851 lines
30 KiB
FortranFixed
851 lines
30 KiB
FortranFixed
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c\BeginDoc
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c
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c\Name: cnaitr
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c
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c\Description:
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c Reverse communication interface for applying NP additional steps to
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c a K step nonsymmetric Arnoldi factorization.
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c
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c Input: OP*V_{k} - V_{k}*H = r_{k}*e_{k}^T
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c
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c with (V_{k}^T)*B*V_{k} = I, (V_{k}^T)*B*r_{k} = 0.
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c
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c Output: OP*V_{k+p} - V_{k+p}*H = r_{k+p}*e_{k+p}^T
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c
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c with (V_{k+p}^T)*B*V_{k+p} = I, (V_{k+p}^T)*B*r_{k+p} = 0.
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c
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c where OP and B are as in cnaupd. The B-norm of r_{k+p} is also
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c computed and returned.
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c
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c\Usage:
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c call cnaitr
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c ( IDO, BMAT, N, K, NP, NB, RESID, RNORM, V, LDV, H, LDH,
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c IPNTR, WORKD, INFO )
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c
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c\Arguments
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c IDO Integer. (INPUT/OUTPUT)
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c Reverse communication flag.
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c -------------------------------------------------------------
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c IDO = 0: first call to the reverse communication interface
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c IDO = -1: compute Y = OP * X where
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c IPNTR(1) is the pointer into WORK for X,
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c IPNTR(2) is the pointer into WORK for Y.
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c This is for the restart phase to force the new
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c starting vector into the range of OP.
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c IDO = 1: compute Y = OP * X where
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c IPNTR(1) is the pointer into WORK for X,
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c IPNTR(2) is the pointer into WORK for Y,
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c IPNTR(3) is the pointer into WORK for B * X.
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c IDO = 2: compute Y = B * X where
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c IPNTR(1) is the pointer into WORK for X,
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c IPNTR(2) is the pointer into WORK for Y.
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c IDO = 99: done
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c -------------------------------------------------------------
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c When the routine is used in the "shift-and-invert" mode, the
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c vector B * Q is already available and do not need to be
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c recomputed in forming OP * Q.
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c
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c BMAT Character*1. (INPUT)
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c BMAT specifies the type of the matrix B that defines the
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c semi-inner product for the operator OP. See cnaupd.
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c B = 'I' -> standard eigenvalue problem A*x = lambda*x
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c B = 'G' -> generalized eigenvalue problem A*x = lambda*M**x
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c
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c N Integer. (INPUT)
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c Dimension of the eigenproblem.
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c
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c K Integer. (INPUT)
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c Current size of V and H.
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c
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c NP Integer. (INPUT)
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c Number of additional Arnoldi steps to take.
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c
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c NB Integer. (INPUT)
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c Blocksize to be used in the recurrence.
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c Only work for NB = 1 right now. The goal is to have a
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c program that implement both the block and non-block method.
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c
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c RESID Complex array of length N. (INPUT/OUTPUT)
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c On INPUT: RESID contains the residual vector r_{k}.
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c On OUTPUT: RESID contains the residual vector r_{k+p}.
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c
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c RNORM Real scalar. (INPUT/OUTPUT)
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c B-norm of the starting residual on input.
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c B-norm of the updated residual r_{k+p} on output.
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c
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c V Complex N by K+NP array. (INPUT/OUTPUT)
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c On INPUT: V contains the Arnoldi vectors in the first K
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c columns.
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c On OUTPUT: V contains the new NP Arnoldi vectors in the next
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c NP columns. The first K columns are unchanged.
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c
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c LDV Integer. (INPUT)
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c Leading dimension of V exactly as declared in the calling
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c program.
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c
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c H Complex (K+NP) by (K+NP) array. (INPUT/OUTPUT)
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c H is used to store the generated upper Hessenberg matrix.
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c
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c LDH Integer. (INPUT)
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c Leading dimension of H exactly as declared in the calling
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c program.
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c
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c IPNTR Integer array of length 3. (OUTPUT)
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c Pointer to mark the starting locations in the WORK for
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c vectors used by the Arnoldi iteration.
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c -------------------------------------------------------------
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c IPNTR(1): pointer to the current operand vector X.
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c IPNTR(2): pointer to the current result vector Y.
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c IPNTR(3): pointer to the vector B * X when used in the
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c shift-and-invert mode. X is the current operand.
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c -------------------------------------------------------------
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c
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c WORKD Complex work array of length 3*N. (REVERSE COMMUNICATION)
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c Distributed array to be used in the basic Arnoldi iteration
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c for reverse communication. The calling program should not
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c use WORKD as temporary workspace during the iteration !!!!!!
