3.3 Divided Differences
ο»Ώ3.3 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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