NAG Library Function Document

nag_real_symm_sparse_eigensystem_iter (f12fbc)

Note: this function uses optional arguments to define choices in the problem specification. If you wish to use default settings for all of the optional arguments, then the option setting function nag_real_symm_sparse_eigensystem_option (f12fdc) need not be called. If, however, you wish to reset some or all of the settings please refer to Section 11 in nag_real_symm_sparse_eigensystem_option (f12fdc) for a detailed description of the specification of the optional arguments.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_real_symm_sparse_eigensystem_iter (f12fbc) is an iterative solver in a suite of functions consisting of nag_real_symm_sparse_eigensystem_init (f12fac), nag_real_symm_sparse_eigensystem_iter (f12fbc), nag_real_symm_sparse_eigensystem_sol (f12fcc), nag_real_symm_sparse_eigensystem_option (f12fdc) and nag_real_symm_sparse_eigensystem_monit (f12fec). It is used to find some of the eigenvalues (and optionally the corresponding eigenvectors) of a standard or generalized eigenvalue problem defined by real symmetric matrices.

2 Specification

#include <nag.h>

#include <nagf12.h>

void	nag_real_symm_sparse_eigensystem_iter (Integer irevcm, double resid[], double v[], double x, double y, double mx, Integer nshift, double comm[], Integer icomm[], NagError *fail)

3 Description

The suite of functions is designed to calculate some of the eigenvalues,

λ

, (and optionally the corresponding eigenvectors,

x

) of a standard eigenvalue problem

A x = λ x

, or of a generalized eigenvalue problem

A x = λ B x

of order

n

, where

n

is large and the coefficient matrices

A

and

B

are sparse, real and symmetric. The suite can also be used to find selected eigenvalues/eigenvectors of smaller scale dense, real and symmetric problems.

nag_real_symm_sparse_eigensystem_iter (f12fbc) is a reverse communication function, based on the ARPACK routine dsaupd, using the Implicitly Restarted Arnoldi iteration method, which for symmetric problems reduces to a variant of the Lanczos method. The method is described in Lehoucq and Sorensen (1996) and Lehoucq (2001) while its use within the ARPACK software is described in great detail in Lehoucq et al. (1998). An evaluation of software for computing eigenvalues of sparse symmetric matrices is provided in Lehoucq and Scott (1996). This suite of functions offers the same functionality as the ARPACK software for real symmetric problems, but the interface design is quite different in order to make the option setting clearer and to simplify the interface of nag_real_symm_sparse_eigensystem_iter (f12fbc).

The setup function nag_real_symm_sparse_eigensystem_init (f12fac) must be called before nag_real_symm_sparse_eigensystem_iter (f12fbc), the reverse communication iterative solver. Options may be set for nag_real_symm_sparse_eigensystem_iter (f12fbc) by prior calls to the option setting function nag_real_symm_sparse_eigensystem_option (f12fdc) and a post-processing function nag_real_symm_sparse_eigensystem_sol (f12fcc) must be called following a successful final exit from nag_real_symm_sparse_eigensystem_iter (f12fbc). nag_real_symm_sparse_eigensystem_monit (f12fec), may be called following certain flagged, intermediate exits from nag_real_symm_sparse_eigensystem_iter (f12fbc) to provide additional monitoring information about the computation.

nag_real_symm_sparse_eigensystem_iter (f12fbc) uses reverse communication, i.e., it returns repeatedly to the calling program with the argument irevcm (see Section 5) set to specified values which require the calling program to carry out one of the following tasks:

–	compute the matrix-vector product $y = OP x$ , where $OP$ is defined by the computational mode;
–	compute the matrix-vector product $y = B x$ ;
–	notify the completion of the computation;
–	allow the calling program to monitor the solution.

The problem type to be solved (standard or generalized), the spectrum of eigenvalues of interest, the mode used (regular, regular inverse, shifted inverse, Buckling or Cayley) and other options can all be set using the option setting function nag_real_symm_sparse_eigensystem_option (f12fdc).

4 References

Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications 23 551–562

Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory

Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications 17 789–821

Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philidelphia

5 Arguments

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than x and y must remain unchanged.

