HERMITE
Hermite polynomial interpolating function and derivative values

HERMITE is a MATLAB library which constructs the Hermite polynomial which interpolates function and derivative values at given points.

In other words, the user supplies n sets of data, (x(i),y(i),yp(i)), and the algorithm determines a polynomial p(x) such that, for 1 <= i <= n

p(x(i)) = y(i)
p'(x(i)) = yp(i)

Note that p(x) is a "global" polynomial, not a piecewise polynomial. Given n data points, p(x) will be a polynomial of degree 2n-1. As the value n increases, the increasing degree of the interpolating polynomial makes it liable to oscillations between the data, and eventually to severe inaccuracy even at the data points.

Generally, the interpolation problem for a large number of data points should be handled differently, for instance by piecewise polynomials.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

HERMITE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CHEBYSHEV, a MATLAB library which computes the Chebyshev interpolant/approximant to a given function over an interval.

DIVDIF, a MATLAB library which computes interpolants by divided differences.

HERMITE_CUBIC, a MATLAB library which can compute the value, derivatives or integral of a Hermite cubic polynomial, or manipulate an interpolating function made up of piecewise Hermite cubic polynomials.

INTERP, a MATLAB library which can be used for parameterizing and interpolating data;

LAGRANGE_INTERP_1D, a MATLAB library 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 library which defines and evaluates radial basis function (RBF) interpolants to 1D data.

SHEPARD_INTERP_1D, a MATLAB library which defines and evaluates Shepard interpolants to 1D data, which are based on inverse distance weighting.

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

TEST_INTERP_1D, a MATLAB library which defines test problems for interpolation of data y(x), depending on a 1D argument.

VANDERMONDE_INTERP_1D, a MATLAB library 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.

Source Code:

• dif_deriv.m, computes the derivative of a polynomial in divided difference form.
• 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 a standard polynomial.
• dif_vals.m, evaluates a divided difference polynomial at a set of points.
• hermite_basis_0.m, evaluates a zero-order Hermite interpolation basis function.
• hermite_basis_1.m, evaluates a first-order Hermite interpolation basis function.
• hermite_demo.m, computes and prints Hermite interpolant information for data.
• hermite_interpolant.m, sets up a divided difference table from Hermite data.
• hermite_interpolant_rule.m, quadrature rule for a Hermite interpolant.
• hermite_interpolant_value.m, evaluates the Hermite interpolant polynomial.
• r8poly_ant_val.m, evaluates the antiderivative of a polynomial in standard form.
• r8poly_degree.m, returns the degree of a polynomial.
• r8poly_print.m, prints out a polynomial.
• r8vec_chebyshev.m, creates a vector of Chebyshev spaced values.
• r8vec_print.m, prints an R8VEC.
• r8vec_uniform_01.m, returns a unit pseudorandom R8VEC.
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• hermite_test.m, a sample calling program.
• hermite_test01.m, three data points, a cubic function.
• hermite_test02.m, three data points, f(x) = a 5-th degree polynomial.
• hermite_test03.m, three random abscissas, f(x)=random 5-th degree polynomial.
• hermite_test04.m, shows the behavior of the interpolant for Runge's example on equally-spaced data.
• hermite_test05.m, shows the behavior of the interpolant for Runge's example on Chebyshev-spaced data.
• hermite_test06.m, tests HERMITE_BASIS_0 and HERMITE_BASIS_1.
• hermite_test07.m, tests HERMITE_INTERPOLANT_RULE.
• hermite_test08.m, tabulates the interpolant and its derivative.
• hermite_test_output.txt, the output file.

You can go up one level to the MATLAB source codes.