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shoot.f
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shoot.f
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C> \file shoot.f
C> \brief Solver routines for eigenvalue shooting and utilities
C> \author S. Scott Collis
C***********************************************************************
C> \author S. Scott Collis
C>
C> \brief Solve eigenvalue problem using Godunov-Conte method
C>
C> \details First order linear boundary value problem solver
C> using Conte's method. Fourth order Runge-Kutta is used for
C> time advancement
C>
C> \param[in] nstep number of integration steps between t0 and tf
C> \param[in] n number of ODEs to integrate
C> \param[in] r number of particular solutions
C> \param[in] yo starting values
C> \param[in] yf ending values
C> \param[in] to Interval start value
C> \param[in] tf Interval end value
C> \param[in,out] c Current eigenvalue
C> \param[in] eigfun Whether to output eigenfunction at the end
C> \param[in] FHOMO Subroutine for homogeneous equation
C> \param[in] FPART Subroutine for particular equation
C> \param[in] INPROD Function to compute inner product
C***********************************************************************
subroutine CONTE(nstep, n, r, yo, yf, to, tf, c, eigfun,
& FHOMO, FPART, INPROD)
C***********************************************************************
integer nstep, n, r
complex yo(n), yf(n)
real to, tf, h
complex c
logical eigfun
complex INPROD
external INPROD, FHOMO, FPART
C***********************************************************************
integer i, m, q, s, mi, mj, qend, IPVT(n-r), icount
real t, tq(0:nstep)
complex B(n-r,0:nstep), err, cold, ctemp
complex U(n,n-r,0:nstep), P(n-r,n-r,0:nstep), z(n,n-r)
complex y(n,0:nstep), v(n,0:nstep), w(n), omega(n,0:nstep)
complex eta(n), A(n,n-r), x(n), FAC(n-r,n-r), det1, det
complex olderr, ut(n,n-r), fd, vmax
complex cm1, cm2, errm1, errm2, qt, At, Bt, Ct, Utemp(n)
real test, testalpha, pi, det2, AI(n,n-r), angle
real aa, bb, cc, fdxr, fdxi, fdyr, fdyi
logical norm
real errtol, maxcount, eigtol
C***********************************************************************
c
c initialize variables
c
errtol = 1.0e-15
maxcount = 20
eigtol = 1.0e-15
c
c set the normalization constraint
c
pi = ACOS(-1.0)
c
c Orthonormalize when angle is greater than testalpha
c angle = [0,90] degrees (10 deg seems good)
c
angle = 10.0
testalpha = angle*pi/180.0
c
c compute the step size
c
h = (tf-to)/nstep
c
c Begin the eigenvalue iteration loop
c
icount = 1
err = 1.0
cm1 = c + 1.0
do while ((abs(err) .ge. errtol .and. icount .le. maxcount) .and.
& (abs(c-cm1) .ge. eigtol) )
c
c q = number of normalizations
c qend = total number of normalizations
c tq = time at normalization q
c
q = 0
tq(0) = to
c
c Set the Homogeneous initial conditions
c
do m = 1, n-r
do i = 1, n
if (i .eq. r+m) then
u(i,m,0) = 1.0
else
u(i,m,0) = 0.0
end if
end do
end do
c
c Set the particular initial conditions
c
do i = 1, r
v(i,0) = yo(i)
end do
do i = r+1, n
v(i,0) = 0.0
end do
c
c Integrate the homo. and particular equations
c
do k = 1, nstep
t = to + h*k
c
c Loop thru all homogeneous solutions
c
do m = 1, n-r
#ifdef USE_RKCK45
call CRKCK45(n, U(1,m,k-1), U(1,m,k), t-h, h, FHOMO)
#else
call CRK4(n, U(1,m,k-1), U(1,m,k), t-h, h, FHOMO)
#endif
end do
c
c Now the particular solution
c
#ifdef USE_RKCK45
call CRKCK45(n, v(1,k-1), v(1,k), t-h, h, FPART)
#else
call CRK4(n, v(1,k-1), v(1,k), t-h, h, FPART)
#endif
c
c Check to see it orthonomalization is required
c
norm = .false.
do mi = 1, n-r
do mj = 1, n-r
if (mi .ne. mj) then
aa = ABS(inprod(n, U(1,mi,k), U(1,mi,k)))
bb = ABS(inprod(n, U(1,mj,k), U(1,mj,k)))
cc = ABS(inprod(n, U(1,mi,k), U(1,mj,k)))
#ifdef USE_ANALYTIC_INPROD
test = ACOS(MIN(1.0,ABS(cc)/SQRT(aa*bb)))
#else
test = ACOS(ABS(cc)/SQRT(aa*bb))
#endif
if (test .le. testalpha) norm = .true.
