1,5

An effective way to approximate the derivative of equally spaced numerical data is to differentiate its Lagrange interpolating polynomial. If y[x] is equally spaced data from x = -n to +n, its Lagrange interpolating polynomial P(x) has degree 2*n+1. Then P'(0) may be expressed as a weighted sum over the y[x]. This is the triangle of coefficients C[n,m] such that P'(0) = (1/d[n]) * Sum_{m=-n}^n C[n,m] y[m]. The denominator d[n] is given by sequence A099996. This is very useful in numerical analysis. For example, when n=1, this gives the centered difference approximation to the derivative.

facs[x_, j_, n_] := Product[If[k == j, 1, x - k], {k, -n, n}]

coefs[x_, n_] := Table[facs[x, j, n]/facs[j, j, n], {j, -n, n}]

d[n_] := Apply[LCM, Table[j, {j, 1, 2*n}]]

dCoefs[x_, n_] := d[n] D[coefs[y, n], y] /. {y -> x}

MatrixForm[Table[dCoefs[0, n], {n, 1, 5}]]

When divided by sequence A099996, this triangle gives the coefficients needed to estimate derivatives from equally spaced numerical data using Lagrange interpolation. The first and last entry of each row of the triangle has absolute value LCM{1, 2, ..., 2*n}/n*binomial(2n, n), as seen in A068553.

Sequence in context: A048729 A003131 A010679 this_sequence A021557 A118540 A165110

Adjacent sequences: A156741 A156742 A156743 this_sequence A156745 A156746 A156747

sign,tabl

Bruce Boghosian (bruce.boghosian(AT)tufts.edu), Feb 14 2009