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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Remark: 1. When : 2. When = :

, = 0; , = 0.

,,

= 1 - 2 - , = - 2, = 0

2,

=1

2+1 = ().

3. , = , -2, , + 1 - 2 - , 2,

When : , = 0; When = : , = 0.

4. , = 2, + 2 - ,(),

When : , = 0; When = : , = 1.

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Example 3.4.1 Use Hermite polynomial that agrees with the data in the table to find an approximation of 1.5

( )

( )

0 1.3 0.6200860 -0.5220232

1 1.6 0.4554022 -0.5698959

2 1.9 0.2818186 -0.5811571

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3rd Degree Hermite Polynomial

? Given distinct 0, 1 and values of and at these numbers.

3

= 1 + 2 - 0 1 - 0

1 - 1 - 0

2

0

+ - 0

1 - 1 - 0

2

0

+ 1 + 2 1 - 1 - 0

0 - 0 - 1

2

1

+ - 1

0 - 0 - 1

2

1

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