CNAUPD(3) MathKeisan ARPACK routine CNAUPD(3)
NAME
CNAUPD - Reverse communication interface for the large-scale (sparse)
complex non-symmetric eigenvalue calculation.
SYNOPSIS
SUBROUTINE CNAUPD(IDO, BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, LDV,
IPARAM, IPNTR, WORKD, WORKL, LWORKL, RWORK, INFO )
INTEGER IDO, N, NEV, NCV, LDV, LWORKL, INFO
INTEGER IPARAM(11), IPNTR(14)
REAL TOL
COMPLEX RESID(N), V(N,NCV), WORKD(3*N), WORKL(LWORK)
REAL RWORK(NCV)
CHARACTER BMAT*1, WHICH*2
PURPOSE
CNAUPD is a reverse communication interface for the Implicitly Restarted
Arnoldi iteration. This is intended to be used to find a few eigenpairs
of a complex linear operator OP with respect to a semi-inner product
defined c by a hermitian positive semi-definite real matrix B. B may be
the identity matrix. NOTE: if both OP and B are real, then dsaupd or
dnaupd should be used.
The computed approximate eigenvalues are called Ritz values and
the corresponding approximate eigenvectors are called Ritz vectors.
CNAUPD is usually called iteratively to solve one of the
following problems:
Mode 1: A*x = lambda*x.
===> OP = A and B = I.
Mode 2: A*x = lambda*M*x, M hermitian positive definite
===> OP = inv[M]*A and B = M.
===> (If M can be factored see remark 3 below)
Mode 3: A*x = lambda*M*x, M hermitian semi-definite
===> OP = inv[A - sigma*M]*M and B = M.
===> shift-and-invert mode
If OP*x = amu*x, then lambda = sigma + 1/amu.
NOTE: The action of w <- inv[A - sigma*M]*v or w <- inv[M]*v
should be accomplished either by a direct method
using a sparse matrix factorization and solving
[A - sigma*M]*w = v or M*w = v,
or through an iterative method for solving these
systems. If an iterative method is used, the
convergence test must be more stringent than
the accuracy requirements for the eigenvalue
approximations.
ARGUMENTS
IDO Integer. (INPUT/OUTPUT)
Reverse communication flag. IDO must be zero on the first
call to CNAUPD. IDO will be set internally to
indicate the type of operation to be performed. Control is
then given back to the calling routine which has the
responsibility to carry out the requested operation and call
CNAUPD with the result. The operand is given in
WORKD(IPNTR(1)), the result must be put in WORKD(IPNTR(2)).
-------------------------------------------------------------
IDO = 0: first call to the reverse communication interface
IDO = -1: compute Y = OP * X where
IPNTR(1) is the pointer into WORKD for X,
IPNTR(2) is the pointer into WORKD for Y.
This is for the initialization phase to force the
starting vector into the range of OP.
IDO = 1: compute Y = OP * X where
IPNTR(1) is the pointer into WORKD for X,
IPNTR(2) is the pointer into WORKD for Y.
In mode 3, the vector B * X is already
available in WORKD(ipntr(3)). It does not
need to be recomputed in forming OP * X.
IDO = 2: compute Y = M * X where
IPNTR(1) is the pointer into WORKD for X,
IPNTR(2) is the pointer into WORKD for Y.
IDO = 3: compute and return the shifts in the first
NP locations of WORKL.
IDO = 99: done
-------------------------------------------------------------
After the initialization phase, when the routine is used in
the "shift-and-invert" mode, the vector M * X is already
available and does not need to be recomputed in forming OP*X.
BMAT Character*1. (INPUT)
BMAT specifies the type of the matrix B that defines the
semi-inner product for the operator OP.
BMAT = 'I' -> standard eigenvalue problem A*x = lambda*x
BMAT = 'G' -> generalized eigenvalue problem A*x = lambda*M*x
N Integer. (INPUT)
Dimension of the eigenproblem.
WHICH Character*2. (INPUT)
'LM' -> want the NEV eigenvalues of largest magnitude.
'SM' -> want the NEV eigenvalues of smallest magnitude.
'LR' -> want the NEV eigenvalues of largest real part.
'SR' -> want the NEV eigenvalues of smallest real part.
'LI' -> want the NEV eigenvalues of largest imaginary part.
'SI' -> want the NEV eigenvalues of smallest imaginary part.
NEV Integer. (INPUT)
Number of eigenvalues of OP to be computed. 0 < NEV < N-1.
