517 lines
18 KiB
FortranFixed
517 lines
18 KiB
FortranFixed
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c-----------------------------------------------------------------------
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c\BeginDoc
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c
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c\Name: ssapps
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c
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c\Description:
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c Given the Arnoldi factorization
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c
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c A*V_{k} - V_{k}*H_{k} = r_{k+p}*e_{k+p}^T,
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c
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c apply NP shifts implicitly resulting in
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c
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c A*(V_{k}*Q) - (V_{k}*Q)*(Q^T* H_{k}*Q) = r_{k+p}*e_{k+p}^T * Q
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c
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c where Q is an orthogonal matrix of order KEV+NP. Q is the product of
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c rotations resulting from the NP bulge chasing sweeps. The updated Arnoldi
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c factorization becomes:
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c
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c A*VNEW_{k} - VNEW_{k}*HNEW_{k} = rnew_{k}*e_{k}^T.
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c
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c\Usage:
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c call ssapps
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c ( N, KEV, NP, SHIFT, V, LDV, H, LDH, RESID, Q, LDQ, WORKD )
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c
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c\Arguments
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c N Integer. (INPUT)
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c Problem size, i.e. dimension of matrix A.
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c
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c KEV Integer. (INPUT)
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c INPUT: KEV+NP is the size of the input matrix H.
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c OUTPUT: KEV is the size of the updated matrix HNEW.
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c
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c NP Integer. (INPUT)
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c Number of implicit shifts to be applied.
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c
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c SHIFT Real array of length NP. (INPUT)
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c The shifts to be applied.
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c
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c V Real N by (KEV+NP) array. (INPUT/OUTPUT)
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c INPUT: V contains the current KEV+NP Arnoldi vectors.
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c OUTPUT: VNEW = V(1:n,1:KEV); the updated Arnoldi vectors
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c are in the first KEV columns of V.
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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 Real (KEV+NP) by 2 array. (INPUT/OUTPUT)
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c INPUT: H contains the symmetric tridiagonal matrix of the
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c Arnoldi factorization with the subdiagonal in the 1st column
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c starting at H(2,1) and the main diagonal in the 2nd column.
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c OUTPUT: H contains the updated tridiagonal matrix in the
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c KEV leading submatrix.
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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 RESID Real array of length (N). (INPUT/OUTPUT)
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c INPUT: RESID contains the the residual vector r_{k+p}.
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c OUTPUT: RESID is the updated residual vector rnew_{k}.
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c
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c Q Real KEV+NP by KEV+NP work array. (WORKSPACE)
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c Work array used to accumulate the rotations during the bulge
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c chase sweep.
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c
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c LDQ Integer. (INPUT)
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c Leading dimension of Q exactly as declared in the calling
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c program.
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c
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c WORKD Real work array of length 2*N. (WORKSPACE)
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c Distributed array used in the application of the accumulated
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c orthogonal matrix Q.
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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 real
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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 ivout ARPACK utility routine that prints integers.
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c second ARPACK utility routine for timing.
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c svout ARPACK utility routine that prints vectors.
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c slamch LAPACK routine that determines machine constants.
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c slartg LAPACK Givens rotation construction routine.
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c slacpy LAPACK matrix copy routine.
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c slaset LAPACK matrix initialization routine.
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c sgemv Level 2 BLAS routine for matrix vector multiplication.
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c saxpy Level 1 BLAS that computes a vector triad.
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c scopy Level 1 BLAS that copies one vector to another.
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c sscal Level 1 BLAS that scales 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\Revision history:
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c 12/16/93: Version ' 2.4'
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c
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c\SCCS Information: @(#)
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c FILE: sapps.F SID: 2.6 DATE OF SID: 3/28/97 RELEASE: 2
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c
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c\Remarks
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c 1. In this version, each shift is applied to all the subblocks of
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c the tridiagonal matrix H and not just to the submatrix that it
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c comes from. This routine assumes that the subdiagonal elements
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c of H that are stored in h(1:kev+np,1) are nonegative upon input
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c and enforce this condition upon output. This version incorporates
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c deflation. See code for documentation.
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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 ssapps
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& ( n, kev, np, shift, v, ldv, h, ldh, resid, q, ldq, workd )
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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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integer kev, ldh, ldq, ldv, n, np
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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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Real
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& h(ldh,2), q(ldq,kev+np), resid(n), shift(np),
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& v(ldv,kev+np), workd(2*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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Real
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& one, zero
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parameter (one = 1.0E+0, zero = 0.0E+0)
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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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integer i, iend, istart, itop, j, jj, kplusp, msglvl
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logical first
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Real
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& a1, a2, a3, a4, big, c, epsmch, f, g, r, s
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save epsmch, first
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c
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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 saxpy, scopy, sscal, slacpy, slartg, slaset, svout,
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& ivout, second, sgemv
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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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Real
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& slamch
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external slamch
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c
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c %----------------------%
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c | Intrinsics Functions |
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c %----------------------%
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c
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intrinsic abs
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c
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c %----------------%
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c | Data statments |
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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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epsmch = slamch('Epsilon-Machine')
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first = .false.
