# hermite_interpolant Hermite polynomial interpolating function and derivative values

hermite_interpolant, a C 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.

### Languages:

hermite_interpolant 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 C library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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

HERMITE_CUBIC, a C 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.

LAGRANGE_INTERP_1D, a C 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, a C library which defines and evaluates radial basis function (RBF) interpolants to multidimensional data.

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

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

### 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:

Last revised on 08 July 2019.