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|
subroutine SplineCub(x, y, d, n, type, A_d, A_sd, qdy, lll)
*
* PURPOSE
* computes a cubic spline interpolation function
* in Hermite form (ie computes the derivatives d(i) of the
* spline in each interpolation point (x(i), y(i)))
*
* ARGUMENTS
* inputs :
* n number of interpolation points (n >= 3)
* x, y the n interpolation points, x must be in strict increasing order
* type type of the spline : currently
* type = 0 correspond to a NOT_A_KNOT spline where it is
* imposed the conditions :
* s'''(x(2)-) = s'''(x(2)+)
* and s'''(x(n-1)-) = s'''(x(n-1)+)
* type = 1 correspond to a NATURAL spline with the conditions :
* s''(x1) = 0
* and s''(xn) = 0
* type = 2 correspond to a CLAMPED spline (d(1) and d(n) are given)
*
* type = 3 correspond to a PERIODIC spline
* outputs :
* d the derivatives in each x(i) i = 1..n
*
* work arrays :
* A_d(1..n), A_sd(1..n-1), qdy(1..n-1)
* lll(1..n-1) (used only in the periodic case)
*
* NOTES
* this routine requires (i) n >= 3 (for natural) n >=4 (for not_a_knot)
* (ii) strict increasing abscissae x(i)
* (iii) y(1) = y(n) in the periodic case
* THESE CONDITIONS MUST BE TESTED IN THE CALLING CODE
*
* AUTHOR
* Bruno Pincon
*
* July 22 2004 : correction of the case not_a_knot which worked only
* for equidistant abscissae
*
implicit none
integer n, type
double precision x(n), y(n), d(n), A_d(n), A_sd(n-1), qdy(n-1),
$ lll(n-1)
include 'constinterp.h'
integer i
double precision r
if (n .eq. 2) then
if (type .ne. CLAMPED) then
d(1) = (y(2) - y(1)) / (x(2) - x(1))
d(2) = d(1)
endif
return
endif
if (n .eq. 3 .and. type .eq. NOT_A_KNOT) then
call derivd(x, y, d, n, 1, FAST)
return
endif
do i = 1, n-1
A_sd(i) = 1.d0 / (x(i+1) - x(i))
qdy(i) = (y(i+1) - y(i)) * A_sd(i)**2
enddo
* compute the coef matrix and r.h.s. for rows 2..n-1
* (which don't relies on the type)
do i = 2, n-1
A_d(i) = 2.d0*( A_sd(i-1) + A_sd(i) )
d(i) = 3.d0 * ( qdy(i-1) + qdy(i) )
enddo
* compute equ 1 and n in function of the type
if (type .eq. NATURAL) then
A_d(1) = 2.d0*A_sd(1)
d(1) = 3.d0 * qdy(1)
A_d(n) = 2.d0*A_sd(n-1)
d(n) = 3.d0 * qdy(n-1)
call TriDiagLDLtSolve(A_d, A_sd, d, n)
else if (type .eq. NOT_A_KNOT) then
! s'''(x(2)-) = s'''(x(2)+)
r = A_sd(2)/A_sd(1)
A_d(1) = A_sd(1)/(1.d0 + r)
d(1) = ((3.d0*r+2.d0)*qdy(1)+r*qdy(2))/(1.d0+r)**2
! s'''(x(n-1)-) = s'''(x(n-1)+)
r = A_sd(n-2)/A_sd(n-1)
A_d(n) = A_sd(n-1)/(1.d0 + r)
d(n) = ((3.d0*r+2.d0)*qdy(n-1)+r*qdy(n-2))/(1.d0+r)**2
call TriDiagLDLtSolve(A_d, A_sd, d, n)
else if (type .eq. CLAMPED) then
! d(1) and d(n) are already known
d(2) = d(2) - d(1)*A_sd(1)
d(n-1) = d(n-1) - d(n)*A_sd(n-1)
call TriDiagLDLtSolve(A_d(2), A_sd(2), d(2), n-2)
else if (type .eq. PERIODIC) then
A_d(1) = 2.d0*( A_sd(1) + A_sd(n-1) )
d(1) = 3.d0 * ( qdy(1) + qdy(n-1) )
lll(1) = A_sd(n-1)
call dset(n-2, 0.d0, lll(2),1) ! mise a zero
lll(n-2) = A_sd(n-2)
call CyclicTriDiagLDLtSolve(A_d, A_sd, lll, d, n-1)
d(n) = d(1)
endif
end ! subroutine SplineCub
subroutine TriDiagLDLtSolve(d, l, b, n)
*
* PURPOSE
* solve a linear system A x = b with a symetric tridiagonal positive definite
* matrix A by using an LDL^t factorization
*
* PARAMETERS
* d(1..n) : on input the diagonal of A
* on output the diagonal of the (diagonal) matrix D
* l(1..n-1) : on input the sub-diagonal of A
* on output the sub-diagonal of L
* b(1..n) : on input contains the r.h.s. b
* on output the solution x
* n : the dimension
*
* CAUTION
* no zero pivot detection
*
implicit none
integer n
integer flag
double precision d(n), l(n-1), b(n)
integer i
double precision temp
do i = 2, n
temp = l(i-1)
l(i-1) = l(i-1) / d(i-1)
d(i) = d(i) - temp * l(i-1)
b(i) = b(i) - l(i-1)*b(i-1)
enddo
b(n) = b(n) / d(n)
do i = n-1, 1, -1
b(i) = b(i)/d(i) - l(i)*b(i+1)
enddo
end ! subroutine TriDiagLDLtSolve
subroutine CyclicTriDiagLDLtSolve(d, lsd, lll, b, n)
*
* PURPOSE
* solve a linear system A x = b with a symetric "nearly" tridiagonal
* positive definite matrix A by using an LDL^t factorization,
* the matrix A has the form :
*
* |x x x| |1 |
* |x x x | |x 1 |
* | x x x | | x 1 |
* | x x x | and so the L is like | x 1 |
* | x x x | | x 1 |
* | x x x| | x 1 |
* |x x x| |x x x x x x 1|
*
* PARAMETERS
* d(1..n) : on input the diagonal of A