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c On input, WORKD(1:N) = B*RESID and is used to save some
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c computation at the first step.
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c
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c INFO Integer. (OUTPUT)
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c = 0: Normal exit.
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c > 0: Size of the spanning invariant subspace of OP found.
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c
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c\EndDoc
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c
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c-----------------------------------------------------------------------
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c
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c\BeginLib
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c
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c\Local variables:
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c xxxxxx Complex
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c
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c\References:
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c 1. D.C. Sorensen, "Implicit Application of Polynomial Filters in
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c a k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992),
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c pp 357-385.
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c 2. R.B. Lehoucq, "Analysis and Implementation of an Implicitly
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c Restarted Arnoldi Iteration", Rice University Technical Report
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c TR95-13, Department of Computational and Applied Mathematics.
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c
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c\Routines called:
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c cgetv0 ARPACK routine to generate the initial vector.
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c ivout ARPACK utility routine that prints integers.
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c second ARPACK utility routine for timing.
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c cmout ARPACK utility routine that prints matrices
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c cvout ARPACK utility routine that prints vectors.
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c clanhs LAPACK routine that computes various norms of a matrix.
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c clascl LAPACK routine for careful scaling of a matrix.
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c slabad LAPACK routine for defining the underflow and overflow
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c limits.
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c slamch LAPACK routine that determines machine constants.
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c slapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
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c cgemv Level 2 BLAS routine for matrix vector multiplication.
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c caxpy Level 1 BLAS that computes a vector triad.
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c ccopy Level 1 BLAS that copies one vector to another .
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c cdotc Level 1 BLAS that computes the scalar product of two vectors.
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c cscal Level 1 BLAS that scales a vector.
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c csscal Level 1 BLAS that scales a complex vector by a real number.
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c scnrm2 Level 1 BLAS that computes the norm of a vector.
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c
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c\Author
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c Danny Sorensen Phuong Vu
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c Richard Lehoucq CRPC / Rice University
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c Dept. of Computational & Houston, Texas
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c Applied Mathematics
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c Rice University
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c Houston, Texas
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c
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c\SCCS Information: @(#)
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c FILE: naitr.F SID: 2.3 DATE OF SID: 8/27/96 RELEASE: 2
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c
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c\Remarks
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c The algorithm implemented is:
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c
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c restart = .false.
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c Given V_{k} = [v_{1}, ..., v_{k}], r_{k};
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c r_{k} contains the initial residual vector even for k = 0;
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c Also assume that rnorm = || B*r_{k} || and B*r_{k} are already
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c computed by the calling program.
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c
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c betaj = rnorm ; p_{k+1} = B*r_{k} ;
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c For j = k+1, ..., k+np Do
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c 1) if ( betaj < tol ) stop or restart depending on j.
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c ( At present tol is zero )
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c if ( restart ) generate a new starting vector.
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c 2) v_{j} = r(j-1)/betaj; V_{j} = [V_{j-1}, v_{j}];
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c p_{j} = p_{j}/betaj
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c 3) r_{j} = OP*v_{j} where OP is defined as in cnaupd
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c For shift-invert mode p_{j} = B*v_{j} is already available.
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c wnorm = || OP*v_{j} ||
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c 4) Compute the j-th step residual vector.
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c w_{j} = V_{j}^T * B * OP * v_{j}
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c r_{j} = OP*v_{j} - V_{j} * w_{j}
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c H(:,j) = w_{j};
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c H(j,j-1) = rnorm
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c rnorm = || r_(j) ||
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c If (rnorm > 0.717*wnorm) accept step and go back to 1)
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c 5) Re-orthogonalization step:
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c s = V_{j}'*B*r_{j}
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c r_{j} = r_{j} - V_{j}*s; rnorm1 = || r_{j} ||
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c alphaj = alphaj + s_{j};
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c 6) Iterative refinement step:
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c If (rnorm1 > 0.717*rnorm) then
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c rnorm = rnorm1
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c accept step and go back to 1)
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c Else
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c rnorm = rnorm1
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c If this is the first time in step 6), go to 5)
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c Else r_{j} lies in the span of V_{j} numerically.