#ifdef VERBOSE
write (*,*) k, mi, mj, 180.0*test/pi,
& 180.0*testalpha/pi, norm
#endif
end if
end do
end do
c
c Perform normalization
c
c if (norm .or. (k .eq. nstep) .or. (mod(k,10) .eq. 0) ) then
if (norm .or. (k .eq. nstep)) then
q = q + 1
tq(q) = t
if (k .eq. nstep) qend = q
c
c Gram-Schmidt
c
w(1) = SQRT(inprod(n, U(1,1,k), U(1,1,k)))
do i = 1, n
z(i,1) = U(i,1,k)/w(1)
end do
do mi = 2, (n-r)
do i = 1, n
eta(i) = U(i,mi,k)
end do
do mj = mi-1, 1, -1
do i = 1, n
eta(i) = eta(i) - inprod(n, U(1,mi,k), z(1,mj))*z(i,mj)
end do
end do
w(mi) = SQRT(inprod(n, eta, eta))
do i = 1, n
z(i,mi) = eta(i)/w(mi)
end do
end do
c
c Now I have the orthonormal basis in z and
c the norms in w so I can compute the P orthonormalization
c matrix
c
do j = 1, n-r
do i = 1, j
if (j .eq. i) then
P(j,i,q) = 1.0/w(j)
else
P(j,i,q) = 0.0
do s = i, j-1
P(j,i,q) = P(j,i,q)-inprod(n,U(1,j,k),z(1,s))/w(j)*
& P(s,i,q)
end do
end if
end do
end do
c
c Check the P matrix
c
if (.false.) then
do i = 1, n
do m = 1, n-r
ut(i,m) = 0.0
do j = 1, n-r
ut(i,m) = ut(i,m) + U(i,j,k)*P(m,j,q)
end do
end do
end do
do i = 1,n
write (*,*) i,(ut(i,m), m = 1, n-r)
end do
write (*,*)
do i = 1,n
write (*,*) i,( z(i,m), m = 1, n-r)
end do
write (*,*)
write (*,*)
end if
c
c Now update the U matrix with the orthonormal values
c
do i = 1, n
do m = 1, n-r
U(i,m,k) = z(i,m)
end do
end do
c
c Form the orthogonal complement of the particular solution
c
do j = 1, n-r
omega(j,q) = inprod(n, v(1,k), U(1,j,k))
end do
do i = 1, n
do j = 1, n-r
v(i,k) = v(i,k) - U(i,j,k)*omega(j,q)
end do
end do
end if
end do
c
c write out the current solutions
c
#ifdef VERBOSE_SOLUTIONS
do k = 1, nstep
t = to + h*k
write (20,10) t,REAL(U(1,1,k)),AIMAG(U(1,1,k)),REAL(U(1,2,k)),
& AIMAG(U(1,2,k)),REAL(v(1,k)),AIMAG(v(1,k))
10 format (1x,7(ES16.8E3,1x))
end do
#endif
c
c check boundary conditions
c
if (.true.) then
c
c strictly enforce the zero BC
c
B(1,qend) = -U(1,2,nstep)/U(1,1,nstep)
B(2,qend) = 1.0
else
c
c strictly enforce the zero slope BC
c
B(1,qend) = 1.0
B(2,qend) = -U(2,1,nstep)/U(2,2,nstep)
end if
#ifdef CHECK_DETERMINATE
c
c Check the determinate (requires IMSL)
c
do i = 1, n
x(i) = yf(i) - v(i,nstep)
do m = 1, n-r
A(i,m) = U(i,m,nstep)
end do
end do
CALL WRCRN ('A', n, n-r, A, n, 0)
c
c compute the determinate
c
call LFTCG( n-r, A, n, fac, n-r, ipvt)
call LFDCG( n-r, fac, n-r, ipvt, det1, det2)
det = det1*10**det2
write (*,*) 'det = ', det
#endif
ctemp = c
if (icount .eq. 1) then
cm2 = c
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
errm2 = err
c = CMPLX( REAL(c)*.9999, AIMAG(c) )
else if (icount .eq. 2) then
cm1 = c
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
errm1 = err
c c = CMPLX( REAL(c), AIMAG(c)*.9999 )
fdxr = REAL(err-errm2) / REAL(c-cm2)
fdxi = AIMAG(err-errm2) / REAL(c-cm2)
fd = CMPLX( fdxr, fdxi )
c = c - err/fd
else
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
qt = (c-cm1)/(cm1-cm2)
At = qt*err-qt*(1.+qt)*errm1+qt**2*errm2
Bt = (2.*qt+1.)*err-(1.+qt)**2*errm1+qt**2*errm2
Ct = (1.+qt)*err
if ( ABS(Bt+SQRT(Bt**2-4.*At*Ct)) .gt.