1: irevcm – Integer *Input/Output

On initial entry:

irevcm = 0

, otherwise an error condition will be raised.

On intermediate re-entry: must be unchanged from its previous exit value. Changing irevcm to any other value between calls will result in an error.

On intermediate exit: has the following meanings.

$irevcm = -1$: The calling program must compute the matrix-vector product $y = OP x$ , where $x$ is stored in x and the result $y$ is placed in y.
$irevcm = 1$: The calling program must compute the matrix-vector product $y = OP x$ . This is similar to the case $irevcm = -1$ except that the result of the matrix-vector product $B x$ (as required in some computational modes) has already been computed and is available in mx.
$irevcm = 2$: The calling program must compute the matrix-vector product $y = B x$ , where $x$ is stored in x and $y$ is placed in y.
$irevcm = 3$: Compute the nshift real and imaginary parts of the shifts where the real parts are to be placed in the first nshift locations of the array y and the imaginary parts are to be placed in the first nshift locations of the array mx. Only complex conjugate pairs of shifts may be applied and the pairs must be placed in consecutive locations. This value of irevcm will only arise if the optional argument $Supplied Shifts$ is set in a prior call to nag_real_symm_sparse_eigensystem_option (f12fdc) which is intended for experienced users only; the default and recommended option is to use exact shifts (see Lehoucq et al. (1998) for details and guidance on the choice of shift strategies).
$irevcm = 4$: Monitoring step: a call to nag_real_symm_sparse_eigensystem_monit (f12fec) can now be made to return the number of Arnoldi iterations, the number of converged Ritz values, their real and imaginary parts, and the corresponding Ritz estimates.

On final exit:

irevcm = 5

: nag_real_symm_sparse_eigensystem_iter (f12fbc) has completed its tasks. The value of fail determines whether the iteration has been successfully completed, or whether errors have been detected. On successful completion nag_real_symm_sparse_eigensystem_sol (f12fcc) must be called to return the requested eigenvalues and eigenvectors (and/or Schur vectors).

Constraint: on initial entry,

irevcm = 0

; on re-entry irevcm must remain unchanged.

2: resid[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array resid must be at least

n

(see nag_real_symm_sparse_eigensystem_init (f12fac)).

On initial entry: need not be set unless the option

Initial Residual

has been set in a prior call to nag_real_symm_sparse_eigensystem_option (f12fdc) in which case resid should contain an initial residual vector, possibly from a previous run.

On intermediate re-entry: must be unchanged from its previous exit. Changing resid to any other value between calls may result in an error exit.

On intermediate exit: contains the current residual vector.

On final exit: contains the final residual vector.

3: v[ $n \times ncv$ ] – doubleInput/Output

The

i

th element of the

j

th basis vector is stored in location

v [n \times (i - 1) + j - 1]

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, ncv

On initial entry: need not be set.

On intermediate re-entry: must be unchanged from its previous exit.

On intermediate exit: contains the current set of Arnoldi basis vectors.

On final exit: contains the final set of Arnoldi basis vectors.

4: x – double **Input/Output

On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of comm.

On intermediate re-entry: is not normally changed.

On intermediate exit: contains the vector

x

when irevcm returns the value

- 1

+ 1

2

On final exit: does not contain useful data.

5: y – double **Input/Output

On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of comm.

On intermediate re-entry: must contain the result of

y = OP x

when irevcm returns the value

- 1

+ 1

. It must contain the real parts of the computed shifts when irevcm returns the value

3

On intermediate exit: does not contain useful data.

On final exit: does not contain useful data.

6: mx – double **Input/Output

On initial entry: need not be set, it is used as a convenient mechanism for accessing elements of comm.

On intermediate re-entry: it must contain the imaginary parts of the computed shifts when irevcm returns the value

3

On intermediate exit: contains the vector

B x

when irevcm returns the value

+ 1

On final exit: does not contain any useful data.

7: nshift – Integer *Output

On intermediate exit: if the option

Supplied Shifts

is set and irevcm returns a value of

3

, nshift returns the number of complex shifts required.

