# INTPOL: Linear, Cubic, Inverse, Derivatives, Integral

### HicEst numeric linear or cubic Akima interpolation of xy data. Roots, Find, 1st and 2nd derivative, integration.

• result = INTPOL(XVector=x_vector, YVector=y_array [, optional keywords] )
result is:
• when LEN(y_array) == LEN(x_vector)
• with LEN(y_array) == 4* LEN(x_vector)
• the Akima method avoids the wiggles of the Spline method
• of interpolated data between Xi and X2
• example with N=10 (note linear interpolation at boundaries):
• xdata = 0.5 1.9 3.0 4.2 4.8 6.2 6.7 8.1 9.3 9.5
• ycubic = 2.9 2.7 -0.8 3.3 -0.1 7.7 3.9 -1.8 1.1 3.5
• INTPOL(Init, XVector=xdata, YVector=ycubic) ! initialization
• yc = INTPOL(Xi=x, XVector=xdata, YVector=ycubic) ! cubic interpolation
• y1 = INTPOL(Xi=x, XVector=xdata, YVector=ycubic, DYdx) ! 1st derivative
• yi = INTPOL(Xi=xdata(1), XVector=xdata, YVector=ycubic, X2=x) ! integral keyword type mini sample keyword sequence is insignificant XVector vec XV=year (required ) a vector with N nodes of the independent variable in rising order DIMENSION year(N) nodes may be unequally spaced YVector arr YV=linear (required ) linear interpolation (XV and YV have same length): REAL :: x_linear(N), y_linear(N) cubic interpolation (YV has 4 times the elements of XV): REAL x_cubic(N), y_cubic(4,N) For x < node(2) or x > node(N-1): Interpolation is always linear. This means that extrapolation is also linear. the outermost 2 nodes determine the slope at boundaries (force boundary slopes by inserting extra nodes close to the first/last node) the Akima polynomial coefficients are in y(N+1, ..., 4*N) after the 1st call with Init Init --- Init (required on 1st call to cubic interpolation ) to initialize the Akima coefficients Xi num X=3.1 interpolate at Xi, default is Xi=0 X2 num x2=3.9 integrate the interpolated data from Xi to X2 Find num find=y find x closest to Xi with interpolated value equal to y DYdx --- dy 1st derivative of interpolated data at Xi DYdx num dy=yprime find x closest to Xi with 1st derivative equal to yprime D2ydx --- d2y 2nd derivative of interpolated data at Xi D2ydy num d2y=y2prime find x closest to Xi with 2nd derivative equal to y2prime ERror LBL ER=99 on error jump to ⇾ label 99

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