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regcoil_splines.f90
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module regcoil_splines
use stel_kinds
implicit none
private
public :: spline, periodic_spline
public :: new_spline, delete_spline
public :: new_periodic_spline, delete_periodic_spline
public :: splint, dsplint, periodic_splint
public :: inter_d_cspl, inter_cspl
public :: fitp_curv1, fitp_curvp1
public :: fitp_curv2, fitp_curvp2
public :: fitp_surf1, fitp_surf2
public :: lf_spline, fitp_curvd
type :: spline
integer :: n
real(dp), dimension (:), pointer :: x, y, y2
end type spline
type :: periodic_spline
integer :: n
real(dp) :: period
real(dp), dimension (:), pointer :: x, y, y2
end type periodic_spline
contains
subroutine new_spline (n, x, y, spl)
implicit none
integer, intent (in) :: n
real(dp), dimension (n), intent (in) :: x, y
type (spline), intent (out) :: spl
real(dp), dimension (n) :: temp
integer :: ierr
spl%n = n
allocate (spl%x(n),spl%y(n))
spl%x = x
spl%y = y
allocate (spl%y2(n))
call fitp_curv1 (n, x, y, 0.0_dp, 0.0_dp, 3, spl%y2, temp, 1.0_dp, ierr)
end subroutine new_spline
subroutine new_periodic_spline (n, x, y, period, spl)
implicit none
integer, intent (in) :: n
real(dp), dimension (n), intent (in) :: x, y
real(dp), intent (in) :: period
type (periodic_spline), intent (out) :: spl
real(dp), dimension (2*n) :: temp
integer :: ierr
spl%n = n
spl%period = period
allocate (spl%x(n),spl%y(n))
spl%x = x
spl%y = y
allocate (spl%y2(n))
call fitp_curvp1 (n,x,y,period,spl%y2,temp,1.0_dp,ierr)
end subroutine new_periodic_spline
subroutine delete_spline (spl)
implicit none
type (spline), intent (in out) :: spl
spl%n = 0
deallocate (spl%x,spl%y)
nullify (spl%x)
nullify (spl%y)
deallocate (spl%y2)
nullify (spl%y2)
end subroutine delete_spline
subroutine delete_periodic_spline (spl)
implicit none
type (periodic_spline), intent (in out) :: spl
spl%n = 0
spl%period = 0.0_dp
deallocate (spl%x,spl%y)
nullify (spl%x)
nullify (spl%y)
deallocate (spl%y2)
nullify (spl%y2)
end subroutine delete_periodic_spline
function splint (x, spl)
implicit none
real(dp), intent (in) :: x
type (spline), intent (in) :: spl
real(dp) :: splint
splint = fitp_curv2 (x, spl%n, spl%x, spl%y, spl%y2, 1.0_dp)
end function splint
function periodic_splint (x, spl)
implicit none
real(dp), intent (in) :: x
type (periodic_spline), intent (in) :: spl
real(dp) :: periodic_splint
periodic_splint = fitp_curvp2 &
(x, spl%n, spl%x, spl%y, spl%period, spl%y2, 1.0_dp)
end function periodic_splint
function dsplint (x, spl)
implicit none
real(dp), intent (in) :: x
type (spline), intent (in) :: spl
real(dp) :: dsplint
dsplint = fitp_curvd (x, spl%n, spl%x, spl%y, spl%y2, 1.0_dp)
end function dsplint
function splintint (x0, x1, spl)
implicit none
real(dp), intent (in) :: x0, x1
type (spline), intent (in) :: spl
real(dp) :: splintint
splintint = fitp_curvi (x0,x1,spl%n,spl%x,spl%y,spl%y2,1.0_dp)
end function splintint
function periodic_splintint (x0, x1, spl)
implicit none
real(dp), intent (in) :: x0, x1
type (periodic_spline), intent (in) :: spl
real(dp) :: periodic_splintint
periodic_splintint = fitp_curvpi &
