NAG Library Routine Document

F12FGF

Note: this routine uses optional parameters to define choices in the problem specification. If you wish to use default settings for all of the optional parameters, then the option setting routine F12FDF need not be called. If, however, you wish to reset some or all of the settings please refer to Section 10 in F12FDF for a detailed description of the specification of the optional parameters.

+− 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

F12FGF is the main solver routine in a suite of routines which includes F12FDF and F12FFF. F12FGF must be called following an initial call to F12FFF and following any calls to F12FDF.

F12FGF returns approximations to selected eigenvalues, and (optionally) the corresponding eigenvectors, of a standard or generalized eigenvalue problem defined by real banded symmetric matrices. The banded matrix must be stored using the LAPACK storage format for real banded nonsymmetric matrices.

2 Specification

SUBROUTINE F12FGF (

KL, KU, AB, LDAB, MB, LDMB, SIGMA, NCONV, D, Z, LDZ, RESID, V, LDV, COMM, ICOMM, IFAIL)

INTEGER	KL, KU, LDAB, LDMB, NCONV, LDZ, LDV, ICOMM(*), IFAIL
REAL (KIND=nag_wp)	AB(LDAB,), MB(LDMB,), SIGMA, D(), Z(LDZ,), RESID(), V(LDV,), COMM(*)

3 Description

The suite of routines is designed to calculate some of the eigenvalues,

λ

, (and optionally the corresponding eigenvectors,

x

) of a standard eigenvalue problem

A x = λ x

, or of a generalized eigenvalue problem

A x = λ B x

of order

n

, where

n

is large and the coefficient matrices

A

and

B

are banded, real and symmetric.

Following a call to the initialization routine F12FFF, F12FGF returns the converged approximations to eigenvalues and (optionally) the corresponding approximate eigenvectors and/or an orthonormal basis for the associated approximate invariant subspace. The eigenvalues (and eigenvectors) are selected from those of a standard or generalized eigenvalue problem defined by real banded symmetric matrices. There is negligible additional computational cost to obtain eigenvectors; an orthonormal basis is always computed, but there is an additional storage cost if both are requested.

The banded matrices

A

and

B

must be stored using the LAPACK storage format for banded nonsymmetric matrices; please refer to Section 3.3.2 in the F07 Chapter Introduction for details on this storage format.

F12FGF is based on the banded driver routines dsbdr1 to dsbdr6 from the ARPACK package, which uses the Implicitly Restarted Lanczos iteration method. The method is described in Lehoucq and Sorensen (1996) and Lehoucq (2001) while its use within the ARPACK software is described in great detail in Lehoucq et al. (1998). This suite of routines offers the same functionality as the ARPACK banded driver software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and to combine the different drivers into a general purpose routine.

F12FGF, is a general purpose forward communication routine that must be called following initialization by F12FFF. F12FGF uses options, set either by default or explicitly by calling F12FDF, to return the converged approximations to selected eigenvalues and (optionally):

–	the corresponding approximate eigenvectors;
–	an orthonormal basis for the associated approximate invariant subspace;
–	both.

4 References

Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562

Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory

Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821

Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia

5 Parameters

1: KL – INTEGERInput

On entry: the number of subdiagonals of the matrices

A

and

B

Constraint:

KL \geq 0

2: KU – INTEGERInput

On entry: the number of superdiagonals of the matrices

A

and

B

. Since

A

and

B

are symmetric, the normal case is

KU = KL

Constraint:

KU \geq 0

3: AB(LDAB, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array AB must be at least

\max (1, N)

(see F12FFF).

On entry: must contain the matrix

A

in LAPACK banded storage format for nonsymmetric matrices (see Section 3.3.4 in the F07 Chapter Introduction).

4: LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F12FGF is called.

Constraint:

LDAB \geq 2 \times KL + KU + 1

5: MB(LDMB, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array MB must be at least

\max (1, N)

(see F12FFF).

On entry: must contain the matrix

B

in LAPACK banded storage format for nonsymmetric matrices (see Section 3.3.4 in the F07 Chapter Introduction).

6: LDMB – INTEGERInput

On entry: the first dimension of the array MB as declared in the (sub)program from which F12FGF is called.

Constraint:

LDMB \geq 2 \times KL + KU + 1

7: SIGMA – REAL (KIND=nag_wp)Input

On entry: if one of the Shifted Inverse (see F12FDF) modes has been selected then SIGMA contains the real shift used; otherwise SIGMA is not referenced.

8: NCONV – INTEGEROutput

On exit: the number of converged eigenvalues.

9: D( $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array D must be at least

NCV

(see F12FFF).

On exit: the first NCONV locations of the array D contain the converged approximate eigenvalues.

