Polynomial interpolation and extrapolation
PROBLEM
We know the value of a function at a set of points A.1<A.2<...<A.N. Estimate the value of a function for arbitrary V.
ALGORITHM
Given arrays A.1,...,A.N and B.1,...,B.N, and given a value V. For polynomial P. of degree N-1 such that AI=A.I; P.AI=B.I, for I=1,...,N, we estimate value Y=P.V
IMPLEMENTATION
Unit: internal function
Global variables: ascending sequence of values A.1,...,A.N, array B.
Parameters: positive integer N, arbitrary value V
Returns:
Couple of values separated by blank - Y, and error estimate Dy
POLINT: procedure expose A. B.
parse arg N, V
Ns = 1; Dif = ABS(V - A.1)
do I = 1 to N
Dift = ABS(V - A.I)
if Dift < Dif
then do; Ns = I; Dif = Dift; end
C.I = B.I; D.I = B.I
end
Y = B.Ns; Ns = Ns - 1;
do M = 1 to N - 1
do I = 1 to N - M
Ho = A.I - V
IpM = I + M; Hp = A.IpM - V
Ip1 = I + 1; W = C.Ip1 - D.I
Den = Ho - Hp
if Den = 0 then
call ERROR "POLINT: Error -",
"two input A's are",
"(to within
roundoff) identical"
Den = W / Den; D.I = Hp * Den
C.I = Ho * Den
end
if 2 * Ns < (N - M)
then do; Nsp1 = Ns + 1; Dy = C.Nsp1; end
else do; Dy = D.Ns; Ns = Ns - 1; end
Y = Y + Dy
end
return Y Dy
ERROR: say ARG(1); exit
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EXAMPLE
The following program
N = 5
do J = 1 to 5; A.J = J; end
B.1 = 1.8; B.2 = 2.27; B.3 = 3.18
B.4 = 4.01; B.5 = 4.91
say POLINT(N, 2.25)
exit
POLINT: procedure expose A. B.
...
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displays on the screen 2.48801270 0.0114501953.
CONNECTIONS
Literature
Olehla M., Tiser J. Prakticke pouziti FORTRANu
Nakladatestvi dopravy a spoju Praha 1976
Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P. Numerical Recipes in C : the art of scientific computing
- 2nd ed. University Press, Cambridge, 1992