& ABS(Bt-SQRT(Bt**2-4.*At*Ct)) ) then
c = ctemp-(ctemp-cm1)*2.*Ct/(Bt+SQRT(Bt**2-4.*At*Ct))
else
c = ctemp-(ctemp-cm1)*2.*Ct/(Bt-SQRT(Bt**2-4.*At*Ct))
end if
cm2 = cm1
cm1 = ctemp
errm2 = errm1
errm1 = err
end if
write (*,30) icount, real(c), aimag(c), real(err), aimag(err)
30 format (1x,i4,2(1x,es16.8e3,1x,es16.8e3,3x))
icount = icount + 1
end do
c
c.... Output Eigenvalue
c
write (*,40) real(ctemp), aimag(ctemp), qend
40 format (/,'Eigenvalue = ',e17.8e3,1x,e17.8e3,2x,i5/)
write (*,45) icount, abs(err), abs(c-cm1)
45 format (' Eigenvalue iterations = ', i5/,' |error| = ',
& es17.8e3/,' |c-cm1| = ', es17.8e3/)
c
c Second Pass to compute the eigenfunctions
c
if (eigfun) then
vmax = 0.0
k = nstep
do i = 1, n
y(i,k) = v(i,k)
do m = 1, n-r
y(i,k) = y(i,k) + U(i,m,k)*B(m,q)
end do
end do
do m = 1, n-r
B(m,q-1) = 0.0
do j = 1, n-r
B(m,q-1) = B(m,q-1) + P(j,m,q)*(B(j,q)-omega(j,q))
end do
end do
do k = nstep-1, 0, -1
t = to + h*k
if ( t .lt. tq(q-1) ) then
q = q - 1
do m = 1, n-r
B(m,q-1) = 0.0
do j = 1, n-r
B(m,q-1) = B(m,q-1) + P(j,m,q)*(B(j,q)-omega(j,q))
end do
end do
end if
do i = 1, n
y(i,k) = v(i,k)
do m = 1, n-r
y(i,k) = y(i,k) + U(i,m,k)*B(m,q-1)
end do
end do
if ( ABS(y(i,k)) .gt. ABS(vmax) ) then
vmax = y(i,k)
end if
end do
c
c Output eigenfunction and derivatives: u, u', u'', u'''
c
do k = 0, nstep
t = to + h*k
write (11,20) t, REAL(y(1,k)/vmax),AIMAG(y(1,k)/vmax)
write (12,20) t, REAL(y(2,k)/vmax),AIMAG(y(2,k)/vmax)
write (13,20) t, REAL(y(3,k)/vmax),AIMAG(y(3,k)/vmax)
write (14,20) t, REAL(y(4,k)/vmax),AIMAG(y(4,k)/vmax)
20 format (1x,3(ES16.8E3,1x))
end do
end if
return
end
C***********************************************************************
subroutine CONTE_IC(nstep, testalpha, n, r, yo, yf, to, tf, c,
& eigfun, FIC, FHOMO, FPART, INPROD)
C***********************************************************************
C
C First order linear boundary value problem solver using Conte's
C method with a non Dirac function initial condition.