TOL Double precision scalar. (INPUT)
Stopping criteria: the relative accuracy of the Ritz value
is considered acceptable if BOUNDS(I) .LE. TOL*ABS(RITZ(I))
where ABS(RITZ(I)) is the magnitude when RITZ(I) is complex.
DEFAULT = dlamch('EPS') (machine precision as computed
by the LAPACK auxiliary subroutine dlamch).
RESID Complex*16 array of length N. (INPUT/OUTPUT)
On INPUT:
If INFO .EQ. 0, a random initial residual vector is used.
If INFO .NE. 0, RESID contains the initial residual vector,
possibly from a previous run.
On OUTPUT:
RESID contains the final residual vector.
NCV Integer. (INPUT)
Number of columns of the matrix V. NCV must satisfy the two
inequalities 1 <= NCV-NEV and NCV <= N.
This will indicate how many Arnoldi vectors are generated
at each iteration. After the startup phase in which NEV
Arnoldi vectors are generated, the algorithm generates
approximately NCV-NEV Arnoldi vectors at each subsequent update
iteration. Most of the cost in generating each Arnoldi vector is
in the matrix-vector operation OP*x. (See remark 4 below.)
V Complex*16 array N by NCV. (OUTPUT)
Contains the final set of Arnoldi basis vectors.
LDV Integer. (INPUT)
Leading dimension of V exactly as declared in the calling program.
IPARAM Integer array of length 11. (INPUT/OUTPUT)
IPARAM(1) = ISHIFT: method for selecting the implicit shifts.
The shifts selected at each iteration are used to filter out
the components of the unwanted eigenvector.
-------------------------------------------------------------
ISHIFT = 0: the shifts are to be provided by the user via
reverse communication. The NCV eigenvalues of
the Hessenberg matrix H are returned in the part
of WORKL array corresponding to RITZ.
ISHIFT = 1: exact shifts with respect to the current
Hessenberg matrix H. This is equivalent to
restarting the iteration from the beginning
after updating the starting vector with a linear
combination of Ritz vectors associated with the
"wanted" eigenvalues.
ISHIFT = 2: other choice of internal shift to be defined.
-------------------------------------------------------------
IPARAM(2) = No longer referenced
IPARAM(3) = MXITER
On INPUT: maximum number of Arnoldi update iterations allowed.
On OUTPUT: actual number of Arnoldi update iterations taken.
IPARAM(4) = NB: blocksize to be used in the recurrence.
The code currently works only for NB = 1.
IPARAM(5) = NCONV: number of "converged" Ritz values.
This represents the number of Ritz values that satisfy
the convergence criterion.
IPARAM(6) = IUPD
No longer referenced. Implicit restarting is ALWAYS used.
IPARAM(7) = MODE
On INPUT determines what type of eigenproblem is being solved.
Must be 1,2,3; See under Description of CNAUPD for the
four modes available.
IPARAM(8) = NP
When ido = 3 and the user provides shifts through reverse
communication (IPARAM(1)=0), _naupd returns NP, the number
of shifts the user is to provide. 0 < NP < NCV-NEV.
IPARAM(9) = NUMOP, IPARAM(10) = NUMOPB, IPARAM(11) = NUMREO,
OUTPUT: NUMOP = total number of OP*x operations,
NUMOPB = total number of B*x operations if BMAT='G',
NUMREO = total number of steps of re-orthogonalization.
IPNTR Integer array of length 14. (OUTPUT)
Pointer to mark the starting locations in the WORKD and WORKL
arrays for matrices/vectors used by the Arnoldi iteration.
-------------------------------------------------------------
IPNTR(1): pointer to the current operand vector X in WORKD.
IPNTR(2): pointer to the current result vector Y in WORKD.
IPNTR(3): pointer to the vector B * X in WORKD when used in
the shift-and-invert mode.
IPNTR(4): pointer to the next available location in WORKL
that is untouched by the program.
IPNTR(5): pointer to the NCV by NCV upper Hessenberg
matrix H in WORKL.
IPNTR(6): pointer to the ritz value array RITZ
IPNTR(7): pointer to the (projected) ritz vector array Q
IPNTR(8): pointer to the error BOUNDS array in WORKL.
IPNTR(14): pointer to the NP shifts in WORKL. See Remark 5 below.
Note: IPNTR(9:13) is only referenced by zneupd. See Remark 2 below.
IPNTR(9): pointer to the NCV RITZ values of the
original system.
IPNTR(10): Not Used
IPNTR(11): pointer to the NCV corresponding error bounds.
IPNTR(12): pointer to the NCV by NCV upper triangular
Schur matrix for H.
