NAG Library Function Document

nag_real_banded_sparse_eigensystem_sol (f12agc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

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[$\mathit{dim}$] – const doubleInput

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 nag_real_banded_sparse_eigensystem_init (f12afc)).

On entry: must contain the matrix $A$ in LAPACK column-ordered banded storage format for nonsymmetric matrices (see Section 3.3.4 in the f07 Chapter Introduction).

4:     mb[$\mathit{dim}$] – const doubleInput

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 nag_real_banded_sparse_eigensystem_init (f12afc)).

On entry: must contain the matrix $B$ in LAPACK column-ordered banded storage format for nonsymmetric matrices (see Section 3.3.4 in the f07 Chapter Introduction).

5:     sigmar – doubleInput

On entry: if one of the $\mathbf{Shifted\; Inverse\; Real}$ modes (see nag_real_sparse_eigensystem_option (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:     sigmai – doubleInput

On entry: if one of the $\mathbf{Shifted\; Inverse\; Real}$ modes (see nag_real_sparse_eigensystem_option (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:     nconv – Integer *Output

On exit: the number of converged eigenvalues.

8:     dr[$\mathit{dim}$] – doubleOutput

Note: the dimension, dim, of the array dr must be at least $\mathbf{nev}+1$ (see nag_real_banded_sparse_eigensystem_init (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:     di[$\mathit{dim}$] – doubleOutput

Note: the dimension, dim, of the array di must be at least $\mathbf{nev}+1$ (see nag_real_banded_sparse_eigensystem_init (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:   z[$\mathit{dim}$] – doubleOutput

Note: the dimension, dim, of the array z must be at least $\mathbf{n}\times \left(\mathbf{nev}+1\right)$ (see nag_real_banded_sparse_eigensystem_init (f12afc)).

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:   resid[$\mathit{dim}$] – doubleInput/Output

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

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

On exit: contains the final residual vector.

12:   v[$\mathit{dim}$] – doubleOutput

Note: the dimension, dim, of the array v must be at least $\max \left(1,\mathbf{n}\times \mathbf{ncv}\right)$ (see nag_real_banded_sparse_eigensystem_init (f12afc)).

On exit: if the option $\mathbf{Vectors}$ (see nag_real_sparse_eigensystem_option (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:   comm[$\mathit{dim}$] – doubleCommunication Array

Note: the dimension, dim, of the array comm must be at least $\max \left(1,\mathbf{lcomm}\right)$ (see nag_real_banded_sparse_eigensystem_init (f12afc)).

On entry: must remain unchanged from the prior call to nag_real_sparse_eigensystem_option (f12adc) and nag_real_banded_sparse_eigensystem_init (f12afc).

On exit: contains no useful information.

14:   icomm[$\mathit{dim}$] – IntegerCommunication Array

Note: the dimension, dim, of the array icomm must be at least $\max \left(1,\mathbf{licomm}\right)$ (see nag_real_banded_sparse_eigensystem_init (f12afc)).

On entry: must remain unchanged from the prior call to nag_real_sparse_eigensystem_option (f12adc) and nag_real_banded_sparse_eigensystem_init (f12afc).

On exit: contains no useful information.

15:   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.

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.

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_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

Program Text (f12agce.c)

10.2  Program Data

Program Data (f12agce.d)

10.3  Program Results

Program Results (f12agce.r)