C Runge-Kutta is used for time advancement but there are
C other options available via compile defines
C
C nstep = number of integration steps between [t0,tf]
C n = number of ODEs to integrate
C r = number of particular solutions
C yo = starting values
C yf = ending values
C to = Interval start value
C tf = Interval end value
C c = Current eigenvalue
C eigfun = Whether to output eigenfunction at the end
C FIC = Subroutine to compute the initial condition
C FHOMO = Subroutine for homogeneous equation
C FPART = Subroutin for particular equation
C INPROD = Function to compute inner product
C
C***********************************************************************
integer nstep, n, r
integer i, m, q, s, mi, mj, qend, IPVT(n-r), icount, north
real t, tq(0:nstep), to, tf, h
complex yo(n), yf(n), B(n-r,0:nstep), err, c, cold, ctemp
complex U(n,n-r,0:nstep), P(n-r,n-r,0:nstep), z(n,n-r)
complex y(n,0:nstep), v(n,0:nstep), w(n-r), omega(n,0:nstep)
complex eta(n),A(n,n-r),x(n),FAC(n-r,n-r), det1, det
complex olderr, ut(n,n-r), fd, vmax
complex cm1, cm2, errm1, errm2, qt, At, Bt, Ct, Utemp(n)
real aa, bb, cc
real test, testalpha, pi, det2, AI(n,n-r)
real fdxr, fdxi, fdyr, fdyi
logical norm, eigfun
complex INPROD
external INPROD, FIC, FHOMO, FPART
real errtol, maxcount, eigtol
#ifdef USE_NR_ODEINT
external NR_CRKQC
#endif
c
c Setup for ZVODE
c
#ifdef USE_VODE
real RWORK(20+(n*(n-r)))
complex ZWORK(15*(n*(n-r)))
integer IWORK(30+(n*(n-r)))
external FHOMO_VODE
real RTOL, ATOL, RPAR
integer ITOL, ITASK, ISTATE, IOPT, LZW, LRW, LIW, MF, IPAR(2)
ITOL = 1
RTOL = 1.0D-9
ATOL = 1.0D-8
ITASK = 1
ISTATE = 1
IOPT = 0
LZW = 15*(n*(n-r))
LRW = 20+(n*(n-r))
LIW = 30+(n*(n-r))
MF = 10
RPAR = 0.0
IPAR(1) = n
IPAR(2) = n-r
#endif
c
c initialize variables
c
errtol = 1.0e-14
maxcount = 20
eigtol = 1.0e-14
c
c set the normalization constraint
c
pi = ACOS(-1.0)
c
c compute the step size
c
h = (tf-to)/nstep
c
c Begin the eigenvalue iteration loop
c
icount = 1
err = 1.0
cm1 = c + 1.0
do while ( ((abs(err) .ge. errtol) .or. (abs(c-cm1) .ge. eigtol))
& .and. (icount .le. maxcount) )
q = 0
tq(0) = tf
c
c Set the initial conditions
c
k = 0
call FIC(n,r,U(1,1,k))
c
c Gram-Schmidt
c
w(1) = SQRT(inprod(n, U(1,1,k), U(1,1,k)))
do i = 1, n
z(i,1) = U(i,1,k)/w(1)
end do
do mi = 2, (n-r)
do i = 1, n
