3.4 Hermite Interpolation 3.5 Cubic Spline Interpolation

[Pages:14]3.4 Hermite Interpolation 3.5 Cubic Spline Interpolation

1

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, ... , .

2

Theorem. If 1 , and 0, ... , , distinct

numbers, the Hermite polynomial 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 .

3.

-0 2... - 2+2 !

2

2+2

(())

is

the

error

term.

3

3rd Degree Hermite Polynomial

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

numbers.

3 =

1

+

2

1

- 0 - 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

4

Hermite Polynomial by Divided Differences

Suppose 0, ... , and , are given at these numbers.

Define 0, ... , 2+1 by

2 = 2+1 = ,

for = 0, ... ,

Construct divided difference table, but use 0 , 1 , . . ,

to set the following undefined divided difference:

0, 1 , 2, 3 , ... , 2, 2+1 .

The Hermite polynomial is

2+1

2+1 = 0 +

0, ... , - 0 ... ( - -1)

=1

5

Divided Difference Notation for Hermite Interpolation

? Divided difference notation: 3 = 0 + 0 - 0 + 0, 0, 1 - 0 2 + [0, 0, 1, 1] - 0 2( - 1)

6

Problems with High Order Polynomial Interpolation

? 21 equal-spaced numbers to interpolate

=

1 1+162

.

The

interpolating

polynomial

oscillates between interpolation points.

7

Cubic Splines

? Idea: Use piecewise polynomial interpolation, i.e, divide the interval into smaller sub-intervals, and construct different low degree polynomial approximations (with small oscillations) on the sub-intervals.

? Challenge: If () are not known, can we still generate interpolating polynomial with continuous derivatives?

8

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download