NAG Library Routine Document

F12ABF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{IREVCM}$ – INTEGERInput/Output

On initial entry: $\mathbf{IREVCM}=0$, otherwise an error condition will be raised.

On intermediate re-entry: must be unchanged from its previous exit value. Changing IREVCM to any other value between calls will result in an error.

On intermediate exit: has the following meanings.

$\mathbf{IREVCM}=-1$

The calling program must compute the matrix-vector product $y=\mathrm{OP}x$, where $x$ is stored in X (by default) or in the array COMM (starting from the location given by the first element of ICOMM) when the option $\mathbf{Pointers}=\mathrm{YES}$ is set in a prior call to F12ADF. The result $y$ is returned in X (by default) or in the array COMM (starting from the location given by the second element of ICOMM) when the option $\mathbf{Pointers}=\mathrm{YES}$ is set. If $B$ is not symmetric semi-definite then the precomputed values in MX should not be used (see the explanation under $\mathbf{IREVCM}=2$).

$\mathbf{IREVCM}=1$

The calling program must compute the matrix-vector product $y=\mathrm{OP}x$. This is similar to the case $\mathbf{IREVCM}=-1$ except that the result of the matrix-vector product $Bx$ (as required in some computational modes) has already been computed and is available in MX (by default) or in the array COMM (starting from the location given by the third element of ICOMM) when the option $\mathbf{Pointers}=\mathrm{YES}$ is set.

$\mathbf{IREVCM}=2$

The calling program must compute the matrix-vector product $y=Bx$, where $x$ is stored as described in the case $\mathbf{IREVCM}=-1$ and $y$ is returned in the location described by the case $\mathbf{IREVCM}=1$. This computation is requested when solving the Generalized problem using either Shifted Inverse Imaginary or Shifted Inverse Real; in these cases $B$ is used as an inner-product space and requires that $B$ be symmetric semi-definite. If neither $A$ nor $B$ is symmetric semi-definite then the problem should be reformulated in a Standard form.

$\mathbf{IREVCM}=3$

Compute the NSHIFT real and imaginary parts of the shifts where the real parts are to be returned in the first NSHIFT locations of the array X and the imaginary parts are to be returned in the first NSHIFT locations of the array MX. Only complex conjugate pairs of shifts may be applied and the pairs must be placed in consecutive locations. This value of IREVCM will only arise if the optional parameter Supplied Shifts is set in a prior call to F12ADF which is intended for experienced users only; the default and recommended option is to use exact shifts (see Lehoucq et al. (1998) for details).

$\mathbf{IREVCM}=4$

Monitoring step: a call to F12AEF can now be made to return the number of Arnoldi iterations, the number of converged Ritz values, their real and imaginary parts, and the corresponding Ritz estimates.

On final exit: $\mathbf{IREVCM}=5$: F12ABF has completed its tasks. The value of IFAIL determines whether the iteration has been successfully completed, or whether errors have been detected. On successful completion F12ACF must be called to return the requested eigenvalues and eigenvectors (and/or Schur vectors).

Constraint: on initial entry, $\mathbf{IREVCM}=0$; on re-entry IREVCM must remain unchanged.

2:     $\mathrm{RESID}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array RESID must be at least $\mathbf{N}$ (see F12AAF).

On initial entry: need not be set unless the option Initial Residual has been set in a prior call to F12ADF in which case RESID should contain an initial residual vector, possibly from a previous run.

On intermediate re-entry: must be unchanged from its previous exit. Changing RESID to any other value between calls may result in an error exit.

On intermediate exit: contains the current residual vector.

On final exit: contains the final residual vector.

3:     $\mathrm{V}\left(\mathbf{LDV},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array V must be at least $\max \left(1,\mathbf{NCV}\right)$ (see F12AAF).

On initial entry: need not be set.

On intermediate re-entry: must be unchanged from its previous exit.

On intermediate exit: contains the current set of Arnoldi basis vectors.

On final exit: contains the final set of Arnoldi basis vectors.

4:     $\mathrm{LDV}$ – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F12ABF is called.

Constraint: $\mathbf{LDV}\ge \mathbf{N}$.

5:     $\mathrm{X}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array X must be at least $\mathbf{N}$ if $\mathbf{Pointers}=\mathrm{NO}$ (default) and at least $1$ if $\mathbf{Pointers}=\mathrm{YES}$ (see F12AAF).

