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expokit.orig.f
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expokit.orig.f
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*----------------------------------------------------------------------|
subroutine DMEXPV( n, m, t, v, w, tol, anorm,
. wsp,lwsp, iwsp,liwsp, matvec, itrace,iflag )
implicit none
integer n, m, lwsp, liwsp, itrace, iflag, iwsp(liwsp)
double precision t, tol, anorm, v(n), w(n), wsp(lwsp)
external matvec
*-----Purpose----------------------------------------------------------|
*
*--- DMEXPV computes w = exp(t*A)*v - Customised for MARKOV CHAINS.
*
* It does not compute the matrix exponential in isolation but
* instead, it computes directly the action of the exponential
* operator on the operand vector. This way of doing so allows
* for addressing large sparse problems.
*
* The method used is based on Krylov subspace projection
* techniques and the matrix under consideration interacts only
* via the external routine `matvec' performing the matrix-vector
* product (matrix-free method).
*
* This is a customised version for Markov Chains. This means that a
* check is done within this code to ensure that the resulting vector
* w is a probability vector, i.e., w must have all its components
* in [0,1], with sum equal to 1. This check is done at some expense
* and the user may try DGEXPV which is cheaper since it ignores
* probability constraints.
*
* IMPORTANT: The check assumes that the transition rate matrix Q
* satisfies Qe = 0, where e=(1,...,1)'. Don't use DMEXPV
* if this condition does not hold. Use DGEXPV instead.
* DMEXPV/DGEXPV require the matrix-vector product
* y = A*x = Q'*x, i.e, the TRANSPOSE of Q times a vector.
* Failure to remember this leads to wrong results.
*
*-----Arguments--------------------------------------------------------|
*
* n : (input) order of the principal matrix A.
*
* m : (input) maximum size for the Krylov basis.
*
* t : (input) time at wich the solution is needed (can be < 0).
*
* v(n) : (input) given operand vector.
*
* w(n) : (output) computed approximation of exp(t*A)*v.
*
* tol : (input/output) the requested acurracy tolerance on w.
* If on input tol=0.0d0 or tol is too small (tol.le.eps)
* the internal value sqrt(eps) is used, and tol is set to
* sqrt(eps) on output (`eps' denotes the machine epsilon).
* (`Happy breakdown' is assumed if h(j+1,j) .le. anorm*tol)
*
* anorm : (input) an approximation of some norm of A.
*
* wsp(lwsp): (workspace) lwsp .ge. n*(m+1)+n+(m+2)^2+4*(m+2)^2+ideg+1
* +---------+-------+---------------+
* (actually, ideg=6) V H wsp for PADE
*
* iwsp(liwsp): (workspace) liwsp .ge. m+2
*
* matvec : external subroutine for matrix-vector multiplication.
* synopsis: matvec( x, y )
* double precision x(*), y(*)
* computes: y(1:n) <- A*x(1:n)
* where A is the principal matrix.
*
* IMPORTANT: DMEXPV requires the product y = Ax = Q'x, i.e.
* the TRANSPOSE of the transition rate matrix.
*
* itrace : (input) running mode. 0=silent, 1=print step-by-step info
*
* iflag : (output) exit flag.
* <0 - bad input arguments
* 0 - no problem
* 1 - maximum number of steps reached without convergence
* 2 - requested tolerance was too high
*
*-----Accounts on the computation--------------------------------------|
* Upon exit, an interested user may retrieve accounts on the
* computations. They are located in the workspace arrays wsp and
* iwsp as indicated below:
*
* location mnemonic description
* -----------------------------------------------------------------|
* iwsp(1) = nmult, number of matrix-vector multiplications used
* iwsp(2) = nexph, number of Hessenberg matrix exponential evaluated
* iwsp(3) = nscale, number of repeated squaring involved in Pade
* iwsp(4) = nstep, number of integration steps used up to completion
* iwsp(5) = nreject, number of rejected step-sizes
* iwsp(6) = ibrkflag, set to 1 if `happy breakdown' and 0 otherwise
* iwsp(7) = mbrkdwn, if `happy brkdown', basis-size when it occured
* -----------------------------------------------------------------|
* wsp(1) = step_min, minimum step-size used during integration
* wsp(2) = step_max, maximum step-size used during integration
* wsp(3) = x_round, maximum among all roundoff errors (lower bound)
* wsp(4) = s_round, sum of roundoff errors (lower bound)
* wsp(5) = x_error, maximum among all local truncation errors
* wsp(6) = s_error, global sum of local truncation errors
* wsp(7) = tbrkdwn, if `happy breakdown', time when it occured
* wsp(8) = t_now, integration domain successfully covered
* wsp(9) = hump, i.e., max||exp(sA)||, s in [0,t] (or [t,0] if t<0)
* wsp(10) = ||w||/||v||, scaled norm of the solution w.
