NAG Library Routine Document

f12auf (complex_band_solve)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{kl}$ – IntegerInput

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

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

2:     $\mathbf{ku}$ – IntegerInput

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

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

3:     $\mathbf{ab}\left(\mathbf{ldab},*\right)$ – Complex (Kind=nag_wp) arrayInput

Note: the second dimension of the array ab must be at least $\max \left(1,\mathbf{n}\right)$ (see f12atf).

On entry: must contain the matrix $A$ in LAPACK banded storage format for non-Hermitian matrices (see Section 3.3.4 in the F07 Chapter Introduction).

4:     $\mathbf{ldab}$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f12auf is called.

Constraint: $\mathbf{ldab}\ge 2\times \mathbf{kl}+\mathbf{ku}+1$.

5:     $\mathbf{mb}\left(\mathbf{ldmb},*\right)$ – Complex (Kind=nag_wp) arrayInput

Note: the second dimension of the array mb must be at least $\max \left(1,\mathbf{n}\right)$ (see f12atf).

On entry: must contain the matrix $B$ in LAPACK banded storage format for non-Hermitian matrices (see Section 3.3.4 in the F07 Chapter Introduction).

6:     $\mathbf{ldmb}$ – IntegerInput

On entry: the first dimension of the array mb as declared in the (sub)program from which f12auf is called.

Constraint: $\mathbf{ldmb}\ge 2\times \mathbf{kl}+\mathbf{ku}+1$.

7:     $\mathbf{sigma}$ – Complex (Kind=nag_wp)Input

On entry: if the Shifted Inverse mode (see f12arf) has been selected then sigma must contain the shift used; otherwise sigma is not referenced. Section 4.2 in the F12 Chapter Introduction describes the use of shift and invert transformations.

8:     $\mathbf{nconv}$ – IntegerOutput

On exit: the number of converged eigenvalues.

9:     $\mathbf{d}\left(*\right)$ – Complex (Kind=nag_wp) arrayOutput

Note: the dimension of the array d must be at least $\mathbf{nev}$ (see f12atf).

On exit: the first nconv locations of the array d contain the converged approximate eigenvalues.

10:   $\mathbf{z}\left(\mathbf{ldz},*\right)$ – Complex (Kind=nag_wp) arrayOutput

Note: the second dimension of the array z must be at least $\mathbf{nev}$ 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 f12arf and f12atf).

On exit: if the default option $\mathbf{Vectors}=\mathrm{RITZ}$ (see f12arf) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d, otherwise z is not referenced. The complex eigenvector associated with an eigenvalue $\mathbf{d}\left(j\right)$ is stored in the corresponding array section of z, namely $\mathbf{z}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$ and $\mathit{j}=1,2,\dots ,\mathbf{nconv}$.

11:   $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f12auf 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, $\mathbf{ldz}\ge 1$.

12:   $\mathbf{resid}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array resid must be at least $\mathbf{n}$ (see f12atf).

On entry: need not be set unless the option Initial Residual has been set in a prior call to f12arf in which case resid must contain an initial residual vector.

On exit: contains the final residual vector. This can be used as the starting residual to improve convergence on the solution of a closely related eigenproblem. This has no relation to the error residual $Ax-\lambda x$ or $Ax-\lambda Bx$.

13:   $\mathbf{v}\left(\mathbf{ldv},*\right)$ – Complex (Kind=nag_wp) arrayOutput

Note: the second dimension of the array v must be at least $\max \left(1,\mathbf{ncv}\right)$ (see f12atf).

On exit: if the option $\mathbf{Vectors}=\mathrm{SCHUR}$ or $\mathrm{RITZ}$ (see f12arf) has been set and a separate array z has been passed (i.e., z does not equal v), then the first nconv columns 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:   $\mathbf{ldv}$ – IntegerInput

On entry: the first dimension of the array v as declared in the (sub)program from which f12auf is called.

Constraint: $\mathbf{ldv}\ge \mathbf{n}$.

15:   $\mathbf{comm}\left(60\right)$ – Complex (Kind=nag_wp) arrayCommunication Array

On entry: must remain unchanged from the prior call to f12arf and f12atf.

16:   $\mathbf{icomm}\left(140\right)$ – Integer arrayCommunication Array

On entry: must remain unchanged from the prior call to f12arf and f12atf.

17:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, $\mathbf{kl}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{kl}\ge 0$.

$\mathbf{ifail}=2$

On entry, $\mathbf{ku}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ku}\ge 0$.

$\mathbf{ifail}=3$

On entry, $\mathbf{ldab}=〈\mathit{\text{value}}〉$, $2\times \mathbf{kl}+\mathbf{ku}+1=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ldab}\ge 2\times \mathbf{kl}+\mathbf{ku}+1$.

$\mathbf{ifail}=5$

The maximum number of iterations $\le 0$, the option Iteration Limit has been set to $〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=6$

The options Generalized and Regular are incompatible.

$\mathbf{ifail}=7$

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

$\mathbf{ifail}=8$

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

$\mathbf{ifail}=9$

On entry, $\mathbf{ldz}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ in f12aff.
Constraint: $\mathbf{ldz}\ge \mathbf{n}$.

$\mathbf{ifail}=10$

On entry, $\mathbf{Vectors}=\text{Select}$, 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. The size of the current Arnoldi factorization $=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=13$

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

$\mathbf{ifail}=14$

During calculation of a Schur form, there was a failure to compute a number of eigenvalues Please contact NAG.

$\mathbf{ifail}=15$

The computed Schur form could not be reordered by an internal call. Please contact NAG.

$\mathbf{ifail}=16$

Error in internal call to compute 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 matrix. 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 matrix. Please contact NAG.

$\mathbf{ifail}=21$

The maximum number of iterations has been reached. The maximum number of $\text{iterations}=〈\mathit{\text{value}}〉$. The number of converged eigenvalues $=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=22$

No shifts could be applied during a cycle of the implicitly restarted Arnoldi iteration.

$\mathbf{ifail}=23$

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

$\mathbf{ifail}=-99$

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

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

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

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f12aufe.f90)

10.2

Program Data

Program Data (f12aufe.d)

10.3

Program Results

Program Results (f12aufe.r)