PSSAUPD(3) MathKeisan PARPACK routine PSSAUPD(3)
NAME
PSSAUPD - Reverse communication interface for the large-scale (sparse)
symmetric eigenvalue calculation.
SYNOPSIS
SUBROUTINE PSSAUPD
( COMM, IDO, BMAT, N, WHICH, NEV, TOL, RESID, NCV, V, LDV, IPARAM,
IPNTR, WORKD, WORKL, LWORKL, INFO )
CHARACTER BMAT*1, WHICH*2
INTEGER IDO, INFO, LDV, LWORKL, N, NCV, NEV
REAL TOL
INTEGER IPARAM(11), IPNTR(11)
REAL RESID(N), V(LDV,NCV), WORKD(3*N), WORKL(LWORKL)
PURPOSE
Reverse communication interface for the Implicitly Restarted Arnoldi
Iteration. For symmetric problems this reduces to a variant of the Lanczos
method. This method has been designed to compute approximations to a
few eigenpairs of a linear operator OP that is real and symmetric
with respect to a real positive semi-definite symmetric matrix B,
i.e.
B*OP = (OP')*B.
Another way to express this condition is
< x,OPy > = < OPx,y > where < z,w > = z'Bw .
In the standard eigenproblem B is the identity matrix.
( A' denotes transpose of A)
The computed approximate eigenvalues are called Ritz values and
the corresponding approximate eigenvectors are called Ritz vectors.
pssaupd is usually called iteratively to solve one of the
following problems:
Mode 1: A*x = lambda*x, A symmetric
===> OP = A and B = I.
Mode 2: A*x = lambda*M*x, A symmetric, M symmetric positive definite
===> OP = inv[M]*A and B = M.
===> (If M can be factored see remark 3 below)
Mode 3: K*x = lambda*M*x, K symmetric, M symmetric positive semi-definite
===> OP = (inv[K - sigma*M])*M and B = M.
===> Shift-and-Invert mode
Mode 4: K*x = lambda*KG*x, K symmetric positive semi-definite,
KG symmetric indefinite
===> OP = (inv[K - sigma*KG])*K and B = K.
===> Buckling mode
Mode 5: A*x = lambda*M*x, A symmetric, M symmetric positive semi-definite
===> OP = inv[A - sigma*M]*[A + sigma*M] and B = M.
===> Cayley transformed mode
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
COMM MPI Communicator for the processor grid. (INPUT)
IDO Integer. (INPUT/OUTPUT)
Reverse communication flag. IDO must be zero on the first
call to pssaupd. 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
pssaupd with the result. The operand is given in
WORKD(IPNTR(1)), the result must be put in WORKD(IPNTR(2)).
(If Mode = 2 see remark 5 below)
-------------------------------------------------------------
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 3,4 and 5, 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 = B * X where
IPNTR(1) is the pointer into WORKD for X,
IPNTR(2) is the pointer into WORKD for Y.
IDO = 3: compute the IPARAM(8) shifts where
IPNTR(11) is the pointer into WORKL for
placing the shifts. See remark 6 below.
IDO = 99: done
-------------------------------------------------------------
BMAT Character*1. (INPUT)
BMAT specifies the type of the matrix B that defines the
semi-inner product for the operator OP.
B = 'I' -> standard eigenvalue problem A*x = lambda*x
B = 'G' -> generalized eigenvalue problem A*x = lambda*B*x
N Integer. (INPUT)
Dimension of the eigenproblem.
WHICH Character*2. (INPUT)
Specify which of the Ritz values of OP to compute.
'LA' - compute the NEV largest (algebraic) eigenvalues.
'SA' - compute the NEV smallest (algebraic) eigenvalues.
'LM' - compute the NEV largest (in magnitude) eigenvalues.
'SM' - compute the NEV smallest (in magnitude) eigenvalues.
'BE' - compute NEV eigenvalues, half from each end of the
spectrum. When NEV is odd, compute one more from the
high end than from the low end.
(see remark 1 below)
NEV Integer. (INPUT)
Number of eigenvalues of OP to be computed. 0 < NEV < N.
TOL Real scalar. (INPUT)
Stopping criterion: the relative accuracy of the Ritz value
is considered acceptable if BOUNDS(I) .LE. TOL*ABS(RITZ(I)).
If TOL .LE. 0. is passed a default is set:
DEFAULT = SLAMCH('EPS') (machine precision as computed
by the LAPACK auxiliary subroutine SLAMCH).
RESID Real 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 (less than or equal to N).