On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of COMM.

On intermediate re-entry: if $\mathbf{Pointers}=\mathrm{YES}$, X need not be set.

If $\mathbf{Pointers}=\mathrm{NO}$, X must contain the result of $y=\mathrm{OP}x$ when IREVCM returns the value $-1$ or $+1$. It must return the real parts of the computed shifts when IREVCM returns the value $3$.

On intermediate exit: if $\mathbf{Pointers}=\mathrm{YES}$, X is not referenced.

If $\mathbf{Pointers}=\mathrm{NO}$, X contains the vector $x$ when IREVCM returns the value $-1$ or $+1$.

On final exit: does not contain useful data.

6:     $\mathrm{MX}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array MX must be at least $\mathbf{N}$ if $\mathbf{Pointers}=\mathrm{NO}$ (default) and at least $1$ if $\mathbf{Pointers}=\mathrm{YES}$ (see F12AAF).

On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of COMM.

On intermediate re-entry: if $\mathbf{Pointers}=\mathrm{YES}$, MX need not be set.

If $\mathbf{Pointers}=\mathrm{NO}$, MX must contain the result of $y=Bx$ when IREVCM returns the value $2$. It must return the imaginary parts of the computed shifts when IREVCM returns the value $3$.

On intermediate exit: if $\mathbf{Pointers}=\mathrm{YES}$, MX is not referenced.

If $\mathbf{Pointers}=\mathrm{NO}$, MX contains the vector $Bx$ when IREVCM returns the value $+1$.

On final exit: does not contain any useful data.

7:     $\mathrm{NSHIFT}$ – INTEGEROutput

On intermediate exit: if the option Supplied Shifts is set and IREVCM returns a value of $3$, NSHIFT returns the number of complex shifts required.

8:     $\mathrm{COMM}\left(*\right)$ – REAL (KIND=nag_wp) arrayCommunication Array

Note: the dimension of the array COMM must be at least $\max \left(1,\mathbf{LCOMM}\right)$ (see F12AAF).

On initial entry: must remain unchanged following a call to the setup routine F12AAF.

On exit: contains data defining the current state of the iterative process.

9:     $\mathrm{ICOMM}\left(*\right)$ – INTEGER arrayCommunication Array

Note: the dimension of the array ICOMM must be at least $\max \left(1,\mathbf{LICOMM}\right)$ (see F12AAF).

On initial entry: must remain unchanged following a call to the setup routine F12AAF.

On exit: contains data defining the current state of the iterative process.

10:   $\mathrm{IFAIL}$ – INTEGERInput/Output

On initial entry: IFAIL must be set to $0$, $-1\text{ or }1$. 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 $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if $\mathbf{IFAIL}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.

On final exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On initial entry, the maximum number of iterations $\le 0$, the option Iteration Limit has been set to a non-positive value.

$\mathbf{IFAIL}=2$

The options Generalized and Regular are incompatible.

$\mathbf{IFAIL}=3$

The option Initial Residual was selected but the starting vector held in RESID is zero.

$\mathbf{IFAIL}=4$

The maximum number of iterations has been reached. Some Ritz values may have converged; a subsequent call to F12ACF will return the number of converged values and the converged values.

$\mathbf{IFAIL}=5$

No shifts could be applied during a cycle of the implicitly restarted Arnoldi iteration. One possibility is to increase the size of NCV relative to NEV (see Section 5 in F12AAF for details of these parameters).

$\mathbf{IFAIL}=6$

Could not build an Arnoldi factorization. Consider changing NCV or NEV in the initialization routine (see Section 5 in F12AAF for details of these parameters).

$\mathbf{IFAIL}=7$

Unexpected error in internal call to compute eigenvalues and corresponding error bounds of the current upper Hessenberg matrix. Please contact NAG.

$\mathbf{IFAIL}=8$

Either the initialization routine F12AAF has not been called prior to the first call of this routine or a communication array has become corrupted.

$\mathbf{IFAIL}=9$

An unexpected error has occurred. Please contact NAG.

$\mathbf{IFAIL}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

$\mathbf{IFAIL}=-999$

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f12abfe.f90)

10.2  Program Data

Program Data (f12abfe.d)

10.3  Program Results

Program Results (f12abfe.r)