(x0,x1,spl%n,spl%x,spl%y,spl%period,spl%y2, 1.0_dp)
end function periodic_splintint
subroutine inter_d_cspl(n,r,data,m,x,dint,ddint)
implicit none
integer, intent(in) :: n, m
real(dp), dimension(n), intent(in) :: r, data
real(dp), dimension(m), intent(in) :: x
real(dp), dimension(m), intent(out) :: dint, ddint
integer, parameter :: max=1000
real(dp), dimension(max) :: ddata, temp
integer :: i,ierr
if (n .gt. max) then
write (*,*) 'error in inter_d_cspl'
write (*,*) 'increase max'
stop
endif
ierr = 0
call fitp_curv1(n,r,data,0.0_dp,0.0_dp,3,ddata,temp,1.0_dp,ierr)
if (ierr .ne. 0) then
if (ierr .eq. 1) then
write (*,*) 'FITPACK: curv1 error: n < 2'
elseif (ierr .eq. 2) then
write (*,*) 'FITPACK: curv1 error: x-values not increasing'
else
write (*,*) 'FITPACK: curv1 error'
endif
stop
endif
do i=1,m
dint(i) = fitp_curv2 (x(i),n,r,data,ddata,1.0_dp)
ddint(i)= fitp_curvd (x(i),n,r,data,ddata,1.0_dp)
enddo
end subroutine inter_d_cspl
subroutine inter_cspl(n,r,data,m,x,dint)
implicit none
integer, intent(in) :: n, m
real(dp), dimension(n), intent(in) :: r, data
real(dp), dimension(m), intent(in) :: x
real(dp), dimension(m), intent(out) :: dint
integer, parameter :: max=1000
real(dp), dimension(max) :: ddata, temp
integer :: i,ierr
if (n .gt. max) then
write (*,*) 'error in inter_cspl'
write (*,*) 'increase max'
stop
endif
ierr = 0
call fitp_curv1(n,r,data,0.0_dp,0.0_dp,3,ddata,temp,1.0_dp,ierr)
if (ierr .ne. 0) then
if (ierr .eq. 1) then
write (*,*) 'FITPACK: curv1 error: n < 2'
elseif (ierr .eq. 2) then
write (*,*) 'FITPACK: curv1 error: x-values not increasing'
else
write (*,*) 'FITPACK: curv1 error'
endif
stop
endif
do i=1,m
dint(i) = fitp_curv2 (x(i),n,r,data,ddata,1.0_dp)
enddo
end subroutine inter_cspl
subroutine inter_getspl (n, x, y, y2)
implicit none
integer, intent(in) :: n
real(dp), dimension(:), intent(in) :: x, y
real(dp), dimension(:), intent(out) :: y2
integer, parameter :: max=1000
real(dp), dimension(max) :: temp
integer :: ierr
if (n .gt. max) then
write (*,*) 'error in inter_getspl'
write (*,*) 'increase max'
stop
endif
ierr = 0
call fitp_curv1(n,x,y,0.0_dp,0.0_dp,3,y2,temp,1.0_dp,ierr)
if (ierr .ne. 0) then
if (ierr .eq. 1) then
write (*,*) 'FITPACK: curv1 error: n < 2'
elseif (ierr .eq. 2) then
write (*,*) 'FITPACK: curv1 error: x-values not increasing'
else
write (*,*) 'FITPACK: curv1 error'
endif
stop
endif
end subroutine inter_getspl
real(dp) function inter_splint (x0, n, x, y, y2)
real(dp), intent(in) :: x0
integer, intent(in) :: n
real(dp), dimension(n), intent(in) :: x, y, y2
inter_splint = fitp_curv2 (x0, n, x, y, y2, 1.0_dp)
end function inter_splint
real(dp) function inter_dsplint (x0, n, x, y, y2)
real(dp), intent(in) :: x0
integer, intent(in) :: n
real(dp), dimension(n), intent(in) :: x, y, y2
inter_dsplint = fitp_curvd (x0, n, x, y, y2, 1.0_dp)
end function inter_dsplint
real(dp) function inter_d2splint (x0, n, x, y, y2)
real(dp), intent(in) :: x0
integer, intent(in) :: n
real(dp), dimension(n), intent(in) :: x, y, y2
real(dp) :: yx(500)
data yx(1)/1.0_dp/
save yx
! This next line is needed because yx has the "save" attribute, and multiple threads may call this function.