10: Z(LDZ, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array Z must be at least

NCV + 1

if the default option

Vectors = RITZ

has been selected and at least

1

if the option

Vectors = NONE

SCHUR

has been selected (see F12FFF).

On exit: if the default option

Vectors = RITZ

(see F12FDF) has been selected then Z contains the final set of eigenvectors corresponding to the eigenvalues held in D. The real eigenvector associated with eigenvalue

i - 1

, for

i = 1, 2, \dots, NCONV

, is stored in the

i

th column of Z.

11: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F12FGF is called.

Constraints:

if the default option $Vectors = Ritz$ has been selected, $LDZ \geq N$ ;
if the option $Vectors = None or Schur$ has been selected, $LDZ \geq 1$ .

12: RESID( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array RESID must be at least

N

(see F12FFF).

On entry: need not be set unless the option Initial Residual has been set in a prior call to F12FDF in which case RESID must contain an initial residual vector.

On exit: contains the final residual vector.

13: V(LDV, $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array V must be at least

\max (1, NCV)

(see F12FFF).

On exit: if the option Vectors (see F12FDF) has been set to Schur or Ritz and a separate array Z has been passed then the first

NCONV \times n

elements of V will contain approximate Schur vectors that span the desired invariant subspace.

The

j

th Schur vector is stored in the

i

th column of V.

14: LDV – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F12FGF is called.

Constraint:

LDV \geq n

15: COMM( $*$ ) – REAL (KIND=nag_wp) arrayCommunication Array

Note: the dimension of the array COMM must be at least

\max (1, LCOMM)

(see F12FFF).

On initial entry: must remain unchanged from the prior call to F12FDF and F12FFF.

On exit: contains no useful information.

16: ICOMM( $*$ ) – INTEGER arrayCommunication Array

Note: the dimension of the array ICOMM must be at least

\max (1, LICOMM)

(see F12FFF).

On initial entry: must remain unchanged from the prior call to F12FBF and F12FDF.

On exit: contains no useful information.

17: 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.

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.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $KL < 0$ .

$IFAIL = 2$: On entry, $KU < 0$ .

$IFAIL = 3$: On entry, $LDAB < 2 \times KL + KU + 1$ .

$IFAIL = 4$: $Iteration Limit < 0$ .

$IFAIL = 5$: The options Generalized and Regular are incompatible.

$IFAIL = 6$: Eigenvalues from Both Ends of the spectrum were requested, but only one eigenvalue (NEV) is requested.

$IFAIL = 7$: The Initial Residual was selected but the starting vector held in RESID is zero.

$IFAIL = 8$: On entry, $LDZ < \max (1, N)$ or $LDZ < 1$ when no vectors are required.

$IFAIL = 9$: On entry, the option $Vectors = Select$ was selected, but this is not yet implemented.

$IFAIL = 10$: The number of eigenvalues found to sufficient accuracy is zero.

$IFAIL = 11$: Could not build a Lanczos factorization. Consider changing NCV or NEV in the initialization routine (see Section 5 in F12FAF for details of these parameters).

$IFAIL = 12$: Unexpected error in internal call to compute eigenvalues and corresponding error bounds of the current symmetric tridiagonal matrix. Please contact NAG.

$IFAIL = 13$: Unexpected error during calculation of a real Schur form: there was a failure to compute all the converged eigenvalues. Please contact NAG.

$IFAIL = 14$: Failure during internal factorization of real banded matrix. Please contact NAG.

$IFAIL = 15$: Failure during internal solution of real banded system. Please contact NAG.

$IFAIL = 16$: The maximum number of iterations has been reached. Some Ritz values may have converged; NCONV returns the number of converged values.

$IFAIL = 17$: No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration. One possibility is to increase the size of NCV relative to NEV (see Section 5 in F12FFF for details of these parameters).

$IFAIL = 18$: An unexpected error has occurred. Please contact NAG.

$IFAIL = 19$: The routine was unable to dynamically allocate sufficient internal workspace. Please contact NAG.

7 Accuracy

The relative accuracy of a Ritz value,

λ

, is considered acceptable if its Ritz estimate

\leq Tolerance \times |λ|

. The default Tolerance used is the machine precision given by X02AJF.

8 Further Comments

None.

9 Example

This example solves

A x = λ x

in regular mode, where

A

is obtained from the standard central difference discretization of the two-dimensional convection-diffusion operator

\frac{d^{2} u}{d x^{2}} + \frac{d^{2} u}{d y^{2}} = ρ \frac{d u}{d x}

on the unit square with zero Dirichlet boundary conditions.

A

is stored in LAPACK banded storage format.

NAG Library Routine DocumentF12FGF

+− 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

NAG Library Routine Document

F12FGF