SUBROUTINE AKIMA (NP,ND,XD,YD,NI,XI, YI) C C PREVIOUSLY THIS ROUTINE WAS CALLED UVIP3P AND WAS IN SINGLE PRECISION C IT WAS DOWNLOADED FROM GAMS AT NIST ON OCT 21 1998 C IT PROVIDES SLIGHTLY BETTER FITTING THAN THE IMSL ROUTINE DCSAKM C THIS IS A SUBJECTIVE COMMENT - IT DOES HOWEVER PREVENT C OSCILLATIONS BETWEEN SPARCE DATA POINTS C C Univariate Interpolation (Improved Akima Method) C C Hiroshi Akima C U.S. Department of Commerce, NTIA/ITS C Version of 89/07/04 C C This subroutine performs univariate interpolation. It is based C on the improved A method developed by Hiroshi Akima, 'A method C of univariate interpolation that has the accuracy of a third- C degree polynomial,' ACM TOMS, vol. xx, pp. xxx-xxx, 19xx. (The C equation numbers referred to in the comments below are those in C the paper.) C C In this method, the interpolating function is a piecewise C function composed of a set of polynomials applicable to C successive intervals of the given data points. This method C uses third-degree polynomials as the default, but the user has C an option to use higher-degree polynomial to reduce undulations C in resulting curves. C C This method has the accuracy of a third-degree polynomial if C the degree of the polynomials for the interpolating function is C set to three. C C The input arguments are C NP = degree of the polynomials for the interpolating C function, C ND = number of input data points C (must be equal to 2 or greater), C XD = array of dimension ND, containing the abscissas of C the input data points C (must be in a monotonic increasing order), C YD = array of dimension ND, containing the ordinates of C the input data points, C NI = number of points for which interpolation is desired C (must be equal to 1 or greater), C XI = array of dimension NI, containing the abscissas of C the desired points. C C The output argument is C YI = array of dimension NI, where the ordinates of the C desired points are to be stored. C C If an integer value smaller than 3 is given to the NP argument, C this subroutine assumes NP = 3. C C The XI array elements need not be monotonic, but this C subroutine interpolates faster if the XI array elements are C given in a monotonic order. C C If the XI array element is less than XD(1) or greater than C XD(ND), this subroutine linearly interpolates the YI value. C C C Specification statement implicit real*8 (a-h,o-z) DIMENSION XD(*),YD(*),XI(*), YI(*) C Error check 10 IF (ND.LE.1) GO TO 90 IF (NI.LE.0) GO TO 91 DO 11 ID=2,ND IF (XD(ID).LE.XD(ID-1)) GO TO 92 11 CONTINUE C Branches off special cases. IF (ND.LE.4) GO TO 50 C General case -- Five data points of more C Calculates some local variables. 20 NP0=MAX(3,NP) NPM1=NP0-1 RENPM1=NPM1 RENNM2=NP0*(NP0-2) C Main calculation for the general case C First (outermost) DO-loop with respect to the desired points 30 DO 39 II=1,NI IF (II.EQ.1) IINTPV=-1 XII=XI(II) C Locates the interval that includes the desired point by binary C search. IF (XII.LE.XD(1)) THEN IINT=0 ELSE IF (XII.LT.XD(ND)) THEN IDMN=1 IDMX=ND IDMD=(IDMN+IDMX)/2 31 IF (XII.GE.XD(IDMD)) THEN IDMN=IDMD ELSE IDMX=IDMD END IF IDMD=(IDMN+IDMX)/2 IF (IDMD.GT.IDMN) GO TO 31 IINT=IDMD ELSE IINT=ND END IF C End of locating the interval of interest C Interpolation or extrapolation in one of the three subcases IF (IINT.LE.0) THEN C Subcase 1 -- Linear extrapolation when the abscissa of the C desired point is equal to that of the first data C point or