* on output the diagonal of the (diagonal) matrix D
* lsd(1..n-2) : on input the sub-diagonal of A (without A(n,n-1))
* on output the sub-diagonal of L (without L(n,n-1))
* lll(1..n-1) : on input the last line of A (without A(n,n))
* on output the last line of L (without L(n,n))
* b(1..n) : on input contains the r.h.s. b
* on output the solution x
* n : the dimension
*
* CAUTION
* no zero pivot detection
*
implicit none
integer n
double precision d(n), lsd(n-2), lll(n-1), b(n)
integer i, j
double precision temp1, temp2
* compute the LDL^t factorization
do i = 1, n-2
temp1 = lsd(i)
temp2 = lll(i)
lsd(i) = lsd(i) / d(i) ! elimination coef L(i,i-1)
lll(i) = lll(i) / d(i) ! elimination coef L(n,i-1)
d(i+1) = d(i+1) - lsd(i) * temp1 ! elimination on line i+1
lll(i+1) = lll(i+1) - lll(i) * temp1 ! elimination on line n
d(n) = d(n) - lll(i) * temp2 ! elimination on line n
enddo
temp2 = lll(n-1)
lll(n-1) = lll(n-1) / d(n-1)
d(n) = d(n) - lll(n-1) * temp2
* solve LDL^t x = b (but use b for x and for the intermediary vectors...)
do i = 2, n-1
b(i) = b(i) - lsd(i-1)*b(i-1)
enddo
do j = 1, n-1
b(n) = b(n) - lll(j)*b(j)
enddo
do i = 1, n
b(i) = b(i) / d(i)
enddo
b(n-1) = b(n-1) - lll(n-1)*b(n)
do i = n-2, 1, -1
b(i) = b(i) - lsd(i)*b(i+1) - lll(i)*b(n)
enddo
end ! subroutine CyclicTriDiagLDLtSolve
integer function isearch(t, x, n)
*
* PURPOSE
* x(1..n) being an array (with strict increasing order and n >=2)
* representing intervals, this routine return i such that :
*
* x(i) <= t <= x(i+1)
*
* and 0 if t is not in [x(1), x(n)]
*
implicit none
integer n
double precision t, x(n)
integer i1, i2, i
if ( x(1) .le. t .and. t .le. x(n)) then
* dichotomic search
i1 = 1
i2 = n
do while ( i2 - i1 .gt. 1 )
i = (i1 + i2)/2
if ( t .le. x(i) ) then
i2 = i
else
i1 = i
endif
enddo
isearch = i1
else
isearch = 0
endif
end
subroutine EvalPWHermite(t, st, dst, d2st, d3st, m, x, y, d, n,
$ outmode)
*
* PURPOSE
* evaluation at the abscissae t(1..m) of the piecewise hermite function
* define by x(1..n), y(1..n), d(1..n) (d being the derivatives at the
* x(i)) together with its derivative, second derivative and third derivative
*
* PARAMETERS
*
* outmode : define what return in case t(j) not in [x(1), x(n)]
implicit none
integer m, n, outmode
double precision t(m), st(m), dst(m), d2st(m), d3st(m),
$ x(n), y(n), d(n)
include 'constinterp.h'
integer i, j
integer isearch, isanan
external isearch, isanan
double precision tt
double precision return_a_nan
external return_a_nan
logical new_call
common /INFO_HERMITE/new_call
new_call = .true.
i = 0
do j = 1, m
tt = t(j)
call fast_int_search(tt, x, n, i) ! recompute i only if necessary
if ( i .ne. 0 ) then
call EvalHermite(tt, x(i), x(i+1), y(i), y(i+1), d(i),
$ d(i+1), st(j), dst(j), d2st(j), d3st(j), i)
else ! t(j) is outside [x(1), x(n)] evaluation depend upon outmode
if (outmode .eq. BY_NAN .or. isanan(tt) .eq. 1) then
st(j) = return_a_nan()
dst(j) = st(j)
d2st(j) = st(j)
d3st(j) = st(j)
elseif (outmode .eq. BY_ZERO) then
st(j) = 0.d0
dst(j) = 0.d0
d2st(j) = 0.d0
d3st(j) = 0.d0
elseif (outmode .eq. C0) then
dst(j) = 0.d0
d2st(j) = 0.d0
d3st(j) = 0.d0
if ( tt .lt. x(1) ) then
st(j) = y(1)
else
st(j) = y(n)
endif
elseif (outmode .eq. LINEAR) then
d2st(j) = 0.d0
d3st(j) = 0.d0
if ( tt .lt. x(1) ) then
dst(j) = d(1)
st(j) = y(1) + (tt - x(1))*d(1)
else
dst(j) = d(n)
st(j) = y(n) + (tt - x(n))*d(n)
endif
else
if (outmode .eq. NATURAL) then
call near_interval(tt, x, n, i)
elseif (outmode .eq. PERIODIC) then
call coord_by_periodicity(tt, x, n, i)
endif
call EvalHermite(tt, x(i), x(i+1), y(i), y(i+1), d(i),
$ d(i+1), st(j), dst(j), d2st(j), d3st(j), i)
endif
endif
enddo
end ! subroutine EvalPWHermite
subroutine EvalHermite(t, xa, xb, ya, yb, da, db, h, dh, ddh,
$ dddh, i)
implicit none
double precision t, xa, xb, ya, yb, da, db, h, dh, ddh, dddh
integer i
logical new_call
common /INFO_HERMITE/new_call
double precision tmxa, dx, p, c2, c3
integer old_i
save old_i, c2, c3
data old_i/0/
if (old_i .ne. i .or. new_call) then
* compute the following Newton form :
* h(t) = ya + da*(t-xa) + c2*(t-xa)^2 + c3*(t-xa)^2*(t-xb)
dx = 1.d0/(xb - xa)
p = (yb - ya) * dx
c2 = (p - da) * dx
c3 = ( (db - p) + (da - p) ) * (dx*dx)
new_call = .false.