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c Set r_{j} = 0 and rnorm = 0; go to 1)
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c EndIf
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c End Do
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c
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c\EndLib
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c
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c-----------------------------------------------------------------------
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c
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subroutine cnaitr
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& (ido, bmat, n, k, np, nb, resid, rnorm, v, ldv, h, ldh,
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& ipntr, workd, info)
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c
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c %----------------------------------------------------%
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c | Include files for debugging and timing information |
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c %----------------------------------------------------%
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c
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include 'debug.h'
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include 'stat.h'
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c
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c %------------------%
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c | Scalar Arguments |
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c %------------------%
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c
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character bmat*1
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integer ido, info, k, ldh, ldv, n, nb, np
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Real
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& rnorm
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c
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c %-----------------%
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c | Array Arguments |
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c %-----------------%
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c
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integer ipntr(3)
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Complex
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& h(ldh,k+np), resid(n), v(ldv,k+np), workd(3*n)
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c
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c %------------%
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c | Parameters |
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c %------------%
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c
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Complex
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& one, zero
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Real
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& rone, rzero
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parameter (one = (1.0E+0, 0.0E+0), zero = (0.0E+0, 0.0E+0),
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& rone = 1.0E+0, rzero = 0.0E+0)
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c
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c %--------------%
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c | Local Arrays |
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c %--------------%
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c
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Real
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& rtemp(2)
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c
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c %---------------%
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c | Local Scalars |
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c %---------------%
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c
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logical first, orth1, orth2, rstart, step3, step4
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integer ierr, i, infol, ipj, irj, ivj, iter, itry, j, msglvl,
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& jj
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Real
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& ovfl, smlnum, tst1, ulp, unfl, betaj,
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& temp1, rnorm1, wnorm
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Complex
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& cnorm
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c
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save first, orth1, orth2, rstart, step3, step4,
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& ierr, ipj, irj, ivj, iter, itry, j, msglvl, ovfl,
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& betaj, rnorm1, smlnum, ulp, unfl, wnorm
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c
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c %----------------------%
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c | External Subroutines |
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c %----------------------%
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c
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external caxpy, ccopy, cscal, csscal, cgemv, cgetv0,
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& slabad, cvout, cmout, ivout, second
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c
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c %--------------------%
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c | External Functions |
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c %--------------------%
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c
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Complex
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& cdotc
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Real
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& slamch, scnrm2, clanhs, slapy2
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external cdotc, scnrm2, clanhs, slamch, slapy2
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c
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c %---------------------%
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c | Intrinsic Functions |
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c %---------------------%
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c
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intrinsic aimag, real, max, sqrt
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c
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c %-----------------%
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c | Data statements |
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c %-----------------%
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c
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data first / .true. /
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c
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c %-----------------------%
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c | Executable Statements |
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c %-----------------------%
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c
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if (first) then
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c
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c %-----------------------------------------%
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c | Set machine-dependent constants for the |
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c | the splitting and deflation criterion. |
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c | If norm(H) <= sqrt(OVFL), |
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c | overflow should not occur. |
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c | REFERENCE: LAPACK subroutine clahqr |
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c %-----------------------------------------%
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c
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unfl = slamch( 'safe minimum' )
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ovfl = real(one / unfl)
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call slabad( unfl, ovfl )
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ulp = slamch( 'precision' )
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smlnum = unfl*( n / ulp )
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first = .false.
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end if
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c
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if (ido .eq. 0) then
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c
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c %-------------------------------%
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c | Initialize timing statistics |
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c | & message level for debugging |
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c %-------------------------------%
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c
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call second (t0)
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msglvl = mcaitr
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c
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c %------------------------------%
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c | Initial call to this routine |
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c %------------------------------%
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c
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info = 0
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step3 = .false.
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step4 = .false.
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rstart = .false.
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orth1 = .false.
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orth2 = .false.
