3.4 Hermite Interpolation 3.5 Cubic Spline Interpolation

3.4 Hermite Interpolation 3.5 Cubic Spline Interpolation

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Illustration. Consider to interpolate tanh() using Lagrange polynomial and nodes 0 = -1.5, 1 = 0, 2 = 1.5.

Now interpolate tanh() using nodes 0 = -1.5, 1 = 0, 2 = 1.5. Moreover, Let 1st derivative of interpolating polynomial agree with derivative of tanh() at these nodes. Remark:This is called Hermite interpolating polynomial.

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Hermite Polynomial

Definition. Suppose 1[, ]. Let 0, ... , be distinct numbers in [, ], the Hermite polynomial

() approximating is that:

1. = , for = 0, ... ,

2.

= ,

for = 0, ... ,

Remark: () and () agree not only function values but also 1st derivative values at , = 0, ... , .

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Osculating Polynomials

Definition 3.8 Let 0, ... , be distinct numbers in ,

and for = 0, ... , , let be a nonnegative integer.

Suppose

that

,

,

where

=

max

0

.

The

osculating polynomial approximating is the polynomial

()

of

least

degree

such

that

=

for each

= 0, ... , and k= 0, ... , .

Remark: the degree of () is at most = = 0 + .

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Theorem 3.9 If 1 , and 0, ... , , distinct numbers, the Hermite polynomial of degree at most 2 + 1 is:

2+1 = ,() + ,()

=0

=0

Where

, = [1 - 2( - ),()]2,() , = - 2,

Moreover, if 2+2 , , then

= 2+1

+

- 0 2 ... - 2 + 2 !

2

2+2 (())

for some , .

Remark:

1. 2+1 is a polynomial of degree at most 2 + 1.

2. ,() is jth Lagrange basis polynomial of degree .

2

2

3.

-0 ... - 2+2 !

2+2 (()) is the error term.

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