eta(i) = U(i,mi,k)
end do
do mj = mi-1, 1, -1
do i = 1, n
eta(i) = eta(i) - inprod(n, U(1,mi,k), z(1,mj))*z(i,mj)
end do
end do
w(mi) = SQRT(inprod(n, eta, eta))
do i = 1, n
z(i,mi) = eta(i)/w(mi)
end do
end do
c
c Now update the U matrix with the orthonormal values
c
do i = 1, n
do m = 1, n-r
U(i,m,k) = z(i,m)
end do
end do
#ifdef CHECK_ORTHOGONALITY
aa = ABS(inprod(n, U(1,1,k), U(1,1,k)))
bb = ABS(inprod(n, U(1,2,k), U(1,2,k)))
cc = ABS(inprod(n, U(1,1,k), U(1,2,k)))
test = ACOS(cc/SQRT(aa*bb))*180./pi
write (*,*) k, mi, mj, test
#endif
c
c Integrate the homo. and particular equations
c
do k = 1, nstep
t = tf - h*k
c
c Loop thru all homogeneous solutions
c
do m = 1, n-r
#ifdef USE_NR_ODEINT
do i = 1, n
Utemp(i) = U(i,m,k-1)
end do
call NR_CODEINT(Utemp,n,t+h,t,1.E-6,-h,1.e-20,nok,nbad,
& FHOMO,NR_CRKQC)
do i = 1, n
U(i,m,k) = Utemp(i)
end do
c#elif USE_VODE
c ISTATE=1
c rt0 = t+h
c rt1 = t
c do i = 1, n
c Utemp(i) = U(i,m,k-1)
c end do
cc write(*,*) "Calling ZVODE...", rt0, rt1
c call ZVODE(FHOMO_VODE,n,Utemp,rt0,rt1,ITOL,RTOL,ATOL,ITASK,
c & ISTATE,IOPT,ZWORK,LZW,RWORK,LRW,IWORK,LIW,JEX,MF,
c & RPAR,IPAR)
c write(*,*) "Finixhed ZVODE...", rt0, rt1
c if (ISTATE .lt. 0) then
c write(*,*) "Error in ZVODE: ISTATE = ", ISTATE
c call exit(1)
c end if
c do i = 1, n
c U(i,m,k) = Utemp(i)
c end do
#elif USE_LSRK14
call CLSRK14(n, U(1,m,k-1), U(1,m,k), t+h, -h, FHOMO)
#elif USE_RKCK45
call CRKCK45( n, U(1,m,k-1), U(1,m,k), t+h, -h, FHOMO )
#else
call CRK4(n, U(1,m,k-1), U(1,m,k), t+h, -h, FHOMO)
#endif
end do
#if 0
if (norm .or. k.eq.1) ISTATE=1
ISTATE=1
rt0 = t+h
rt1 = t
do m = 1, n-r
do i = 1, n
U(i,m,k) = U(i,m,k-1)
end do
end do
c write(*,*) "Calling ZVODE...", rt0, rt1, n*(n-r)
call ZVODE(FHOMO_VODE,n*(n-r),U(1,1,k),rt0,rt1,ITOL,RTOL,ATOL,
& ITASK,ISTATE,IOPT,ZWORK,LZW,RWORK,LRW,IWORK,LIW,
& JEX,MF,RPAR,IPAR)
if (ISTATE .lt. 0) then
write(*,*) "Error in ZVODE: ISTATE = ", ISTATE
call exit(1)
end if
#else
c write(*,*) "Hack...fix it"
c stop
#endif
c
c Test to see if normalization is required
c
norm = .false.
ta = pi*testalpha/180.0
do mi = 1, n-r
do mj = 1, n-r
if (mi .ne. mj) then
aa = ABS(inprod(n, U(1,mi,k), U(1,mi,k)))
bb = ABS(inprod(n, U(1,mj,k), U(1,mj,k)))
cc = ABS(inprod(n, U(1,mi,k), U(1,mj,k)))
test = acos(cc/SQRT(aa*bb))
if (test .lt. ta) norm = .true.