endif
old_i = i
* eval h(t), h'(t), h"(t) and h"'(t), by a generalised Horner 's scheme
tmxa = t - xa
h = c2 + c3*(t - xb)
dh = h + c3*tmxa
ddh = 2.d0*( dh + c3*tmxa )
dddh = 6.d0*c3
h = da + h*tmxa
dh = h + dh*tmxa
h = ya + h*tmxa
end ! subroutine EvalHermite
subroutine fast_int_search(xx, x, nx, i)
implicit none
integer nx, i
double precision xx, x(nx)
integer isearch
external isearch
if ( i .eq. 0 ) then
i = isearch(xx, x, nx)
elseif ( .not. (x(i) .le. xx .and. xx .le. x(i+1))) then
i = isearch(xx, x, nx)
endif
end
subroutine coord_by_periodicity(t, x, n, i)
*
* PURPOSE
* recompute t such that t in [x(1), x(n)] by periodicity :
* and then the interval i of this new t
*
implicit none
integer n, i
double precision t, x(n)
integer isearch
external isearch
double precision r, dx
dx = x(n) - x(1)
r = (t - x(1)) / dx
if (r .ge. 0.d0) then
t = x(1) + (r - aint(r))*dx
else
r = abs(r)
t = x(n) - (r - aint(r))*dx
endif
! some cautions in case of roundoff errors (is necessary ?)
if (t .lt. x(1)) then
t = x(1)
i = 1
else if (t .gt. x(n)) then
t = x(n)
i = n-1
else
i = isearch(t, x, n)
endif
end ! subroutine coord_by_periodicity
subroutine near_grid_point(xx, x, nx, i)
*
* calcule le point de la grille le plus proche ... a detailler
*
implicit none
integer i, nx
double precision xx, x(nx)
if (xx .lt. x(1)) then
i = 1
xx = x(1)
else ! xx > x(nx)
i = nx-1
xx = x(nx)
endif
end
subroutine near_interval(xx, x, nx, i)
*
* idem sans modifier xx
*
implicit none
integer i, nx
double precision xx, x(nx)
if (xx .lt. x(1)) then
i = 1
else ! xx > x(nx)
i = nx-1
endif
end
subroutine proj_by_per(t, xmin, xmax)
*
* PURPOSE
* recompute t such that t in [xmin, xmax] by periodicity.
*
implicit none
double precision t, xmin, xmax
double precision r, dx
dx = xmax - xmin
r = (t - xmin) / dx
if (r .ge. 0.d0) then
t = xmin + (r - aint(r))*dx
else
r = abs(r)
t = xmax - (r - aint(r))*dx
endif
! some cautions in case of roundoff errors (is necessary ?)
if (t .lt. xmin) then
t = xmin
else if (t .gt. xmax) then
t = xmax
endif
end ! subroutine proj_by_per
subroutine proj_on_grid(xx, xmin, xmax)
*
implicit none
double precision xx, xmin, xmax
if (xx .lt. xmin) then
xx = xmin
else
xx = xmax
endif
end
double precision function return_a_nan()
implicit none
logical first
double precision a, b
save first
data first /.true./
data a /1.d0/, b /1.d0/
save a
if (first) then
first = .false.
a = (a - b) / (a - b)
endif
return_a_nan = a
end
subroutine BiCubicSubSpline(x, y, u, nx, ny, C, p, q, r, type)
*
* PURPOSE
* compute bicubic subsplines
*
implicit none
integer nx, ny, type
double precision x(nx), y(ny), u(nx, ny), C(4,4,nx-1,ny-1),
$ p(nx, ny), q(nx, ny), r(nx, ny)
integer i, j
include 'constinterp.h'
if (type .eq. MONOTONE) then
* approximation des derivees par SUBROUTINE DPCHIM(N,X,F,D,INCFD)
! p = du/dx
do j = 1, ny
call dpchim(nx, x, u(1,j), p(1,j), 1)
enddo
! q = du/dy
do i = 1, nx
call dpchim(ny, y, u(i,1), q(i,1), nx)
enddo
! r = d2 u/ dx dy approchee via dq / dx
do j = 1, ny
call dpchim(nx, x, q(1,j), r(1,j), 1)
enddo
elseif (type .eq. FAST .or. type .eq. FAST_PERIODIC) then
* approximation des derivees partielles par methode simple
! p = du/dx
do j = 1, ny
call derivd(x, u(1,j), p(1,j), nx, 1, type)
enddo
! q = du/dy
do i = 1, nx
call derivd(y, u(i,1), q(i,1), ny, nx, type)
enddo
! r = d2 u/ dx dy approchee via dq / dx
do j = 1, ny
call derivd(x, q(1,j), r(1,j), nx, 1, type)
enddo
endif
* calculs des coefficients dans les bases (x-x(i))^k (y-y(j))^l 0<= k,l <= 3
* pour evaluation rapide via Horner par la suite
call coef_bicubic(u, p, q, r, x, y, nx, ny, C)
end ! subroutine BiCubicSubSpline
subroutine BiCubicSpline(x, y, u, nx, ny, C, p, q, r,
$ A_d, A_sd, d, ll, qdu, u_temp, type)
*
* PURPOSE
* compute bicubic splines
*
implicit none
integer nx, ny, type
double precision x(nx), y(ny), u(nx, ny), C(4,4,nx-1,ny-1),
