8:     QLDQ* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Q must be at least max1,N if JOBZ='V', and at least 1 otherwise.
On exit: if JOBZ='V', the n by n orthogonal matrix used in the reduction to tridiagonal form.
If JOBZ='N', Q is not referenced.
9:     LDQ – INTEGERInput
On entry: the first dimension of the array Q as declared in the (sub)program from which F08HBF (DSBEVX) is called.
Constraints:
  • if JOBZ='V', LDQ max1,N ;
  • otherwise LDQ1.
10:   VL – REAL (KIND=nag_wp)Input
11:   VU – REAL (KIND=nag_wp)Input
On entry: if RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues.
If RANGE='A' or 'I', VL and VU are not referenced.
Constraint: if RANGE='V', VL<VU.
12:   IL – INTEGERInput
13:   IU – INTEGERInput
On entry: if RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If RANGE='A' or 'V', IL and IU are not referenced.
Constraints:
  • if RANGE='I' and N=0, IL=1 and IU=0;
  • if RANGE='I' and N>0, 1 IL IU N .
14:   ABSTOL – REAL (KIND=nag_wp)Input
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 a,b  of width less than or equal to
ABSTOL+ε maxa,b ,  
where ε  is the machine precision. If ABSTOL is less than or equal to zero, then ε T1  will be used in its place, where T is the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2 × X02AMF   , not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2 × X02AMF   . See Demmel and Kahan (1990).
15:   M – INTEGEROutput
On exit: the total number of eigenvalues found. 0MN.
If RANGE='A', M=N.
If RANGE='I', M=IU-IL+1.
16:   WN – REAL (KIND=nag_wp) arrayOutput
On exit: the first M elements contain the selected eigenvalues in ascending order.
17:   ZLDZ* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array Z must be at least max1,M if JOBZ='V', and at least 1 otherwise.
On exit: if JOBZ='V', then
  • if INFO=0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with Wi;
  • if an eigenvector fails to converge (INFO>0), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in JFAIL.
If JOBZ='N', Z is not referenced.
Note:  you must ensure that at least max1,M columns are supplied in the array Z; if RANGE='V', the exact value of M is not known in advance and an upper bound of at least N must be used.
18:   LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08HBF (DSBEVX) is called.
Constraints:
  • if JOBZ='V', LDZ max1,N ;
  • otherwise LDZ1.
19:   WORK7×N – REAL (KIND=nag_wp) arrayWorkspace
20:   IWORK5×N – INTEGER arrayWorkspace
21:   JFAIL* – INTEGER arrayOutput
Note: the dimension of the array JFAIL must be at least max1,N.
On exit: if JOBZ='V', then
  • if INFO=0, the first M elements of JFAIL are zero;
  • if INFO>0, JFAIL contains the indices of the eigenvectors that failed to converge.
If JOBZ='N', JFAIL is not referenced.
22:   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, then i eigenvectors failed to converge. Their indices are stored in array JFAIL. Please see ABSTOL.

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

F08HBF (DSBEVX) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F08HBF (DSBEVX) 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 kd n2  if JOBZ='N', and is proportional to n3  if JOBZ='V' and RANGE='A', otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.
The complex analogue of this routine is F08HPF (ZHBEVX).

10  Example

This example finds the eigenvalues in the half-open interval -3,3 , and the corresponding 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 .  

10.1  Program Text

Program Text (f08hbfe.f90)

10.2  Program Data

Program Data (f08hbfe.d)

10.3  Program Results

Program Results (f08hbfe.r)


F08HBF (DSBEVX) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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