c write (*,*) k, mi, mj, test, testalpha, norm
end if
end do
end do
c
c Perform normalization
c
if ( norm .or. (k .eq. nstep) ) then
q = q + 1
c write (*,*) 'Normalization = ', q
tq(q) = t
if (k .eq. nstep) then
qend = q
c write (*,*) 'qend = ',qend
end if
c
c Gram-Schmidt
c
w(1) = SQRT(inprod(n, U(1,1,k), U(1,1,k)))
do i = 1, n
z(i,1) = U(i,1,k)/w(1)
end do
do mi = 2, (n-r)
do i = 1, n
eta(i) = U(i,mi,k)
end do
do mj = mi-1, 1, -1
do i = 1, n
eta(i) = eta(i) - inprod(n, U(1,mi,k),
& z(1,mj))*z(i,mj)
end do
end do
w(mi) = SQRT(inprod(n, eta, eta))
do i = 1, n
z(i,mi) = eta(i)/w(mi)
end do
end do
c
c Now I have the orthonormal basis in z and
c the norms in w so I can compute the P orthonormalization
c matrix
c
do j = 1, n-r
do i = 1, j
if (j .eq. i) then
P(j,i,q) = 1.0/w(j)
else
P(j,i,q) = 0.0
do s = i, j-1
P(j,i,q) = P(j,i,q)-inprod(n,U(1,j,k),z(1,s))/w(j)*
& P(s,i,q)
end do
end if
end do
end do
c
c Check the P matrix
c
if (.false.) then
do i = 1, n
do m = 1, n-r
ut(i,m) = 0.0
do j = 1, n-r
ut(i,m) = ut(i,m) + U(i,j,k)*P(m,j,q)
end do
end do
end do
do i = 1,n
write (*,*) i,(ut(i,m), m = 1, n-r)
end do
write (*,*)
do i = 1,n
write (*,*) i,( z(i,m), m = 1, n-r)
end do
write (*,*)
write (*,*)
end if
c
c Now update the U matrix with the orthonormal values
c
do i = 1, n
do m = 1, n-r
U(i,m,k) = z(i,m)
end do
end do
end if
end do
c
c check boundary conditions
c
if (.true.) then
c
c strictly enforce the zero BC
c
B(1,qend) = -U(1,2,nstep)/U(1,1,nstep)
B(2,qend) = 1.0
c olderr = err
c err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
else
c
c strictly enforce the zero slope BC
c
B(1,qend) = 1.0
B(2,qend) = -U(2,1,nstep)/U(2,2,nstep)
c olderr = err
c err = U(1,1,nstep)*B(1,qend) + U(1,2,nstep)*B(2,qend)
end if
ctemp = c
if (icount .eq. 1) then
cm2 = c
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
errm2 = err
c = CMPLX( REAL(c)*.9999, AIMAG(c) )
else if (icount .eq. 2) then
cm1 = c
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
errm1 = err
c c = CMPLX( REAL(c), AIMAG(c)*.9999 )
fdxr = REAL(err-errm2) / REAL(c-cm2)
fdxi = AIMAG(err-errm2) / REAL(c-cm2)
fd = CMPLX( fdxr, fdxi )
c = c - err/fd
else
err = U(2,1,nstep)*B(1,qend) + U(2,2,nstep)*B(2,qend)
qt = (c-cm1)/(cm1-cm2)
At = qt*err-qt*(1.+qt)*errm1+qt**2*errm2
Bt = (2.*qt+1.)*err-(1.+qt)**2*errm1+qt**2*errm2
Ct = (1.+qt)*err
if ( ABS(Bt+SQRT(Bt**2-4.*At*Ct)) .gt.
& ABS(Bt-SQRT(Bt**2-4.*At*Ct)) ) then
c = ctemp-(ctemp-cm1)*2.*Ct/(Bt+SQRT(Bt**2-4.*At*Ct))
else
c = ctemp-(ctemp-cm1)*2.*Ct/(Bt-SQRT(Bt**2-4.*At*Ct))
end if
cm2 = cm1
cm1 = ctemp
errm2 = errm1
errm1 = err
end if
write (*,30) icount,real(ctemp),aimag(ctemp),real(err),
& aimag(err)
30 format (1x,i4,2(e17.8,e17.8,3x))
icount = icount + 1
end do
c
c.... Output Eigenvalue
c
icount = icount - 1
write (*,40) real(ctemp), aimag(ctemp), qend
40 format (/,'Eigenvalue = ',e17.8,1x,e17.8,2x,i5/)
write (*,45) icount, abs(err), abs(c-cm1)
45 format (' Eigenvalue iterations = ', i5/,' |error| = ',
& es17.8/,' |c-cm1| = ', es17.8/)
c
c Second Pass to compute the eigenfunctions
c
if (eigfun) then
q = qend
vmax = 0.0
k = nstep
do i = 1, n
y(i,k) = v(i,k)
do m = 1, n-r
y(i,k) = y(i,k) + U(i,m,k)*B(m,q)
end do
end do
do m = 1, n-r
B(m,q-1) = 0.0
do j = 1, n-r
B(m,q-1) = B(m,q-1) + P(j,m,q)*B(j,q)
end do
end do
do k = nstep-1, 0, -1
t = tf - h*k
if ( t .gt. tq(q-1) ) then
q = q - 1
do m = 1, n-r
B(m,q-1) = 0.0
do j = 1, n-r
B(m,q-1) = B(m,q-1) + P(j,m,q)*B(j,q)
end do
end do
end if
do i = 1, n
y(i,k) = 0.0
do m = 1, n-r
y(i,k) = y(i,k) + U(i,m,k)*B(m,q-1)
end do
end do
