NAG FL Interface
f08haf (dsbev)

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FL Specification Language:

1 Purpose

f08haf computes all the eigenvalues and, optionally, all the eigenvectors of a real n×n symmetric band matrix A of bandwidth (2kd+1) .

2 Specification

Fortran Interface
Subroutine f08haf ( jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, info)
Integer, Intent (In) :: n, kd, ldab, ldz
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), z(ldz,*)
Real (Kind=nag_wp), Intent (Out) :: w(n), work(3*n-2)
Character (1), Intent (In) :: jobz, uplo
C Header Interface
#include <nag.h>
void  f08haf_ (const char *jobz, const char *uplo, const Integer *n, const Integer *kd, double ab[], const Integer *ldab, double w[], double z[], const Integer *ldz, double work[], Integer *info, const Charlen length_jobz, const Charlen length_uplo)
The routine may be called by the names f08haf, nagf_lapackeig_dsbev or 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
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.
Only eigenvalues are computed.
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 Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: kd Integer Input
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: ab(ldab,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array ab must be at least max(1,n).
On entry: the upper or lower triangle of the n×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 ab(kd+1+i-j,j)​ for ​max(1,j-kd)ij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab(1+i-j,j)​ for ​jimin(n,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 Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f08haf is called.
Constraint: ldabkd+1.
7: w(n) Real (Kind=nag_wp) array Output
On exit: the eigenvalues in ascending order.
8: z(ldz,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array z must be at least max(1,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 w(i).
If jobz='N', z is not referenced.
9: ldz Integer Input
On entry: the first dimension of the array z as declared in the (sub)program from which f08haf is called.
  • if jobz='V', ldz max(1,n) ;
  • otherwise ldz1.
10: work(3×n-2) Real (Kind=nag_wp) array Workspace
11: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
The algorithm failed to converge; value 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 = O(ε) A2 ,  
and ε is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08haf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08haf 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.

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)