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end if
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itop = 1
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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 = msapps
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c
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kplusp = kev + np
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c
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c %----------------------------------------------%
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c | Initialize Q to the identity matrix of order |
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c | kplusp used to accumulate the rotations. |
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c %----------------------------------------------%
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c
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call slaset ('All', kplusp, kplusp, zero, one, q, ldq)
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c
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c %----------------------------------------------%
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c | Quick return if there are no shifts to apply |
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c %----------------------------------------------%
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c
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if (np .eq. 0) go to 9000
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c
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c %----------------------------------------------------------%
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c | Apply the np shifts implicitly. Apply each shift to the |
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c | whole matrix and not just to the submatrix from which it |
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c | comes. |
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c %----------------------------------------------------------%
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c
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do 90 jj = 1, np
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c
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istart = itop
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c
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c %----------------------------------------------------------%
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c | Check for splitting and deflation. Currently we consider |
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c | an off-diagonal element h(i+1,1) negligible if |
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c | h(i+1,1) .le. epsmch*( |h(i,2)| + |h(i+1,2)| ) |
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c | for i=1:KEV+NP-1. |
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c | If above condition tests true then we set h(i+1,1) = 0. |
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c | Note that h(1:KEV+NP,1) are assumed to be non negative. |
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c %----------------------------------------------------------%
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c
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20 continue
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c
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c %------------------------------------------------%
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c | The following loop exits early if we encounter |
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c | a negligible off diagonal element. |
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c %------------------------------------------------%
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c
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do 30 i = istart, kplusp-1
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big = abs(h(i,2)) + abs(h(i+1,2))
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if (h(i+1,1) .le. epsmch*big) then
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if (msglvl .gt. 0) then
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call ivout (logfil, 1, i, ndigit,
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& '_sapps: deflation at row/column no.')
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call ivout (logfil, 1, jj, ndigit,
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& '_sapps: occured before shift number.')
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call svout (logfil, 1, h(i+1,1), ndigit,
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& '_sapps: the corresponding off diagonal element')
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end if
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h(i+1,1) = zero
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iend = i
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go to 40
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end if
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30 continue
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iend = kplusp
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40 continue
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c
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if (istart .lt. iend) then
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c
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c %--------------------------------------------------------%
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c | Construct the plane rotation G'(istart,istart+1,theta) |
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c | that attempts to drive h(istart+1,1) to zero. |
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c %--------------------------------------------------------%
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c
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f = h(istart,2) - shift(jj)
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g = h(istart+1,1)
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call slartg (f, g, c, s, r)
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c
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c %-------------------------------------------------------%
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c | Apply rotation to the left and right of H; |
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c | H <- G' * H * G, where G = G(istart,istart+1,theta). |
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c | This will create a "bulge". |
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c %-------------------------------------------------------%
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c
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a1 = c*h(istart,2) + s*h(istart+1,1)
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a2 = c*h(istart+1,1) + s*h(istart+1,2)
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a4 = c*h(istart+1,2) - s*h(istart+1,1)
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a3 = c*h(istart+1,1) - s*h(istart,2)
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h(istart,2) = c*a1 + s*a2
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h(istart+1,2) = c*a4 - s*a3
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h(istart+1,1) = c*a3 + s*a4
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c
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c %----------------------------------------------------%
