542 lines
18 KiB
Fortran
542 lines
18 KiB
Fortran
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!=======================================================================
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!
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MODULE INTERPOLATION
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!
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! This module contains interpolation routines using the
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! following approaches:
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!
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!
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! * cubic spline interpolation : CUBIC_SPLINE_INTERP(F,X,N_X,XX)
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!
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! * Lagrange interpolation : LAG_NP_INTERP(X,F,XX) with N = 3-5
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!
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!
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! where F is the array representing the function to interpolate,
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! X the abscissae array and XX the interpolation point
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!
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USE ACCURACY_REAL
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!
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CONTAINS
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!
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!=======================================================================
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!
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FUNCTION CUBIC_SPLINE_INTERP(F,X,N_X,XX)
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!
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! This function takes a user-defined function F
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! and computes its value at XX
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!
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!
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! Input parameters:
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!
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! * F : array defining f
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! * X : array defining the abscissae of f
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! * N_X : size of the X and F arrays
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! * XX : value at which F is computed
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!
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!
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! Output variables :
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!
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! * FUNC : value F(XX)
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 4 Jun 2020
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!
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!
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USE DIMENSION_CODE
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USE REAL_NUMBERS, ONLY : SMALL
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!
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IMPLICIT NONE
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!
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INTEGER, INTENT(IN) :: N_X
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!
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INTEGER :: I,INTERP,NN
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INTEGER :: LOGF
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!
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REAL (WP), INTENT(IN) :: F(NSIZE),X(NSIZE)
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REAL (WP), INTENT(IN) :: XX
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!
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REAL (WP) :: CUBIC_SPLINE_INTERP
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REAL (WP) :: DIFF,RES
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!
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REAL (WP) :: ABS
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!
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INTERP = 1 !
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LOGF = 6 ! index of log file
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!
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! Checking if XX is part of the X array
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!
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DO I = 1, N_X !
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DIFF = ABS(X(I) - XX) !
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IF(DIFF < SMALL) THEN !
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INTERP = 0 !
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NN = I !
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GO TO 10 !
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END IF !
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END DO !
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10 CONTINUE !
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!
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! Interpolation whenever necessary
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!
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IF(INTERP == 1) THEN !
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CALL INTERP_NR(LOGF,X,F,N_X,XX,RES) !
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CUBIC_SPLINE_INTERP = RES !
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ELSE !
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CUBIC_SPLINE_INTERP = F(NN) !
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END IF !
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!
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END FUNCTION CUBIC_SPLINE_INTERP
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!
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!=======================================================================
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!
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SUBROUTINE INTERP_NR(LOGF,X,F,N_POINTS,XG,G)
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!
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! This subroutine interpolates a function F(X) at point XG using
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! the cubic spline interpolation of
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!
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! "Numerical Recipes in Fortran 77 second edition"
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!
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! from W. H. Press, S. A. Teukolsky, W. T. Vetterling
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! and B. P. Flannery, p. 109
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!
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!
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! Input parameters:
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!
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! LOGF : Fortran unit for log file
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! X : x coordinates of the input function F
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! F : y coordinates of the input function F
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! N_POINTS : number of points of the input function
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! XG : x point at which the interpolation is made
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!
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!
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! Output parameters:
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!
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! G : interpolated value of F at x = XG
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!
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!
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! Intermediate parameters:
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!
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! YP1 : value of first derivative of interpolating function at point 1
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! YPN : value of first derivative of interpolating function at point N
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!
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! If YP1 and/or YPN are larger or equal than 1.0D+30, the routine will set them
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! at the boundary condition of a natural spline, with second derivative on
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! that boundary.
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 17 Dec 2020
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!
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!
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IMPLICIT NONE
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!
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INTEGER, INTENT(IN) :: N_POINTS,LOGF
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!
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REAL (WP), INTENT(IN) :: X(N_POINTS),F(N_POINTS)
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REAL (WP), INTENT(IN) :: XG
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REAL (WP), INTENT(OUT) :: G
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!
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REAL (WP) :: F2(N_POINTS)
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REAL (WP) :: YP1,YPN
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!
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! Construction of the cubic spline used for the interpolation
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!
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YP1 = 2.0E+30_WP !
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YPN = 2.0E+30_WP !
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!
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CALL SPLINE(X,F,N_POINTS,YP1,YPN,F2) !
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!
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! Interpolation at x = XG
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!
