# divdif

divdif, a MATLAB code which creates, prints and manipulates divided difference polynomials.

Divided difference polynomials are a systematic method of computing polynomial approximations to scattered data. The representations are compact, and may easily be updated with new data, rebased at zero, or analyzed to produce the standard form polynomial, integral or derivative polynomials.

Other routines are available to convert the divided difference representation to standard polynomial format. This is a natural way to determine the coefficients of the polynomial that interpolates a given set of data, for instance.

One surprisingly simple but useful routine is available to take a set of roots and compute the divided difference or standard form polynomial that passes through those roots.

Finally, the Newton-Cotes quadrature formulas can be derived using divided difference methods, so a few routines are given which can compute the weights and abscissas of open or closed rules for an arbitrary number of nodes.

### Languages:

divdif is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

bernstein_polynomial, a MATLAB code which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

chebyshev_polynomial, a MATLAB code which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

differ, a MATLAB code which determines the finite difference coefficients necessary in order to combine function values at known locations to compute an approximation of given accuracy to a derivative of a given order.

hermite_polynomial, a MATLAB code which computes the Hermite interpolant, a polynomial that matches function values and derivatives.

interp, a MATLAB code which can be used for parameterizing and interpolating data;

lagrange_interp_1d, a MATLAB code which defines and evaluates the Lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).

rbf_interp_1d, a MATLAB code which defines and evaluates radial basis function (RBF) interpolants to 1D data.

shepard_interp_1d, a MATLAB code which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.

spline, a MATLAB code which includes many routines to construct and evaluate spline interpolants and approximants.

test_approx, a MATLAB code which defines a number of test problems for approximation and interpolation.

test_interp_1d, a MATLAB code which defines test problems for interpolation of data y(x), depending on a 1D argument.

vandermonde_interp_1d, a MATLAB code which finds a polynomial interpolant to a function of 1D data by setting up and solving a linear system for the polynomial coefficients, involving the Vandermonde matrix.

### Reference:

1. Philip Davis,
Interpolation and Approximation,
Dover, 1975,
ISBN: 0-486-62495-1,
LC: QA221.D33
2. Carl deBoor,
A Practical Guide to Splines,
Springer, 2001,
ISBN: 0387953663,
LC: QA1.A647.v27.
3. Jean-Paul Berrut, Lloyd Trefethen,
Barycentric Lagrange Interpolation,
SIAM Review,
Volume 46, Number 3, September 2004, pages 501-517.
4. FM Larkin,
Root Finding by Divided Differences,
Numerische Mathematik,
Volume 37, pages 93-104, 1981.