$ p(nx, ny), q(nx, ny), r(nx, ny), A_d(*),
$ A_sd(*), d(ny), ll(*), qdu(*), u_temp(ny)
include 'constinterp.h'
integer i, j
! compute du/dx
do j = 1, ny
call SplineCub(x, u(1,j), p(1,j), nx, type, A_d, A_sd, qdu, ll)
enddo
! compute du/dy
do i = 1, nx
call dcopy(ny, u(i,1), nx, u_temp, 1)
call SplineCub(y, u_temp, d, ny, type, A_d, A_sd, qdu, ll)
call dcopy(ny, d, 1, q(i,1), nx)
enddo
! compute ddu/dxdy
call SplineCub(x, q(1,1), r(1,1), nx, type, A_d, A_sd, qdu, ll)
call SplineCub(x, q(1,ny), r(1,ny), nx, type, A_d, A_sd, qdu, ll)
do i = 1, nx
call dcopy(ny, p(i,1), nx, u_temp, 1)
d(1) = r(i,1)
d(ny) = r(i,ny)
call SplineCub(y, u_temp, d, ny, CLAMPED, A_d, A_sd, qdu, ll)
call dcopy(ny-2, d(2), 1, r(i,2), nx)
enddo
* calculs des coefficients dans les bases (x-x(i))^k (y-y(j))^l 0<= k,l <= 3
* pour evaluation rapide via Horner par la suite
call coef_bicubic(u, p, q, r, x, y, nx, ny, C)
end ! subroutine BiCubicSpline
subroutine derivd(x, u, du, n, inc, type)
*
* PURPOSE
* given functions values u(i) at points x(i), i = 1, ..., n
* this subroutine computes approximations du(i) of the derivative
* at the points x(i).
*
* METHOD
* For i in [2,n-1], the "centered" formula of order 2 is used :
* d(i) = derivative at x(i) of the interpolation polynomial
* of the points {(x(j),u(j)), j in [i-1,i+1]}
*
* For i=1 and n, if type = FAST_PERIODIC (in which case u(n)=u(1)) then
* the previus "centered" formula is also used else (type = FAST), d(1)
* is the derivative at x(1) of the interpolation polynomial of
* {(x(j),u(j)), j in [1,3]} and the same method is used for d(n)
*
* ARGUMENTS
* inputs :
* n integer : number of point (n >= 2)
* x, u double precision : the n points, x must be in strict increasing order
* type integer : FAST (the function is non periodic) or FAST_PERIODIC
* (the function is periodic), in this last case u(n) must be equal to u(1))
* inc integer : to deal easily with 2d applications, u(i) is in fact
* u(1,i) with u declared as u(inc,*) to avoid the direct management of
* the increment inc (the i th value given with u(1 + inc*(i-1) ...)
* outputs :
* d the derivatives in each x(i) i = 1..n
*
* NOTES
* this routine requires (i) n >= 2
* (ii) strict increasing abscissae x(i)
* (iii) u(1)=u(n) if type = FAST_PERIODIC
* ALL THESE CONDITIONS MUST BE TESTED IN THE CALLING CODE
*
* AUTHOR
* Bruno Pincon
*
implicit none
integer n, inc, type
double precision x(n), u(inc,*), du(inc,*)
include 'constinterp.h'
double precision dx_l, du_l, dx_r, du_r, w_l, w_r
integer i, k
if (n .eq. 2) then ! special case used linear interp
du(1,1) = (u(1,2) - u(1,1))/(x(2)-x(1))
du(1,2) = du(1,1)
return
endif
if (type .eq. FAST_PERIODIC) then
dx_r = x(n)-x(n-1)
du_r = (u(1,1) - u(1,n-1)) / dx_r
do i = 1, n-1
dx_l = dx_r
du_l = du_r
dx_r = x(i+1) - x(i)
du_r = (u(1,i+1) - u(1,i))/dx_r
w_l = dx_r/(dx_l + dx_r)
w_r = 1.d0 - w_l
du(1,i) = w_l*du_l + w_r*du_r
enddo
du(1,n) = du(1,1)
else if (type .eq. FAST) then
dx_l = x(2) - x(1)
du_l = (u(1,2) - u(1,1))/dx_l
dx_r = x(3) - x(2)
du_r = (u(1,3) - u(1,2))/dx_r
w_l = dx_r/(dx_l + dx_r)
w_r = 1.d0 - w_l
du(1,1) = (1.d0 + w_r)*du_l - w_r*du_r
du(1,2) = w_l*du_l + w_r*du_r
do i = 3, n-1
dx_l = dx_r
du_l = du_r
dx_r = x(i+1) - x(i)
du_r = (u(1,i+1) - u(1,i))/dx_r
w_l = dx_r/(dx_l + dx_r)
w_r = 1.d0 - w_l
du(1,i) = w_l*du_l + w_r*du_r
enddo
du(1,n) = (1.d0 + w_l)*du_r - w_l*du_l
endif
end
subroutine coef_bicubic(u, p, q, r, x, y, nx, ny, C)
*
* PURPOSE
* compute for each polynomial (i,j)-patch (defined on
* [x(i),x(i+1)]x[y(i),y(i+1)]) the following base
* representation :
* i,j _4_ _4_ i,j k-1 l-1
* u (x,y) = >__ >__ C (x-x(i)) (y-y(j))
* k=1 l=1 k,l
*
* from the "Hermite" representation (values of u, p = du/dx,
* q = du/dy, r = ddu/dxdy at the 4 vertices (x(i),y(j)),
* (x(i+1),y(j)), (x(i+1),y(j+1)), (x(i),y(j+1)).