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j = k + 1
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ipj = 1
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irj = ipj + n
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ivj = irj + n
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end if
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c
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c %-------------------------------------------------%
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c | When in reverse communication mode one of: |
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c | STEP3, STEP4, ORTH1, ORTH2, RSTART |
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c | will be .true. when .... |
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c | STEP3: return from computing OP*v_{j}. |
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c | STEP4: return from computing B-norm of OP*v_{j} |
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c | ORTH1: return from computing B-norm of r_{j+1} |
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c | ORTH2: return from computing B-norm of |
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c | correction to the residual vector. |
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c | RSTART: return from OP computations needed by |
|
||
|
|
c | cgetv0. |
|
||
|
|
c %-------------------------------------------------%
|
||
|
|
c
|
||
|
|
if (step3) go to 50
|
||
|
|
if (step4) go to 60
|
||
|
|
if (orth1) go to 70
|
||
|
|
if (orth2) go to 90
|
||
|
|
if (rstart) go to 30
|
||
|
|
c
|
||
|
|
c %-----------------------------%
|
||
|
|
c | Else this is the first step |
|
||
|
|
c %-----------------------------%
|
||
|
|
c
|
||
|
|
c %--------------------------------------------------------------%
|
||
|
|
c | |
|
||
|
|
c | A R N O L D I I T E R A T I O N L O O P |
|
||
|
|
c | |
|
||
|
|
c | Note: B*r_{j-1} is already in WORKD(1:N)=WORKD(IPJ:IPJ+N-1) |
|
||
|
|
c %--------------------------------------------------------------%
|
||
|
|
|
||
|
|
1000 continue
|
||
|
|
c
|
||
|
|
if (msglvl .gt. 1) then
|
||
|
|
call ivout (logfil, 1, j, ndigit,
|
||
|
|
& '_naitr: generating Arnoldi vector number')
|
||
|
|
call svout (logfil, 1, rnorm, ndigit,
|
||
|
|
& '_naitr: B-norm of the current residual is')
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c | STEP 1: Check if the B norm of j-th residual |
|
||
|
|
c | vector is zero. Equivalent to determine whether |
|
||
|
|
c | an exact j-step Arnoldi factorization is present. |
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c
|
||
|
|
betaj = rnorm
|
||
|
|
if (rnorm .gt. rzero) go to 40
|
||
|
|
c
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c | Invariant subspace found, generate a new starting |
|
||
|
|
c | vector which is orthogonal to the current Arnoldi |
|
||
|
|
c | basis and continue the iteration. |
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c
|
||
|
|
if (msglvl .gt. 0) then
|
||
|
|
call ivout (logfil, 1, j, ndigit,
|
||
|
|
& '_naitr: ****** RESTART AT STEP ******')
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %---------------------------------------------%
|
||
|
|
c | ITRY is the loop variable that controls the |
|
||
|
|
c | maximum amount of times that a restart is |
|
||
|
|
c | attempted. NRSTRT is used by stat.h |
|
||
|
|
c %---------------------------------------------%
|
||
|
|
c
|
||
|
|
betaj = rzero
|
||
|
|
nrstrt = nrstrt + 1
|
||
|
|
itry = 1
|
||
|
|
20 continue
|
||
|
|
rstart = .true.
|
||
|
|
ido = 0
|
||
|
|
30 continue
|
||
|
|
c
|
||
|
|
c %--------------------------------------%
|
||
|
|
c | If in reverse communication mode and |
|
||
|
|
c | RSTART = .true. flow returns here. |
|
||
|
|
c %--------------------------------------%
|
||
|
|
c
|
||
|
|
call cgetv0 (ido, bmat, itry, .false., n, j, v, ldv,
|
||
|
|
& resid, rnorm, ipntr, workd, ierr)
|
||
|
|
if (ido .ne. 99) go to 9000
|
||
|
|
if (ierr .lt. 0) then
|
||
|
|
itry = itry + 1
|
||
|
|
if (itry .le. 3) go to 20
|
||
|
|
c
|
||
|
|
c %------------------------------------------------%
|
||
|
|
c | Give up after several restart attempts. |
|
||
|
|
c | Set INFO to the size of the invariant subspace |
|
||
|
|
c | which spans OP and exit. |
|
||
|
|
c %------------------------------------------------%
|
||
|
|
c
|
||
|
|
info = j - 1
|
||
|
|
call second (t1)
|
||
|
|
tcaitr = tcaitr + (t1 - t0)
|
||
|
|
ido = 99
|
||
|
|
go to 9000
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
40 continue
|
||
|
|
c
|
||
|
|
c %---------------------------------------------------------%
|
||
|
|
c | STEP 2: v_{j} = r_{j-1}/rnorm and p_{j} = p_{j}/rnorm |
|
||
|
|
c | Note that p_{j} = B*r_{j-1}. In order to avoid overflow |
|
||
|
|
c | when reciprocating a small RNORM, test against lower |
|
||
|
|
c | machine bound. |
|
||
|
|
c %---------------------------------------------------------%
|
||
|
|
c
|
||
|
|
call ccopy (n, resid, 1, v(1,j), 1)
|
||
|
|
if ( rnorm .ge. unfl) then
|
||
|
|
temp1 = rone / rnorm
|
||
|
|
call csscal (n, temp1, v(1,j), 1)
|
||
|
|
call csscal (n, temp1, workd(ipj), 1)
|
||
|
|
else
|
||
|
|
c
|
||
|
|
c %-----------------------------------------%
|
||
|
|
c | To scale both v_{j} and p_{j} carefully |
|
||
|
|
c | use LAPACK routine clascl |
|
||
|
|
c %-----------------------------------------%
|
||
|
|
c
|
||
|
|
call clascl ('General', i, i, rnorm, rone,
|
||
|
|
& n, 1, v(1,j), n, infol)
|
||
|
|
call clascl ('General', i, i, rnorm, rone,
|
||
|
|
& n, 1, workd(ipj), n, infol)
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %------------------------------------------------------%
|
||
|
|
c | STEP 3: r_{j} = OP*v_{j}; Note that p_{j} = B*v_{j} |
|
||
|
|
c | Note that this is not quite yet r_{j}. See STEP 4 |
|
||
|
|
c %------------------------------------------------------%
|
||
|
|
c
|
||
|
|
step3 = .true.