!$OMP threadprivate(yx)
integer :: i
if (yx(1) .ne. 0.0_dp) then
do i=1,500
yx(i) = 0.0_dp
enddo
endif
inter_d2splint = fitp_curv2 (x0, n, x, y2, yx, 1e5_dp)
end function inter_d2splint
subroutine inter_getpspl (n, x, p, y, y2)
implicit none
integer, intent(in) :: n
real(dp), dimension(n), intent(in) :: x, y
real(dp), dimension(n), intent(out) :: y2
real(dp), intent(in) :: p
integer, parameter :: max=1000
real(dp), dimension(max) :: temp
integer :: ierr
if (n .gt. max) then
write (*,*) 'error in inter_getpspl'
write (*,*) 'increase max'
stop
endif
ierr=0
call fitp_curvp1(n,x,y,p,y2,temp,1.0_dp,ierr)
if (ierr .ne. 0) then
if (ierr .eq. 1) then
write (*,*) 'FITPACK: curvp1 error: n < 2'
elseif (ierr .eq. 2) then
write (*,*) 'FITPACK: curvp1 error: p <= x(n)-x(1)'
elseif (ierr .eq. 3) then
write (*,*) 'FITPACK: curvp1 error: x-values not increasing'
else
write (*,*) 'FITPACK: curv1 error'
endif
stop
endif
end subroutine inter_getpspl
real(dp) function inter_psplint (x0, n, x, p, y, y2)
real(dp), intent(in) :: x0
real(dp), intent(in) :: p
integer, intent(in) :: n
real(dp), dimension(n), intent(in) :: x, y, y2
inter_psplint = fitp_curvp2 (x0, n, x, y, p, y2, 1.0_dp)
end function inter_psplint
real(dp) function inter_pdsplint (x0, n, x, p, y, y2)
real(dp), intent(in) :: x0
real(dp), intent(in) :: p
integer, intent(in) :: n
real(dp), dimension(n), intent(in) :: x, y, y2
inter_pdsplint = fitp_curvpd (x0, n, x, y, p, y2, 1.0_dp)
end function inter_pdsplint
! From inet!cs.utexas.edu!cline Tue Oct 31 17:10:31 CST 1989
! Received: from mojave.cs.utexas.edu by cs.utexas.edu (5.59/1.44)
! id AA29509; Tue, 31 Oct 89 17:11:51 CST
! Posted-Date: Tue, 31 Oct 89 17:10:31 CST
! Message-Id: <[email protected]>
! Received: by mojave.cs.utexas.edu (14.5/1.4-Client)
! id AA04442; Tue, 31 Oct 89 17:10:34 cst
! Date: Tue, 31 Oct 89 17:10:31 CST
! X-Mailer: Mail User's Shell (6.5 4/17/89)
! From: [email protected] (Alan Cline)
! To: [email protected]
! Subject: New FITPACK Subset for netlib
!
!
! This new version of FITPACK distributed by netlib is about 20% of
! the total package in terms of characters, lines of code, and num-
! ber of subprograms. However, these 25 subprograms represent about
! 95% of usages of the package. What has been omitted are such ca-
! pabilities as:
! 1. Automatic tension determination,
! 2. Derivatives, arclengths, and enclosed areas for planar
! curves,
! 3. Three dimensional curves,
! 4. Special surface fitting using equispacing assumptions,
! 5. Surface fitting in annular, wedge, polar, toroidal, lunar,
! and spherical geometries,
! 6. B-splines in tension generation and usage,
! 7. General surface fitting in three dimensional space.
!
! (The code previously circulated in netlib is less than 10% of the
! total package and is more than a decade old. Its usage is dis-
! couraged.)
!
! Please note: Two versions of the subroutine snhcsh are included.
! Both serve the same purpose: obtaining approximations to certain
! hyperbolic trigonometric-like functions. The first is less accu-
! rate (but more efficient) than the second. Installers should se-
! lect the one with the precision they desire.
!
! Interested parties can obtain the entire package on disk or tape
! from Pleasant Valley Software, 8603 Altus Cove, Austin TX (USA),
! 78759 at a cost of $495 US. A 340 page manual is available for
! $30 US per copy. The package includes examples and machine
! readable documentation.
subroutine fitp_curv1 (n,x,y,slp1,slpn,islpsw,yp,temp,sigma,ierr)
implicit none
integer, intent(in) :: n, islpsw
integer, intent(out) :: ierr
real(dp), dimension(n), intent(in) :: x, y
real(dp), dimension(n), intent(out) :: yp, temp
real(dp), intent(in) :: slp1,slpn,sigma
!
! coded by alan kaylor cline
! from fitpack -- january 26, 1987
! a curve and surface fitting package
! a product of pleasant valley software
! 8603 altus cove, austin, texas 78759, usa
!