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CALL SPLINT(X,F,F2,N_POINTS,XG,G,*10) !
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!
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RETURN !
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!
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10 WRITE(LOGF,11) !
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STOP !
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!
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! Formats
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!
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11 FORMAT(//,10X,'<<<<< WRONG VALUE FOR XA IN SUBROUTINE ', &
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' SPLINT >>>>>',//)
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!
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END SUBROUTINE INTERP_NR
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!
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!=======================================================================
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!
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SUBROUTINE SPLINE(X,Y,N,YP1,YPN,Y2)
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!
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! This subroutine constructs the second derivative of the
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! interpolating function for the input function Y = f(X).
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!
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! Taken from "Numerical Recipes in Fortran 77 second edition"
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!
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! from W. H. Press, S. A. Teukolsky, W. T. Vetterling
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! and B. P. Flannery, p. 109
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!
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!
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! Input parameters:
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!
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! X : x coordinates of the input function
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! Y : y coordinates of the input function
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! N : number of points of the input function
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! YP1 : value of first derivative of interpolating function at point 1
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! YPN : value of first derivative of interpolating function at point N
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!
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!
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! Output parameters:
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!
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! Y2 : second derivative of the interpolating function on X grid
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!
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!
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! If YP1 and/or YPN are larger or equal than 1.0D+30, the routine will set them
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! at the boundary condition of a natural spline, with second derivative on
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! that boundary.
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!
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! ---> This is the double precision version <---
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!
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!
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! Last modified (DS) : 21 Dec 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,THREE,HALF,MIC
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USE DIMENSION_CODE, ONLY : NZ_MAX
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!
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IMPLICIT NONE
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!
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INTEGER, INTENT(IN) :: N
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!
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INTEGER :: I,K
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!
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REAL (WP), INTENT(IN) :: X(N),Y(N)
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REAL (WP), INTENT(IN) :: YP1,YPN
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REAL (WP), INTENT(OUT) :: Y2(N)
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!
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REAL (WP) :: U(NZ_MAX)
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REAL (WP) :: SIG,P,QN,UN
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!
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! Lower boundary condition
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!
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IF(YP1 > 0.99E+30_WP) THEN !
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Y2(1) = ZERO !
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U(1) = ZERO !
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ELSE !
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Y2(1) = - HALF !
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U(1) = ( THREE / (X(2)-X(1)) ) * & !
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( (Y(2) - Y(1)) / (X(2) - X(1)) - YP1 ) !
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END IF !
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!
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! Decomposition loop of the tridiagonal algorithm
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!
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DO I = 2, N-1 !
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SIG = (X(I) - X(I-1)) / (X(I+1) - X(I-1)) !
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P = SIG * Y2(I-1) + TWO !
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Y2(I) = (SIG - ONE) / P !
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U(I) = ( 6.0E0_WP * ( (Y(I+1) - Y(I)) / (X(I+1) - X(I)) - &!
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(Y(I) - Y(I-1)) / (X(I) - X(I-1)) &!
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) / (X(I+1) - X(I-1)) - SIG * U(I-1) &!
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) / P !
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IF(U(I) < MIC) U(I) = ZERO !
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END DO !
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!
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! Upper boundary condition
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!
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IF(YPN > 0.99E+30_WP) THEN !
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QN = ZERO !
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UN = ZERO !
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ELSE !
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QN = HALF !
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UN = ( THREE / (X(N) - X(N-1)) ) * & !
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( YPN - (Y(N) - Y(N-1)) / (X(N) - X(N-1)) ) !
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END IF !
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!
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Y2(N) = (UN - QN * U(N-1)) / (QN * Y2(N-1) + ONE) !
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!
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! Backsubstitution loop of the tridiagonal algorithm
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!
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DO K = N-1, 1, -1 !
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Y2(K) = Y2(K) * Y2(K+1) + U(K) !
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END DO !
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!
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END SUBROUTINE SPLINE
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!
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!=======================================================================
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!
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SUBROUTINE SPLINT(XA,YA,Y2A,N,X,Y,*)
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!
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! This subroutine performs a cubic spline interpolation Y
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! of YA(XA) at point X
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!
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! Taken from "Numerical Recipes in Fortran 77 second edition"
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!
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! from W. H. Press, S. A. Teukolsky, W. T. Vetterling
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! and B. P. Flannery, p. 110
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!
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!
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! Input parameters:
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!