if ( ABS(y(i,k)) .gt. ABS(vmax) ) then
vmax = y(i,k)
end if
end do
c
c Output eigenfunction and derivatives: u, u', u'', u'''
c
do k = 0, nstep
t = tf - h*k
write (11,20) t, REAL(y(1,k)/vmax),AIMAG(y(1,k)/vmax)
write (12,20) t, REAL(y(2,k)/vmax),AIMAG(y(2,k)/vmax)
write (13,20) t, REAL(y(3,k)/vmax),AIMAG(y(3,k)/vmax)
write (14,20) t, REAL(y(4,k)/vmax),AIMAG(y(4,k)/vmax)
20 format (1x,3(ES16.8E3,1x))
end do
end if
return
end
C***********************************************************************
C> \brief Advance one time step using fourth order (real) Runge-Kutta
C> \param[in] neq number of equations
C> \param[in] yo initial value
C> \param[out] yf final value
C> \param[in] to intial time
C> \param[in] h time step
C> \param[in] FUNC function to integrate
C***********************************************************************
subroutine SRK4(neq, yo, yf, to, h, FUNC)
C***********************************************************************
C
C Advance one time step using fourth order (real) Runge-Kutta
C
C***********************************************************************
external FUNC
integer neq
real to, h
real yo(neq), yf(neq)
real f(neq), k1(neq), k2(neq), k3(neq), k4(neq), q(neq)
call FUNC(neq, yo, to, f)
do j = 1 , neq
k1(j) = h*f(j)
q(j) = yo(j) + 0.5*k1(j)
end do
call FUNC(neq, q, to+0.5*h, f)
do j = 1 , neq
k2(j) = h*f(j)
q(j) = yo(j) + 0.5*k2(j)
end do
call FUNC(neq, q, to+0.5*h, f)
do j = 1 , neq
k3(j) = h*f(j)
q(j) = yo(j) + k3(j)
end do
call FUNC(neq, q, to+h, f)
do j = 1 , neq
k4(j) = h*f(j)
yf(j) = yo(j)+k1(j)/6.+(k2(j)+k3(j))/3.+k4(j)/6.
end do
return
end
C***********************************************************************
C> \brief Advance one time step using fourth order (real) Runge-Kutta
C> \param[in] neq number of equations
C> \param[in] yo initial value
C> \param[out] yf final value
C> \param[in] to intial time
C> \param[in] h time step
C> \param[in] FUNC function to integrate
C***********************************************************************
subroutine CRK4(neq, yo, yf, to, h, FUNC)
C***********************************************************************
C
C Advance one time step using fourth order (complex) Runge-Kutta
C
c***********************************************************************
external FUNC
integer neq
real to, h
complex yo(neq), yf(neq)
complex f(neq), k1(neq), k2(neq), k3(neq), k4(neq), q(neq)
call FUNC(neq, yo, to, f)
do j = 1 , neq
k1(j) = h*f(j)
q(j) = yo(j) + 0.5*k1(j)
end do
call FUNC(neq, q, to+0.5*h, f)
do j = 1 , neq
k2(j) = h*f(j)
q(j) = yo(j) + 0.5*k2(j)
end do
call FUNC(neq, q, to+0.5*h, f)
do j = 1 , neq
k3(j) = h*f(j)
q(j) = yo(j) + k3(j)
end do
call FUNC(neq, q, to+h, f)
do j = 1 , neq
k4(j) = h*f(j)
yf(j) = yo(j)+k1(j)/6.+(k2(j)+k3(j))/3.+k4(j)/6.
end do
return
end
C***********************************************************************
SUBROUTINE SPLINE (N,X,Y,FDP)
C***********************************************************************
C
C.... Note: this routine is in the public domain and available
C at https://web.stanford.edu/class/me200c/
C
C-----THIS SUBROUTINE COMPUTES THE SECOND DERIVATIVES NEEDED
C-----IN CUBIC SPLINE INTERPOLATION. THE INPUT DATA ARE:
C-----N = NUMBER OF DATA POINTS
C-----X = ARRAY CONTAINING THE VALUES OF THE INDEPENDENT VARIABLE
C----- (ASSUMED TO BE IN ASCENDING ORDER)
C-----Y = ARRAY CONTAINING THE VALUES OF THE FUNCTION AT THE
C----- DATA POINTS GIVEN IN THE X ARRAY
C-----THE OUTPUT IS THE ARRAY FDP WHICH CONTAINS THE SECOND
C-----DERIVATIVES OF THE INTERPOLATING CUBIC SPLINE.
