Let a function f(t) be specified at n+1 points t _i , i=0, , n. A straightforward way to interpolate f(t) between the known values makes use of polynomials. The Lagrange basis polynomials L _i (t) are defined by

One can easily see that they satisfy Li(t _i )=1, i=0, , n, and L _i (t _j )=0 if i ? j. From this it is directly clear that the polynomial

indeed attains the desired values at the grid points t _i . The degree of p(t) is n, which is is in general the minimum value for polynomial interpolation at n+1 points. It can be proven that the Lagrange interpolation error can be expressed as

for some ? in the smallest interval containing t ₀, t _n and t.

If not only function values but also derivatives are prescribed, interpolation can be performed in a similar way, using the Hermite basis polynomials H _{1, i} and H _{2, i} . The first one is defined as

The second one is defined as

The basis polynomials H _{1, i} (t) and H _{2, i} (t) satisfy the properties H _{1, i} (t _j )= ? _i,j, , H _{2, i} (t _j )=0, and ( ? _ij=0 if i ? j and ? _ii=1). From this we can directly...