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! XA : x coordinates of the input function
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! YA : y coordinates of the input function
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! Y2A : y coordinates second derivative of the interpolating function
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! (output of subroutine SPLINE)
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! N : number of points of the input function
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! X : x value at which interpolation is made
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!
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!
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! Output parameters:
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!
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! Y : cubic-spline interpolated value
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!
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!
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! ---> This is the double precision version <---
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!
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USE REAL_NUMBERS, ONLY : ZERO
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!
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IMPLICIT NONE
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!
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INTEGER, INTENT(IN) :: N
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!
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INTEGER :: KLO,KHI,K
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!
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REAL (WP), INTENT(IN) :: XA(N),YA(N),Y2A(N)
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REAL (WP), INTENT(IN) :: X
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REAL (WP), INTENT(OUT) :: Y
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!
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REAL (WP) :: H,A,B
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!
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! Starting values for the points of the XA grid bracketing
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! the value X at which we interpolate YA
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!
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KLO = 1 !
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KHI = N !
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!
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! Bisection algorithm to find the exact values of KLO and KHI
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!
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1 IF(KHI - KLO > 1) THEN !
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K = (KHI + KLO) / 2 !
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IF(XA(K) > X) THEN !
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KHI = K !
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ELSE !
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KLO = K !
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END IF !
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GO TO 1 !
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END IF !
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!
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! Now, KLO < X < KHI
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!
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H = XA(KHI) - XA(KLO) !
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!
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IF(H == ZERO) RETURN 1 !
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!
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! Evaluation of the cubic spline polynomial
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!
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A = (XA(KHI) - X) / H !
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B = (X - XA(KLO)) / H !
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Y = A * YA(KLO) + B * YA(KHI) + & !
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( (A**3 - A) * Y2A(KLO) + (B**3 - B) * Y2A(KHI) ) * & !
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(H**2) / 6.0E0_WP !
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!
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END SUBROUTINE SPLINT
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!
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!=======================================================================
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!
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FUNCTION LAG_3P_INTERP(X,A,XX)
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!
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! This function computes a 3-point Lagrange interpolation
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!
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! Reference: (1) https://en.wikipedia.org/wiki/Lagrange_polynomial
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!
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! Input parameters :
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!
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! * X : array containing the abscissae of A
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! * A : function to interpolate
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! * XX : interpolation point
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 4 Jun 2020
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!
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!
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IMPLICIT NONE
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!
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REAL (WP), INTENT(IN) :: X(3),A(3)
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REAL (WP), INTENT(IN) :: XX
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REAL (WP) :: LAG_3P_INTERP
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!
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REAL (WP) :: L1,L2,L3
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!
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! Computing the Lagrange polynomials in XX
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!
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L1 = (XX-X(2))/(X(1)-X(2)) * (XX-X(3))/(X(1)-X(3)) !
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L2 = (XX-X(1))/(X(2)-X(1)) * (XX-X(3))/(X(2)-X(3)) !
|
||
|
|
L3 = (XX-X(1))/(X(3)-X(1)) * (XX-X(2))/(X(3)-X(2)) !
|
||
|
|
!
|
||
|
|
! Computing the interpolated value
|
||
|
|
!
|
||
|
|
LAG_3P_INTERP = A(1) * L1 + A(2) * L2 + A(3) * L3 !
|
||
|
|
!
|
||
|
|
END FUNCTION LAG_3P_INTERP
|
||
|
|
!
|
||
|
|
!=======================================================================
|
||
|
|
!
|
||
|
|
FUNCTION LAG_4P_INTERP(X,A,XX)
|
||
|
|
!
|
||
|
|
! This function computes a 4-point Lagrange interpolation
|
||
|
|
!
|
||
|
|
! Reference: (1) https://en.wikipedia.org/wiki/Lagrange_polynomial
|
||
|
|
!
|
||
|
|
! Input parameters :
|
||
|
|
!
|
||
|
|
! * X : array containing the abscissae of A
|
||
|
|
! * A : function to interpolate
|
||
|
|
! * XX : interpolation point
|
||
|
|
!
|
||
|
|
!
|
||
|
|
!
|
||
|
|
! Author : D. Sébilleau
|
||
|
|
!
|
||
|
|
! Last modified : 4 Jun 2020
|
||
|
|
!
|
||
|
|
!
|
||
|
|
IMPLICIT NONE
|
||
|
|
!