* -----------------------------------------------------------------|
* The `hump' is a measure of the conditioning of the problem. The
* matrix exponential is well-conditioned if hump = 1, whereas it is
* poorly-conditioned if hump >> 1. However the solution can still be
* relatively fairly accurate even when the hump is large (the hump
* is an upper bound), especially when the hump and the scaled norm
* of w [this is also computed and returned in wsp(10)] are of the
* same order of magnitude (further details in reference below).
* Markov chains are usually well-conditioned problems.
*
*----------------------------------------------------------------------|
*-----The following parameters may also be adjusted herein-------------|
*
integer mxstep, mxreject, ideg
double precision delta, gamma
parameter( mxstep = 500,
. mxreject = 0,
. ideg = 6,
. delta = 1.2d0,
. gamma = 0.9d0 )
* mxstep : maximum allowable number of integration steps.
* The value 0 means an infinite number of steps.
*
* mxreject: maximum allowable number of rejections at each step.
* The value 0 means an infinite number of rejections.
*
* ideg : the Pade approximation of type (ideg,ideg) is used as
* an approximation to exp(H). The value 0 switches to the
* uniform rational Chebyshev approximation of type (14,14)
*
* delta : local truncation error `safety factor'
*
* gamma : stepsize `shrinking factor'
*
*----------------------------------------------------------------------|
* Roger B. Sidje (rbs@maths.uq.edu.au)
* EXPOKIT: Software Package for Computing Matrix Exponentials.
* ACM - Transactions On Mathematical Software, 24(1):130-156, 1998
*----------------------------------------------------------------------|
*
integer i, j, k1, mh, mx, iv, ih, j1v, ns, ifree, lfree, iexph,
. ireject,ibrkflag,mbrkdwn, nmult, nreject, nexph, nscale,
. nstep
double precision sgn, t_out, tbrkdwn, step_min,step_max, err_loc,
. s_error, x_error, t_now, t_new, t_step, t_old,
. xm, beta, break_tol, p1, p2, p3, eps, rndoff,
. vnorm, avnorm, hj1j, hij, hump, SQR1,
. roundoff, s_round, x_round
intrinsic AINT,ABS,DBLE,LOG10,MAX,MIN,NINT,SIGN,SQRT
double precision DDOT, DNRM2, DASUM
*--- check restrictions on input parameters ...
iflag = 0
if ( lwsp.lt.n*(m+2)+5*(m+2)**2+ideg+1 ) iflag = -1
if ( liwsp.lt.m+2 ) iflag = -2
if ( m.ge.n .or. m.le.0 ) iflag = -3
if ( iflag.ne.0 ) stop 'bad sizes (in input of DMEXPV)'
*
*--- initialisations ...
*
k1 = 2
mh = m + 2
iv = 1
ih = iv + n*(m+1) + n
ifree = ih + mh*mh
lfree = lwsp - ifree + 1
ibrkflag = 0
mbrkdwn = m
nmult = 0
nreject = 0
nexph = 0
nscale = 0
sgn = SIGN( 1.0d0,t )
t_out = ABS( t )
tbrkdwn = 0.0d0
step_min = t_out
step_max = 0.0d0
nstep = 0
s_error = 0.0d0
s_round = 0.0d0
x_error = 0.0d0
x_round = 0.0d0
t_now = 0.0d0
t_new = 0.0d0
p1 = 4.0d0/3.0d0
1 p2 = p1 - 1.0d0
p3 = p2 + p2 + p2
eps = ABS( p3-1.0d0 )
if ( eps.eq.0.0d0 ) go to 1
if ( tol.le.eps ) tol = SQRT( eps )
rndoff = eps*anorm
break_tol = 1.0d-7
*>>> break_tol = tol
*>>> break_tol = anorm*tol
call DCOPY( n, v,1, w,1 )
beta = DNRM2( n, w,1 )
vnorm = beta
hump = beta
*
*--- obtain the very first stepsize ...
*
SQR1 = SQRT( 0.1d0 )
xm = 1.0d0/DBLE( m )
p1 = tol*(((m+1)/2.72D0)**(m+1))*SQRT(2.0D0*3.14D0*(m+1))
t_new = (1.0d0/anorm)*(p1/(4.0d0*beta*anorm))**xm
p1 = 10.0d0**(NINT( LOG10( t_new )-SQR1 )-1)
t_new = AINT( t_new/p1 + 0.55d0 ) * p1
*
*--- step-by-step integration ...
*
100 if ( t_now.ge.t_out ) goto 500
nstep = nstep + 1
t_step = MIN( t_out-t_now, t_new )
p1 = 1.0d0/beta
do i = 1,n
wsp(iv + i-1) = p1*w(i)
enddo
do i = 1,mh*mh
wsp(ih+i-1) = 0.0d0
enddo
*
*--- Arnoldi loop ...