*
implicit none
integer nx, ny
double precision u(nx, ny), p(nx, ny), q(nx, ny), r(nx, ny),
$ x(nx), y(ny), C(4,4,nx-1,ny-1)
integer i, j
double precision a, b, cc, d, dx, dy
do j = 1, ny-1
dy = 1.d0/(y(j+1) - y(j))
do i = 1, nx-1
dx = 1.d0/(x(i+1) - x(i))
C(1,1,i,j) = u(i,j)
C(2,1,i,j) = p(i,j)
C(1,2,i,j) = q(i,j)
C(2,2,i,j) = r(i,j)
a = (u(i+1,j) - u(i,j))*dx
C(3,1,i,j) = (3.d0*a -2.d0*p(i,j) - p(i+1,j))*dx
C(4,1,i,j) = (p(i+1,j) + p(i,j) - 2.d0*a)*(dx*dx)
a = (u(i,j+1) - u(i,j))*dy
C(1,3,i,j) = (3.d0*a -2.d0*q(i,j) - q(i,j+1))*dy
C(1,4,i,j) = (q(i,j+1) + q(i,j) - 2.d0*a)*(dy*dy)
a = (q(i+1,j) - q(i,j))*dx
C(3,2,i,j) = (3.d0*a - r(i+1,j) - 2.d0*r(i,j))*dx
C(4,2,i,j) = (r(i+1,j) + r(i,j) - 2.d0*a)*(dx*dx)
a = (p(i,j+1) - p(i,j))*dy
C(2,3,i,j) = (3.d0*a - r(i,j+1) - 2.d0*r(i,j))*dy
C(2,4,i,j) = (r(i,j+1) + r(i,j) - 2.d0*a)*(dy*dy)
a = (u(i+1,j+1)+u(i,j)-u(i+1,j)-u(i,j+1))*(dx*dx*dy*dy)
$ - (p(i,j+1)-p(i,j))*(dx*dy*dy)
$ - (q(i+1,j)-q(i,j))*(dx*dx*dy)
$ + r(i,j)*(dx*dy)
b = (p(i+1,j+1)+p(i,j)-p(i+1,j)-p(i,j+1))*(dx*dy*dy)
$ - (r(i+1,j)-r(i,j))*(dx*dy)
cc = (q(i+1,j+1)+q(i,j)-q(i+1,j)-q(i,j+1))*(dx*dx*dy)
$ - (r(i,j+1)-r(i,j))*(dx*dy)
d = (r(i+1,j+1)+r(i,j)-r(i+1,j)-r(i,j+1))*(dx*dy)
C(3,3,i,j) = 9.d0*a - 3.d0*b - 3.d0*cc + d
C(3,4,i,j) =(-6.d0*a + 2.d0*b + 3.d0*cc - d)*dy
C(4,3,i,j) =(-6.d0*a + 3.d0*b + 2.d0*cc - d)*dx
C(4,4,i,j) =( 4.d0*a - 2.d0*b - 2.d0*cc + d)*dx*dy
enddo
enddo
end
double precision function EvalBicubic(xx, yy, xk, yk, Ck)
implicit none
double precision xx, yy, xk, yk, Ck(4,4)
double precision dx, dy, u
integer i
dx = xx - xk
dy = yy - yk
u = 0.d0
do i = 4, 1, -1
u = Ck(i,1) + dy*(Ck(i,2) + dy*(Ck(i,3) + dy*Ck(i,4))) + u*dx
enddo
EvalBicubic = u
end ! function EvalBicubic
subroutine EvalBicubic_with_grad(xx, yy, xk, yk, Ck,
$ u, dudx, dudy)
implicit none
double precision xx, yy, xk, yk, Ck(4,4), u, dudx, dudy
double precision dx, dy
integer i
dx = xx - xk
dy = yy - yk
u = 0.d0
dudx = 0.d0
dudy = 0.d0
do i = 4, 1, -1
u = Ck(i,1) + dy*(Ck(i,2) + dy*(Ck(i,3) + dy*Ck(i,4))) + u*dx
dudx = Ck(2,i)+dx*(2.d0*Ck(3,i) + dx*3.d0*Ck(4,i)) + dudx*dy
dudy = Ck(i,2)+dy*(2.d0*Ck(i,3) + dy*3.d0*Ck(i,4)) + dudy*dx
enddo
end ! subroutine EvalBicubic_with_grad
subroutine EvalBicubic_with_grad_and_hes(xx, yy, xk, yk, Ck,
$ u, dudx, dudy,
$ d2udx2, d2udxy, d2udy2)
implicit none
double precision xx, yy, xk, yk, Ck(4,4), u, dudx, dudy, d2udx2,
$ d2udxy, d2udy2
double precision dx, dy
integer i
dx = xx - xk
dy = yy - yk
u = 0.d0
dudx = 0.d0
dudy = 0.d0
d2udx2 = 0.d0
d2udy2 = 0.d0
d2udxy = 0.d0
do i = 4, 1, -1
u = Ck(i,1) + dy*(Ck(i,2) + dy*(Ck(i,3) + dy*Ck(i,4))) + u*dx
dudx = Ck(2,i)+dx*(2.d0*Ck(3,i) + dx*3.d0*Ck(4,i)) + dudx*dy
dudy = Ck(i,2)+dy*(2.d0*Ck(i,3) + dy*3.d0*Ck(i,4)) + dudy*dx
d2udx2 = 2.d0*Ck(3,i) + dx*6.d0*Ck(4,i) + d2udx2*dy
d2udy2 = 2.d0*Ck(i,3) + dy*6.d0*Ck(i,4) + d2udy2*dx
enddo
d2udxy = Ck(2,2)+dy*(2.d0*Ck(2,3)+dy*3.d0*Ck(2,4))
. + dx*(2.d0*(Ck(3,2)+dy*(2.d0*Ck(3,3)+dy*3.d0*Ck(3,4)))
. + dx*(3.d0*(Ck(4,2)+dy*(2.d0*Ck(4,3)+dy*3.d0*Ck(4,4)))))
end ! subroutine EvalBicubic_with_grad_and_hes
subroutine BiCubicInterp(x, y, C, nx, ny, x_eval, y_eval, z_eval,
$ m, outmode)
*
* PURPOSE
* bicubic interpolation :
* the grid is defined by x(1..nx), y(1..ny)
