PDNAUPD(3) MathKeisan PARPACK routine PDNAUPD(3)
NAME
PDNAUPD - Reverse communication interface for the large-scale (sparse)
non-symmetric eigenvalue calculation.
SYNOPSIS
SUBROUTINE PDNAUPD
( 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
DOUBLE PRECISION TOL
INTEGER IPARAM(11), IPNTR(14)
DOUBLE PRECISION RESID(N), V(LDV,NCV), WORKD(3*N), WORKL(LWORKL)
PURPOSE
Reverse communication interface for the Implicitly Restarted Arnoldi
iteration. PDNAUPD computes approximations to a few eigenpairs
of a linear operator "OP" with respect to a semi-inner product defined by
a symmetric positive semi-definite real matrix B. B may be the identity
matrix. NOTE: If the linear operator "OP" is real and symmetric
with respect to the real positive semi-definite symmetric matrix B,
i.e. B*OP = (OP')*B, then subroutine dsaupd should be used instead.
The computed approximate eigenvalues are called Ritz values and
the corresponding approximate eigenvectors are called Ritz vectors.
pdnaupd 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 symmetric 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 symmetric semi-definite
===> OP = Real_Part{ inv[A - sigma*M]*M } and B = M.
===> shift-and-invert mode (in real arithmetic)
If OP*x = amu*x, then
amu = 1/2 * [ 1/(lambda-sigma) + 1/(lambda-conjg(sigma)) ].
Note: If sigma is real, i.e. imaginary part of sigma is zero;
Real_Part{ inv[A - sigma*M]*M } == inv[A - sigma*M]*M
amu == 1/(lambda-sigma).
Mode 4: A*x = lambda*M*x, M symmetric semi-definite
===> OP = Imaginary_Part{ inv[A - sigma*M]*M } and B = M.
===> shift-and-invert mode (in real arithmetic)
If OP*x = amu*x, then
amu = 1/2i * [ 1/(lambda-sigma) - 1/(lambda-conjg(sigma)) ].
Both mode 3 and 4 give the same enhancement to eigenvalues close to
the (complex) shift sigma. However, as lambda goes to infinity,
the operator OP in mode 4 dampens the eigenvalues more strongly than
does OP defined in mode 3.
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 pdnaupd . 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
pdnaupd 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 and 4, 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) real and imaginary parts
of the shifts where INPTR(14) is the pointer
into WORKL for placing the shifts. See Remark
5 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.
BMAT = 'I' -> standard eigenvalue problem A*x = lambda*x
BMAT = 'G' -> generalized eigenvalue problem A*x = lambda*B*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 criterion: 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 Double precision 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 2 <= 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.
NOTE: 2 <= NCV-NEV in order that complex conjugate pairs of Ritz
values are kept together. (See remark 4 below)
V Double precision 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 restart
the Arnoldi iteration in an implicit fashion.
-------------------------------------------------------------
ISHIFT = 0: the shifts are provided by the user via
reverse communication. The real and imaginary
parts of the NCV eigenvalues of the Hessenberg
matrix H are returned in the part of the WORKL
array corresponding to RITZR and RITZI. See remark
5 below.
ISHIFT = 1: exact shifts with respect to the current
Hessenberg matrix H. This is equivalent to
restarting the iteration with a starting vector
that is a linear combination of approximate Schur
vectors associated with the "wanted" Ritz values.
-------------------------------------------------------------
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 1,2,3,4; See under PURPOSE of pdnaupd for the
four modes available.
IPARAM(8) = NP
When ido = 3 and the user provides shifts through reverse
communication (IPARAM(1)=0), pdnaupd returns NP, the number
of shifts the user is to provide. 0 < NP <=NCV-NEV. See Remark
5 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 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 real part of the ritz value array
RITZR in WORKL.
IPNTR(7): pointer to the imaginary part of the ritz value array
RITZI in WORKL.
IPNTR(8): pointer to the Ritz estimates in array WORKL associated
with the Ritz values located in RITZR and RITZI in WORKL.
IPNTR(14): pointer to the NP shifts in WORKL. See Remark 5 below.
Note: IPNTR(9:13) is only referenced by dneupd . See Remark 2 below.
IPNTR(9): pointer to the real part of the NCV RITZ values of the
original system.
IPNTR(10): pointer to the imaginary part of the NCV RITZ values of
the original system.
IPNTR(11): pointer to the NCV corresponding error bounds.
IPNTR(12): pointer to the NCV by NCV upper quasi-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
pdneupd if RVEC = .TRUE. See Remark 2 below.
-------------------------------------------------------------
WORKD Double precision 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 an invariant subspace
associated with the converged Ritz values is desired, see remark
2 below, subroutine dneupd uses this output.
See Data Distribution Note below.
WORKL Double precision 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 + 6*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-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 1,2,3,4.
= -11: IPARAM(7) = 1 and BMAT = 'G' are incompatable.
= -12: IPARAM(1) must be equal to 0 or 1.
= -9999: Could not build an Arnoldi factorization.
IPARAM(5) returns the size of the current Arnoldi
factorization.
Remarks
1. The computed Ritz values are approximate eigenvalues of OP. The
selection of WHICH should be made with this in mind when
Mode = 3 and 4. After convergence, approximate eigenvalues of the
original problem may be obtained with the ARPACK subroutine dneupd .
2. If a basis for the invariant subspace corresponding to the converged Ritz
values is needed, the user must call dneupd immediately following
completion of pdnaupd . 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 requrement is that NCV > NEV + 2.
However, it is recommended that NCV .ge. 2*NEV+1. 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) real and imaginary parts of the shifts in locations
real part imaginary part
----------------------- --------------
1 WORKL(IPNTR(14)) WORKL(IPNTR(14)+NP)
2 WORKL(IPNTR(14)+1) WORKL(IPNTR(14)+NP+1)
. .
. .
. .
NP WORKL(IPNTR(14)+NP-1) WORKL(IPNTR(14)+2*NP-1).
Only complex conjugate pairs of shifts may be applied and the pairs
must be placed in consecutive locations. The real part of the
eigenvalues of the current upper Hessenberg matrix are located in
WORKL(IPNTR(6)) through WORKL(IPNTR(6)+NCV-1) and the imaginary part
in WORKL(IPNTR(7)) through WORKL(IPNTR(7)+NCV-1). They are ordered
according to the order defined by WHICH. The complex conjugate
pairs are kept together and the associated Ritz estimates are located in
WORKL(IPNTR(8)), WORKL(IPNTR(8)+1), ... , WORKL(IPNTR(8)+NCV-1).
MathKeisan PDNAUPD(3)