NAG Library Function Document

nag_real_symm_banded_sparse_eigensystem_sol (f12fgc)

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 $B$. Since $A$ and $B$ are symmetric, the normal case is $\mathbf{ku}=\mathbf{kl}$.

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

3:    $\mathbf{ab}\left[\mathit{dim}\right]$ – 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_symm_banded_sparse_eigensystem_init (f12ffc)).

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:    $\mathbf{mb}\left[\mathit{dim}\right]$ – 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_symm_banded_sparse_eigensystem_init (f12ffc)).

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:    $\mathbf{sigma}$ – doubleInput

On entry: if one of the $\mathbf{Shifted\; Inverse}$ (see nag_real_symm_sparse_eigensystem_option (f12fdc)) modes has been selected then sigma contains the real shift used; otherwise sigma is not referenced.

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

On exit: the number of converged eigenvalues.

7:    $\mathbf{d}\left[\mathit{dim}\right]$ – doubleOutput

Note: the dimension, dim, of the array d must be at least $\mathbf{ncv}$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).

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

8:    $\mathbf{z}\left[\mathbf{n}\times \left(\mathbf{nev}+1\right)\right]$ – doubleOutput

On exit: if the default option $\mathbf{Vectors}=\mathrm{RITZ}$ (see nag_real_symm_sparse_eigensystem_option (f12fdc)) has been selected then z contains the final set of eigenvectors corresponding to the eigenvalues held in d. The real eigenvector associated with eigenvalue $\mathit{i}-1$, for $\mathit{i}=1,2,\dots ,\mathbf{nconv}$, is stored at locations $\mathbf{z}\left[\mathit{i}-1\times n+\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,n$.

9:    $\mathbf{resid}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, of the array resid must be at least $\mathbf{n}$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).

On entry: need not be set unless the option $\mathbf{Initial\; Residual}$ has been set in a prior call to nag_real_symm_sparse_eigensystem_option (f12fdc) in which case resid must contain an initial residual vector.

On exit: contains the final residual vector.

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

On exit: if the option $\mathbf{Vectors}$ (see nag_real_symm_sparse_eigensystem_option (f12fdc)) has been set to Schur or Ritz and 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:  $\mathbf{comm}\left[\mathit{dim}\right]$ – doubleCommunication Array

Note: the dimension, dim, of the array comm must be at least $\max \left(1,\mathbf{lcomm}\right)$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).

On initial entry: must remain unchanged from the prior call to nag_real_symm_sparse_eigensystem_option (f12fdc) and nag_real_symm_banded_sparse_eigensystem_init (f12ffc).

On exit: contains no useful information.

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

Note: the dimension, dim, of the array icomm must be at least $\max \left(1,\mathbf{licomm}\right)$ (see nag_real_symm_banded_sparse_eigensystem_init (f12ffc)).

On initial entry: must remain unchanged from the prior call to nag_real_symm_sparse_eigensystem_iter (f12fbc) and nag_real_symm_sparse_eigensystem_option (f12fdc).

On exit: contains no useful information.

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

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

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

NE_BOTH_ENDS_1

Eigenvalues from both ends of the spectrum were requested, but the number of eigenvalues (nev in nag_real_symm_banded_sparse_eigensystem_init (f12ffc)) requested is one.

NE_INITIALIZATION

Either an initial call to the setup function has not been made or the communication arrays have 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_INT_2

The maximum number of iterations has been reached: there have been $〈\mathit{\text{value}}〉$ iterations. There are $〈\mathit{\text{value}}〉$ converged eigenvalues.

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_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 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_INVALID_OPTION

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

NE_MAX_ITER

During calculation of a tridiagonal form, there was a failure to compute $〈\mathit{\text{value}}〉$ eigenvalues in a total of $〈\mathit{\text{value}}〉$ iterations.

NE_NO_LANCZOS_FAC

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

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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 banded matrix. Please contact NAG.

NE_REAL_BAND_SOL

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

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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f12fgce.c)

10.2

Program Data

Program Data (f12fgce.d)

10.3

Program Results

Program Results (f12fgce.r)