NAG CL Interface
f12agc (real_​band_​solve)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{kl}$ – Integer Input

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

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

2: $\mathbf{ku}$ – Integer Input

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 double Input

Note: the dimension, dim, of the array ab must be at least $\max \left(1,\mathbf{n}\times \left(2\times \mathbf{kl}+\mathbf{ku}+1\right)\right)$ (see f12afc).

On entry: must contain the matrix $A$ in LAPACK column-ordered banded storage format for nonsymmetric matrices (see Section 3.4.4 in the F07 Chapter Introduction).

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

Note: the dimension, dim, of the array mb must be at least $\max \left(1,\mathbf{n}\times \left(2\times \mathbf{kl}+\mathbf{ku}+1\right)\right)$ (see f12afc).

On entry: must contain the matrix $B$ in LAPACK column-ordered banded storage format for nonsymmetric matrices (see Section 3.4.4 in the F07 Chapter Introduction).

5: $\mathbf{sigmar}$ – double Input

On entry: if one of the $\mathbf{Shifted\; Inverse\; Real}$ modes (see f12adc) 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.

6: $\mathbf{sigmai}$ – double Input

On entry: if one of the $\mathbf{Shifted\; Inverse\; Real}$ modes (see f12adc) 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.

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

On exit: the number of converged eigenvalues.

8: $\mathbf{dr}\left[\mathit{dim}\right]$ – double Output

Note: the dimension, dim, of the array dr must be at least $\mathbf{nev}+1$ (see f12afc).

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.

9: $\mathbf{di}\left[\mathit{dim}\right]$ – double Output

Note: the dimension, dim, of the array di must be at least $\mathbf{nev}+1$ (see f12afc).

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.

10: $\mathbf{z}\left[\mathbf{n}\times \mathbf{nev}\right]$ – double Output

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 array segments. The first segment holds the real part of the eigenvector and the second 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.

For example, if $\mathbf{di}\left[0\right]$ is nonzero, the first eigenvector has real parts stored in locations $\mathbf{z}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{n}-1$ and imaginary parts stored in $\mathbf{z}\left[\mathit{i}\right]$, for $\mathit{i}=\mathbf{n},\dots ,2\mathbf{n}-1$.

11: $\mathbf{resid}\left[\mathit{dim}\right]$ – double Input/Output

Note: the dimension, dim, of the array resid must be at least $\mathbf{n}$ (see f12afc).

On entry: need not be set unless the option $\mathbf{Initial\; Residual}$ has been set in a prior call to f12adc in which case resid must contain an initial residual vector.

On exit: contains the final residual vector.

12: $\mathbf{v}\left[\mathbf{n}\times \mathbf{ncv}\right]$ – double Output

On exit: if the option $\mathbf{Vectors}$ (see f12adc) has been set to Schur or Ritz then the first nconv sections of v, of length $n$, will contain approximate Schur vectors that span the desired invariant subspace.

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

13: $\mathbf{comm}\left[\mathit{dim}\right]$ – double Communication Array

Note: the actual argument supplied must be the array comm supplied to the initialization routine f12afc.

On entry: must remain unchanged from the prior call to f12adc and f12afc.

On exit: contains no useful information.

14: $\mathbf{icomm}\left[\mathit{dim}\right]$ – Integer Communication Array

Note: the actual argument supplied must be the array icomm supplied to the initialization routine f12afc.

On entry: must remain unchanged from the prior call to f12adc and f12afc.

On exit: contains no useful information.

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

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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_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: $\mathbf{ku}\ge 0$.

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

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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_INVALID_OPTION

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

NE_MAX_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_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 8 in the Introduction to the NAG Library CL Interface for further information.

NE_NO_SHIFTS_APPLIED

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

NE_OPT_INCOMPAT

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

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 real 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_TRANSFORM_OVFL

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

NE_ZERO_EIGS_FOUND

The number of eigenvalues found to sufficient accuracy is zero.

NE_ZERO_INIT_RESID

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

NE_ZERO_SHIFT

The option $\mathbf{Shifted\; Inverse\; Imaginary}$ has been selected and $\mathbf{sigmai}=$ zero on entry; sigmai must be nonzero for this mode of operation.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results