|
||
|
|
REAL (WP) :: X(4),A(4)
|
||
|
|
REAL (WP) :: XX
|
||
|
|
REAL (WP) :: LAG_4P_INTERP
|
||
|
|
REAL (WP) :: L1,L2,L3,L4
|
||
|
|
!
|
||
|
|
! Computing the Lagrange polynomials in XX
|
||
|
|
!
|
||
|
|
L1 = (XX-X(2))/(X(1)-X(2)) * (XX-X(3))/(X(1)-X(3)) * & !
|
||
|
|
(XX-X(4))/(X(1)-X(4)) !
|
||
|
|
L2 = (XX-X(1))/(X(2)-X(1)) * (XX-X(3))/(X(2)-X(3)) * & !
|
||
|
|
(XX-X(4))/(X(2)-X(4)) !
|
||
|
|
L3 = (XX-X(1))/(X(3)-X(1)) * (XX-X(2))/(X(3)-X(2)) * & !
|
||
|
|
(XX-X(4))/(X(3)-X(4)) !
|
||
|
|
L4 = (XX-X(1))/(X(4)-X(1)) * (XX-X(2))/(X(4)-X(2)) * & !
|
||
|
|
(XX-X(3))/(X(4)-X(3)) !
|
||
|
|
!
|
||
|
|
! Computing the interpolated value
|
||
|
|
!
|
||
|
|
LAG_4P_INTERP = A(1)*L1 + A(2)*L2 + A(3)*L3 + A(4)*L4 !
|
||
|
|
!
|
||
|
|
END FUNCTION LAG_4P_INTERP
|
||
|
|
!
|
||
|
|
!=======================================================================
|
||
|
|
!
|
||
|
|
FUNCTION LAG_5P_INTERP(X,A,XX)
|
||
|
|
!
|
||
|
|
! This function computes a 5-point Lagrange interpolation
|
||
|
|
!
|
||
|
|
! Reference: (1) https://en.wikipedia.org/wiki/Lagrange_polynomial
|
||
|
|
!
|
||
|
|
! Input parameters :
|
||
|
|
!
|
||
|
|
! * X : array containing the abscissae of A
|
||
|
|
! * A : function to interpolate
|
||
|
|
! * XX : interpolation point
|
||
|
|
!
|
||
|
|
!
|
||
|
|
!
|
||
|
|
! Author : D. Sébilleau
|
||
|
|
!
|
||
|
|
! Last modified : 4 Jun 2020
|
||
|
|
!
|
||
|
|
!
|
||
|
|
IMPLICIT NONE
|
||
|
|
!
|
||
|
|
REAL (WP) :: X(5),A(5)
|
||
|
|
REAL (WP) :: XX
|
||
|
|
REAL (WP) :: LAG_5P_INTERP
|
||
|
|
REAL (WP) :: L1,L2,L3,L4,L5
|
||
|
|
!
|
||
|
|
! Computing the Lagrange polynomials in XX
|
||
|
|
!
|
||
|
|
L1 = (XX-X(2))/(X(1)-X(2)) * (XX-X(3))/(X(1)-X(3)) * & !
|
||
|
|
(XX-X(4))/(X(1)-X(4)) * (XX-X(5))/(X(1)-X(5)) !
|
||
|
|
L2 = (XX-X(1))/(X(2)-X(1)) * (XX-X(3))/(X(2)-X(3)) * & !
|
||
|
|
(XX-X(4))/(X(2)-X(4)) * (XX-X(5))/(X(2)-X(5)) !
|
||
|
|
L3 = (XX-X(1))/(X(3)-X(1)) * (XX-X(2))/(X(3)-X(2)) * & !
|
||
|
|
(XX-X(4))/(X(3)-X(4)) * (XX-X(5))/(X(3)-X(5)) !
|
||
|
|
L4 = (XX-X(1))/(X(4)-X(1)) * (XX-X(2))/(X(4)-X(2)) * & !
|
||
|
|
(XX-X(3))/(X(4)-X(3)) * (XX-X(5))/(X(4)-X(5)) !
|
||
|
|
L5 = (XX-X(1))/(X(5)-X(1)) * (XX-X(2))/(X(5)-X(2)) * & !
|
||
|
|
(XX-X(3))/(X(5)-X(3)) * (XX-X(4))/(X(5)-X(4)) !
|
||
|
|
!
|
||
|
|
! Computing the interpolated value
|
||
|
|
!
