3.3 Divided Differences

3.3 Divided Differences

1

Representing th Lagrange Polynomial

? If is the th degree Lagrange interpolating polynomial that agrees with at the points {0, 1, ... , }, express in the form: = 0 + 1 - 0 + 2 - 0 - 1 + + - 0 - 1 - 2 ... - -1

? ? How to find constants 0,..., ?

2

Divided Differences

? Zeroth divided difference:

= ()

? First divided difference:

, +1

=

+1 +1

- -

? Second divided difference:

, +1, +2

=

+1, +2 +2

- -

, +1

? th divided difference:

, +1, ... , +

=

+1, +2, ... , + - , +1, ... , +-1

+ -

3

? Finding constants 0,..., .

1. = 0: 0 = 0 = 0 = 0

2. = 1: 0 + 1 1 - 0 = 1 =

1

1

=

1 1

- 0 - 0

= 0, 1

3. In general: = 0, 1, ... , for = 0, ... ,

4

Newton's Interpolatory Divided Difference Formula

= 0 + 0, 1 - 0 + 0, 1, 2 - 0 - 1 + + 0, ... , - 0 - 1 ... ( - -1)

Or = 0

+ [[0, ... , ] - 0 ... ( - -1)]

=1

5

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