DIMENSION X(N),Y(N),A(N),B(N),C(N),R(N),FDP(N)
C-----COMPUTE THE COEFFICIENTS AND THE RHS OF THE EQUATIONS.
C-----THIS ROUTINE USES THE CANTILEVER CONDITION. THE PARAMETER
C-----ALAMDA (LAMBDA) IS SET TO 1. BUT THIS CAN BE USER-MODIFIED.
C-----A,B,C ARE THE THREE DIAGONALS OF THE TRIDIAGONAL SYSTEM;
C-----R IS THE RIGHT HAND SIDE. THESE ARE NOW ASSEMBLED.
ALAMDA = 1.
NM2 = N - 2
NM1 = N - 1
C(1) = X(2) - X(1)
DO 1 I=2,NM1
C(I) = X(I+1) - X(I)
A(I) = C(I-1)
B(I) = 2.*(A(I) + C(I))
R(I) = 6.*((Y(I+1) - Y(I))/C(I) - (Y(I) - Y(I-1))/C(I-1))
1 CONTINUE
B(2) = B(2) + ALAMDA * C(1)
B(NM1) = B(NM1) + ALAMDA * C(NM1)
C-----AT THIS POINT WE COULD CALL A TRIDIAGONAL SOLVER SUBROUTINE
C-----BUT THE NOTATION IS CLUMSY SO WE WILL SOLVE DIRECTLY. THE
C-----NEXT SECTION SOLVES THE SYSTEM WE HAVE JUST SET UP.
DO 2 I=3,NM1
T = A(I)/B(I-1)
B(I) = B(I) - T * C(I-1)
R(I) = R(I) - T * R(I-1)
2 CONTINUE
FDP(NM1) = R(NM1)/B(NM1)
DO 3 I=2,NM2
NMI = N - I
FDP(NMI) = (R(NMI) - C(NMI)*FDP(NMI+1))/B(NMI)
3 CONTINUE
FDP(1) = ALAMDA * FDP(2)
FDP(N) = ALAMDA * FDP(NM1)
C-----WE NOW HAVE THE DESIRED DERIVATIVES SO WE RETURN TO THE
C-----MAIN PROGRAM.
RETURN
END
C***********************************************************************
SUBROUTINE SPEVAL (N,X,Y,FDP,XX,F)
C***********************************************************************
C
C.... Note: this routine is in the public domain and available
C at https://web.stanford.edu/class/me200c/
C
C-----THIS SUBROUTINE EVALUATES THE CUBIC SPLINE GIVEN
C-----THE 2ND DERIVATIVE COMPUTED BY SUBROUTINE SPLINE.
C-----THE INPUT PARAMETERS N,X,Y,FDP HAVE THE SAME
C-----MEANING AS IN SPLINE.
C-----XX = VALUE OF INDEPENDENT VARIABLE FOR WHICH
C----- AN INTERPOLATED VALUE IS REQUESTED
C-----F = THE INTERPOLATED RESULT
DIMENSION X(N),Y(N),FDP(N)
C-----THE FIRST JOB IS TO FIND THE PROPER INTERVAL.
#if USE_NR_HUNT
c
c Search using bisection with a good guess
c
I = IOLD
IF (XX.EQ.X(1)) THEN
I = 1
ELSE IF (XX.EQ.X(N)) THEN
I = N
ELSE
call HUNT (X,N,XX,I)
END IF
IOLD = I
#elif 1
I = IOLD
IF (XX.EQ.X(1)) THEN
I = 1
ELSE IF (XX.EQ.X(N)) THEN
I = N
ELSE
call BISECT (X,N,XX,I)
ENDiF
IOLD = I
#else
c
c This is really a slow way of searching