|
||
|
|
nopx = nopx + 1
|
||
|
|
call second (t2)
|
||
|
|
call ccopy (n, v(1,j), 1, workd(ivj), 1)
|
||
|
|
ipntr(1) = ivj
|
||
|
|
ipntr(2) = irj
|
||
|
|
ipntr(3) = ipj
|
||
|
|
ido = 1
|
||
|
|
c
|
||
|
|
c %-----------------------------------%
|
||
|
|
c | Exit in order to compute OP*v_{j} |
|
||
|
|
c %-----------------------------------%
|
||
|
|
c
|
||
|
|
go to 9000
|
||
|
|
50 continue
|
||
|
|
c
|
||
|
|
c %----------------------------------%
|
||
|
|
c | Back from reverse communication; |
|
||
|
|
c | WORKD(IRJ:IRJ+N-1) := OP*v_{j} |
|
||
|
|
c | if step3 = .true. |
|
||
|
|
c %----------------------------------%
|
||
|
|
c
|
||
|
|
call second (t3)
|
||
|
|
tmvopx = tmvopx + (t3 - t2)
|
||
|
|
|
||
|
|
step3 = .false.
|
||
|
|
c
|
||
|
|
c %------------------------------------------%
|
||
|
|
c | Put another copy of OP*v_{j} into RESID. |
|
||
|
|
c %------------------------------------------%
|
||
|
|
c
|
||
|
|
call ccopy (n, workd(irj), 1, resid, 1)
|
||
|
|
c
|
||
|
|
c %---------------------------------------%
|
||
|
|
c | STEP 4: Finish extending the Arnoldi |
|
||
|
|
c | factorization to length j. |
|
||
|
|
c %---------------------------------------%
|
||
|
|
c
|
||
|
|
call second (t2)
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
nbx = nbx + 1
|
||
|
|
step4 = .true.
|
||
|
|
ipntr(1) = irj
|
||
|
|
ipntr(2) = ipj
|
||
|
|
ido = 2
|
||
|
|
c
|
||
|
|
c %-------------------------------------%
|
||
|
|
c | Exit in order to compute B*OP*v_{j} |
|
||
|
|
c %-------------------------------------%
|
||
|
|
c
|
||
|
|
go to 9000
|
||
|
|
else if (bmat .eq. 'I') then
|
||
|
|
call ccopy (n, resid, 1, workd(ipj), 1)
|
||
|
|
end if
|
||
|
|
60 continue
|
||
|
|
c
|
||
|
|
c %----------------------------------%
|
||
|
|
c | Back from reverse communication; |
|
||
|
|
c | WORKD(IPJ:IPJ+N-1) := B*OP*v_{j} |
|
||
|
|
c | if step4 = .true. |
|
||
|
|
c %----------------------------------%
|
||
|
|
c
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
call second (t3)
|
||
|
|
tmvbx = tmvbx + (t3 - t2)
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
step4 = .false.