* the known values are z(1..nx,1..ny), (z(i,j) being the value
* at point (x(i),y(j)))
* the interpolation is done on the points x_eval,y_eval(1..m)
* z_eval(k) is the result of the bicubic interpolation of
* (x_eval(k), y_eval(k))
*
implicit none
integer nx, ny, m, outmode
double precision x(nx), y(ny), C(4,4,nx-1,ny-1),
$ x_eval(m), y_eval(m), z_eval(m)
double precision xx, yy
integer i, j, k
include 'constinterp.h'
integer isanan
double precision return_a_nan, EvalBicubic
external isanan, return_a_nan, EvalBicubic
i = 0
j = 0
do k = 1, m
xx = x_eval(k)
call fast_int_search(xx, x, nx, i)
yy = y_eval(k)
call fast_int_search(yy, y, ny, j)
if (i .ne. 0 .and. j .ne. 0) then
z_eval(k) = EvalBicubic(xx, yy, x(i), y(j), C(1,1,i,j))
elseif (outmode .eq. BY_NAN .or. isanan(xx) .eq. 1
$ .or. isanan(yy) .eq. 1) then
z_eval(k) = return_a_nan()
elseif (outmode .eq. BY_ZERO) then
z_eval(k) = 0.d0
elseif (outmode .eq. PERIODIC) then
if (i .eq. 0) call coord_by_periodicity(xx, x, nx, i)
if (j .eq. 0) call coord_by_periodicity(yy, y, ny, j)
z_eval(k) = EvalBicubic(xx, yy, x(i), y(j), C(1,1,i,j))
elseif (outmode .eq. C0) then
if (i .eq. 0) call near_grid_point(xx, x, nx, i)
if (j .eq. 0) call near_grid_point(yy, y, ny, j)
z_eval(k) = EvalBicubic(xx, yy, x(i), y(j), C(1,1,i,j))
elseif (outmode .eq. NATURAL) then
if (i .eq. 0) call near_interval(xx, x, nx, i)
if (j .eq. 0) call near_interval(yy, y, ny, j)
z_eval(k) = EvalBicubic(xx, yy, x(i), y(j), C(1,1,i,j))
endif
enddo
end
subroutine BiCubicInterpWithGrad(x, y, C, nx, ny, x_eval, y_eval,
$ z_eval, dzdx_eval, dzdy_eval,m,
$ outmode)
*
* PURPOSE
* bicubic interpolation :
* the grid is defined by x(1..nx), y(1..ny)
* the known values are z(1..nx,1..ny), (z(i,j) being the value
* at point (x(i),y(j)))
* the interpolation is done on the points x_eval,y_eval(1..m)
* z_eval(k) is the result of the bicubic interpolation of
* (x_eval(k), y_eval(k)) and dzdx_eval(k), dzdy_eval(k) is the gradient
*
implicit none
integer nx, ny, m, outmode
double precision x(nx), y(ny), C(4,4,nx-1,ny-1),
$ x_eval(m), y_eval(m), z_eval(m),
$ dzdx_eval(m), dzdy_eval(m)
double precision xx, yy
integer k, i, j
include 'constinterp.h'
integer isanan
double precision return_a_nan, EvalBicubic
external isanan, return_a_nan, EvalBicubic
logical change_dzdx, change_dzdy
i = 0
j = 0
do k = 1, m
xx = x_eval(k)
call fast_int_search(xx, x, nx, i)
yy = y_eval(k)
call fast_int_search(yy, y, ny, j)
if (i .ne. 0 .and. j .ne. 0) then
call Evalbicubic_with_grad(xx, yy, x(i), y(j), C(1,1,i,j),
$ z_eval(k),
$ dzdx_eval(k), dzdy_eval(k))
elseif ( outmode .eq. BY_NAN .or. isanan(xx) .eq. 1
$ .or. isanan(yy) .eq. 1) then
z_eval(k) = return_a_nan()
dzdx_eval(k) = z_eval(k)
dzdy_eval(k) = z_eval(k)
elseif (outmode .eq. BY_ZERO ) then
z_eval(k) = 0.d0
dzdx_eval(k) = 0.d0
dzdy_eval(k) = 0.d0
elseif (outmode .eq. PERIODIC) then
if (i .eq. 0) call coord_by_periodicity(xx, x, nx, i)
if (j .eq. 0) call coord_by_periodicity(yy, y, ny, j)
call Evalbicubic_with_grad(xx, yy, x(i), y(j), C(1,1,i,j),
$ z_eval(k),
$ dzdx_eval(k), dzdy_eval(k))
elseif (outmode .eq. C0) then
if (i .eq. 0) then
call near_grid_point(xx, x, nx, i)
change_dzdx = .true.
else
change_dzdx = .false.
endif
if (j .eq. 0) then
call near_grid_point(yy, y, ny, j)
change_dzdy = .true.
