3.4 Hermite Interpolation 3.5 Cubic Spline Interpolation
3.4 Hermite Interpolation 3.5 Cubic Spline Interpolation
1
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.
2
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, ... , .
3
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 + .
4
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.
5
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.
6
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
7
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
8
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