*
j1v = iv + n
do 200 j = 1,m
nmult = nmult + 1
call matvec( wsp(j1v-n), wsp(j1v) )
do i = 1,j
hij = DDOT( n, wsp(iv+(i-1)*n),1, wsp(j1v),1 )
call DAXPY( n, -hij, wsp(iv+(i-1)*n),1, wsp(j1v),1 )
wsp(ih+(j-1)*mh+i-1) = hij
enddo
hj1j = DNRM2( n, wsp(j1v),1 )
*--- if `happy breakdown' go straightforward at the end ...
if ( hj1j.le.break_tol ) then
print*,'happy breakdown: mbrkdwn =',j,' h =',hj1j
k1 = 0
ibrkflag = 1
mbrkdwn = j
tbrkdwn = t_now
t_step = t_out-t_now
goto 300
endif
wsp(ih+(j-1)*mh+j) = hj1j
call DSCAL( n, 1.0d0/hj1j, wsp(j1v),1 )
j1v = j1v + n
200 continue
nmult = nmult + 1
call matvec( wsp(j1v-n), wsp(j1v) )
avnorm = DNRM2( n, wsp(j1v),1 )
*
*--- set 1 for the 2-corrected scheme ...
*
300 continue
wsp(ih+m*mh+m+1) = 1.0d0
*
*--- loop while ireject<mxreject until the tolerance is reached ...
*
ireject = 0
401 continue
*
*--- compute w = beta*V*exp(t_step*H)*e1 ..
*
nexph = nexph + 1
mx = mbrkdwn + k1
if ( ideg.ne.0 ) then
*--- irreducible rational Pade approximation ...
call DGPADM( ideg, mx, sgn*t_step, wsp(ih),mh,
. wsp(ifree),lfree, iwsp, iexph, ns, iflag )
iexph = ifree + iexph - 1
nscale = nscale + ns
else
*--- uniform rational Chebyshev approximation ...
iexph = ifree
do i = 1,mx
wsp(iexph+i-1) = 0.0d0
enddo
wsp(iexph) = 1.0d0
call DNCHBV(mx,sgn*t_step,wsp(ih),mh,wsp(iexph),wsp(ifree+mx))
endif
402 continue
*
*--- error estimate ...
*
if ( k1.eq.0 ) then
err_loc = tol
else
p1 = ABS( wsp(iexph+m) ) * beta
p2 = ABS( wsp(iexph+m+1) ) * beta * avnorm
if ( p1.gt.10.0d0*p2 ) then
err_loc = p2
xm = 1.0d0/DBLE( m )
elseif ( p1.gt.p2 ) then
err_loc = (p1*p2)/(p1-p2)
xm = 1.0d0/DBLE( m )
else
err_loc = p1
xm = 1.0d0/DBLE( m-1 )
endif
endif
*
*--- reject the step-size if the error is not acceptable ...
*
if ( (k1.ne.0) .and. (err_loc.gt.delta*t_step*tol) .and.
. (mxreject.eq.0 .or. ireject.lt.mxreject) ) then
t_old = t_step
t_step = gamma * t_step * (t_step*tol/err_loc)**xm
p1 = 10.0d0**(NINT( LOG10( t_step )-SQR1 )-1)
t_step = AINT( t_step/p1 + 0.55d0 ) * p1
if ( itrace.ne.0 ) then
print*,'t_step =',t_old
print*,'err_loc =',err_loc
print*,'err_required =',delta*t_old*tol
print*,'stepsize rejected, stepping down to:',t_step
endif
ireject = ireject + 1
nreject = nreject + 1
if ( mxreject.ne.0 .and. ireject.gt.mxreject ) then
print*,"Failure in DMEXPV: ---"
print*,"The requested tolerance is too high."
Print*,"Rerun with a smaller value."
iflag = 2
return
endif
goto 401
endif
*
*--- now update w = beta*V*exp(t_step*H)*e1 and the hump ...
*
mx = mbrkdwn + MAX( 0,k1-1 )
call DGEMV( 'n', n,mx,beta,wsp(iv),n,wsp(iexph),1,0.0d0,w,1 )
beta = DNRM2( n, w,1 )
hump = MAX( hump, beta )
*
*--- Markov model constraints ...
*
j = 0
do i = 1,n
if ( w(i).lt.0.0d0 ) then
w(i) = 0.0d0
j = j + 1
endif
enddo
p1 = DASUM( n, w,1 )
if ( j.gt.0 ) call DSCAL( n, 1.0d0/p1, w,1 )
roundoff = DABS( 1.0d0-p1 ) / DBLE( n )
*
*--- suggested value for the next stepsize ...
