BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

48 Petronela Paraschiv
interpolate functions φ( x)
have continuous derivates until a random order
(Elias et al., 2004; Tannous et al., 1996; Wagenaar & von Emmerik, 1996).
The interpolation of a discreet set of data [x, y] supposes the
determination of a function f(x) so that f(x) = y, in order to complete the set of
data in any other point x 0 ≠ x 1 . The function f(x) represents the best
approximation of the set of data, in this case not being necessary for the
determined function to pass through all the given points, but it has to be “the
best approximation” after a certain error criterion imposed. Therefore, the
method of the smallest squares gives, for instance, the best approximation in the
sense of minimizing the squre distances from the given points and the
approximation points.
3. Conclusions
1. Applying an interpolation function, linear, spline cubic or polynomial,
for the whole set of data or only for some periods of the values, determines the
interpolation function coefficients so that the interpolation curve to pass through
all given value points.
2. The cubic spline curve is an even curve, defined by a set of
polynomials of the third degree. The curve between each pair of points is a
polynomial of the third degree calculated in such a way as to lead to even
transitions from a polynomial of the third degree to the other.
3. For the situation in which two curves need to be compared, one
experimental given by relative values, and one theoretical given by absolute
values, the interpolation may allow finding a convenient function of
interpolation, the same for both curves to be compared, in which the mathematic
function coefficients differ from ne curve to the other.
4. The interpolation function found needs to satisfy the condition that it
represents the best approximation for any other two curves, experimental and
theoretical, that express the variation of the same physical phenomenon.
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