c-----------------------------------------------------------------------
c\BeginDoc
c
c\Name: dnapps
c
c\Description:
c Given the Arnoldi factorization
c
c A*V_{k} - V_{k}*H_{k} = r_{k+p}*e_{k+p}^T,
c
c apply NP implicit shifts resulting in
c
c A*(V_{k}*Q) - (V_{k}*Q)*(Q^T* H_{k}*Q) = r_{k+p}*e_{k+p}^T * Q
c
c where Q is an orthogonal matrix which is the product of rotations
c and reflections resulting from the NP bulge chage sweeps.
c The updated Arnoldi factorization becomes:
c
c A*VNEW_{k} - VNEW_{k}*HNEW_{k} = rnew_{k}*e_{k}^T.
c
c\Usage:
c call dnapps
c ( N, KEV, NP, SHIFTR, SHIFTI, V, LDV, H, LDH, RESID, Q, LDQ,
c WORKL, WORKD )
c
c\Arguments
c N Integer. (INPUT)
c Problem size, i.e. size of matrix A.
c
c KEV Integer. (INPUT/OUTPUT)
c KEV+NP is the size of the input matrix H.
c KEV is the size of the updated matrix HNEW. KEV is only
c updated on ouput when fewer than NP shifts are applied in
c order to keep the conjugate pair together.
c
c NP Integer. (INPUT)
c Number of implicit shifts to be applied.
c
c SHIFTR, Double precision array of length NP. (INPUT)
c SHIFTI Real and imaginary part of the shifts to be applied.
c Upon, entry to dnapps, the shifts must be sorted so that the
c conjugate pairs are in consecutive locations.
c
c V Double precision N by (KEV+NP) array. (INPUT/OUTPUT)
c On INPUT, V contains the current KEV+NP Arnoldi vectors.
c On OUTPUT, V contains the updated KEV Arnoldi vectors
c in the first KEV columns of V.
c
c LDV Integer. (INPUT)
c Leading dimension of V exactly as declared in the calling
c program.
c
c H Double precision (KEV+NP) by (KEV+NP) array. (INPUT/OUTPUT)
c On INPUT, H contains the current KEV+NP by KEV+NP upper
c Hessenber matrix of the Arnoldi factorization.
c On OUTPUT, H contains the updated KEV by KEV upper Hessenberg
c matrix in the KEV leading submatrix.
c
c LDH Integer. (INPUT)
c Leading dimension of H exactly as declared in the calling
c program.
c
c RESID Double precision array of length N. (INPUT/OUTPUT)
c On INPUT, RESID contains the the residual vector r_{k+p}.
c On OUTPUT, RESID is the update residual vector rnew_{k}
c in the first KEV locations.
c
c Q Double precision KEV+NP by KEV+NP work array. (WORKSPACE)
c Work array used to accumulate the rotations and reflections
c during the bulge chase sweep.
c
c LDQ Integer. (INPUT)
c Leading dimension of Q exactly as declared in the calling
c program.
c
c WORKL Double precision work array of length (KEV+NP). (WORKSPACE)
c Private (replicated) array on each PE or array allocated on
c the front end.
c
c WORKD Double precision work array of length 2*N. (WORKSPACE)
c Distributed array used in the application of the accumulated
c orthogonal matrix Q.
c
c\EndDoc
c
c-----------------------------------------------------------------------
c
c\BeginLib
c
c\Local variables:
c xxxxxx real
c
c\References:
c 1. D.C. Sorensen, "Implicit Application of Polynomial Filters in
c a k-Step Arnoldi Method", SIAM J. Matr. Anal. Apps., 13 (1992),
c pp 357-385.
c
c\Routines called:
c ivout ARPACK utility routine that prints integers.
c second ARPACK utility routine for timing.
c dmout ARPACK utility routine that prints matrices.
c dvout ARPACK utility routine that prints vectors.
c dlabad LAPACK routine that computes machine constants.
c dlacpy LAPACK matrix copy routine.
c dlamch LAPACK routine that determines machine constants.
c dlanhs LAPACK routine that computes various norms of a matrix.
c dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully.
c dlarf LAPACK routine that applies Householder reflection to
c a matrix.
c dlarfg LAPACK Householder reflection construction routine.
c dlartg LAPACK Givens rotation construction routine.
c dlaset LAPACK matrix initialization routine.
c dgemv Level 2 BLAS routine for matrix vector multiplication.
c daxpy Level 1 BLAS that computes a vector triad.
c dcopy Level 1 BLAS that copies one vector to another .
c dscal Level 1 BLAS that scales a vector.
c
c\Author
c Danny Sorensen Phuong Vu
c Richard Lehoucq CRPC / Rice University
c Dept. of Computational & Houston, Texas
c Applied Mathematics
c Rice University
c Houston, Texas
c
c\Revision history:
c xx/xx/92: Version ' 2.1'
c
c\SCCS Information: @(#)
c FILE: napps.F SID: 2.3 DATE OF SID: 4/20/96 RELEASE: 2
c
c\Remarks
c 1. In this version, each shift is applied to all the sublocks of
c the Hessenberg matrix H and not just to the submatrix that it
c comes from. Deflation as in LAPACK routine dlahqr (QR algorithm
c for upper Hessenberg matrices ) is used.
c The subdiagonals of H are enforced to be non-negative.
c
c\EndLib
c
c-----------------------------------------------------------------------
c
subroutine dnapps
& ( n, kev, np, shiftr, shifti, v, ldv, h, ldh, resid, q, ldq,
& workl, workd )
c
c %----------------------------------------------------%
c | Include files for debugging and timing information |
c %----------------------------------------------------%
c
include 'debug.h'
include 'stat.h'
c
c %------------------%
c | Scalar Arguments |
c %------------------%
c
integer kev, ldh, ldq, ldv, n, np
c
c %-----------------%
c | Array Arguments |
c %-----------------%
c
Double precision
& h(ldh,kev+np), resid(n), shifti(np), shiftr(np),
& v(ldv,kev+np), q(ldq,kev+np), workd(2*n), workl(kev+np)
c
c %------------%
c | Parameters |
c %------------%
c
Double precision
& one, zero
parameter (one = 1.0D+0, zero = 0.0D+0)
c
c %------------------------%
c | Local Scalars & Arrays |
c %------------------------%
c
integer i, iend, ir, istart, j, jj, kplusp, msglvl, nr
logical cconj, first
Double precision
& c, f, g, h11, h12, h21, h22, h32, ovfl, r, s, sigmai,
& sigmar, smlnum, ulp, unfl, u(3), t, tau, tst1
save first, ovfl, smlnum, ulp, unfl
c
c %----------------------%
c | External Subroutines |
c %----------------------%
c
external daxpy, dcopy, dscal, dlacpy, dlarfg, dlarf,
& dlaset, dlabad, second, dlartg
c
c %--------------------%
c | External Functions |
c %--------------------%
c
Double precision
& dlamch, dlanhs, dlapy2
external dlamch, dlanhs, dlapy2
c
c %----------------------%
c | Intrinsics Functions |
c %----------------------%
c
intrinsic abs, max, min
c
c %----------------%
c | Data statments |
c %----------------%
c
data first / .true. /
c
c %-----------------------%
c | Executable Statements |
c %-----------------------%
c
if (first) then
c
c %-----------------------------------------------%
c | Set machine-dependent constants for the |
c | stopping criterion. If norm(H) <= sqrt(OVFL), |
c | overflow should not occur. |
c | REFERENCE: LAPACK subroutine dlahqr |
c %-----------------------------------------------%
c
unfl = dlamch( 'safe minimum' )
ovfl = one / unfl
call dlabad( unfl, ovfl )
ulp = dlamch( 'precision' )
smlnum = unfl*( n / ulp )
first = .false.