*
t_new = gamma * t_step * (t_step*tol/err_loc)**xm
p1 = 10.0d0**(NINT( LOG10( t_new )-SQR1 )-1)
t_new = AINT( t_new/p1 + 0.55d0 ) * p1
err_loc = MAX( err_loc, roundoff )
err_loc = MAX( err_loc, rndoff )
*
*--- update the time covered ...
*
t_now = t_now + t_step
*
*--- display and keep some information ...
*
if ( itrace.ne.0 ) then
print*,'integration',nstep,'---------------------------------'
print*,'scale-square =',ns
print*,'wnorm =',beta
print*,'step_size =',t_step
print*,'err_loc =',err_loc
print*,'roundoff =',roundoff
print*,'next_step =',t_new
endif
step_min = MIN( step_min, t_step )
step_max = MAX( step_max, t_step )
s_error = s_error + err_loc
s_round = s_round + roundoff
x_error = MAX( x_error, err_loc )
x_round = MAX( x_round, roundoff )
if ( mxstep.eq.0 .or. nstep.lt.mxstep ) goto 100
iflag = 1
500 continue
iwsp(1) = nmult
iwsp(2) = nexph
iwsp(3) = nscale
iwsp(4) = nstep
iwsp(5) = nreject
iwsp(6) = ibrkflag
iwsp(7) = mbrkdwn
wsp(1) = step_min
wsp(2) = step_max
wsp(3) = x_round
wsp(4) = s_round
wsp(5) = x_error
wsp(6) = s_error
wsp(7) = tbrkdwn
wsp(8) = sgn*t_now
wsp(9) = hump/vnorm
wsp(10) = beta/vnorm
END
*----------------------------------------------------------------------|
*----------------------------------------------------------------------|
subroutine DGPADM( ideg,m,t,H,ldh,wsp,lwsp,ipiv,iexph,ns,iflag )
implicit none
integer ideg, m, ldh, lwsp, iexph, ns, iflag, ipiv(m)
double precision t, H(ldh,m), wsp(lwsp)
*-----Purpose----------------------------------------------------------|
*
* Computes exp(t*H), the matrix exponential of a general matrix in
* full, using the irreducible rational Pade approximation to the
* exponential function exp(x) = r(x) = (+/-)( I + 2*(q(x)/p(x)) ),
* combined with scaling-and-squaring.
*
*-----Arguments--------------------------------------------------------|
*
* ideg : (input) the degre of the diagonal Pade to be used.
* a value of 6 is generally satisfactory.
*
* m : (input) order of H.
*
* H(ldh,m) : (input) argument matrix.
*
* t : (input) time-scale (can be < 0).
*
* wsp(lwsp) : (workspace/output) lwsp .ge. 4*m*m+ideg+1.
*
* ipiv(m) : (workspace)
*
*>>>> iexph : (output) number such that wsp(iexph) points to exp(tH)
* i.e., exp(tH) is located at wsp(iexph ... iexph+m*m-1)
* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
* NOTE: if the routine was called with wsp(iptr),
* then exp(tH) will start at wsp(iptr+iexph-1).
*
* ns : (output) number of scaling-squaring used.
*
* iflag : (output) exit flag.
* 0 - no problem
* <0 - problem
*
*----------------------------------------------------------------------|
* Roger B. Sidje (rbs@maths.uq.edu.au)
* EXPOKIT: Software Package for Computing Matrix Exponentials.
* ACM - Transactions On Mathematical Software, 24(1):130-156, 1998
*----------------------------------------------------------------------|
*
integer mm,i,j,k,ih2,ip,iq,iused,ifree,iodd,icoef,iput,iget
double precision hnorm,scale,scale2,cp,cq
intrinsic INT,ABS,DBLE,LOG,MAX
*--- check restrictions on input parameters ...
mm = m*m
iflag = 0
if ( ldh.lt.m ) iflag = -1
if ( lwsp.lt.4*mm+ideg+1 ) iflag = -2
if ( iflag.ne.0 ) stop 'bad sizes (in input of DGPADM)'
*
*--- initialise pointers ...
*
icoef = 1
ih2 = icoef + (ideg+1)
ip = ih2 + mm
iq = ip + mm
ifree = iq + mm
*
*--- scaling: seek ns such that ||t*H/2^ns|| < 1/2;
* and set scale = t/2^ns ...
*
do i = 1,m
wsp(i) = 0.0d0
enddo
do j = 1,m
do i = 1,m
wsp(i) = wsp(i) + ABS( H(i,j) )
enddo
enddo
hnorm = 0.0d0
do i = 1,m
hnorm = MAX( hnorm,wsp(i) )
enddo
hnorm = ABS( t*hnorm )
if ( hnorm.eq.0.0d0 ) stop 'Error - null H in input of DGPADM.'
ns = MAX( 0,INT(LOG(hnorm)/LOG(2.0d0))+2 )
scale = t / DBLE(2**ns)
scale2 = scale*scale
*
*--- compute Pade coefficients ...
