The most simple case of interpolation is a straight line between two points. (Extending
this line beyond the two points is referred to as extrapolation.) Three points that do not
lie on a common straight line need to be connected by a curve. Then, a unique solution for a connecting curve is provided
by a parabola (having three parameters which can be determined using the three data points). Analogously, for four points
which neither can be connected by a single straight line nor a single parabola, the unique connecting curve is a cubic function
(polynomial of 3\(^{\text{rd}}\) degree having four parameters). In general terms:
Theorem: interpolating polynomial
(without proof)
For \(n+1\) given points \((x_0,y_0),(x_1,y_1),\dots,(x_n,y_n)\)
with \(x_i\neq x_j\) for \(i\neq j\) there exists exactly one polynomial \(p(x)\) of
degree \(\leq n\) with \(p(x_i)=y_i\) for all \(i\in\{0,1,2,\dots,n\}\). We call \(p(x)\) the interpolating polynomial for the given points.
Example:
Consider the following points: \((x_0,y_0) = (0,1)\);
\((x_1,y_1) = (1,6)\); \((x_2,y_2) = (2,15)\). They define a parabola, i.e. a quadratic polynomial
\(p(x)=a+bx+cx^2\). The coefficients are found from the three conditions \(p(0)=1, p(1)=6, p(2)=15\),
which leads to the solution \(p(x)=1+3x+2x^2\).
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© J. Carstensen (Comp. Math.)