Polynomial interpolation

Given distinct nodes $x_0,\dots ,x_n$ and values $f(x_i)$, there is a unique polynomial of degree at most $n$ taking those values.

An interpolation node is one of the distinct argument values at which an interpolating function is required to match prescribed data.

The interpolating polynomial is

For distinct interpolation nodes $x_0,\dots ,x_n$, the polynomial ${\ell }_i(x)={\prod }_{j\ne i}(x-x_j)/(x_i-x_j)$ has ${\ell }_i(x_j)={\delta }_{ij}$. These cardinal polynomials form a basis of the polynomials of degree at most $n$, and give $p(x)={\sum }_{i}p(x_i){\ell }_i(x)$.

For distinct nodes,It is the leading coefficient of the interpolating polynomial through those nodes and obeys

The displayed integral is $(e^{t\beta }-e^{t\gamma })/(\beta -\gamma )$ for distinct exponents and $t{e}^{t\gamma }$ when they coincide. The latter is the continuous limit, so a zero denominator is a removable singularity. This quantity appears in variation-of-constants formula estimates of matrix evolution and avoids omitting an equal-exponent case.

For $h=fg$,

Newton's nested interpolation form isA triangular divided-difference table computes all its coefficients using exactly $n(n+1)/2$ divisions.