*
i = ideg+1
j = 2*ideg+1
wsp(icoef) = 1.0d0
do k = 1,ideg
wsp(icoef+k) = (wsp(icoef+k-1)*DBLE( i-k ))/DBLE( k*(j-k) )
enddo
*
*--- H2 = scale2*H*H ...
*
call DGEMM( 'n','n',m,m,m,scale2,H,ldh,H,ldh,0.0d0,wsp(ih2),m )
*
*--- initialize p (numerator) and q (denominator) ...
*
cp = wsp(icoef+ideg-1)
cq = wsp(icoef+ideg)
do j = 1,m
do i = 1,m
wsp(ip + (j-1)*m + i-1) = 0.0d0
wsp(iq + (j-1)*m + i-1) = 0.0d0
enddo
wsp(ip + (j-1)*(m+1)) = cp
wsp(iq + (j-1)*(m+1)) = cq
enddo
*
*--- Apply Horner rule ...
*
iodd = 1
k = ideg - 1
100 continue
iused = iodd*iq + (1-iodd)*ip
call DGEMM( 'n','n',m,m,m, 1.0d0,wsp(iused),m,
. wsp(ih2),m, 0.0d0,wsp(ifree),m )
do j = 1,m
wsp(ifree+(j-1)*(m+1)) = wsp(ifree+(j-1)*(m+1))+wsp(icoef+k-1)
enddo
ip = (1-iodd)*ifree + iodd*ip
iq = iodd*ifree + (1-iodd)*iq
ifree = iused
iodd = 1-iodd
k = k-1
if ( k.gt.0 ) goto 100
*
*--- Obtain (+/-)(I + 2*(p\q)) ...
*
if ( iodd .eq. 1 ) then
call DGEMM( 'n','n',m,m,m, scale,wsp(iq),m,
. H,ldh, 0.0d0,wsp(ifree),m )
iq = ifree
else
call DGEMM( 'n','n',m,m,m, scale,wsp(ip),m,
. H,ldh, 0.0d0,wsp(ifree),m )
ip = ifree
endif
call DAXPY( mm, -1.0d0,wsp(ip),1, wsp(iq),1 )
call DGESV( m,m, wsp(iq),m, ipiv, wsp(ip),m, iflag )
if ( iflag.ne.0 ) stop 'Problem in DGESV (within DGPADM)'
call DSCAL( mm, 2.0d0, wsp(ip), 1 )
do j = 1,m
wsp(ip+(j-1)*(m+1)) = wsp(ip+(j-1)*(m+1)) + 1.0d0
enddo
iput = ip
if ( ns.eq.0 .and. iodd.eq.1 ) then
call DSCAL( mm, -1.0d0, wsp(ip), 1 )
goto 200
endif
*
*-- squaring : exp(t*H) = (exp(t*H))^(2^ns) ...
*
iodd = 1
do k = 1,ns
iget = iodd*ip + (1-iodd)*iq
iput = (1-iodd)*ip + iodd*iq
call DGEMM( 'n','n',m,m,m, 1.0d0,wsp(iget),m, wsp(iget),m,
. 0.0d0,wsp(iput),m )
iodd = 1-iodd
enddo
200 continue
iexph = iput
END
*----------------------------------------------------------------------|
*----------------------------------------------------------------------|
subroutine DSPADM( ideg,m,t,H,ldh,wsp,lwsp,ipiv,iexph,ns,iflag )
implicit none
integer ideg, m, ldh, lwsp, iexph, ns, iflag, ipiv(m)
double precision t, H(ldh,m), wsp(lwsp)
*-----Purpose----------------------------------------------------------|
*
* Computes exp(t*H), the matrix exponential of a symmetric matrix
* in full, using the irreducible rational Pade approximation to the
* exponential function exp(x) = r(x) = (+/-)( I + 2*(q(x)/p(x)) ),
* combined with scaling-and-squaring.
*
*-----Arguments--------------------------------------------------------|
*
* ideg : (input) the degre of the diagonal Pade to be used.
* a value of 6 is generally satisfactory.
*
* m : (input) order of H.
*
* H(ldh,m) : (input) argument matrix (both lower and upper parts).
*
* t : (input) time-scale (can be < 0).
*
* wsp(lwsp) : (workspace/output) lwsp .ge. 4*m*m+ideg+1.
*
* ipiv(m) : (workspace)
*
*>>>> iexph : (output) number such that wsp(iexph) points to exp(tH)
* i.e., exp(tH) is located at wsp(iexph ... iexph+m*m-1)
* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
* NOTE: if the routine was called with wsp(iptr),
* then exp(tH) will start at wsp(iptr+iexph-1).
*
* ns : (output) number of scaling-squaring used.
*
* iflag : (output) exit flag.
