**for**

**Background for the Hermite
Polynomial**

The cubic Hermite
polynomial p(x) has the
interpolative properties and both
the function values and their derivatives are known at the
endpoints of the
interval . Hermite
polynomials were studied by the French Mathematician Charles
Hermite (1822-1901), and are referred to as a "clamped
cubic," where "clamped" refers to the slope at the endpoints being
fixed. This situation is illustrated in the figure
below.

**Theorem (Cubic Hermite
Polynomial).** If is
continuous on the interval ,
there exists a unique cubic polynomial such
that

,

,

,

.

**Proof ****Hermite
Polynomial Interpolation** **Hermite
Polynomial Interpolation**

**Remark.** The cubic
Hermite polynomial is a generalization of both the Taylor polynomial
and Lagrange polynomial, and it is referred to as an "osculating
polynomial." Hermite polynomials can be generalized to
higher degrees by requiring that the use of more
nodes
and the extension to agreement at higher
derivatives for and . The
details are found in advanced texts on numerical
analysis

**Computer
Programs** **Hermite
Polynomial Interpolation** **Hermite
Polynomial Interpolation**

**Example 1.** Find the
cubic Hermite polynomial or "clamped cubic" that
satisfies

**More Background. The Clamped Cubic
Spline**

A clamped
cubic spline is obtained by forming a piecewise cubic
function which passes through the given set of knots with
the condition the function values, their derivatives and second
derivatives of adjacent cubics agree at the interior
nodes. The endpoint conditions are ,
where are
given.

**Example 2.** Find the
"clamped cubic spline" that satisfies

**Solution
2.**

**More Background. The Natural Cubic
Spline**

A natural
cubic spline is obtained by forming a piecewise cubic
function which passes through the given set of knots
with the condition the function values, their derivatives and second
derivatives of adjacent cubics agree at the interior
nodes. The endpoint conditions are .
The natural cubic spline is said to be "a relaxed curve."

**Example 3.** Find the
"natural cubic spline" that satisfies

**Solution
3.**

**Old Lab Project (****Hermite
polynomial interpolation** **Hermite
polynomial
interpolation****).** Internet
hyperlinks to an old lab project.

**Research Experience for
Undergraduates**

**Hermite
Polynomial Interpolation** **Hermite
Polynomial Interpolation** Internet
hyperlinks to web sites and a bibliography of
articles.

**Download this
Mathematica Notebook****
****Hermite
Polynomial Interpolation**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004