|
||
|
|
c
|
||
|
|
c %-------------------------------------%
|
||
|
|
c | The following is needed for STEP 5. |
|
||
|
|
c | Compute the B-norm of OP*v_{j}. |
|
||
|
|
c %-------------------------------------%
|
||
|
|
c
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
cnorm = cdotc (n, resid, 1, workd(ipj), 1)
|
||
|
|
wnorm = sqrt( slapy2(real(cnorm),aimag(cnorm)) )
|
||
|
|
else if (bmat .eq. 'I') then
|
||
|
|
wnorm = scnrm2(n, resid, 1)
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %-----------------------------------------%
|
||
|
|
c | Compute the j-th residual corresponding |
|
||
|
|
c | to the j step factorization. |
|
||
|
|
c | Use Classical Gram Schmidt and compute: |
|
||
|
|
c | w_{j} <- V_{j}^T * B * OP * v_{j} |
|
||
|
|
c | r_{j} <- OP*v_{j} - V_{j} * w_{j} |
|
||
|
|
c %-----------------------------------------%
|
||
|
|
c
|
||
|
|
c
|
||
|
|
c %------------------------------------------%
|
||
|
|
c | Compute the j Fourier coefficients w_{j} |
|
||
|
|
c | WORKD(IPJ:IPJ+N-1) contains B*OP*v_{j}. |
|
||
|
|
c %------------------------------------------%
|
||
|
|
c
|
||
|
|
call cgemv ('C', n, j, one, v, ldv, workd(ipj), 1,
|
||
|
|
& zero, h(1,j), 1)
|
||
|
|
c
|
||
|
|
c %--------------------------------------%
|
||
|
|
c | Orthogonalize r_{j} against V_{j}. |
|
||
|
|
c | RESID contains OP*v_{j}. See STEP 3. |
|
||
|
|
c %--------------------------------------%
|
||
|
|
c
|
||
|
|
call cgemv ('N', n, j, -one, v, ldv, h(1,j), 1,
|
||
|
|
& one, resid, 1)
|
||
|
|
c
|
||
|
|
if (j .gt. 1) h(j,j-1) = cmplx(betaj, rzero)
|
||
|
|
c
|
||
|
|
call second (t4)
|
||
|
|
c
|
||
|
|
orth1 = .true.
|
||
|
|
c
|
||
|
|
call second (t2)
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
nbx = nbx + 1
|
||
|
|
call ccopy (n, resid, 1, workd(irj), 1)
|
||
|
|
ipntr(1) = irj
|
||
|
|
ipntr(2) = ipj
|
||
|
|
ido = 2
|
||
|
|
c
|
||
|
|
c %----------------------------------%
|
||
|
|
c | Exit in order to compute B*r_{j} |
|
||
|
|
c %----------------------------------%
|
||
|
|
c
|
||
|
|
go to 9000
|
||
|
|
else if (bmat .eq. 'I') then
|
||
|
|
call ccopy (n, resid, 1, workd(ipj), 1)
|
||
|
|
end if
|
||
|
|
70 continue
|
||
|
|
c
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c | Back from reverse communication if ORTH1 = .true. |
|
||
|
|
c | WORKD(IPJ:IPJ+N-1) := B*r_{j}. |
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
call second (t3)
|
||
|
|
tmvbx = tmvbx + (t3 - t2)
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
orth1 = .false.
|
||
|
|
c
|
||
|
|
c %------------------------------%
|
||
|
|
c | Compute the B-norm of r_{j}. |
|
||
|
|
c %------------------------------%
|
||
|
|
c
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
cnorm = cdotc (n, resid, 1, workd(ipj), 1)
|
||
|
|
rnorm = sqrt( slapy2(real(cnorm),aimag(cnorm)) )
|
||
|
|
else if (bmat .eq. 'I') then
|
||
|
|
rnorm = scnrm2(n, resid, 1)
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %-----------------------------------------------------------%
|
||
|
|
c | STEP 5: Re-orthogonalization / Iterative refinement phase |
|
||
|
|
c | Maximum NITER_ITREF tries. |
|
||
|
|
c | |
|
||
|
|
c | s = V_{j}^T * B * r_{j} |
|
||
|
|
c | r_{j} = r_{j} - V_{j}*s |
|
||
|
|
c | alphaj = alphaj + s_{j} |
|
||
|
|
c | |
|
||
|
|
c | The stopping criteria used for iterative refinement is |
|
||
|
|
c | discussed in Parlett's book SEP, page 107 and in Gragg & |
|
||
|
|
c | Reichel ACM TOMS paper; Algorithm 686, Dec. 1990. |
|
||
|
|
c | Determine if we need to correct the residual. The goal is |
|
||
|
|
c | to enforce ||v(:,1:j)^T * r_{j}|| .le. eps * || r_{j} || |
|
||
|
|
c | The following test determines whether the sine of the |
|
||
|
|
c | angle between OP*x and the computed residual is less |
