NAG Library Routine Document

F12AGF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     KL – INTEGERInput

On entry: the number of subdiagonals of the matrices $A$ and $B$.

Constraint: $\mathbf{KL}\ge 0$.

2:     KU – INTEGERInput

On entry: the number of superdiagonals of the matrices $A$ and $B$.

Constraint: $\mathbf{KU}\ge 0$.

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

Note: the second dimension of the array AB must be at least $\max \left(1,\mathbf{N}\right)$ (see F12AFF).

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 F12AGF is called.

Constraint: $\mathbf{LDAB}\ge 2\times \mathbf{KL}+\mathbf{KU}+1$.

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

Note: the second dimension of the array MB must be at least $\max \left(1,\mathbf{N}\right)$ (see F12AFF).

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 F12AGF is called.

Constraint: $\mathbf{LDMB}\ge 2\times \mathbf{KL}+\mathbf{KU}+1$.

7:     SIGMAR – REAL (KIND=nag_wp)Input

On entry: if one of the Shifted Inverse Real modes (see F12ADF) have been selected then SIGMAR must contain the real part of the shift used; otherwise SIGMAR is not referenced. Section 4.3.4 in the F12 Chapter Introduction describes the use of shift and inverse transformations.

8:     SIGMAI – REAL (KIND=nag_wp)Input

On entry: if one of the Shifted Inverse Real modes (see F12ADF) have been selected then SIGMAI must contain the imaginary part of the shift used; otherwise SIGMAI is not referenced. Section 4.3.4 in the F12 Chapter Introduction describes the use of shift and inverse transformations.

9:     NCONV – INTEGEROutput

On exit: the number of converged eigenvalues.

10:   DR($*$) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array DR must be at least $\mathbf{NEV}+1$ (see F12AFF).

On exit: the first NCONV locations of the array DR contain the real parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by NEV since complex values occur as conjugate pairs and the second in the pair can be returned in position $\mathbf{NEV}+1$ of the array.

11:   DI($*$) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array DI must be at least $\mathbf{NEV}+1$ (see F12AFF).

On exit: the first NCONV locations of the array DI contain the imaginary parts of the converged approximate eigenvalues. The number of eigenvalues returned may be one more than the number requested by NEV since complex values occur as conjugate pairs and the second in the pair can be returned in position $\mathbf{NEV}+1$ of the array.

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

Note: the second dimension of the array Z must be at least $\mathbf{NEV}+1$ if the default option $\mathbf{Vectors}=\text{Ritz}$ has been selected and at least $1$ if the option $\mathbf{Vectors}=\text{None or Schur}$ has been selected (see F12AAF).

On exit: if the default option $\mathbf{Vectors}=\text{Ritz}$ has been selected then Z contains the final set of eigenvectors corresponding to the eigenvalues held in DR and DI. The complex eigenvector associated with the eigenvalue with positive imaginary part is stored in two consecutive columns. The first column holds the real part of the eigenvector and the second column holds the imaginary part. The eigenvector associated with the eigenvalue with negative imaginary part is simply the complex conjugate of the eigenvector associated with the positive imaginary part.

13:   LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F12AGF is called.

Constraints:

• if the default option $\mathbf{Vectors}=\text{Ritz}$ has been selected, $\mathbf{LDZ}\ge \mathbf{N}$;
• if the option $\mathbf{Vectors}=\text{None or Schur}$ has been selected, $\mathbf{LDZ}\ge 1$.

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

Note: the dimension of the array RESID must be at least $\mathbf{N}$ (see F12AFF).

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

On exit: contains the final residual vector.

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

Note: the second dimension of the array V must be at least $\max \left(1,\mathbf{NCV}\right)$ (see F12AFF).

On exit: if the option Vectors (see F12ADF) has been set to Schur or Ritz then the first $\mathbf{NCONV}\times n$ elements of V will contain approximate Schur vectors that span the desired invariant subspace.

The $i$th Schur vector is stored in the $i$th column of V.

16:   LDV – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F12AGF is called.

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

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

On entry: must remain unchanged from the prior call to F12ADF and F12AFF.

On exit: contains no useful information.

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

On entry: must remain unchanged from the prior call to F12ADF and F12AFF.

On exit: contains no useful information.

19:   IFAIL – INTEGERInput/Output

On 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, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On 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 entry, $\mathbf{KL}<0$.

$\mathbf{IFAIL}=2$

On entry, $\mathbf{KU}<0$.

$\mathbf{IFAIL}=3$

On entry, $\mathbf{LDAB}<2\times \mathbf{KL}+\mathbf{KU}+1$.

$\mathbf{IFAIL}=4$

On entry, the option $\mathbf{Shifted\; Inverse\; Imaginary}$ was selected, and $\mathbf{SIGMAI}=\text{zero}$, but SIGMAI must be nonzero for this computational mode.

$\mathbf{IFAIL}=5$

$\mathbf{Iteration\; Limit}<0$.

$\mathbf{IFAIL}=6$

The options Generalized and Regular are incompatible.

$\mathbf{IFAIL}=7$

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

$\mathbf{IFAIL}=8$

Either the initialization routine F12AFF has not been called prior to the first call of this routine or a communication array has become corrupted.

$\mathbf{IFAIL}=9$

On entry, $\mathbf{LDZ}<\mathbf{N}$ or $\mathbf{LDZ}<1$ when no vectors are required.

$\mathbf{IFAIL}=10$

On entry, the option $\mathbf{Vectors}=\text{Select}$ was selected, but this is not yet implemented.

$\mathbf{IFAIL}=11$

The number of eigenvalues found to sufficient accuracy is zero.

$\mathbf{IFAIL}=12$

Could not build an Arnoldi factorization. Consider changing NCV or NEV in the initialization routine (see Section 5 in F12AFF for details of these parameters).

$\mathbf{IFAIL}=13$

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

$\mathbf{IFAIL}=14$

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

$\mathbf{IFAIL}=15$

Unexpected error: the computed Schur form could not be reordered by an internal call. Please contact NAG.

$\mathbf{IFAIL}=16$

Unexpected error in internal call while calculating eigenvectors. Please contact NAG.

$\mathbf{IFAIL}=17$

Failure during internal factorization of real banded matrix. Please contact NAG.

$\mathbf{IFAIL}=18$

Failure during internal solution of real banded system. Please contact NAG.

$\mathbf{IFAIL}=19$

Failure during internal factorization of complex banded matrix. Please contact NAG.

$\mathbf{IFAIL}=20$

Failure during internal solution of complex banded system. Please contact NAG.

$\mathbf{IFAIL}=21$

The maximum number of iterations has been reached. Some Ritz values may have converged; NCONV returns the number of converged values.

$\mathbf{IFAIL}=22$

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 F12AFF for details of these parameters).

$\mathbf{IFAIL}=23$

Overflow occurred during transformation of Ritz values to those of the original problem.

$\mathbf{IFAIL}=24$

The routine was unable to dynamically allocate sufficient internal workspace. Please contact NAG.

$\mathbf{IFAIL}=25$

An unexpected error has occurred. Please contact NAG.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f12agfe.f90)

9.2  Program Data

Program Data (f12agfe.d)

9.3  Program Results

Program Results (f12agfe.r)