NAG Library Routine Document

F02FHF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F02FHF finds the eigenvalues of the generalized band symmetric eigenvalue problem Ax=λBx, where A and B are symmetric band matrices and B is positive-definite.

2 Specification

SUBROUTINE F02FHF (	N, MA, A, LDA, MB, B, LDB, D, WORK, LWORK, IFAIL)
INTEGER	N, MA, LDA, MB, LDB, LWORK, IFAIL
*double precision*	A(LDA,N), B(LDB,N), D(N), WORK(LWORK)

3 Description

The generalized band symmetric eigenvalue problem Ax=λBx, where A is a symmetric band matrix of band width 2mA+1 and B is a positive-definite symmetric band matrix of band width 2mB+1, is solved by a variant of the method of Crawford (see Crawford (1973)).

F02FHF first transforms the problem Ax=λBx to a standard band symmetric eigenvalue problem Cy=λy, where C is a band symmetric matrix of band width 2mA+1, using F01BUF and F01BVF. This step involves the implicit inversion of the matrix B and so this routine should be used with caution if B is ill-conditioned with respect to inversion.

The eigenvalues of the standard problem Cy=λy are then obtained by reducing C to tridiagonal form and then applying the QL variant of the QR algorithm to the tridiagonal form, using F08HEF (DSBTRD) and F08JFF (DSTERF). The above-mentioned routines should be consulted for further information on the methods used.

Once the eigenvalues have been found by this routine, selected eigenvectors may be obtained by repeated calls to F02SDF with the original matrices A and B as data.

The routine assumes that mA≥mB and hence if the band width of A is actually smaller than that of B, then A must be filled out with additional zero diagonals.

4 References

Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44

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 Parameters

1: N – INTEGERInput

On entry: n, the order of the matrices A and B.

Constraint: N≥1.

2: MA – INTEGERInput

On entry: mA, the number of superdiagonals within the band of A. Normally mA≪n.

Constraint: 0≤MA≤N-1.

3: A(LDA,N) – double precision arrayInput/Output

On entry: the upper triangle of the n by n symmetric band matrix A, with the diagonal of the matrix stored in the mA+1th row of the array, and the mA superdiagonals within the band stored in the first mA rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if n=6 and m=2, the storage space is

* * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66

Elements in the top left corner of the array need not be set. The following code assigns the matrix elements within the band to the correct elements of the array:

       MA1 = MA + 1
       DO 20 J = 1, N
          DO 10 I = MAX(1,J-MA1+1), J
             A(I-J+MA1,J) = matrix (I,J)
    10    CONTINUE
    20 CONTINUE

On exit: A is overwritten by the corresponding elements of C.

4: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F02FHF is called.

Constraint: LDA≥MA+1.

5: MB – INTEGERInput

On entry: mB, the number of superdiagonals within the band of B.

Constraint: 0≤MB≤MA.

6: B(LDB,N) – double precision arrayInput/Output

On entry: the upper triangle of the n by n symmetric positive-definite band matrix B, with the diagonal of the matrix stored in the mB+1th row of the array, and the mB superdiagonals within the band stored in the first mB rows of the array. Each column of the matrix is stored in the corresponding column of the array.

On exit: B is overwritten.

7: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F02FHF is called.

Constraint: LDB≥MB+1.

8: D(N) – double precision arrayOutput

On exit: the eigenvalues in descending order of magnitude.

9: WORK(LWORK) – double precision arrayWorkspace

10: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F02FHF is called.

Constraint: LWORK≥maxN,3×MA+MB×MA+MB+1.

11: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to 0, -1 or 1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

On exit: IFAIL=0 unless the routine detects an error (see Section 6).

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.