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!! |
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!! Copyright (c) 2006 The University of Notre Dame. All Rights Reserved. |
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!! |
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!! The University of Notre Dame grants you ("Licensee") a |
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!! non-exclusive, royalty free, license to use, modify and |
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!! redistribute this software in source and binary code form, provided |
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!! that the following conditions are met: |
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!! |
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!! 1. Acknowledgement of the program authors must be made in any |
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!! publication of scientific results based in part on use of the |
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!! program. An acceptable form of acknowledgement is citation of |
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!! the article in which the program was described (Matthew |
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!! A. Meineke, Charles F. Vardeman II, Teng Lin, Christopher |
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!! J. Fennell and J. Daniel Gezelter, "OOPSE: An Object-Oriented |
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!! Parallel Simulation Engine for Molecular Dynamics," |
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!! J. Comput. Chem. 26, pp. 252-271 (2005)) |
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!! |
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!! 2. Redistributions of source code must retain the above copyright |
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!! notice, this list of conditions and the following disclaimer. |
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!! |
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!! 3. Redistributions in binary form must reproduce the above copyright |
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!! notice, this list of conditions and the following disclaimer in the |
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!! documentation and/or other materials provided with the |
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!! distribution. |
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!! |
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!! This software is provided "AS IS," without a warranty of any |
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!! kind. All express or implied conditions, representations and |
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!! warranties, including any implied warranty of merchantability, |
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!! fitness for a particular purpose or non-infringement, are hereby |
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!! excluded. The University of Notre Dame and its licensors shall not |
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!! be liable for any damages suffered by licensee as a result of |
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!! using, modifying or distributing the software or its |
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!! derivatives. In no event will the University of Notre Dame or its |
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!! licensors be liable for any lost revenue, profit or data, or for |
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!! direct, indirect, special, consequential, incidental or punitive |
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!! damages, however caused and regardless of the theory of liability, |
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!! arising out of the use of or inability to use software, even if the |
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!! University of Notre Dame has been advised of the possibility of |
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!! such damages. |
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!! |
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!! |
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!! interpolation.F90 |
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!! |
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!! Created by Charles F. Vardeman II on 03 Apr 2006. |
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!! |
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!! PURPOSE: Generic Spline interpolation routines. These routines |
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!! assume that we are on a uniform grid for precomputation of |
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!! spline parameters. |
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!! |
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!! @author Charles F. Vardeman II |
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!! @version $Id: interpolation.F90,v 1.6 2006-04-17 21:49:12 gezelter Exp $ |
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|
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|
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module interpolation |
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use definitions |
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use status |
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implicit none |
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PRIVATE |
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|
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character(len = statusMsgSize) :: errMSG |
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|
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type, public :: cubicSpline |
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logical :: isUniform = .false. |
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integer :: np = 0 |
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real(kind=dp) :: dx_i |
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real (kind=dp), pointer,dimension(:) :: x => null() |
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real (kind=dp), pointer,dimension(:,:) :: c => null() |
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end type cubicSpline |
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|
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public :: newSpline |
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public :: deleteSpline |
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public :: lookupSpline |
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public :: lookupUniformSpline |
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public :: lookupNonuniformSpline |
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public :: lookupUniformSpline1d |
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|
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contains |
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|
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|
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subroutine newSpline(cs, x, y, yp1, ypn, isUniform) |
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|
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!************************************************************************ |
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! |
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! newSpline solves for slopes defining a cubic spline. |
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! |
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! Discussion: |
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! |
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! A tridiagonal linear system for the unknown slopes S(I) of |
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! F at x(I), I=1,..., N, is generated and then solved by Gauss |
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! elimination, with S(I) ending up in cs%C(2,I), for all I. |
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! |
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! Reference: |
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! |
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! Carl DeBoor, |
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! A Practical Guide to Splines, |
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! Springer Verlag. |
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! |
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! Parameters: |
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! |
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! Input, real x(N), the abscissas or X values of |
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! the data points. The entries of x are assumed to be |
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! strictly increasing. |
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! |
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! Input, real y(I), contains the function value at x(I) for |
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! I = 1, N. |
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! |
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! Input, real yp1 contains the slope at x(1) |
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! Input, real ypn contains the slope at x(N) |
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! |
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! On output, the slopes at x(I) have been stored in |
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! cs%C(2,I), for I = 1 to N. |
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|
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implicit none |
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|
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type (cubicSpline), intent(inout) :: cs |
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real( kind = DP ), intent(in) :: x(:), y(:) |
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real( kind = DP ), intent(in) :: yp1, ypn |