less. C Estimates the first derivative when the interval is not the C same as the one for the previous desired point. -- C cf. Equation (8) IF (IINT.NE.IINTPV) THEN IINTPV=IINT X0=XD(1) X1=XD(2)-X0 X2=XD(3)-X0 X3=XD(4)-X0 Y0=YD(1) Y1=YD(2)-Y0 Y2=YD(3)-Y0 Y3=YD(4)-Y0 DLT=X1*X2*X3*(X2-X1)*(X3-X2)*(X3-X1) A1=(((X2*X3)**2)*(X3-X2)*Y1 1 +((X3*X1)**2)*(X1-X3)*Y2 2 +((X1*X2)**2)*(X2-X1)*Y3)/DLT END IF C Evaluates the YI value. YI(II)=Y0+A1*(XII-X0) C End of Subcase 1 ELSE IF (IINT.GE.ND) THEN C Subcase 2 -- Linear extrapolation when the abscissa of the C desired point is equal to that of the last data C point or greater. C Estimates the first derivative when the interval is not the C same as the one for the previous desired point. -- C cf. Equation (8) IF (IINT.NE.IINTPV) THEN IINTPV=IINT X0=XD(ND) X1=XD(ND-1)-X0 X2=XD(ND-2)-X0 X3=XD(ND-3)-X0 Y0=YD(ND) Y1=YD(ND-1)-Y0 Y2=YD(ND-2)-Y0 Y3=YD(ND-3)-Y0 DLT=X1*X2*X3*(X2-X1)*(X3-X2)*(X3-X1) A1=(((X2*X3)**2)*(X3-X2)*Y1 1 +((X3*X1)**2)*(X1-X3)*Y2 2 +((X1*X2)**2)*(X2-X1)*Y3)/DLT END IF C Evaluates the YI value. YI(II)=Y0+A1*(XII-X0) C End of Subcase 2 ELSE C Subcase 3 -- Interpolation when the abscissa of the desired C point is between those of the first and last C data points. C Calculates the coefficients of the third-degree polynomial (for C NP.LE.3) or the factors for the higher-degree polynomials (for C NP.GT.3), when the interval is not the same as the one for the C previous desired point. IF (IINT.NE.IINTPV) THEN IINTPV=IINT C The second DO-loop with respect to the two endpoints of the C interval DO 37 IEPT=1,2 C Calculates the estimate of the first derivative at an endpoint. C Initial setting for calculation ID0=IINT+IEPT-1 X0=XD(ID0) Y0=YD(ID0) SMPEF=0.0d0 SMWTF=0.0d0 SMPEI=0.0d0 SMWTI=0.0d0 C The third (innermost) DO-loop with respect to the four primary C estimate of the first derivative DO 36 IPE=1,4 C Selects point numbers of four consecutive data points for C calculating the primary estimate of the first derivative. IF (IPE.EQ.1) THEN ID1=ID0-3 ID2=ID0-2 ID3=ID0-1 ELSE IF (IPE.EQ.2) THEN ID1=ID0+1 ELSE IF (IPE.EQ.3) THEN ID2=ID0+2 ELSE ID3=ID0+3 END IF C Checks if any point number falls outside the legitimate range C (between 1 and ND). Skips calculation of the primary estimate C if any does. IF (ID1.LT.1.OR.ID2.LT.1.OR.ID3.LT.1.OR. 1 ID1.GT.ND.OR.ID2.GT.ND.OR.ID3.GT.ND) 2 GO TO 36 C Calculates the primary estimate of the first derivative -- C cf. Equation (8) X1=XD(ID1)-X0 X2=XD(ID2)-X0 X3=XD(ID3)-X0 Y1=YD(ID1)-Y0 Y2=YD(ID2)-Y0 Y3=YD(ID3)-Y0 DLT=X1*X2*X3*(X2-X1)*(X3-X2)*(X3-X1) PE=(((X2*X3)**2)*(X3-X2)*Y1 1 +((X3*X1)**2)*(X1-X3)*Y2 2 +((X1*X2)**2)*(X2-X1)*Y3)/DLT C Calculates the volatility factor, VOL, and distance factor, C SXX, for the primary estimate. -- cf. Equations (9) and (11) SX=X1+X2+X3 SY=Y1+Y2+Y3 SXX=X1*X1+X2*X2+X3*X3 SXY=X1*Y1+X2*Y2+X3*Y3 DNM=4.0d0*SXX-SX*SX B0=(SXX*SY-SX*SXY)/DNM B1=(4.0d0*SXY-SX*SY)/DNM DY0=-B0 DY1=Y1-(B0+B1*X1) DY2=Y2-(B0+B1*X2) DY3=Y3-(B0+B1*X3) VOL=DY0*DY0+DY1*DY1+DY2*DY2+DY3*DY3 C Calculates the EPSLN value, which is used to decide whether or C not the volatility factor, VOL, is essentially zero. EPSLN=(YD(ID0)**2+YD(ID1)**2 1 +YD(ID2)**2+YD(ID3)**2)*1.0D-12 C Accumulates the weighted primary estimates. -- C cf. Equations (13) and (14) IF (VOL.GT.EPSLN) THEN C - For finite weight. WT=1.0d0/(VOL*SXX) SMPEF=SMPEF+PE*WT SMWTF=SMWTF+WT ELSE C - For infinite weight. SMPEI=SMPEI+PE SMWTI=SMWTI+1.0d0 