else
change_dzdy = .false.
endif
call Evalbicubic_with_grad(xx, yy, x(i), y(j), C(1,1,i,j),
$ z_eval(k),
$ dzdx_eval(k), dzdy_eval(k))
if (change_dzdx) dzdx_eval(k) = 0.d0
if (change_dzdy) dzdy_eval(k) = 0.d0
elseif (outmode .eq. NATURAL) then
if (i .eq. 0) call near_interval(xx, x, nx, i)
if (j .eq. 0) call near_interval(yy, y, ny, j)
call Evalbicubic_with_grad(xx, yy, x(i), y(j), C(1,1,i,j),
$ z_eval(k),
$ dzdx_eval(k), dzdy_eval(k))
endif
enddo
end
subroutine BiCubicInterpWithGradAndHes(x, y, C, nx, ny,
$ x_eval, y_eval, z_eval,
$ dzdx_eval, dzdy_eval,
$ d2zdx2_eval, d2zdxy_eval, d2zdy2_eval,
$ m, outmode)
*
* PURPOSE
* bicubic interpolation :
* the grid is defined by x(1..nx), y(1..ny)
* the known values are z(1..nx,1..ny), (z(i,j) being the value
* at point (x(i),y(j)))
* the interpolation is done on the points x_eval,y_eval(1..m)
* z_eval(k) is the result of the bicubic interpolation of
* (x_eval(k), y_eval(k)), [dzdx_eval(k), dzdy_eval(k)] is the gradient
* and [d2zdx2(k), d2zdxy(k), d2zdy2(k)] the Hessean
*
implicit none
integer nx, ny, m, outmode
double precision x(nx), y(ny), C(4,4,nx-1,ny-1),
$ x_eval(m), y_eval(m), z_eval(m), dzdx_eval(m),
$ dzdy_eval(m), d2zdx2_eval(m), d2zdxy_eval(m), d2zdy2_eval(m)
double precision xx, yy
integer k, i, j
include 'constinterp.h'
integer isanan
double precision return_a_nan, EvalBicubic
external isanan, return_a_nan, EvalBicubic
logical change_dzdx, change_dzdy
i = 0
j = 0
do k = 1, m
xx = x_eval(k)
call fast_int_search(xx, x, nx, i)
yy = y_eval(k)
call fast_int_search(yy, y, ny, j)
if (i .ne. 0 .and. j .ne. 0) then
call Evalbicubic_with_grad_and_hes(xx, yy, x(i), y(j),
$ C(1,1,i,j), z_eval(k),
$ dzdx_eval(k), dzdy_eval(k),
$ d2zdx2_eval(k), d2zdxy_eval(k), d2zdy2_eval(k))
elseif ( outmode .eq. BY_NAN .or. isanan(xx) .eq. 1
$ .or. isanan(yy) .eq. 1) then
z_eval(k) = return_a_nan()
dzdx_eval(k) = z_eval(k)
dzdy_eval(k) = z_eval(k)
d2zdx2_eval(k) = z_eval(k)
d2zdxy_eval(k) = z_eval(k)
d2zdy2_eval(k) = z_eval(k)
elseif (outmode .eq. BY_ZERO ) then
z_eval(k) = 0.d0
dzdx_eval(k) = 0.d0
dzdy_eval(k) = 0.d0
d2zdx2_eval(k) = 0.d0
d2zdxy_eval(k) = 0.d0
d2zdy2_eval(k) = 0.d0
elseif (outmode .eq. PERIODIC) then
if (i .eq. 0) call coord_by_periodicity(xx, x, nx, i)
if (j .eq. 0) call coord_by_periodicity(yy, y, ny, j)
call Evalbicubic_with_grad_and_hes(xx, yy, x(i), y(j),
$ C(1,1,i,j), z_eval(k),
$ dzdx_eval(k), dzdy_eval(k),
$ d2zdx2_eval(k), d2zdxy_eval(k), d2zdy2_eval(k))
elseif (outmode .eq. C0) then
if (i .eq. 0) then
call near_grid_point(xx, x, nx, i)
change_dzdx = .true.
else
change_dzdx = .false.
endif
if (j .eq. 0) then
call near_grid_point(yy, y, ny, j)
change_dzdy = .true.
else
change_dzdy = .false.
endif
call Evalbicubic_with_grad_and_hes(xx, yy, x(i), y(j),
$ C(1,1,i,j), z_eval(k),
$ dzdx_eval(k), dzdy_eval(k),
$ d2zdx2_eval(k), d2zdxy_eval(k), d2zdy2_eval(k))
if (change_dzdx) then
dzdx_eval(k) = 0.d0
d2zdx2_eval(k) = 0.d0
d2zdxy_eval(k) = 0.d0
endif
if (change_dzdy) then
dzdy_eval(k) = 0.d0
d2zdxy_eval(k) = 0.d0
d2zdy2_eval(k) = 0.d0
endif
elseif (outmode .eq. NATURAL) then
if (i .eq. 0) call near_interval(xx, x, nx, i)
if (j .eq. 0) call near_interval(yy, y, ny, j)
call Evalbicubic_with_grad_and_hes(xx, yy, x(i), y(j),
$ C(1,1,i,j), z_eval(k),
$ dzdx_eval(k), dzdy_eval(k),
$ d2zdx2_eval(k), d2zdxy_eval(k), d2zdy2_eval(k))
endif
enddo
end
subroutine driverdb3val(xp, yp, zp, fp, np, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work,
$ xmin, xmax, ymin, ymax, zmin, zmax,
$ outmode)
*
* PURPOSE
* driver on to db3val
*
implicit none
integer np, nx, ny, nz, kx, ky, kz, outmode
double precision xp(np), yp(np), zp(np), fp(np),
$ tx(*), ty(*), tz(*), bcoef(*), work(*),
$ xmin, xmax, ymin, ymax, zmin, zmax
integer k
logical flag_x, flag_y, flag_z
double precision x, y, z
include 'constinterp.h'
integer isanan
double precision return_a_nan, db3val
external isanan, return_a_nan, db3val
do k = 1, np
x = xp(k)
if ( xmin .le. x .and. x .le. xmax ) then
flag_x = .true.