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c | Accumulate the rotation in the matrix Q; Q <- Q*G |
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c %----------------------------------------------------%
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c
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do 60 j = 1, min(istart+jj,kplusp)
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a1 = c*q(j,istart) + s*q(j,istart+1)
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q(j,istart+1) = - s*q(j,istart) + c*q(j,istart+1)
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q(j,istart) = a1
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60 continue
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c
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c
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c %----------------------------------------------%
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c | The following loop chases the bulge created. |
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c | Note that the previous rotation may also be |
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c | done within the following loop. But it is |
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c | kept separate to make the distinction among |
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c | the bulge chasing sweeps and the first plane |
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c | rotation designed to drive h(istart+1,1) to |
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c | zero. |
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c %----------------------------------------------%
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c
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do 70 i = istart+1, iend-1
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c
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c %----------------------------------------------%
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c | Construct the plane rotation G'(i,i+1,theta) |
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c | that zeros the i-th bulge that was created |
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c | by G(i-1,i,theta). g represents the bulge. |
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c %----------------------------------------------%
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c
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f = h(i,1)
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g = s*h(i+1,1)
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c
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c %----------------------------------%
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c | Final update with G(i-1,i,theta) |
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c %----------------------------------%
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c
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h(i+1,1) = c*h(i+1,1)
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call slartg (f, g, c, s, r)
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c
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c %-------------------------------------------%
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c | The following ensures that h(1:iend-1,1), |
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c | the first iend-2 off diagonal of elements |
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c | H, remain non negative. |
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c %-------------------------------------------%
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c
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if (r .lt. zero) then
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r = -r
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c = -c
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s = -s
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end if
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c
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c %--------------------------------------------%
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c | Apply rotation to the left and right of H; |
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c | H <- G * H * G', where G = G(i,i+1,theta) |
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c %--------------------------------------------%
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c
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h(i,1) = r
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c
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a1 = c*h(i,2) + s*h(i+1,1)
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a2 = c*h(i+1,1) + s*h(i+1,2)
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a3 = c*h(i+1,1) - s*h(i,2)
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a4 = c*h(i+1,2) - s*h(i+1,1)
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c
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|
h(i,2) = c*a1 + s*a2
|
||
|
|
h(i+1,2) = c*a4 - s*a3
|
||
|
|
h(i+1,1) = c*a3 + s*a4
|
||
|
|
c
|
||
|
|
c %----------------------------------------------------%
|
||
|
|
c | Accumulate the rotation in the matrix Q; Q <- Q*G |
|
||
|
|
c %----------------------------------------------------%
|
||
|
|
c
|
||
|
|
do 50 j = 1, min( i+jj, kplusp )
|
||
|
|
a1 = c*q(j,i) + s*q(j,i+1)
|
||
|
|
q(j,i+1) = - s*q(j,i) + c*q(j,i+1)
|
||
|
|
q(j,i) = a1
|
||
|
|
50 continue
|
||
|
|
c
|
||
|
|
70 continue
|
||
|
|
c
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %--------------------------%
|
||
|
|
c | Update the block pointer |
|
||
|
|
c %--------------------------%
|
||
|
|
c
|
||
|
|
istart = iend + 1
|
||
|
|
c
|
||
|
|
c %------------------------------------------%
|
||
|
|
c | Make sure that h(iend,1) is non-negative |
|
||
|
|
c | If not then set h(iend,1) <-- -h(iend,1) |
|
||
|
|
c | and negate the last column of Q. |
|
||
|
|
c | We have effectively carried out a |
|
||
|
|
c | similarity on transformation H |
|
||
|
|
c %------------------------------------------%
|
||
|
|
c
|
||
|
|
if (h(iend,1) .lt. zero) then
|
||
|
|
h(iend,1) = -h(iend,1)
|
||
|
|
call sscal(kplusp, -one, q(1,iend), 1)
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
c %--------------------------------------------------------%
|
||
|
|
c | Apply the same shift to the next block if there is any |
|
||
|
|
c %--------------------------------------------------------%
|
||
|
|
c
|
||
|
|
if (iend .lt. kplusp) go to 20
|
||
|
|
c
|
||
|
|
c %-----------------------------------------------------%
|
||
|
|
c | Check if we can increase the the start of the block |
|
||
|
|
c %-----------------------------------------------------%
|
||
|
|
c
|
||
|
|
do 80 i = itop, kplusp-1
|
||
|
|
if (h(i+1,1) .gt. zero) go to 90
|
||
|
|
itop = itop + 1
|
||
|
|
80 continue
|
||
|
|
c
|
||
|
|
c %-----------------------------------%
|
||
|
|
c | Finished applying the jj-th shift |
|
||
|
|
c %-----------------------------------%
|
||
|
|
c
|
||
|
|
90 continue
|
||
|
|
c
|
||
|
|
c %------------------------------------------%
|
||
|
|
c | All shifts have been applied. Check for |
|
||
|
|
c | more possible deflation that might occur |
|
||
|
|
c | after the last shift is applied. |
|
||
|
|
c %------------------------------------------%
|
||
|
|
c
|
||
|
|
do 100 i = itop, kplusp-1
|
||
|
|
big = abs(h(i,2)) + abs(h(i+1,2))
|
||
|
|
if (h(i+1,1) .le. epsmch*big) then
|
||
|
|
if (msglvl .gt. 0) then
|
||
|
|
call ivout (logfil, 1, i, ndigit,
|
||
|
|
& '_sapps: deflation at row/column no.')