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logical, intent(in) :: isUniform |
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real( kind = DP ) :: g, divdif1, divdif3, dx |
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integer :: i, alloc_error, np |
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|
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alloc_error = 0 |
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|
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if (cs%np .ne. 0) then |
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call handleWarning("interpolation::newSpline", & |
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"cubicSpline struct was already created") |
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call deleteSpline(cs) |
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end if |
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|
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! make sure the sizes match |
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|
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np = size(x) |
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|
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if ( size(y) .ne. np ) then |
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call handleError("interpolation::newSpline", & |
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"Array size mismatch") |
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end if |
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|
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cs%np = np |
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cs%isUniform = isUniform |
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|
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allocate(cs%x(np), stat=alloc_error) |
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if(alloc_error .ne. 0) then |
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call handleError("interpolation::newSpline", & |
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"Error in allocating storage for x") |
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endif |
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|
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allocate(cs%c(4,np), stat=alloc_error) |
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if(alloc_error .ne. 0) then |
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call handleError("interpolation::newSpline", & |
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"Error in allocating storage for c") |
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endif |
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|
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do i = 1, np |
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cs%x(i) = x(i) |
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cs%c(1,i) = y(i) |
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enddo |
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|
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! Set the first derivative of the function to the second coefficient of |
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! each of the endpoints |
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|
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cs%c(2,1) = yp1 |
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cs%c(2,np) = ypn |
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|
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! |
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! Set up the right hand side of the linear system. |
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! |
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|
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do i = 2, cs%np - 1 |
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cs%c(2,i) = 3.0_DP * ( & |
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(x(i) - x(i-1)) * (cs%c(1,i+1) - cs%c(1,i)) / (x(i+1) - x(i)) + & |
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(x(i+1) - x(i)) * (cs%c(1,i) - cs%c(1,i-1)) / (x(i) - x(i-1))) |
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end do |
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|
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! |
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! Set the diagonal coefficients. |
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! |
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cs%c(4,1) = 1.0_DP |
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do i = 2, cs%np - 1 |
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cs%c(4,i) = 2.0_DP * ( x(i+1) - x(i-1) ) |
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end do |
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cs%c(4,cs%np) = 1.0_DP |
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! |
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! Set the off-diagonal coefficients. |
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! |
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cs%c(3,1) = 0.0_DP |
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do i = 2, cs%np |
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cs%c(3,i) = x(i) - x(i-1) |
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end do |
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! |
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! Forward elimination. |
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! |
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do i = 2, cs%np - 1 |
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g = -cs%c(3,i+1) / cs%c(4,i-1) |
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cs%c(4,i) = cs%c(4,i) + g * cs%c(3,i-1) |
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cs%c(2,i) = cs%c(2,i) + g * cs%c(2,i-1) |
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end do |
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! |
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! Back substitution for the interior slopes. |
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! |
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do i = cs%np - 1, 2, -1 |
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cs%c(2,i) = ( cs%c(2,i) - cs%c(3,i) * cs%c(2,i+1) ) / cs%c(4,i) |
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end do |
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! |
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! Now compute the quadratic and cubic coefficients used in the |
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! piecewise polynomial representation. |
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! |
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do i = 1, cs%np - 1 |
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dx = x(i+1) - x(i) |
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divdif1 = ( cs%c(1,i+1) - cs%c(1,i) ) / dx |
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divdif3 = cs%c(2,i) + cs%c(2,i+1) - 2.0_DP * divdif1 |
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cs%c(3,i) = ( divdif1 - cs%c(2,i) - divdif3 ) / dx |
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cs%c(4,i) = divdif3 / ( dx * dx ) |
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end do |
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|
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cs%c(3,cs%np) = 0.0_DP |
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cs%c(4,cs%np) = 0.0_DP |
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|
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cs%dx_i = 1.0_DP / dx |
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|
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return |
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end subroutine newSpline |
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|
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subroutine deleteSpline(this) |
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|
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type(cubicSpline) :: this |
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|
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if(associated(this%x)) then |
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deallocate(this%x) |
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this%x => null() |
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end if |
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if(associated(this%c)) then |
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deallocate(this%c) |
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this%c => null() |
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end if |
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|
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this%np = 0 |
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|
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end subroutine deleteSpline |
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|
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subroutine lookupNonuniformSpline(cs, xval, yval) |
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|
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!************************************************************************* |
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! |
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! lookupNonuniformSpline evaluates a piecewise cubic Hermite interpolant. |
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! |
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! Discussion: |
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! |
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! newSpline must be called first, to set up the |
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! spline data from the raw function and derivative data. |
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! |
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! Modified: |
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! |
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! 06 April 1999 |