This will indicate how many Lanczos vectors are generated
at each iteration. After the startup phase in which NEV
Lanczos vectors are generated, the algorithm generates
NCV-NEV Lanczos vectors at each subsequent update iteration.
Most of the cost in generating each Lanczos vector is in the
matrix-vector product OP*x. (See remark 4 below).
V Real N by NCV array. (OUTPUT)
The NCV columns of V contain the Lanczos 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 restart
the Arnoldi iteration in an implicit fashion.
-------------------------------------------------------------
ISHIFT = 0: the shifts are provided by the user via
reverse communication. The NCV eigenvalues of
the current tridiagonal matrix T are returned in
the part of WORKL array corresponding to RITZ.
See remark 6 below.
ISHIFT = 1: exact shifts with respect to the reduced
tridiagonal matrix T. This is equivalent to
restarting the iteration with a starting vector
that is a linear combination of Ritz vectors
associated with the "wanted" Ritz values.
-------------------------------------------------------------
IPARAM(2) = LEVEC
No longer referenced. See remark 2 below.
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 1,2,3,4,5; See under PURPOSE of pssaupd for the
five modes available.
IPARAM(8) = NP
When ido = 3 and the user provides shifts through reverse
communication (IPARAM(1)=0), pssaupd returns NP, the number
of shifts the user is to provide. 0 < NP <=NCV-NEV. See Remark
6 below.
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 11. (OUTPUT)
Pointer to mark the starting locations in the WORKD and WORKL
arrays for matrices/vectors used by the Lanczos 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 2 tridiagonal matrix T in WORKL.
IPNTR(6): pointer to the NCV RITZ values array in WORKL.
IPNTR(7): pointer to the Ritz estimates in array WORKL associated
with the Ritz values located in RITZ in WORKL.
IPNTR(11): pointer to the NP shifts in WORKL. See Remark 6 below.
Note: IPNTR(8:10) is only referenced by psseupd. See Remark 2.
IPNTR(8): pointer to the NCV RITZ values of the original system.
IPNTR(9): pointer to the NCV corresponding error bounds.
IPNTR(10): pointer to the NCV by NCV matrix of eigenvectors
of the tridiagonal matrix T. Only referenced by
psseupd if RVEC = .TRUE. See Remarks.
-------------------------------------------------------------
WORKD Real 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. Upon termination
WORKD(1:N) contains B*RESID(1:N). If the Ritz vectors are desired
subroutine psseupd uses this output.
See Data Distribution Note below.
WORKL Real 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 NCV**2 + 8*NCV .
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 must be greater than NEV and less than or equal to N.
= -4: The maximum number of Arnoldi update iterations allowed
must be greater than zero.
= -5: WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.
= -6: BMAT must be one of 'I' or 'G'.
= -7: Length of private work array WORKL is not sufficient.
= -8: Error return from trid. eigenvalue calculation;
Informatinal error from LAPACK routine ssteqr.
= -9: Starting vector is zero.
= -10: IPARAM(7) must be 1,2,3,4,5.
= -11: IPARAM(7) = 1 and BMAT = 'G' are incompatable.
= -12: IPARAM(1) must be equal to 0 or 1.
= -13: NEV and WHICH = 'BE' are incompatable.
= -9999: Could not build an Arnoldi factorization.
IPARAM(5) returns the size of the current Arnoldi
factorization. The user is advised to check that
enough workspace and array storage has been allocated.
Remarks
1. The converged Ritz values are always returned in ascending
algebraic order. The computed Ritz values are approximate
eigenvalues of OP. The selection of WHICH should be made
with this in mind when Mode = 3,4,5. After convergence,
approximate eigenvalues of the original problem may be obtained
with the ARPACK subroutine psseupd.
2. If the Ritz vectors corresponding to the converged Ritz values
are needed, the user must call psseupd immediately following completion
of pssaupd. This is new starting with version 2.1 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 requrement is that NCV > NEV.
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.
5. If IPARAM(7) = 2 then in the Reverse commuication interface the user
must do the following. When IDO = 1, Y = OP * X is to be computed.
When IPARAM(7) = 2 OP = inv(B)*A. After computing A*X the user
must overwrite X with A*X. Y is then the solution to the linear set
of equations B*Y = A*X.
6. When IPARAM(1) = 0, and IDO = 3, the user needs to provide the
NP = IPARAM(8) shifts in locations:
1 WORKL(IPNTR(11))
2 WORKL(IPNTR(11)+1)
.
.
.
NP WORKL(IPNTR(11)+NP-1).
The eigenvalues of the current tridiagonal matrix are located in
WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1). They are in 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 PSSAUPD(3)