! this subroutine determines the parameters necessary to
! compute an interpolatory spline under tension through
! a sequence of functional values. the slopes at the two
! ends of the curve may be specified or omitted. for actual
! computation of points on the curve it is necessary to call
! the function curv2.
!
! on input--
!
! n is the number of values to be interpolated (n.ge.2).
!
! x is an array of the n increasing abscissae of the
! functional values.
!
! y is an array of the n ordinates of the values, (i. e.
! y(k) is the functional value corresponding to x(k) ).
!
! slp1 and slpn contain the desired values for the first
! derivative of the curve at x(1) and x(n), respectively.
! the user may omit values for either or both of these
! parameters and signal this with islpsw.
!
! islpsw contains a switch indicating which slope data
! should be used and which should be estimated by this
! subroutine,
! = 0 if slp1 and slpn are to be used,
! = 1 if slp1 is to be used but not slpn,
! = 2 if slpn is to be used but not slp1,
! = 3 if both slp1 and slpn are to be estimated
! internally.
!
! yp is an array of length at least n.
!
! temp is an array of length at least n which is used for
! scratch storage.
!
! and
!
! sigma contains the tension factor. this value indicates
! the curviness desired. if abs(sigma) is nearly zero
! (e.g. .001) the resulting curve is approximately a
! cubic spline. if abs(sigma) is large (e.g. 50.) the
! resulting curve is nearly a polygonal line. if sigma
! equals zero a cubic spline results. a standard value
! for sigma is approximately 1. in absolute value.
!
! on output--
!
! yp contains the values of the second derivative of the
! curve at the given nodes.
!
! ierr contains an error flag,
! = 0 for normal return,
! = 1 if n is less than 2,
! = 2 if x-values are not strictly increasing.
!
! and
!
! n, x, y, slp1, slpn, islpsw and sigma are unaltered.
!
! this subroutine references package modules ceez, terms,
! and snhcsh.
!
!-----------------------------------------------------------
integer :: i, ibak, nm1, np1
real(dp) :: sdiag1, diag1, delxnm, dx1, diag, sdiag2, dx2, diag2
real(dp) :: delxn, slpp1, delx1, sigmap, c3, c2, c1, slppn, delx2
nm1 = n-1
np1 = n+1
ierr = 0
if (n .le. 1) go to 8
if (x(n) .le. x(1)) go to 9
!
! denormalize tension factor
!
sigmap = abs(sigma)*real(n-1,dp)/(x(n)-x(1))
!
! approximate end slopes
!
if (islpsw .ge. 2) go to 1
slpp1 = slp1
go to 2
1 delx1 = x(2)-x(1)
delx2 = delx1+delx1
if (n .gt. 2) delx2 = x(3)-x(1)
if (delx1 .le. 0. .or. delx2 .le. delx1) go to 9
call fitp_ceez (delx1,delx2,sigmap,c1,c2,c3,n)
slpp1 = c1*y(1)+c2*y(2)
if (n .gt. 2) slpp1 = slpp1+c3*y(3)
2 if (islpsw .eq. 1 .or. islpsw .eq. 3) go to 3
slppn = slpn
go to 4
3 delxn = x(n)-x(nm1)
delxnm = delxn+delxn
if (n .gt. 2) delxnm = x(n)-x(n-2)
if (delxn .le. 0. .or. delxnm .le. delxn) go to 9
call fitp_ceez (-delxn,-delxnm,sigmap,c1,c2,c3,n)
slppn = c1*y(n)+c2*y(nm1)
if (n .gt. 2) slppn = slppn+c3*y(n-2)
!
! set up right hand side and tridiagonal system for yp and
! perform forward elimination
!
4 delx1 = x(2)-x(1)
if (delx1 .le. 0.) go to 9
dx1 = (y(2)-y(1))/delx1
call fitp_terms (diag1,sdiag1,sigmap,delx1)
yp(1) = (dx1-slpp1)/diag1
temp(1) = sdiag1/diag1
if (n .eq. 2) go to 6
do i = 2,nm1
delx2 = x(i+1)-x(i)
if (delx2 .le. 0.) go to 9
dx2 = (y(i+1)-y(i))/delx2
call fitp_terms (diag2,sdiag2,sigmap,delx2)
diag = diag1+diag2-sdiag1*temp(i-1)
yp(i) = (dx2-dx1-sdiag1*yp(i-1))/diag
temp(i) = sdiag2/diag
dx1 = dx2
diag1 = diag2
sdiag1 = sdiag2
end do
6 diag = diag1-sdiag1*temp(nm1)
yp(n) = (slppn-dx1-sdiag1*yp(nm1))/diag
!