END IF 36 CONTINUE C End of the third DO-loop C Calculates the final estimate of the first derivative. -- C cf. Equation (14) IF (SMWTI.LT.0.5d0) THEN C - When no infinite weights exist. YP=SMPEF/SMWTF ELSE C - When infinite weights exist. YP=SMPEI/SMWTI END IF IF (IEPT.EQ.1) THEN YP0=YP ELSE YP1=YP END IF C End of the calculation of the estimate of the first derivative C at an endpoint 37 CONTINUE C End of the second DO-loop IF (NP0.LE.3) THEN C Calculates the coefficients of the third-degree polynomial C (when NP.LE.3). -- cf. Equation (4) DX=XD(IINT+1)-XD(IINT) DY=YD(IINT+1)-YD(IINT) A0=YD(IINT) A1=YP0 YP1=YP1-YP0 YP0=YP0-DY/DX A2=-(3.0d0*YP0+YP1)/DX A3= (2.0d0*YP0+YP1)/(DX*DX) ELSE C Calculates the factors for the higher-degree polynomials C (when NP.GT.3). -- cf. Equation (20) DX=XD(IINT+1)-XD(IINT) DY=YD(IINT+1)-YD(IINT) T0=YP0*DX-DY T1=YP1*DX-DY AA0= (T0+RENPM1*T1)/RENNM2 AA1=-(RENPM1*T0+T1)/RENNM2 END IF END IF C End of the calculation of the coefficients of the third-degree C polynomial (when NP.LE.3) or the factors for the higher-degree C polynomials (when NP.GT.3), when the interval is not the same C as the one for the previous desired point. C Evaluates the YI value. IF (NP0.LE.3) THEN C - With a third-degree polynomial. -- cf. Equation (3) XX=XII-XD(IINT) YI(II)=A0+XX*(A1+XX*(A2+XX*A3)) ELSE C - With a higher-degree polynomial. -- cf. Equation (19) U=(XII-XD(IINT))/DX UC=1.0d0-U V=AA0*((U**NP0)-U)+AA1*((UC**NP0)-UC) YI(II)=YD(IINT)+DY*U+V END IF C End of Subcase 3 END IF 39 CONTINUE C End of the first DO-loop C End of general case RETURN C Special cases -- Four data points or less C Preliminary processing for the special cases 50 X0=XD(1) Y0=YD(1) X1=XD(2)-X0 Y1=YD(2)-Y0 IF (ND.EQ.2) GO TO 60 X2=XD(3)-X0 Y2=YD(3)-Y0 IF (ND.EQ.3) GO TO 70 X3=XD(4)-X0 Y3=YD(4)-Y0 GO TO 80 C Special Case 1 -- Two data points C (Linear interpolation and extrapolation) 60 A1=Y1/X1 DO 61 II=1,NI YI(II)=Y0+A1*(XI(II)-X0) 61 CONTINUE C End of Special Case 1 RETURN C Special Case 2 -- Three data points C (Quadratic interpolation and linear extrapolation) 70 DLT=X1*X2*(X2-X1) A1=(X2*X2*Y1-X1*X1*Y2)/DLT A2=(X1*Y2-X2*Y1)/DLT A12=2.0d0*A2*X2+A1 DO 71 II=1,NI XX=XI(II)-X0 IF (XX.LE.0.0d0) THEN YI(II)=Y0+A1*XX ELSE IF (XX.LT.X2) THEN YI(II)=Y0+XX*(A1+XX*A2) ELSE YI(II)=Y0+Y2+A12*(XX-X2) END IF 71 CONTINUE C End of Special Case 2 RETURN C Special Case 3 -- Four data points C (Cubic interpolation and linear extrapolation) 80 DLT=X1*X2*X3*(X2-X1)*(X3-X2)*(X3-X1) A1=(((X2*X3)**2)*(X3-X2)*Y1 1 +((X3*X1)**2)*(X1-X3)*Y2 2 +((X1*X2)**2)*(X2-X1)*Y3)/DLT A2=(X2*X3*(X2*X2-X3*X3)*Y1 1 +X3*X1*(X3*X3-X1*X1)*Y2 2 +X1*X2*(X1*X1-X2*X2)*Y3)/DLT A3=(X2*X3*(X3-X2)*Y1 1 +X3*X1*(X1-X3)*Y2 2 +X1*X2*(X2-X1)*Y3)/DLT A13=(3.0d0*A3*X3+2.0d0*A2)*X3+A1 DO 81 II=1,NI XX=XI(II)-X0 IF (XX.LE.0.0d0) THEN YI(II)=Y0+A1*XX ELSE IF (XX.LT.X3) THEN YI(II)=Y0+XX*(A1+XX*(A2+XX*A3)) ELSE YI(II)=Y0+Y3+A13*(XX-X3) END IF 81 CONTINUE C End of Special Case 3 RETURN C Error exit 90 WRITE (*,99090) ND GO TO 99 91 WRITE (*,99091) NI GO TO 99 92 WRITE (*,99092) ID,XD(ID-1),XD(ID) 99 WRITE (*,99099) RETURN C Format statements for error messages 99090 FORMAT (1X/ ' *** Insufficient data points.' 1 7X,'ND =',I3) 99091 FORMAT (1X/ ' *** No desired points.' 1 7X,'NI =',I3) 99092 FORMAT (1X/ ' *** Two data points identical or out of ', 1 'sequence.'/ 2 7X,'ID, XD(ID-1), XD(ID) =',I5,2F10.3) 99099 FORMAT (' Error detected in the akima subroutine'/) END