* 0 - no problem
* <0 - problem
*
*----------------------------------------------------------------------|
* Roger B. Sidje (rbs@maths.uq.edu.au)
* EXPOKIT: Software Package for Computing Matrix Exponentials.
* ACM - Transactions On Mathematical Software, 24(1):130-156, 1998
*----------------------------------------------------------------------|
*
integer mm,i,j,k,ih2,ip,iq,iused,ifree,iodd,icoef,iput,iget
double precision hnorm,scale,scale2,cp,cq
intrinsic INT,ABS,DBLE,LOG,MAX
*--- check restrictions on input parameters ...
mm = m*m
iflag = 0
if ( ldh.lt.m ) iflag = -1
if ( lwsp.lt.4*mm+ideg+1 ) iflag = -2
if ( iflag.ne.0 ) stop 'bad sizes (in input of DSPADM)'
*
*--- initialise pointers ...
*
icoef = 1
ih2 = icoef + (ideg+1)
ip = ih2 + mm
iq = ip + mm
ifree = iq + mm
*
*--- scaling: seek ns such that ||t*H/2^ns|| < 1/2;
* and set scale = t/2^ns ...
*
do i = 1,m
wsp(i) = 0.0d0
enddo
do j = 1,m
do i = 1,m
wsp(i) = wsp(i) + ABS( H(i,j) )
enddo
enddo
hnorm = 0.0d0
do i = 1,m
hnorm = MAX( hnorm,wsp(i) )
enddo
hnorm = ABS( t*hnorm )
if ( hnorm.eq.0.0d0 ) stop 'Error - null H in input of DSPADM.'
ns = MAX( 0,INT(LOG(hnorm)/LOG(2.0d0))+2 )
scale = t / DBLE(2**ns)
scale2 = scale*scale
*
*--- compute Pade coefficients ...
*
i = ideg+1
j = 2*ideg+1
wsp(icoef) = 1.0d0
do k = 1,ideg
wsp(icoef+k) = (wsp(icoef+k-1)*DBLE( i-k ))/DBLE( k*(j-k) )
enddo
*
*--- H2 = scale2*H*H ...
*
call DGEMM( 'n','n',m,m,m,scale2,H,ldh,H,ldh,0.0d0,wsp(ih2),m )
*
*--- initialize p (numerator) and q (denominator) ...
*
cp = wsp(icoef+ideg-1)
cq = wsp(icoef+ideg)
do j = 1,m
do i = 1,m
wsp(ip + (j-1)*m + i-1) = 0.0d0
wsp(iq + (j-1)*m + i-1) = 0.0d0
enddo
wsp(ip + (j-1)*(m+1)) = cp
wsp(iq + (j-1)*(m+1)) = cq
enddo
*
*--- Apply Horner rule ...
*
iodd = 1
k = ideg - 1
100 continue
iused = iodd*iq + (1-iodd)*ip
call DGEMM( 'n','n',m,m,m, 1.0d0,wsp(iused),m,
. wsp(ih2),m, 0.0d0,wsp(ifree),m )
do j = 1,m
wsp(ifree+(j-1)*(m+1)) = wsp(ifree+(j-1)*(m+1))+wsp(icoef+k-1)
enddo
ip = (1-iodd)*ifree + iodd*ip
iq = iodd*ifree + (1-iodd)*iq
ifree = iused
iodd = 1-iodd
k = k-1
if ( k.gt.0 ) goto 100
*
*--- Obtain (+/-)(I + 2*(p\q)) ...
*
if ( iodd .eq. 1 ) then
call DGEMM( 'n','n',m,m,m, scale,wsp(iq),m,
. H,ldh, 0.0d0,wsp(ifree),m )
iq = ifree
else
call DGEMM( 'n','n',m,m,m, scale,wsp(ip),m,
. H,ldh, 0.0d0,wsp(ifree),m )
ip = ifree
endif
call DAXPY( mm, -1.0d0,wsp(ip),1, wsp(iq),1 )
call DSYSV( 'U',m,m,wsp(iq),m,ipiv,wsp(ip),m,wsp(ih2),mm,iflag )
if ( iflag.ne.0 ) stop 'Problem in DSYSV (within DSPADM)'
call DSCAL( mm, 2.0d0, wsp(ip), 1 )
do j = 1,m
wsp(ip+(j-1)*(m+1)) = wsp(ip+(j-1)*(m+1)) + 1.0d0
enddo
iput = ip
if ( ns.eq.0 .and. iodd.eq.1 ) then
call DSCAL( mm, -1.0d0, wsp(ip), 1 )
goto 200
endif
*
*-- squaring : exp(t*H) = (exp(t*H))^(2^ns) ...