! perform back substitution
!
do i = 2,n
ibak = np1-i
yp(ibak) = yp(ibak)-temp(ibak)*yp(ibak+1)
end do
return
!
! too few points
!
8 ierr = 1
return
!
! x-values not strictly increasing
!
9 ierr = 2
return
end subroutine fitp_curv1
subroutine fitp_curvs (n,x,y,d,isw,s,eps,ys,ysp,sigma,temp,ierr)
implicit none
integer, intent(in) :: n, isw
integer, intent(out) :: ierr
real(dp), dimension(n), intent(in) :: x, y, d
real(dp), dimension(n,9), intent(out) :: temp
real(dp), dimension(n), intent(out) :: ys, ysp
real(dp), intent(in) :: sigma,s,eps
!
! coded by alan kaylor cline
! from fitpack -- january 26, 1987
! a curve and surface fitting package
! a product of pleasant valley software
! 8603 altus cove, austin, texas 78759, usa
!
! this subroutine determines the parameters necessary to
! compute a smoothing spline under tension. for a given
! increasing sequence of abscissae (x(i)), i = 1,..., n and
! associated ordinates (y(i)), i = 1,..., n, the function
! determined minimizes the summation from i = 1 to n-1 of
! the square of the second derivative of f plus sigma
! squared times the difference of the first derivative of f
! and (f(x(i+1))-f(x(i)))/(x(i+1)-x(i)) squared, over all
! functions f with two continuous derivatives such that the
! summation of the square of (f(x(i))-y(i))/d(i) is less
! than or equal to a given constant s, where (d(i)), i = 1,
! ..., n are a given set of observation weights. the
! function determined is a spline under tension with third
! derivative discontinuities at (x(i)), i = 2,..., n-1. for
! actual computation of points on the curve it is necessary
! to call the function curv2. the determination of the curve
! is performed by subroutine curvss, the subroutine curvs
! only decomposes the workspace for curvss.
!
! on input--
!
! n is the number of values to be smoothed (n.ge.2).
!
! x is an array of the n increasing abscissae of the
! values to be smoothed.
!
! y is an array of the n ordinates of the values to be
! smoothed, (i. e. y(k) is the functional value
! corresponding to x(k) ).
!
! d is a parameter containing the observation weights.
! this may either be an array of length n or a scalar
! (interpreted as a constant). the value of d
! corresponding to the observation (x(k),y(k)) should
! be an approximation to the standard deviation of error.
!
! isw contains a switch indicating whether the parameter
! d is to be considered a vector or a scalar,
! = 0 if d is an array of length n,
! = 1 if d is a scalar.
!
! s contains the value controlling the smoothing. this
! must be non-negative. for s equal to zero, the
! subroutine does interpolation, larger values lead to
! smoother funtions. if parameter d contains standard
! deviation estimates, a reasonable value for s is
! float(n).
!
! eps contains a tolerance on the relative precision to
! which s is to be interpreted. this must be greater than
! or equal to zero and less than or equal to one. a
! reasonable value for eps is sqrt(2./float(n)).
!
! ys is an array of length at least n.
!
! ysp is an array of length at least n.
!
! sigma contains the tension factor. this value indicates
! the degree to which the first derivative part of the
! smoothing functional is emphasized. if sigma is nearly
! zero (e. g. .001) the resulting curve is approximately a
! cubic spline. if sigma is large (e. g. 50.) the
! resulting curve is nearly a polygonal line. if sigma
! equals zero a cubic spline results. a standard value for
! sigma is approximately 1.
!
! and
!
! temp is an array of length at least 9*n which is used
! for scratch storage.
!
! on output--
!
! ys contains the smoothed ordinate values.
!
! ysp contains the values of the second derivative of the
! smoothed curve at the given nodes.
!
! ierr contains an error flag,
! = 0 for normal return,
! = 1 if n is less than 2,
! = 2 if s is negative,
! = 3 if eps is negative or greater than one,
! = 4 if x-values are not strictly increasing,
! = 5 if a d-value is non-positive.
!
! and
!
! n, x, y, d, isw, s, eps, and sigma are unaltered.
!