8: comm[ $\dim$ ] – doubleCommunication Array

Note: the dimension, dim, of the array comm must be at least

\max (1, lcomm)

(see nag_real_symm_sparse_eigensystem_init (f12fac)).

On initial entry: must remain unchanged following a call to the setup function nag_real_symm_sparse_eigensystem_init (f12fac).

On exit: contains data defining the current state of the iterative process.

9: icomm[ $\dim$ ] – IntegerCommunication Array

Note: the dimension, dim, of the array icomm must be at least

\max (1, licomm)

(see nag_real_symm_sparse_eigensystem_init (f12fac)).

On initial entry: must remain unchanged following a call to the setup function nag_real_symm_sparse_eigensystem_init (f12fac).

On exit: contains data defining the current state of the iterative process.

10: 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 $⟨value⟩$ had an illegal value.
NE_BOTH_ENDS_1: Eigenvalues from both ends of the spectrum were requested, but the number of eigenvalues (see nev in nag_real_symm_sparse_eigensystem_init (f12fac)) requested is one.
NE_INT: The maximum number of iterations $\leq 0$ , the option $Iteration Limit$ has been set to $⟨value⟩$ .
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_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_MAX_ITER: The maximum number of iterations has been reached. The maximum number of $iterations = ⟨value⟩$ . The number of converged eigenvalues $= ⟨value⟩$ . The post-processing function nag_real_symm_sparse_eigensystem_sol (f12fcc) may be called to recover the converged eigenvalues at this point. Alternatively, the maximum number of iterations may be increased by a call to the option setting function nag_real_symm_sparse_eigensystem_option (f12fdc) and the reverse communication loop restarted. A large number of iterations may indicate a poor choice for the values of nev and ncv; it is advisable to experiment with these values to reduce the number of iterations (see nag_real_symm_sparse_eigensystem_init (f12fac)).
NE_NO_LANCZOS_FAC: Could not build a Lanczos factorization. The size of the current Lanczos factorization $= ⟨value⟩$ .
NE_NO_SHIFTS_APPLIED: No shifts could be applied during a cycle of the implicitly restarted Lanczos iteration.
NE_OPT_INCOMPAT: The options $Generalized$ and $Regular$ are incompatible.
NE_ZERO_INIT_RESID: The option $Initial Residual$ was selected but the starting vector held in resid is zero.

7 Accuracy

The relative accuracy of a Ritz value,

λ

, is considered acceptable if its Ritz estimate

\leq Tolerance \times |λ|

. The default

Tolerance

used is the machine precision given by nag_machine_precision (X02AJC).

8 Parallelism and Performance

nag_real_symm_sparse_eigensystem_iter (f12fbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_real_symm_sparse_eigensystem_iter (f12fbc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

For this function two examples are presented, with a main program and two example problems given in Example 1 (ex1) and Example 2 (ex2).

Example 1 (ex1)

The example solves

A x = λ x

in shift-invert mode, where

A

is obtained from the standard central difference discretization of the one-dimensional Laplacian operator

\frac{\partial^{2} u}{\partial x^{2}}

with zero Dirichlet boundary conditions. Eigenvalues closest to the shift

σ = 0

are sought.

Example 2 (ex2)

This example illustrates the use of nag_real_symm_sparse_eigensystem_iter (f12fbc) to compute the leading terms in the singular value decomposition of a real general matrix

A

. The example finds a few of the largest singular values (

σ

) and corresponding right singular values (

ν

) for the matrix

A

by solving the symmetric problem:

(A^{T} A) ν = σ ν .

Here

A

is the

m

n

real matrix derived from the simplest finite difference discretization of the two-dimensional kernal

k (s, t) d t

where

k (s, t) = \{\begin{matrix} s (t - 1) & if ​ 0 \leq s \leq t \leq 1 \\ t (s - 1) & if ​ 0 \leq t < s \leq 1 \end{matrix} .

Note: this formulation is appropriate for the case

m \geq n

. Reverse the rules of

A

and

A^{T}

in the case of

m < n

NAG Library Function Documentnag_real_symm_sparse_eigensystem_iter (f12fbc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_real_symm_sparse_eigensystem_iter (f12fbc)