*
iodd = 1
do k = 1,ns
iget = iodd*ip + (1-iodd)*iq
iput = (1-iodd)*ip + iodd*iq
call DGEMM( 'n','n',m,m,m, 1.0d0,wsp(iget),m, wsp(iget),m,
. 0.0d0,wsp(iput),m )
iodd = 1-iodd
enddo
200 continue
iexph = iput
END
*----------------------------------------------------------------------|
*----------------------------------------------------------------------|
subroutine ZGPADM(ideg,m,t,H,ldh,wsp,lwsp,ipiv,iexph,ns,iflag)
implicit none
double precision t
integer ideg, m, ldh, lwsp, iexph, ns, iflag, ipiv(m)
complex*16 H(ldh,m), wsp(lwsp)
*-----Purpose----------------------------------------------------------|
*
* Computes exp(t*H), the matrix exponential of a general complex
* matrix in full, using the irreducible rational Pade approximation
* to the exponential exp(z) = r(z) = (+/-)( I + 2*(q(z)/p(z)) ),
* combined with scaling-and-squaring.
*
*-----Arguments--------------------------------------------------------|
*
* ideg : (input) the degre of the diagonal Pade to be used.
* a value of 6 is generally satisfactory.
*
* m : (input) order of H.
*
* H(ldh,m) : (input) argument matrix.
*
* t : (input) time-scale (can be < 0).
*
* wsp(lwsp) : (workspace/output) lwsp .ge. 4*m*m+ideg+1.
*
* ipiv(m) : (workspace)
*
*>>>> iexph : (output) number such that wsp(iexph) points to exp(tH)
* i.e., exp(tH) is located at wsp(iexph ... iexph+m*m-1)
* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
* NOTE: if the routine was called with wsp(iptr),
* then exp(tH) will start at wsp(iptr+iexph-1).
*
* ns : (output) number of scaling-squaring used.
*
* iflag : (output) exit flag.
* 0 - no problem
* <0 - problem
*
*----------------------------------------------------------------------|
* Roger B. Sidje (rbs@maths.uq.edu.au)
* EXPOKIT: Software Package for Computing Matrix Exponentials.
* ACM - Transactions On Mathematical Software, 24(1):130-156, 1998
*----------------------------------------------------------------------|
*
integer i,j,k,icoef,mm,ih2,iodd,iused,ifree,iq,ip,iput,iget
double precision hnorm
complex*16 cp, cq, scale, scale2, ZERO, ONE
parameter( ZERO=(0.0d0,0.0d0), ONE=(1.0d0,0.0d0) )
intrinsic ABS, CMPLX, DBLE, INT, LOG, MAX
*--- check restrictions on input parameters ...
mm = m*m
iflag = 0
if ( ldh.lt.m ) iflag = -1
if ( lwsp.lt.4*mm+ideg+1 ) iflag = -2
if ( iflag.ne.0 ) stop 'bad sizes (in input of ZGPADM)'
*
*--- initialise pointers ...
*
icoef = 1
ih2 = icoef + (ideg+1)
ip = ih2 + mm
iq = ip + mm
ifree = iq + mm
*
*--- scaling: seek ns such that ||t*H/2^ns|| < 1/2;
* and set scale = t/2^ns ...
*
do i = 1,m
wsp(i) = ZERO
enddo
do j = 1,m
do i = 1,m
wsp(i) = wsp(i) + ABS( H(i,j) )
enddo
enddo
hnorm = 0.0d0
do i = 1,m
hnorm = MAX( hnorm,DBLE(wsp(i)) )
enddo
hnorm = ABS( t*hnorm )
if ( hnorm.eq.0.0d0 ) stop 'Error - null H in input of ZGPADM.'
ns = MAX( 0,INT(LOG(hnorm)/LOG(2.0d0))+2 )
scale = CMPLX( t/DBLE(2**ns),0.0d0 )
scale2 = scale*scale
*
*--- compute Pade coefficients ...
*
i = ideg+1
j = 2*ideg+1
wsp(icoef) = ONE
do k = 1,ideg
wsp(icoef+k) = (wsp(icoef+k-1)*DBLE( i-k ))/DBLE( k*(j-k) )
enddo
*
*--- H2 = scale2*H*H ...
*
call ZGEMM( 'n','n',m,m,m,scale2,H,ldh,H,ldh,ZERO,wsp(ih2),m )
*
*--- initialise p (numerator) and q (denominator) ...
*
cp = wsp(icoef+ideg-1)
cq = wsp(icoef+ideg)
do j = 1,m
do i = 1,m
wsp(ip + (j-1)*m + i-1) = ZERO
wsp(iq + (j-1)*m + i-1) = ZERO
enddo
wsp(ip + (j-1)*(m+1)) = cp
wsp(iq + (j-1)*(m+1)) = cq
enddo
*
*--- Apply Horner rule ...