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! |
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! Reference: |
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! |
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! Conte and de Boor, |
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! Algorithm PCUBIC, |
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! Elementary Numerical Analysis, |
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! 1973, page 234. |
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! |
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! Parameters: |
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! |
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implicit none |
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|
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type (cubicSpline), intent(in) :: cs |
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real( kind = DP ), intent(in) :: xval |
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real( kind = DP ), intent(out) :: yval |
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real( kind = DP ) :: dx |
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integer :: i, j |
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! |
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! Find the interval J = [ cs%x(J), cs%x(J+1) ] that contains |
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! or is nearest to xval. |
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! |
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j = cs%np - 1 |
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|
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do i = 1, cs%np - 2 |
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|
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if ( xval < cs%x(i+1) ) then |
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j = i |
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exit |
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end if |
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|
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end do |
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! |
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! Evaluate the cubic polynomial. |
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! |
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dx = xval - cs%x(j) |
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|
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yval = cs%c(1,j) + dx * ( cs%c(2,j) + dx * ( cs%c(3,j) + dx * cs%c(4,j) ) ) |
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|
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return |
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end subroutine lookupNonuniformSpline |
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|
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subroutine lookupUniformSpline(cs, xval, yval) |
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|
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!************************************************************************* |
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! |
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! lookupUniformSpline evaluates a piecewise cubic Hermite interpolant. |
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! |
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! Discussion: |
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! |
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! newSpline must be called first, to set up the |
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! spline data from the raw function and derivative data. |
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! |
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! Modified: |
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! |
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! 06 April 1999 |
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! |
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! Reference: |
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! |
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! Conte and de Boor, |
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! Algorithm PCUBIC, |
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! Elementary Numerical Analysis, |
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! 1973, page 234. |
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! |
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! Parameters: |
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! |
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implicit none |
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|
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type (cubicSpline), intent(in) :: cs |
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real( kind = DP ), intent(in) :: xval |
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real( kind = DP ), intent(out) :: yval |
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real( kind = DP ) :: a, b, c, d, dx |
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integer :: i, j |
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! |
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! Find the interval J = [ cs%x(J), cs%x(J+1) ] that contains |
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! or is nearest to xval. |
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|
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j = MAX(1, MIN(cs%np, idint((xval-cs%x(1)) * cs%dx_i) + 1)) |
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|
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dx = xval - cs%x(j) |
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|
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a = cs%c(1,j) |
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b = cs%c(2,j) |
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c = cs%c(3,j) |
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d = cs%c(4,j) |
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|
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yval = c + dx * d |
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yval = b + dx * yval |
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yval = a + dx * yval |
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|
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return |
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end subroutine lookupUniformSpline |
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|
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subroutine lookupUniformSpline1d(cs, xval, yval, dydx) |
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|
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implicit none |
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|
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type (cubicSpline), intent(in) :: cs |
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real( kind = DP ), intent(in) :: xval |
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real( kind = DP ), intent(out) :: yval, dydx |
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real( kind = DP ) :: a, b, c, d, dx |
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integer :: i, j |
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|
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! Find the interval J = [ cs%x(J), cs%x(J+1) ] that contains |
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! or is nearest to xval. |
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|
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j = MAX(1, MIN(cs%np, idint((xval-cs%x(1)) * cs%dx_i) + 1)) |
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|
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dx = xval - cs%x(j) |
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|
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a = cs%c(1,j) |
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b = cs%c(2,j) |
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c = cs%c(3,j) |
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d = cs%c(4,j) |
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|
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yval = c + dx * d |
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yval = b + dx * yval |
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yval = a + dx * yval |
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|
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dydx = 2.0d0 * c + 3.0d0 * d * dx |
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dydx = b + dx * dydx |
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|
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return |
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end subroutine lookupUniformSpline1d |
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|
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subroutine lookupSpline(cs, xval, yval) |
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|
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type (cubicSpline), intent(in) :: cs |
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real( kind = DP ), intent(inout) :: xval |
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real( kind = DP ), intent(inout) :: yval |
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|
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if (cs%isUniform) then |
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call lookupUniformSpline(cs, xval, yval) |
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else |
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call lookupNonuniformSpline(cs, xval, yval) |
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endif |
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|
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return |
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end subroutine lookupSpline |
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|
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end module interpolation |