end if
c
c %-------------------------------%
c | Initialize timing statistics |
c | & message level for debugging |
c %-------------------------------%
c
call second (t0)
msglvl = mnapps
kplusp = kev + np
c
c %--------------------------------------------%
c | Initialize Q to the identity to accumulate |
c | the rotations and reflections |
c %--------------------------------------------%
c
call dlaset ('All', kplusp, kplusp, zero, one, q, ldq)
c
c %----------------------------------------------%
c | Quick return if there are no shifts to apply |
c %----------------------------------------------%
c
if (np .eq. 0) go to 9000
c
c %----------------------------------------------%
c | Chase the bulge with the application of each |
c | implicit shift. Each shift is applied to the |
c | whole matrix including each block. |
c %----------------------------------------------%
c
cconj = .false.
do 110 jj = 1, np
sigmar = shiftr(jj)
sigmai = shifti(jj)
c
if (msglvl .gt. 2 ) then
call ivout (logfil, 1, [jj], ndigit,
& '_napps: shift number.')
call dvout (logfil, 1, [sigmar], ndigit,
& '_napps: The real part of the shift ')
call dvout (logfil, 1, [sigmai], ndigit,
& '_napps: The imaginary part of the shift ')
end if
c
c %-------------------------------------------------%
c | The following set of conditionals is necessary |
c | in order that complex conjugate pairs of shifts |
c | are applied together or not at all. |
c %-------------------------------------------------%
c
if ( cconj ) then
c
c %-----------------------------------------%
c | cconj = .true. means the previous shift |
c | had non-zero imaginary part. |
c %-----------------------------------------%
c
cconj = .false.
go to 110
else if ( jj .lt. np .and. abs( sigmai ) .gt. zero ) then
c
c %------------------------------------%
c | Start of a complex conjugate pair. |
c %------------------------------------%
c
cconj = .true.
else if ( jj .eq. np .and. abs( sigmai ) .gt. zero ) then
c
c %----------------------------------------------%
c | The last shift has a nonzero imaginary part. |
c | Don't apply it; thus the order of the |
c | compressed H is order KEV+1 since only np-1 |
c | were applied. |
c %----------------------------------------------%
c
kev = kev + 1
go to 110
end if
istart = 1
20 continue
c
c %--------------------------------------------------%
c | if sigmai = 0 then |
c | Apply the jj-th shift ... |
c | else |
c | Apply the jj-th and (jj+1)-th together ... |
c | (Note that jj < np at this point in the code) |
c | end |
c | to the current block of H. The next do loop |
c | determines the current block ; |
c %--------------------------------------------------%
c
do 30 i = istart, kplusp-1
c
c %----------------------------------------%
c | Check for splitting and deflation. Use |
c | a standard test as in the QR algorithm |
c | REFERENCE: LAPACK subroutine dlahqr |
c %----------------------------------------%
c
tst1 = abs( h( i, i ) ) + abs( h( i+1, i+1 ) )
if( tst1.eq.zero )
& tst1 = dlanhs( '1', kplusp-jj+1, h, ldh, workl )
if( abs( h( i+1,i ) ).le.max( ulp*tst1, smlnum ) ) then
if (msglvl .gt. 0) then
call ivout (logfil, 1, [i], ndigit,
& '_napps: matrix splitting at row/column no.')
call ivout (logfil, 1, [jj], ndigit,
& '_napps: matrix splitting with shift number.')
call dvout (logfil, 1, h(i+1,i), ndigit,
& '_napps: off diagonal element.')