*
iodd = 1
k = ideg - 1
100 continue
iused = iodd*iq + (1-iodd)*ip
call ZGEMM( 'n','n',m,m,m, ONE,wsp(iused),m,
. wsp(ih2),m, ZERO,wsp(ifree),m )
do j = 1,m
wsp(ifree+(j-1)*(m+1)) = wsp(ifree+(j-1)*(m+1))+wsp(icoef+k-1)
enddo
ip = (1-iodd)*ifree + iodd*ip
iq = iodd*ifree + (1-iodd)*iq
ifree = iused
iodd = 1-iodd
k = k-1
if ( k.gt.0 ) goto 100
*
*--- Obtain (+/-)(I + 2*(p\q)) ...
*
if ( iodd.ne.0 ) then
call ZGEMM( 'n','n',m,m,m, scale,wsp(iq),m,
. H,ldh, ZERO,wsp(ifree),m )
iq = ifree
else
call ZGEMM( 'n','n',m,m,m, scale,wsp(ip),m,
. H,ldh, ZERO,wsp(ifree),m )
ip = ifree
endif
call ZAXPY( mm, -ONE,wsp(ip),1, wsp(iq),1 )
call ZGESV( m,m, wsp(iq),m, ipiv, wsp(ip),m, iflag )
if ( iflag.ne.0 ) stop 'Problem in ZGESV (within ZGPADM)'
call ZDSCAL( mm, 2.0d0, wsp(ip), 1 )
do j = 1,m
wsp(ip+(j-1)*(m+1)) = wsp(ip+(j-1)*(m+1)) + ONE
enddo
iput = ip
if ( ns.eq.0 .and. iodd.ne.0 ) then
call ZDSCAL( mm, -1.0d0, wsp(ip), 1 )
goto 200
endif
*
*-- squaring : exp(t*H) = (exp(t*H))^(2^ns) ...
*
iodd = 1
do k = 1,ns
iget = iodd*ip + (1-iodd)*iq
iput = (1-iodd)*ip + iodd*iq
call ZGEMM( 'n','n',m,m,m, ONE,wsp(iget),m, wsp(iget),m,
. ZERO,wsp(iput),m )
iodd = 1-iodd
enddo
200 continue
iexph = iput
END
*----------------------------------------------------------------------|
subroutine ZHPADM(ideg,m,t,H,ldh,wsp,lwsp,ipiv,iexph,ns,iflag)
implicit none
double precision t
integer ideg, m, ldh, lwsp, iexph, ns, iflag, ipiv(m)
complex*16 H(ldh,m), wsp(lwsp)
*-----Purpose----------------------------------------------------------|
*
* Computes exp(t*H), the matrix exponential of an Hermitian matrix
* in full, using the irreducible rational Pade approximation to the
* exponential function exp(z) = r(z) = (+/-)( I + 2*(q(z)/p(z)) ),
* combined with scaling-and-squaring.
*
*-----Arguments--------------------------------------------------------|
*
* ideg : (input) the degre of the diagonal Pade to be used.
* a value of 6 is generally satisfactory.
*
* m : (input) order of H.
*
* H(ldh,m) : (input) argument matrix (both lower and upper parts).
*
* t : (input) time-scale (can be < 0).
*
* wsp(lwsp) : (workspace/output) lwsp .ge. 4*m*m+ideg+1.
*
* ipiv(m) : (workspace)
*
*>>>> iexph : (output) number such that wsp(iexph) points to exp(tH)
* i.e., exp(tH) is located at wsp(iexph ... iexph+m*m-1)
* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
* NOTE: if the routine was called with wsp(iptr),
* then exp(tH) will start at wsp(iptr+iexph-1).
*
* ns : (output) number of scaling-squaring used.
*
* iflag : (output) exit flag.
* 0 - no problem
* <0 - problem
*
*----------------------------------------------------------------------|
* Roger B. Sidje (rbs@maths.uq.edu.au)
* EXPOKIT: Software Package for Computing Matrix Exponentials.
* ACM - Transactions On Mathematical Software, 24(1):130-156, 1998
*----------------------------------------------------------------------|
*
integer i,j,k,icoef,mm,ih2,iodd,iused,ifree,iq,ip,iput,iget
double precision hnorm
complex*16 cp, cq, scale, scale2, ZERO, ONE
parameter( ZERO=(0.0d0,0.0d0), ONE=(1.0d0,0.0d0) )
intrinsic ABS, CMPLX, DBLE, INT, LOG, MAX
*--- check restrictions on input parameters ...
mm = m*m
iflag = 0
if ( ldh.lt.m ) iflag = -1
if ( lwsp.lt.4*mm+ideg+1 ) iflag = -2
if ( iflag.ne.0 ) stop 'bad sizes (in input of ZHPADM)'
*
*--- initialise pointers ...
*
icoef = 1
ih2 = icoef + (ideg+1)
ip = ih2 + mm
iq = ip + mm