NAG Library Function Document

nag_complex_banded_eigensystem_solve (f12auc)

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:    $\mathbf{ab}\left[\mathit{dim}\right]$ – const ComplexInput

Note: the dimension, dim, of the array ab must be at least $\mathbf{n}\times \left(2\times \mathbf{kl}+\mathbf{ku}+1\right)$ (see nag_complex_banded_eigensystem_init (f12atc)).

On entry: must contain the matrix $A$ in LAPACK column-ordered banded storage format for non-Hermitian matrices; that is, element $a_{ij}$ is stored in $\mathbf{ab}\left[\left(j-1\right)\times \left(2\times \mathbf{kl}+\mathbf{ku}+1\right)+\mathbf{kl}+\mathbf{ku}+i-j\right]$, which may be written as $\mathbf{ab}\left[\left(2\times j-1\right)\times \mathbf{kl}+j\times \mathbf{ku}+i-1\right]$, for $\max \left(1,j-\mathbf{ku}\right)\le i\le \min \left(n,j+\mathbf{kl}\right)$ and $j=1,2,\dots ,n$, (see Section 3.3.4 in the f07 Chapter Introduction).

4:    $\mathbf{mb}\left[\mathit{dim}\right]$ – const ComplexInput

Note: the dimension, dim, of the array mb must be at least $\mathbf{n}\times \left(2\times \mathbf{kl}+\mathbf{ku}+1\right)$ (see nag_complex_banded_eigensystem_init (f12atc)).

On entry: must contain the matrix $B$ in LAPACK column-ordered banded storage format for non-Hermitian matrices; that is, element $a_{ij}$ is stored in $\mathbf{mb}\left[\left(j-1\right)\times \left(2\times \mathbf{kl}+\mathbf{ku}+1\right)+\mathbf{kl}+\mathbf{ku}+i-j\right]$, which may be written as $\mathbf{mb}\left[\left(2\times j-1\right)\times \mathbf{kl}+j\times \mathbf{ku}+i-1\right]$, for $\max \left(1,j-\mathbf{ku}\right)\le i\le \min \left(n,j+\mathbf{kl}\right)$ and $j=1,2,\dots ,n$, (see Section 3.3.4 in the f07 Chapter Introduction).

5:    $\mathbf{sigma}$ – ComplexInput

On entry: if the $\mathbf{Shifted\; Inverse}$ mode (see nag_complex_sparse_eigensystem_option (f12arc)) 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.

6:    $\mathbf{nconv}$ – Integer *Output

On exit: the number of converged eigenvalues.

7:    $\mathbf{d}\left[\mathbf{nev}\right]$ – ComplexOutput

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

8:    $\mathbf{z}\left[\mathit{dim}\right]$ – ComplexOutput

Note: the dimension, dim, of the array z must be at least $\mathbf{n}\times \mathbf{nev}$ if the default option $\mathbf{Vectors}=\mathrm{RITZ}$ (see nag_complex_sparse_eigensystem_option (f12arc)) has been selected (see nag_complex_banded_eigensystem_init (f12atc)).

On exit: if the default option $\mathbf{Vectors}=\mathrm{RITZ}$ (see nag_complex_sparse_eigensystem_option (f12arc)) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d, otherwise z is not referenced and may be NULL. The complex eigenvector associated with an eigenvalue $\mathbf{d}\left[j\right]$ is stored in the corresponding array section of z, namely $\mathbf{z}\left[\mathbf{n}\times \left(\mathit{j}-1\right)+\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$ and $\mathit{j}=1,2,\dots ,\mathbf{nconv}$.

9:    $\mathbf{resid}\left[\mathbf{n}\right]$ – ComplexInput/Output

On entry: need not be set unless the option $\mathbf{Initial\; Residual}$ has been set in a prior call to nag_complex_sparse_eigensystem_option (f12arc) 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 $Ax-\lambda Bx$.

10:  $\mathbf{v}\left[\mathbf{n}\times \mathbf{ncv}\right]$ – ComplexOutput

On exit: if the option $\mathbf{Vectors}=\mathrm{SCHUR}$ or $\mathrm{RITZ}$ (see nag_complex_sparse_eigensystem_option (f12arc)) has been set and a separate array z has been passed (i.e., z does not equal v), then the first nconv sections of v, of length $n$, will contain approximate Schur vectors that span the desired invariant subspace.

The $j$th Schur vector is stored in locations $\mathbf{v}\left[\mathbf{n}\times \left(\mathit{j}-1\right)+\mathit{i}-1\right]$, for $\mathit{j}=1,2,\dots ,\mathbf{nconv}$ and $\mathit{i}=1,2,\dots ,n$.

11:  $\mathbf{comm}\left[60\right]$ – ComplexCommunication Array

On entry: must remain unchanged from the prior call to nag_complex_sparse_eigensystem_option (f12arc) and nag_complex_banded_eigensystem_init (f12atc).

12:  $\mathbf{icomm}\left[140\right]$ – IntegerCommunication Array

On entry: must remain unchanged from the prior call to nag_complex_sparse_eigensystem_option (f12arc) and nag_complex_banded_eigensystem_init (f12atc).

13:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_COMP_BAND_FAC

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

NE_COMP_BAND_SOL

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

NE_EIGENVALUES

The number of eigenvalues found to sufficient accuracy is zero.

NE_INITIALIZATION

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

NE_INT

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

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

NE_INTERNAL_EIGVAL_FAIL

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

NE_INTERNAL_EIGVEC_FAIL

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

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.

NE_INVALID_OPTION

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

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

NE_NO_ARNOLDI_FAC

Could not build an Arnoldi factorization. The size of the current Arnoldi factorization $=〈\mathit{\text{value}}〉$.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

NE_NO_SHIFTS_APPLIED

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

NE_OPT_INCOMPAT

The options $\mathbf{Generalized}$ and $\mathbf{Regular}$ are incompatible.

NE_OVERFLOW

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

NE_REAL_BAND_FAC

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

NE_REAL_BAND_SOL

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

NE_SCHUR_EIG_FAIL

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

NE_SCHUR_REORDER

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

NE_TOO_MANY_ITER

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

NE_ZERO_RESID

The option $\mathbf{Initial\; Residual}$ was selected but the starting vector held in resid is zero.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f12auce.c)

10.2  Program Data

Program Data (f12auce.d)

10.3  Program Results

Program Results (f12auce.r)