! this subroutine references package modules curvss, terms,
! and snhcsh.
!
!-----------------------------------------------------------
!
! decompose temp into nine arrays and call curvss
!
call fitp_curvss (n,x,y,d,isw,s,eps,ys,ysp,sigma,temp(1,1), &
temp(1,2),temp(1,3),temp(1,4),temp(1,5), &
temp(1,6),temp(1,7),temp(1,8),temp(1,9), &
ierr)
end subroutine fitp_curvs
real(dp) function fitp_curv2 (t,n,x,y,yp,sigma)
implicit none
integer, intent(in) :: n
real(dp), dimension(n), intent(in) :: x, y, yp
real(dp), intent(in) :: t, sigma
!
! coded by alan kaylor cline
! from fitpack -- january 26, 1987
! a curve and surface fitting package
! a product of pleasant valley software
! 8603 altus cove, austin, texas 78759, usa
!
! this function interpolates a curve at a given point
! using a spline under tension. the subroutine curv1 should
! be called earlier to determine certain necessary
! parameters.
!
! on input--
!
! t contains a real value to be mapped onto the interpo-
! lating curve.
!
! n contains the number of points which were specified to
! determine the curve.
!
! x and y are arrays containing the abscissae and
! ordinates, respectively, of the specified points.
!
! yp is an array of second derivative values of the curve
! at the nodes.
!
! and
!
! sigma contains the tension factor (its sign is ignored).
!
! the parameters n, x, y, yp, and sigma should be input
! unaltered from the output of curv1.
!
! on output--
!
! curv2 contains the interpolated value.
!
! none of the input parameters are altered.
!
! this function references package modules intrvl and
! snhcsh.
!
!-----------------------------------------------------------
integer :: i, im1
real(dp) :: ss, sigdel, dummy, s1, s2, sum, sigmap
real(dp) :: del1, del2, dels
!
! determine interval
!
im1 = fitp_intrvl(t,x,n)
i = im1+1
!
! denormalize tension factor
!
sigmap = abs(sigma)*real(n-1,dp)/(x(n)-x(1))
!
! set up and perform interpolation
!
del1 = t-x(im1)
del2 = x(i)-t
dels = x(i)-x(im1)
sum = (y(i)*del1+y(im1)*del2)/dels
if (sigmap .ne. 0.) go to 1
fitp_curv2 = sum-del1*del2*(yp(i)*(del1+dels)+yp(im1)*(del2+dels))/(6.*dels)
return
1 sigdel = sigmap*dels
call fitp_snhcsh (ss,dummy,sigdel,-1)
call fitp_snhcsh (s1,dummy,sigmap*del1,-1)
call fitp_snhcsh (s2,dummy,sigmap*del2,-1)
fitp_curv2 = sum+(yp(i)*del1*(s1-ss)+yp(im1)*del2*(s2-ss))/(sigdel*sigmap*(1.+ss))
return
end function fitp_curv2
real(dp) function fitp_curvd (t,n,x,y,yp,sigma)
implicit none
integer, intent(in) :: n
real(dp), dimension(n), intent(in) :: x, y, yp
real(dp), intent(in) :: t, sigma
!
! coded by alan kaylor cline
! from fitpack -- january 26, 1987
! a curve and surface fitting package
! a product of pleasant valley software
! 8603 altus cove, austin, texas 78759, usa
!
! this function differentiates a curve at a given point
! using a spline under tension. the subroutine curv1 should
! be called earlier to determine certain necessary
! parameters.
!
! on input--
!
! t contains a real value at which the derivative is to be
! determined.
!
! n contains the number of points which were specified to
! determine the curve.
!
! x and y are arrays containing the abscissae and
! ordinates, respectively, of the specified points.
!
! yp is an array of second derivative values of the curve
! at the nodes.
!
! and
!
! sigma contains the tension factor (its sign is ignored).
!
! the parameters n, x, y, yp, and sigma should be input
! unaltered from the output of curv1.
!
! on output--
!
! curvd contains the derivative value.
!
! none of the input parameters are altered.
!
! this function references package modules intrvl and
! snhcsh.
!
!-----------------------------------------------------------
integer :: i, im1
real(dp) :: ss, sigdel, dummy, c1, c2, sum, sigmap
real(dp) :: del1, del2, dels
!
! determine interval
!
im1 = fitp_intrvl(t,x,n)
i = im1+1
!
