INTPOL: linear interpolation, inverse interpolation, cubic Akima interpolation, differentiation, integration (H)

- result = INTPOL(XVector=x_vector, YVector=y_array [, optional keywords] )
result is:
- y interpolated
  at Xi in the absence of DYdx, D2ydx, or Find
- first derivative
  at Xi with DYdx
- second derivative
  at Xi with D2ydx (discontinuous at the nodes)
- interpolated roots
  : find the x values closest to Xi for:
  - given y: Find=y
  - given y': DYdx=yprime
  - given y": D2ydx=y2prime
linear interpolation
when LEN(y_array) == LEN(x_vector)
cubic Akima interpolation
with LEN(y_array) == 4*LEN(x_vector)
- the Akima method avoids the wiggles of the Spline method
numeric integration
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

data and cubic interpolation, 1st derivative and 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

INTPOL: linear interpolation, inverse interpolation, cubic Akima interpolation, differentiation, integration (H)

required

required

slope at boundaries

required on 1st call to cubic interpolation

default is Xi=0