|
||
|
|
c | than or equal to 0.717. |
|
||
|
|
c %-----------------------------------------------------------%
|
||
|
|
c
|
||
|
|
if ( rnorm .gt. 0.717*wnorm ) go to 100
|
||
|
|
c
|
||
|
|
iter = 0
|
||
|
|
nrorth = nrorth + 1
|
||
|
|
c
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c | Enter the Iterative refinement phase. If further |
|
||
|
|
c | refinement is necessary, loop back here. The loop |
|
||
|
|
c | variable is ITER. Perform a step of Classical |
|
||
|
|
c | Gram-Schmidt using all the Arnoldi vectors V_{j} |
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c
|
||
|
|
80 continue
|
||
|
|
c
|
||
|
|
if (msglvl .gt. 2) then
|
||
|
|
rtemp(1) = wnorm
|
||
|
|
rtemp(2) = rnorm
|
||
|
|
call svout (logfil, 2, rtemp, ndigit,
|
||
|
|
& '_naitr: re-orthogonalization; wnorm and rnorm are')
|
||
|
|
call cvout (logfil, j, h(1,j), ndigit,
|
||
|
|
& '_naitr: j-th column of H')
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %----------------------------------------------------%
|
||
|
|
c | Compute V_{j}^T * B * r_{j}. |
|
||
|
|
c | WORKD(IRJ:IRJ+J-1) = v(:,1:J)'*WORKD(IPJ:IPJ+N-1). |
|
||
|
|
c %----------------------------------------------------%
|
||
|
|
c
|
||
|
|
call cgemv ('C', n, j, one, v, ldv, workd(ipj), 1,
|
||
|
|
& zero, workd(irj), 1)
|
||
|
|
c
|
||
|
|
c %---------------------------------------------%
|
||
|
|
c | Compute the correction to the residual: |
|
||
|
|
c | r_{j} = r_{j} - V_{j} * WORKD(IRJ:IRJ+J-1). |
|
||
|
|
c | The correction to H is v(:,1:J)*H(1:J,1:J) |
|
||
|
|
c | + v(:,1:J)*WORKD(IRJ:IRJ+J-1)*e'_j. |
|
||
|
|
c %---------------------------------------------%
|
||
|
|
c
|
||
|
|
call cgemv ('N', n, j, -one, v, ldv, workd(irj), 1,
|
||
|
|
& one, resid, 1)
|
||
|
|
call caxpy (j, one, workd(irj), 1, h(1,j), 1)
|
||
|
|
c
|
||
|
|
orth2 = .true.
|
||
|
|
call second (t2)
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
nbx = nbx + 1
|
||
|
|
call ccopy (n, resid, 1, workd(irj), 1)
|
||
|
|
ipntr(1) = irj
|
||
|
|
ipntr(2) = ipj
|
||
|
|
ido = 2
|
||
|
|
c
|
||
|
|
c %-----------------------------------%
|
||
|
|
c | Exit in order to compute B*r_{j}. |
|
||
|
|
c | r_{j} is the corrected residual. |
|
||
|
|
c %-----------------------------------%
|
||
|
|
c
|
||
|
|
go to 9000
|
||
|
|
else if (bmat .eq. 'I') then
|
||
|
|
call ccopy (n, resid, 1, workd(ipj), 1)
|
||
|
|
end if
|
||
|
|
90 continue
|
||
|
|
c
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c | Back from reverse communication if ORTH2 = .true. |
|
||
|
|
c %---------------------------------------------------%
|
||
|
|
c
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
call second (t3)
|
||
|
|
tmvbx = tmvbx + (t3 - t2)
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %-----------------------------------------------------%
|
||
|
|
c | Compute the B-norm of the corrected residual r_{j}. |
|
||
|
|
c %-----------------------------------------------------%
|
||
|
|
c
|
||
|
|
if (bmat .eq. 'G') then
|
||
|
|
cnorm = cdotc (n, resid, 1, workd(ipj), 1)
|
||
|
|
rnorm1 = sqrt( slapy2(real(cnorm),aimag(cnorm)) )
|
||
|
|
else if (bmat .eq. 'I') then
|
||
|
|
rnorm1 = scnrm2(n, resid, 1)
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
if (msglvl .gt. 0 .and. iter .gt. 0 ) then
|
||
|
|
call ivout (logfil, 1, j, ndigit,
|
||
|
|
& '_naitr: Iterative refinement for Arnoldi residual')
|
||
|
|
if (msglvl .gt. 2) then
|
||
|
|
rtemp(1) = rnorm
|
||
|
|
rtemp(2) = rnorm1
|
||
|
|
call svout (logfil, 2, rtemp, ndigit,
|
||
|
|
& '_naitr: iterative refinement ; rnorm and rnorm1 are')
|
||
|
|
end if
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %-----------------------------------------%
|
||
|
|
c | Determine if we need to perform another |
|
||
|
|