else
flag_x = .false.
endif
y = yp(k)
if ( ymin .le. y .and. y .le. ymax ) then
flag_y = .true.
else
flag_y = .false.
endif
z = zp(k)
if ( zmin .le. z .and. z .le. zmax ) then
flag_z = .true.
else
flag_z = .false.
endif
if ( flag_x .and. flag_y .and. flag_z ) then
fp(k) = db3val(x, y, z, 0, 0, 0, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
elseif (outmode .eq. BY_NAN .or. isanan(x) .eq. 1
$ .or. isanan(y) .eq. 1
$ .or. isanan(z) .eq. 1) then
fp(k) = return_a_nan()
elseif (outmode .eq. BY_ZERO) then
fp(k) = 0.d0
else
if (outmode .eq. PERIODIC) then
if (.not. flag_x) call proj_by_per(x, xmin, xmax)
if (.not. flag_y) call proj_by_per(y, ymin, ymax)
if (.not. flag_z) call proj_by_per(z, zmin, zmax)
elseif (outmode .eq. C0) then
if (.not. flag_x) call proj_on_grid(x, xmin, xmax)
if (.not. flag_y) call proj_on_grid(y, ymin, ymax)
if (.not. flag_z) call proj_on_grid(z, zmin, zmax)
endif
fp(k) = db3val(x, y, z, 0, 0, 0, tx, ty, tz, nx, ny, nz,
$ kx, ky, kz, bcoef, work)
endif
enddo
end
subroutine driverdb3valwithgrad(xp, yp, zp, fp,
$ dfpdx, dfpdy, dfpdz, np, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work,
$ xmin, xmax, ymin, ymax, zmin, zmax,
$ outmode)
*
* PURPOSE
* driver on to db3val with gradient computing
*
implicit none
integer np, nx, ny, nz, kx, ky, kz, outmode
double precision xp(np), yp(np), zp(np), fp(np),
$ dfpdx(np), dfpdy(np), dfpdz(np),
$ tx(*), ty(*), tz(*), bcoef(*), work(*),
$ xmin, xmax, ymin, ymax, zmin, zmax
integer k
logical flag_x, flag_y, flag_z
double precision x, y, z
include 'constinterp.h'
integer isanan
double precision return_a_nan, db3val
external isanan, return_a_nan, db3val
do k = 1, np
x = xp(k)
if ( xmin .le. x .and. x .le. xmax ) then
flag_x = .true.
else
flag_x = .false.
endif
y = yp(k)
if ( ymin .le. y .and. y .le. ymax ) then
flag_y = .true.
else
flag_y = .false.
endif
z = zp(k)
if ( zmin .le. z .and. z .le. zmax ) then
flag_z = .true.
else
flag_z = .false.
endif
if ( flag_x .and. flag_y .and. flag_z ) then
fp(k) = db3val(x, y, z, 0, 0, 0, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
dfpdx(k) = db3val(x, y, z, 1, 0, 0, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
dfpdy(k) = db3val(x, y, z, 0, 1, 0, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
dfpdz(k) = db3val(x, y, z, 0, 0, 1, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
elseif (outmode .eq. BY_NAN .or. isanan(x) .eq. 1
$ .or. isanan(y) .eq. 1
$ .or. isanan(z) .eq. 1) then
fp(k) = return_a_nan()
dfpdx(k) = fp(k)
dfpdy(k) = fp(k)
dfpdz(k) = fp(k)
elseif (outmode .eq. BY_ZERO) then
fp(k) = 0.d0
dfpdx(k) = 0.d0
dfpdy(k) = 0.d0
dfpdz(k) = 0.d0
elseif (outmode .eq. PERIODIC) then
if (.not. flag_x) call proj_by_per(x, xmin, xmax)
if (.not. flag_y) call proj_by_per(y, ymin, ymax)
if (.not. flag_z) call proj_by_per(z, zmin, zmax)
fp(k) = db3val(x, y, z, 0, 0, 0, tx, ty, tz, nx, ny, nz,
$ kx, ky, kz, bcoef, work)
dfpdx(k) = db3val(x, y, z, 1, 0, 0, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
dfpdy(k) = db3val(x, y, z, 0, 1, 0, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
dfpdz(k) = db3val(x, y, z, 0, 0, 1, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
elseif (outmode .eq. C0) then
if (.not. flag_x) call proj_on_grid(x, xmin, xmax)
if (.not. flag_y) call proj_on_grid(y, ymin, ymax)
if (.not. flag_z) call proj_on_grid(z, zmin, zmax)
fp(k) = db3val(x, y, z, 0, 0, 0, tx, ty, tz, nx, ny, nz,
$ kx, ky, kz, bcoef, work)
if ( flag_x ) then
dfpdx(k) = 0.d0
else
dfpdx(k) = db3val(x, y, z, 1, 0, 0, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
endif
if ( flag_y ) then
dfpdy(k) = 0.d0
else
dfpdy(k) = db3val(x, y, z, 0, 1, 0, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
endif
if ( flag_z ) then
dfpdz(k) = 0.d0
else
dfpdz(k) = db3val(x, y, z, 0, 0, 1, tx, ty, tz,
$ nx, ny, nz, kx, ky, kz, bcoef, work)
endif
endif
enddo
end
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