IPNTR(13): pointer to the NCV by NCV matrix of eigenvectors
of the upper Hessenberg matrix H. Only referenced by
zneupd if RVEC = .TRUE. See Remark 2 below.
-------------------------------------------------------------
WORKD Complex*16 work array of length 3*N. (REVERSE COMMUNICATION)
Distributed array to be used in the basic Arnoldi iteration
for reverse communication. The user should not use WORKD
as temporary workspace during the iteration !!!!!!!!!!
See Data Distribution Note below.
WORKL Complex*16 work array of length LWORKL. (OUTPUT/WORKSPACE)
Private (replicated) array on each PE or array allocated on
the front end. See Data Distribution Note below.
LWORKL Integer. (INPUT)
LWORKL must be at least 3*NCV**2 + 5*NCV.
RWORK Double precision work array of length NCV (WORKSPACE)
Private (replicated) array on each PE or array allocated on
the front end.
INFO Integer. (INPUT/OUTPUT)
If INFO .EQ. 0, a randomly initial residual vector is used.
If INFO .NE. 0, RESID contains the initial residual vector,
possibly from a previous run.
Error flag on output.
= 0: Normal exit.
= 1: Maximum number of iterations taken.
All possible eigenvalues of OP has been found. IPARAM(5)
returns the number of wanted converged Ritz values.
= 2: No longer an informational error. Deprecated starting
with release 2 of ARPACK.
= 3: No shifts could be applied during a cycle of the
Implicitly restarted Arnoldi iteration. One possibility
is to increase the size of NCV relative to NEV.
See remark 4 below.
= -1: N must be positive.
= -2: NEV must be positive.
= -3: NCV-NEV >= 2 and less than or equal to N.
= -4: The maximum number of Arnoldi update iteration
must be greater than zero.
= -5: WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'
= -6: BMAT must be one of 'I' or 'G'.
= -7: Length of private work array is not sufficient.
= -8: Error return from LAPACK eigenvalue calculation;
= -9: Starting vector is zero.
= -10: IPARAM(7) must be 1,2,3.
= -11: IPARAM(7) = 1 and BMAT = 'G' are incompatible.
= -12: IPARAM(1) must be equal to 0 or 1.
= -9999: Could not build an Arnoldi factorization.
User input error highly likely. Please
check actual array dimensions and layout.
IPARAM(5) returns the size of the current Arnoldi
factorization.
emarks
1. The computed Ritz values are approximate eigenvalues of OP. The
selection of WHICH should be made with this in mind when using
Mode = 3. When operating in Mode = 3 setting WHICH = 'LM' will
compute the NEV eigenvalues of the original problem that are
closest to the shift SIGMA . After convergence, approximate eigenvalues
of the original problem may be obtained with the ARPACK subroutine zneupd.
2. If a basis for the invariant subspace corresponding to the converged Ritz
values is needed, the user must call zneupd immediately following
completion of CNAUPD. This is new starting with release 2 of ARPACK.
3. If M can be factored into a Cholesky factorization M = LL'
then Mode = 2 should not be selected. Instead one should use
Mode = 1 with OP = inv(L)*A*inv(L'). Appropriate triangular
linear systems should be solved with L and L' rather
than computing inverses. After convergence, an approximate
eigenvector z of the original problem is recovered by solving
L'z = x where x is a Ritz vector of OP.
4. At present there is no a-priori analysis to guide the selection
of NCV relative to NEV. The only formal requirement is that
NCV > NEV + 1. However, it is recommended that NCV .ge. 2*NEV. If
many problems of the same type are to be solved, one should experiment
with increasing NCV while keeping NEV fixed for a given test problem.
This will usually decrease the required number of OP*x operations but it
also increases the work and storage required to maintain the orthogonal
basis vectors. The optimal "cross-over" with respect to CPU time
is problem dependent and must be determined empirically.
See Chapter 8 of Reference 2 for further information.
5. When IPARAM(1) = 0, and IDO = 3, the user needs to provide the
NP = IPARAM(8) complex shifts in locations
WORKL(IPNTR(14)), WORKL(IPNTR(14)+1), ... , WORKL(IPNTR(14)+NP).
Eigenvalues of the current upper Hessenberg matrix are located in
WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1). They are ordered
according to the order defined by WHICH. The associated Ritz estimates
are located in WORKL(IPNTR(8)), WORKL(IPNTR(8)+1), ... ,
WORKL(IPNTR(8)+NCV-1).
MathKeisan CNAUPD(3)