! denormalize tension factor
!
sigmap = abs(sigma)*real(n-1,dp)/(x(n)-x(1))
!
! set up and perform differentiation
!
del1 = t-x(im1)
del2 = x(i)-t
dels = x(i)-x(im1)
sum = (y(i)-y(im1))/dels
if (sigmap .ne. 0.) go to 1
fitp_curvd = sum+(yp(i)*(2.*del1*del1-del2*(del1+dels))- &
yp(im1)*(2.*del2*del2-del1*(del2+dels))) &
/(6.*dels)
return
1 sigdel = sigmap*dels
call fitp_snhcsh (ss,dummy,sigdel,-1)
call fitp_snhcsh (dummy,c1,sigmap*del1,1)
call fitp_snhcsh (dummy,c2,sigmap*del2,1)
fitp_curvd = sum+(yp(i)*(c1-ss)-yp(im1)*(c2-ss))/(sigdel*sigmap*(1.+ss))
return
end function fitp_curvd
real(dp) function fitp_curvi (xl,xu,n,x,y,yp,sigma)
implicit none
integer, intent(in) :: n
real(dp), intent(in) :: xl, xu, sigma
real(dp), dimension(n), intent(in) :: x, y, yp
!
! coded by alan kaylor cline
! from fitpack -- january 26, 1987
! a curve and surface fitting package
! a product of pleasant valley software
! 8603 altus cove, austin, texas 78759, usa
!
! this function integrates a curve specified by a spline
! under tension between two given limits. the subroutine
! curv1 should be called earlier to determine necessary
! parameters.
!
! on input--
!
! xl and xu contain the upper and lower limits of inte-
! gration, respectively. (sl need not be less than or
! equal to xu, curvi (xl,xu,...) .eq. -curvi (xu,xl,...) ).
!
! n contains the number of points which were specified to
! determine the curve.
!
! x and y are arrays containing the abscissae and
! ordinates, respectively, of the specified points.
!
! yp is an array from subroutine curv1 containing
! the values of the second derivatives at the nodes.
!
! and
!
! sigma contains the tension factor (its sign is ignored).
!
! the parameters n, x, y, yp, and sigma should be input
! unaltered from the output of curv1.
!
! on output--
!
! curvi contains the integral value.
!
! none of the input parameters are altered.
!
! this function references package modules intrvl and
! snhcsh.
!
!-----------------------------------------------------------
integer :: i, ilp1, ilm1, il, ium1, iu
real(dp) :: delu1, delu2, c2, ss, cs, cu2, cl1, cl2, cu1
real(dp) :: dell1, dell2, deli, c1, ssign, sigmap
real(dp) :: xxl, xxu, t1, t2, dummy, dels, sum, del1, del2
!
! denormalize tension factor
!
sigmap = abs(sigma)*real(n-1,dp)/(x(n)-x(1))
!
! determine actual upper and lower bounds
!
xxl = xl
xxu = xu
ssign = 1.
if (xl .lt. xu) go to 1
xxl = xu
xxu = xl
ssign = -1.
if (xl .gt. xu) go to 1
!
! return zero if xl .eq. xu
!
fitp_curvi = 0.
return
!
! search for proper intervals
!
1 ilm1 = fitp_intrvl (xxl,x,n)
il = ilm1+1
ium1 = fitp_intrvl (xxu,x,n)
iu = ium1+1
if (il .eq. iu) go to 8
!
! integrate from xxl to x(il)
!
sum = 0.
if (xxl .eq. x(il)) go to 3
del1 = xxl-x(ilm1)
del2 = x(il)-xxl
dels = x(il)-x(ilm1)
t1 = (del1+dels)*del2/(2.*dels)
t2 = del2*del2/(2.*dels)
sum = t1*y(il)+t2*y(ilm1)
if (sigma .eq. 0.) go to 2
call fitp_snhcsh (dummy,c1,sigmap*del1,2)
call fitp_snhcsh (dummy,c2,sigmap*del2,2)
call fitp_snhcsh (ss,cs,sigmap*dels,3)
sum = sum+((dels*dels*(cs-ss/2.)-del1*del1*(c1-ss/2.)) &
*yp(il)+del2*del2*(c2-ss/2.)*yp(ilm1))/ &
(sigmap*sigmap*dels*(1.+ss))
go to 3
2 sum = sum-t1*t1*dels*yp(il)/6. &
-t2*(del1*(del2+dels)+dels*dels)*yp(ilm1)/12.