end if
iend = i
h(i+1,i) = zero
go to 40
end if
30 continue
iend = kplusp
40 continue
c
if (msglvl .gt. 2) then
call ivout (logfil, 1, [istart], ndigit,
& '_napps: Start of current block ')
call ivout (logfil, 1, [iend], ndigit,
& '_napps: End of current block ')
end if
c
c %------------------------------------------------%
c | No reason to apply a shift to block of order 1 |
c %------------------------------------------------%
c
if ( istart .eq. iend ) go to 100
c
c %------------------------------------------------------%
c | If istart + 1 = iend then no reason to apply a |
c | complex conjugate pair of shifts on a 2 by 2 matrix. |
c %------------------------------------------------------%
c
if ( istart + 1 .eq. iend .and. abs( sigmai ) .gt. zero )
& go to 100
c
h11 = h(istart,istart)
h21 = h(istart+1,istart)
if ( abs( sigmai ) .le. zero ) then
c
c %---------------------------------------------%
c | Real-valued shift ==> apply single shift QR |
c %---------------------------------------------%
c
f = h11 - sigmar
g = h21
c
do 80 i = istart, iend-1
c
c %-----------------------------------------------------%
c | Contruct the plane rotation G to zero out the bulge |
c %-----------------------------------------------------%
c
call dlartg (f, g, c, s, r)
if (i .gt. istart) then
c
c %-------------------------------------------%
c | The following ensures that h(1:iend-1,1), |
c | the first iend-2 off diagonal of elements |
c | H, remain non negative. |
c %-------------------------------------------%
c
if (r .lt. zero) then
r = -r
c = -c
s = -s
end if
h(i,i-1) = r
h(i+1,i-1) = zero
end if
c
c %---------------------------------------------%
c | Apply rotation to the left of H; H <- G'*H |
c %---------------------------------------------%
c
do 50 j = i, kplusp
t = c*h(i,j) + s*h(i+1,j)
h(i+1,j) = -s*h(i,j) + c*h(i+1,j)
h(i,j) = t
50 continue
c
c %---------------------------------------------%
c | Apply rotation to the right of H; H <- H*G |
c %---------------------------------------------%
c
do 60 j = 1, min(i+2,iend)
t = c*h(j,i) + s*h(j,i+1)
h(j,i+1) = -s*h(j,i) + c*h(j,i+1)
h(j,i) = t
60 continue
c
c %----------------------------------------------------%
c | Accumulate the rotation in the matrix Q; Q <- Q*G |
c %----------------------------------------------------%
c
do 70 j = 1, min( j+jj, kplusp )
t = 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) = t
70 continue
c
c %---------------------------%
c | Prepare for next rotation |
c %---------------------------%
c
if (i .lt. iend-1) then
f = h(i+1,i)
g = h(i+2,i)
end if
80 continue
c
c %-----------------------------------%
c | Finished applying the real shift. |
c %-----------------------------------%
c
else
c
c %----------------------------------------------------%
c | Complex conjugate shifts ==> apply double shift QR |
c %----------------------------------------------------%
c
h12 = h(istart,istart+1)
h22 = h(istart+1,istart+1)
h32 = h(istart+2,istart+1)
c
c %---------------------------------------------------------%
c | Compute 1st column of (H - shift*I)*(H - conj(shift)*I) |
c %---------------------------------------------------------%
c
s = 2.0*sigmar
t = dlapy2 ( sigmar, sigmai )
u(1) = ( h11 * (h11 - s) + t * t ) / h21 + h12
u(2) = h11 + h22 - s
u(3) = h32
c
do 90 i = istart, iend-1
c
nr = min ( 3, iend-i+1 )
c
c %-----------------------------------------------------%
c | Construct Householder reflector G to zero out u(1). |
c | G is of the form I - tau*( 1 u )' * ( 1 u' ). |
c %-----------------------------------------------------%
c
call dlarfg ( nr, u(1), u(2), 1, tau )
c
if (i .gt. istart) then
h(i,i-1) = u(1)
h(i+1,i-1) = zero
if (i .lt. iend-1) h(i+2,i-1) = zero
end if
u(1) = one
c
c %--------------------------------------%
c | Apply the reflector to the left of H |
c %--------------------------------------%
c
call dlarf ('Left', nr, kplusp-i+1, u, 1, tau,
& h(i,i), ldh, workl)
c
c %---------------------------------------%
c | Apply the reflector to the right of H |