|
||
|
|
call svout (logfil, 1, h(i+1,1), ndigit,
|
||
|
|
& '_sapps: the corresponding off diagonal element')
|
||
|
|
end if
|
||
|
|
h(i+1,1) = zero
|
||
|
|
end if
|
||
|
|
100 continue
|
||
|
|
c
|
||
|
|
c %-------------------------------------------------%
|
||
|
|
c | Compute the (kev+1)-st column of (V*Q) and |
|
||
|
|
c | temporarily store the result in WORKD(N+1:2*N). |
|
||
|
|
c | This is not necessary if h(kev+1,1) = 0. |
|
||
|
|
c %-------------------------------------------------%
|
||
|
|
c
|
||
|
|
if ( h(kev+1,1) .gt. zero )
|
||
|
|
& call sgemv ('N', n, kplusp, one, v, ldv,
|
||
|
|
& q(1,kev+1), 1, zero, workd(n+1), 1)
|
||
|
|
c
|
||
|
|
c %-------------------------------------------------------%
|
||
|
|
c | Compute column 1 to kev of (V*Q) in backward order |
|
||
|
|
c | taking advantage that Q is an upper triangular matrix |
|
||
|
|
c | with lower bandwidth np. |
|
||
|
|
c | Place results in v(:,kplusp-kev:kplusp) temporarily. |
|
||
|
|
c %-------------------------------------------------------%
|
||
|
|
c
|
||
|
|
do 130 i = 1, kev
|
||
|
|
call sgemv ('N', n, kplusp-i+1, one, v, ldv,
|
||
|
|
& q(1,kev-i+1), 1, zero, workd, 1)
|
||
|
|
call scopy (n, workd, 1, v(1,kplusp-i+1), 1)
|
||
|
|
130 continue
|
||
|
|
c
|
||
|
|
c %-------------------------------------------------%
|
||
|
|
c | Move v(:,kplusp-kev+1:kplusp) into v(:,1:kev). |
|
||
|
|
c %-------------------------------------------------%
|
||
|
|
c
|
||
|
|
call slacpy ('All', n, kev, v(1,np+1), ldv, v, ldv)
|
||
|
|
c
|
||
|
|
c %--------------------------------------------%
|
||
|
|
c | Copy the (kev+1)-st column of (V*Q) in the |
|
||
|
|
c | appropriate place if h(kev+1,1) .ne. zero. |
|
||
|
|
c %--------------------------------------------%
|
||
|
|
c
|
||
|
|
if ( h(kev+1,1) .gt. zero )
|
||
|
|
& call scopy (n, workd(n+1), 1, v(1,kev+1), 1)
|
||
|
|
c
|
||
|
|
c %-------------------------------------%
|
||
|
|
c | Update the residual vector: |
|
||
|
|
c | r <- sigmak*r + betak*v(:,kev+1) |
|
||
|
|
c | where |
|
||
|
|
c | sigmak = (e_{kev+p}'*Q)*e_{kev} |
|
||
|
|
c | betak = e_{kev+1}'*H*e_{kev} |
|
||
|
|
c %-------------------------------------%
|
||
|
|
c
|
||
|
|
call sscal (n, q(kplusp,kev), resid, 1)
|
||
|
|
if (h(kev+1,1) .gt. zero)
|
||
|
|
& call saxpy (n, h(kev+1,1), v(1,kev+1), 1, resid, 1)
|
||
|
|
c
|
||
|
|
if (msglvl .gt. 1) then
|
||
|
|
call svout (logfil, 1, q(kplusp,kev), ndigit,
|
||
|
|
& '_sapps: sigmak of the updated residual vector')
|
||
|
|
call svout (logfil, 1, h(kev+1,1), ndigit,
|
||
|
|
& '_sapps: betak of the updated residual vector')
|
||
|
|
call svout (logfil, kev, h(1,2), ndigit,
|
||
|
|
& '_sapps: updated main diagonal of H for next iteration')
|
||
|
|
if (kev .gt. 1) then
|
||
|
|
call svout (logfil, kev-1, h(2,1), ndigit,
|
||
|
|
& '_sapps: updated sub diagonal of H for next iteration')
|
||
|
|
end if
|
||
|
|
end if
|
||
|
|
c
|
||
|
|
call second (t1)
|
||
|
|
tsapps = tsapps + (t1 - t0)
|
||
|
|
c
|
||
|
|
9000 continue
|
||
|
|
return
|
||
|
|
c
|
||
|
|
c %---------------%
|
||
|
|
c | End of ssapps |
|
||
|
|
c %---------------%
|
||
|
|
c
|
||
|
|
end
|