!
! integrate over interior intervals
!
3 if (iu-il .eq. 1) go to 6
ilp1 = il+1
do i = ilp1,ium1
dels = x(i)-x(i-1)
sum = sum+(y(i)+y(i-1))*dels/2.
if (sigma .eq. 0.) go to 4
call fitp_snhcsh (ss,cs,sigmap*dels,3)
sum = sum+(yp(i)+yp(i-1))*dels*(cs-ss/2.)/(sigmap*sigmap*(1.+ss))
go to 5
4 sum = sum-(yp(i)+yp(i-1))*dels*dels*dels/24.
5 continue
end do
!
! integrate from x(iu-1) to xxu
!
6 if (xxu .eq. x(ium1)) go to 10
del1 = xxu-x(ium1)
del2 = x(iu)-xxu
dels = x(iu)-x(ium1)
t1 = del1*del1/(2.*dels)
t2 = (del2+dels)*del1/(2.*dels)
sum = sum+t1*y(iu)+t2*y(ium1)
if (sigma .eq. 0.) go to 7
call fitp_snhcsh (dummy,c1,sigmap*del1,2)
call fitp_snhcsh (dummy,c2,sigmap*del2,2)
call fitp_snhcsh (ss,cs,sigmap*dels,3)
sum = sum+(yp(iu)*del1*del1*(c1-ss/2.)+yp(ium1)* &
(dels*dels*(cs-ss/2.)-del2*del2*(c2-ss/2.))) &
/(sigmap*sigmap*dels*(1.+ss))
go to 10
7 sum = sum-t1*(del2*(del1+dels)+dels*dels)*yp(iu)/12.-t2*t2*dels*yp(ium1)/6.
go to 10
!
! integrate from xxl to xxu
!
8 delu1 = xxu-x(ium1)
delu2 = x(iu)-xxu
dell1 = xxl-x(ium1)
dell2 = x(iu)-xxl
dels = x(iu)-x(ium1)
deli = xxu-xxl
t1 = (delu1+dell1)*deli/(2.*dels)
t2 = (delu2+dell2)*deli/(2.*dels)
sum = t1*y(iu)+t2*y(ium1)
if (sigma .eq. 0.) go to 9
call fitp_snhcsh (dummy,cu1,sigmap*delu1,2)
call fitp_snhcsh (dummy,cu2,sigmap*delu2,2)
call fitp_snhcsh (dummy,cl1,sigmap*dell1,2)
call fitp_snhcsh (dummy,cl2,sigmap*dell2,2)
call fitp_snhcsh (ss,dummy,sigmap*dels,-1)
sum = sum+(yp(iu)*(delu1*delu1*(cu1-ss/2.) &
-dell1*dell1*(cl1-ss/2.)) &
+yp(ium1)*(dell2*dell2*(cl2-ss/2.) &
-delu2*delu2*(cu2-ss/2.)))/ &
(sigmap*sigmap*dels*(1.+ss))
go to 10
9 sum = sum-t1*(delu2*(dels+delu1)+dell2*(dels+dell1))* &
yp(iu)/12. &
-t2*(dell1*(dels+dell2)+delu1*(dels+delu2))* &
yp(ium1)/12.
!
! correct sign and return
!
10 fitp_curvi = ssign*sum
return
end function fitp_curvi
subroutine fitp_curvp1 (n,x,y,p,yp,temp,sigma,ierr)
implicit none
integer, intent(in) :: n
integer, intent(out) :: ierr
real(dp), intent(in) :: sigma, p
real(dp), dimension(:), intent(in) :: x, y
real(dp), dimension(:), intent(out) :: yp, temp
!! real(dp) x(n),y(n),p,yp(n),temp(2*n),sigma
! real(dp) x(n),y(n),p,yp(n),temp(1),sigma
!
! coded by alan kaylor cline
! from fitpack -- january 26, 1987
! a curve and surface fitting package
! a product of pleasant valley software
! 8603 altus cove, austin, texas 78759, usa
!
! this subroutine determines the parameters necessary to
! compute a periodic interpolatory spline under tension
! through a sequence of functional values. for actual ends
! of the curve may be specified or omitted. for actual
! computation of points on the curve it is necessary to call
! the function curvp2.
!
! on input--
!
! n is the number of values to be interpolated (n.ge.2).
!