c | step of re-orthogonalization. |
|
||
|
|
c %-----------------------------------------%
|
||
|
|
c
|
||
|
|
if ( rnorm1 .gt. 0.717*rnorm ) then
|
||
|
|
c
|
||
|
|
c %---------------------------------------%
|
||
|
|
c | No need for further refinement. |
|
||
|
|
c | The cosine of the angle between the |
|
||
|
|
c | corrected residual vector and the old |
|
||
|
|
c | residual vector is greater than 0.717 |
|
||
|
|
c | In other words the corrected residual |
|
||
|
|
c | and the old residual vector share an |
|
||
|
|
c | angle of less than arcCOS(0.717) |
|
||
|
|
c %---------------------------------------%
|
||
|
|
c
|
||
|
|
rnorm = rnorm1
|
||
|
|
c
|
||
|
|
else
|
||
|
|
c
|
||
|
|
c %-------------------------------------------%
|
||
|
|
c | Another step of iterative refinement step |
|
||
|
|
c | is required. NITREF is used by stat.h |
|
||
|
|
c %-------------------------------------------%
|
||
|
|
c
|
||
|
|
nitref = nitref + 1
|
||
|
|
rnorm = rnorm1
|
||
|
|
iter = iter + 1
|
||
|
|
if (iter .le. 1) go to 80
|
||
|
|
c
|
||
|
|
c %-------------------------------------------------%
|
||
|
|
c | Otherwise RESID is numerically in the span of V |
|
||
|
|
c %-------------------------------------------------%
|
||
|
|
c
|
||
|
|
do 95 jj = 1, n
|
||
|
|
resid(jj) = zero
|
||
|
|
95 continue
|
||
|
|
rnorm = rzero
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %----------------------------------------------%
|
||
|
|
c | Branch here directly if iterative refinement |
|
||
|
|
c | wasn't necessary or after at most NITER_REF |
|
||
|
|
c | steps of iterative refinement. |
|
||
|
|
c %----------------------------------------------%
|
||
|
|
c
|
||
|
|
100 continue
|
||
|
|
c
|
||
|
|
rstart = .false.
|
||
|
|
orth2 = .false.
|
||
|
|
c
|
||
|
|
call second (t5)
|
||
|
|
titref = titref + (t5 - t4)
|
||
|
|
c
|
||
|
|
c %------------------------------------%
|
||
|
|
c | STEP 6: Update j = j+1; Continue |
|
||
|
|
c %------------------------------------%
|
||
|
|
c
|
||
|
|
j = j + 1
|
||
|
|
if (j .gt. k+np) then
|
||
|
|
call second (t1)
|
||
|
|
tcaitr = tcaitr + (t1 - t0)
|
||
|
|
ido = 99
|
||
|
|
do 110 i = max(1,k), k+np-1
|
||
|
|
c
|
||
|
|
c %--------------------------------------------%
|
||
|
|
c | Check for splitting and deflation. |
|
||
|
|
c | Use a standard test as in the QR algorithm |
|
||
|
|
c | REFERENCE: LAPACK subroutine clahqr |
|
||
|
|
c %--------------------------------------------%
|
||
|
|
c
|
||
|
|
tst1 = slapy2(real(h(i,i)),aimag(h(i,i)))
|
||
|
|
& + slapy2(real(h(i+1,i+1)), aimag(h(i+1,i+1)))
|
||
|
|
if( tst1.eq.real(zero) )
|
||
|
|
& tst1 = clanhs( '1', k+np, h, ldh, workd(n+1) )
|
||
|
|
if( slapy2(real(h(i+1,i)),aimag(h(i+1,i))) .le.
|
||
|
|
& max( ulp*tst1, smlnum ) )
|
||
|
|
& h(i+1,i) = zero
|
||
|
|
110 continue
|
||
|
|
c
|
||
|
|
if (msglvl .gt. 2) then
|
||
|
|
call cmout (logfil, k+np, k+np, h, ldh, ndigit,
|
||
|
|
& '_naitr: Final upper Hessenberg matrix H of order K+NP')
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
go to 9000
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %--------------------------------------------------------%
|
||
|
|
c | Loop back to extend the factorization by another step. |
|
||
|
|
c %--------------------------------------------------------%
|
||
|
|
c
|
||
|
|
go to 1000
|
||
|
|
c
|
||
|
|
c %---------------------------------------------------------------%
|
||
|
|
c | |
|
||
|
|
c | E N D O F M A I N I T E R A T I O N L O O P |
|
||
|
|
c | |
|
||
|
|
c %---------------------------------------------------------------%
|
||
|
|
c
|
||
|
|
9000 continue
|
||
|
|
return
|
||
|
|
c
|
||
|
|
c %---------------%
|
||
|
|
c | End of cnaitr |
|
||
|
|
c %---------------%
|
||
|
|
c
|
||
|
|
end
|