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 NewtonCotes 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.
Licensing:
The computer code and data files made available on this web page
are distributed under
the MIT license
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.
divdif_test
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:

Philip Davis,
Interpolation and Approximation,
Dover, 1975,
ISBN: 0486624951,
LC: QA221.D33

Carl deBoor,
A Practical Guide to Splines,
Springer, 2001,
ISBN: 0387953663,
LC: QA1.A647.v27.

JeanPaul Berrut, Lloyd Trefethen,
Barycentric Lagrange Interpolation,
SIAM Review,
Volume 46, Number 3, September 2004, pages 501517.

FM Larkin,
Root Finding by Divided Differences,
Numerische Mathematik,
Volume 37, pages 93104, 1981.
Source Code:

cheby_t_zero.m,
returns zeroes of the Chebyshev polynomial T(N)(X).

cheby_u_zero.m,
returns zeroes of the Chebyshev polynomial U(N)(X).

data_to_dif.m,
computes a divided difference table.

data_to_dif_display.m,
computes a divided difference table and shows how.

data_to_r8poly.m,
computes the coefficients of a polynomial interpolating data.

data_to_table.m,
computes the full divided difference table for given data.

dif_antideriv.m,
computes the antiderivative of a divided difference polynomial.

dif_append.m,
adds a pair of data values to a divided difference table.

dif_basis.m,
computes all Lagrange basis polynomials in divided difference form.

dif_basis_deriv.m,
computes divided difference tables for the first derivatives
of all Lagrange basis polynomials.

dif_basis_derivk.m,
computes divided difference tables for the Kth derivatives
of all Lagrange basis polynomials.

dif_basis_i.m,
computes the Ith Lagrange basis polynomial in divided difference form.

dif_deriv_table.m,
computes the derivative of a polynomial in divided difference form.

dif_print.m,
prints the polynomial represented by a divided difference table.

dif_shift_x.m,
replaces one abscissa of a divided difference table with a new one.

dif_shift_zero.m,
shifts a divided difference table so that all abscissas are zero.

dif_to_r8poly.m,
converts a divided difference table to standard polynomial form.

dif_value.m,
evaluates a divided difference polynomial at a point.

lagrange_rule.m,
computes the weights of a Lagrange interpolation rule.

lagrange_sum.m,
carries out a Lagrange interpolation rule.

lagrange_value.m,
applies a naive form of Lagrange interpolation.

nc_rule.m,
computes the weights of a NewtonCotes quadrature rule.

ncc_rule.m,
computes the coefficients of a NewtonCotes closed quadrature rule.

nco_rule.m,
computes the coefficients of a NewtonCotes open quadrature rule.

r8mat_print.m,
prints an R8MAT;

r8mat_print_some.m,
prints some of an R8MAT;

r8mat_transpose_print.m,
prints the transpose of an R8MAT;

r8mat_transpose_print_some.m,
prints some of the transpose of an R8MAT;

r8poly_ant_coef.m,
integrates a polynomial in standard form.

r8poly_ant_val.m,
evaluates the antiderivative of a polynomial in standard form.

r8poly_basis.m,
computes all Lagrange basis polynomials in standard form.

r8poly_basis_1.m,
computes the Ith Lagrange basis polynomial in standard form.

r8poly_deriv_coef.m,
computes the coefficients of the derivative of a polynomial.

r8poly_deriv_value.m,
evaluates the derivative of a polynomial.

r8poly_order.m,
returns the order of a polynomial.

r8poly_print.m,
prints out a polynomial.

r8poly_shift.m,
adjusts the coefficients of a polynomial for a new argument.

r8poly_value_horner.m,
evaluates a polynomial in standard form.

r8vec_is_distinct.m,
is true if the entries in an R8VEC are distinct.

r8vec_print.m,
prints an R8VEC.

r8vec_transpose_print.m,
prints an R8VEC, transposed.

r8vec2_print.m,
prints an R8VEC2.

roots_to_dif.m,
sets up a divided difference table for a polynomial from its roots.

roots_to_r8poly.m,
converts polynomial roots to polynomial coefficients.
Last modified on 14 November 2019.