|
||
|
|
LAG_5P_INTERP= A(1)*L1 + A(2)*L2 + A(3)*L3 + A(4)*L4 + A(5)*L5!
|
||
|
|
!
|
||
|
|
END FUNCTION LAG_5P_INTERP
|
||
|
|
!
|
||
|
|
!=======================================================================
|
||
|
|
!
|
||
|
|
FUNCTION LAG_6P_INTERP(X,A,XX)
|
||
|
|
!
|
||
|
|
! This function computes a 6-point Lagrange interpolation
|
||
|
|
!
|
||
|
|
! Reference: (1) https://en.wikipedia.org/wiki/Lagrange_polynomial
|
||
|
|
!
|
||
|
|
! Input parameters :
|
||
|
|
!
|
||
|
|
! * X : array containing the abscissae of A
|
||
|
|
! * A : function to interpolate
|
||
|
|
! * XX : interpolation point
|
||
|
|
!
|
||
|
|
!
|
||
|
|
!
|
||
|
|
! Author : D. Sébilleau
|
||
|
|
!
|
||
|
|
! Last modified : 15 Sep 2020
|
||
|
|
!
|
||
|
|
!
|
||
|
|
IMPLICIT NONE
|
||
|
|
!
|
||
|
|
REAL (WP) :: X(6),A(6)
|
||
|
|
REAL (WP) :: XX
|
||
|
|
REAL (WP) :: LAG_6P_INTERP
|
||
|
|
REAL (WP) :: L1,L2,L3,L4,L5,L6
|
||
|
|
!
|
||
|
|
! Computing the Lagrange polynomials in XX
|
||
|
|
!
|
||
|
|
L1 = (XX-X(2))/(X(1)-X(2)) * (XX-X(3))/(X(1)-X(3)) * & !
|
||
|
|
(XX-X(4))/(X(1)-X(4)) * (XX-X(5))/(X(1)-X(5)) * & !
|
||
|
|
(XX-X(6))/(X(1)-X(6)) !
|
||
|
|
!
|
||
|
|
L2 = (XX-X(1))/(X(2)-X(1)) * (XX-X(3))/(X(2)-X(3)) * & !
|
||
|
|
(XX-X(4))/(X(2)-X(4)) * (XX-X(5))/(X(2)-X(5)) * & !
|
||
|
|
(XX-X(6))/(X(2)-X(6)) !
|
||
|
|
!
|
||
|
|
L3 = (XX-X(1))/(X(3)-X(1)) * (XX-X(2))/(X(3)-X(2)) * & !
|
||
|
|
(XX-X(4))/(X(3)-X(4)) * (XX-X(5))/(X(3)-X(5)) * & !
|
||
|
|
(XX-X(6))/(X(3)-X(6)) !
|
||
|
|
!
|
||
|
|
L4 = (XX-X(1))/(X(4)-X(1)) * (XX-X(2))/(X(4)-X(2)) * & !
|
||
|
|
(XX-X(3))/(X(4)-X(3)) * (XX-X(5))/(X(4)-X(5)) * & !
|
||
|
|
(XX-X(6))/(X(4)-X(6)) !
|
||
|
|
!
|
||
|
|
L5 = (XX-X(1))/(X(5)-X(1)) * (XX-X(2))/(X(5)-X(2)) * & !
|
||
|
|
(XX-X(3))/(X(5)-X(3)) * (XX-X(4))/(X(5)-X(4)) * & !
|
||
|
|
(XX-X(6))/(X(5)-X(6)) !
|
||
|
|
!
|
||
|
|
L6 = (XX-X(1))/(X(6)-X(1)) * (XX-X(2))/(X(6)-X(2)) * & !
|
||
|
|
(XX-X(3))/(X(6)-X(3)) * (XX-X(4))/(X(6)-X(4)) * & !
|
||
|
|
(XX-X(5))/(X(6)-X(5)) !
|
||
|
|
!
|
||
|
|
! Computing the interpolated value
|
||
|
|
!
|
||
|
|
LAG_6P_INTERP = A(1) * L1 + A(2) * L2 + A(3) * L3 + & !
|
||
|
|
A(4) * L4 + A(5) * L5 + A(6) * L6 !
|
||
|
|
!
|
||
|
|
END FUNCTION LAG_6P_INTERP
|
||
|
|
!
|
||
|
|
END MODULE INTERPOLATION
|