c %---------------------------------------%
c
ir = min ( i+3, iend )
call dlarf ('Right', ir, nr, u, 1, tau,
& h(1,i), ldh, workl)
c
c %-----------------------------------------------------%
c | Accumulate the reflector in the matrix Q; Q <- Q*G |
c %-----------------------------------------------------%
c
call dlarf ('Right', kplusp, nr, u, 1, tau,
& q(1,i), ldq, workl)
c
c %----------------------------%
c | Prepare for next reflector |
c %----------------------------%
c
if (i .lt. iend-1) then
u(1) = h(i+1,i)
u(2) = h(i+2,i)
if (i .lt. iend-2) u(3) = h(i+3,i)
end if
c
90 continue
c
c %--------------------------------------------%
c | Finished applying a complex pair of shifts |
c | to the current block |
c %--------------------------------------------%
c
end if
c
100 continue
c
c %---------------------------------------------------------%
c | Apply the same shift to the next block if there is any. |
c %---------------------------------------------------------%
c
istart = iend + 1
if (iend .lt. kplusp) go to 20
c
c %---------------------------------------------%
c | Loop back to the top to get the next shift. |
c %---------------------------------------------%
c
110 continue
c
c %--------------------------------------------------%
c | Perform a similarity transformation that makes |
c | sure that H will have non negative sub diagonals |
c %--------------------------------------------------%
c
do 120 j=1,kev
if ( h(j+1,j) .lt. zero ) then
call dscal( kplusp-j+1, -one, h(j+1,j), ldh )
call dscal( min(j+2, kplusp), -one, h(1,j+1), 1 )
call dscal( min(j+np+1,kplusp), -one, q(1,j+1), 1 )
end if
120 continue
c
do 130 i = 1, kev
c
c %--------------------------------------------%
c | Final check for splitting and deflation. |
c | Use a standard test as in the QR algorithm |
c | REFERENCE: LAPACK subroutine dlahqr |
c %--------------------------------------------%
c
tst1 = abs( h( i, i ) ) + abs( h( i+1, i+1 ) )
if( tst1.eq.zero )
& tst1 = dlanhs( '1', kev, h, ldh, workl )
if( h( i+1,i ) .le. max( ulp*tst1, smlnum ) )
& h(i+1,i) = zero
130 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 needed in the residual update since we |
c | cannot GUARANTEE that the corresponding entry |
c | of H would be zero as in exact arithmetic. |
c %-------------------------------------------------%
c
if (h(kev+1,kev) .gt. zero)
& call dgemv ('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 of the upper Hessenberg structure of Q. |
c %----------------------------------------------------------%
c
do 140 i = 1, kev
call dgemv ('N', n, kplusp-i+1, one, v, ldv,
& q(1,kev-i+1), 1, zero, workd, 1)
call dcopy (n, workd, 1, v(1,kplusp-i+1), 1)
140 continue
c
c %-------------------------------------------------%
c | Move v(:,kplusp-kev+1:kplusp) into v(:,1:kev). |
c %-------------------------------------------------%
c
call dlacpy ('A', n, kev, v(1,kplusp-kev+1), ldv, v, ldv)
c
c %--------------------------------------------------------------%
c | Copy the (kev+1)-st column of (V*Q) in the appropriate place |
c %--------------------------------------------------------------%
c
if (h(kev+1,kev) .gt. zero)
& call dcopy (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_{kplusp}'*Q)*e_{kev} |
c | betak = e_{kev+1}'*H*e_{kev} |
c %-------------------------------------%
c
call dscal (n, q(kplusp,kev), resid, 1)
if (h(kev+1,kev) .gt. zero)
& call daxpy (n, h(kev+1,kev), v(1,kev+1), 1, resid, 1)
c
if (msglvl .gt. 1) then
call dvout (logfil, 1, q(kplusp,kev), ndigit,
& '_napps: sigmak = (e_{kev+p}^T*Q)*e_{kev}')
call dvout (logfil, 1, h(kev+1,kev), ndigit,
& '_napps: betak = e_{kev+1}^T*H*e_{kev}')
call ivout (logfil, 1, [kev], ndigit,
& '_napps: Order of the final Hessenberg matrix ')
if (msglvl .gt. 2) then
call dmout (logfil, kev, kev, h, ldh, ndigit,
& '_napps: updated Hessenberg matrix H for next iteration')
end if
c
end if
c
9000 continue
call second (t1)
tnapps = tnapps + (t1 - t0)
c
return
c
c %---------------%
c | End of dnapps |
c %---------------%
c
end
