97nach4
TRANSCRIPT
-
7/28/2019 97nach4
1/83
1
Chapter 4
Interpolation and
Approximation
-
7/28/2019 97nach4
2/83
2
4.1 Lagrange Interpolation
The basic interpolation problem can be posedin one of two ways:
-
7/28/2019 97nach4
3/83
3
-
7/28/2019 97nach4
4/83
4
exist
unique
-
7/28/2019 97nach4
5/83
5
-
7/28/2019 97nach4
6/83
6
Example 4.1
e-1/2
-
7/28/2019 97nach4
7/837
-
7/28/2019 97nach4
8/838
Discussion
The construction presented in this section iscalled Lagrange interpolation.
How good is interpolation at approximating a
function? (Sections 4.3, 4.11) Consider another example:
If we use a fourth-degree interpolating polynomialto approximate this function, the results are asshown in Figure 4.3 (a).
-
7/28/2019 97nach4
9/83
9
Error for n=8n= 16
n= 8n= 4
-
7/28/2019 97nach4
10/83
10
Discussion
There are circumstances in which polynomial interpolation asapproximation will work very well, and other circumstances in whichit will not.
The Lagrange form of the interpolating polynomial is not well suitedfor actual computations, and there is an alternative constructionthat is far superior to it.
-
7/28/2019 97nach4
11/83
11
4.2 Newton Interpolation and
Divided Differences
The disadvantage of the Lagrange form
If we decide to add a point to the set of nodes,we have to completely re-compute all of the
functions.
Here we introduce an alternative form of the
polynomial: the Newton form It can allow us to easily write in terms of
-
7/28/2019 97nach4
12/83
12
Newton Interpolation
-
7/28/2019 97nach4
13/83
13
=0
-
7/28/2019 97nach4
14/83
14
Example 4.2
-
7/28/2019 97nach4
15/83
15
Discussion
The coefficients are called divided differences. We can use divided-difference table to find them.
-
7/28/2019 97nach4
16/83
16
Example 4.3
-
7/28/2019 97nach4
17/83
17
-
7/28/2019 97nach4
18/83
18
Example 4.3 (Con.)
-
7/28/2019 97nach4
19/83
19
Table 4.5
-
7/28/2019 97nach4
20/83
20
4.3 Interpolation Error
-
7/28/2019 97nach4
21/83
21
4.5 Application: More
Approximations to the Derivative
dependson x
-
7/28/2019 97nach4
22/83
22
4.5 Application: More
Approximations to the Derivative
The interpolating polynomial in Lagrange form is
The error is given as in (4.20), thus
We get
-
7/28/2019 97nach4
23/83
23
We can use above equations to get:
-
7/28/2019 97nach4
24/83
24
4.7 Piecewise Polynomial
Interpolation If we keep the order of the polynomial fixed and use
different polynomials over different intervals, with thelength of the intervals getting smaller and smaller, theninterpolation can be a very accurate and powerful
approximation tool.
For example:
-
7/28/2019 97nach4
25/83
25
-
7/28/2019 97nach4
26/83
26
-
7/28/2019 97nach4
27/83
27
Example 4.6
-
7/28/2019 97nach4
28/83
28
-
7/28/2019 97nach4
29/83
29
-
7/28/2019 97nach4
30/83
30
4.8 An Introduction to Splines
4.8.1 Definition of the problem
-
7/28/2019 97nach4
31/83
31
Discussion
From the definition:
d: degree of approximation
Related to the number of unknown coefficients (the degrees offreedom)
N: degree of smoothness
Related to the number of constraints
-
7/28/2019 97nach4
32/83
32
Discussion
We can make the first term vanish by setting
This establishes a relationship between the polynomialdegree of the spline and the smoothness degree.
For example: cubic splines
If we consider the common case of cubic splines, then
d=3 and N=2.
-
7/28/2019 97nach4
33/83
33
4.8.2 Cubic B-Splines
B-Spline: assume an uniform grid
-
7/28/2019 97nach4
34/83
34
-
7/28/2019 97nach4
35/83
35
Cubic B-Splines
How do we know that B(x) is a cubic spline
function?
Computer the one-sided derivatives at the knots:
and similarly for the second derivative.
If the one-sided values are equal to each other,
then the first and second derivatives are continuous,and hence B is a cubic spline.
Note that B is only locally defined, meaning that it
is nonzero on only a small interval.
-
7/28/2019 97nach4
36/83
36
-
7/28/2019 97nach4
37/83
37
A Spline Approximation
We can use B to construct a spline approximation to anarbitrary function f.
Define the sequence of functions
-
7/28/2019 97nach4
38/83
38
xi=0.4h=0.05
xi=0.75h=0.05
-
7/28/2019 97nach4
39/83
39
n+1 equations inn+3 unknowns
-
7/28/2019 97nach4
40/83
40
A Spline Approximation
Now, we need to come up with two additional constraintsin order to eliminate two of the unknowns.
Two common choices are The natural spline:
A simple construction
Leads to higher error near the end points
The complete spline: Better approximation properties
Do not actually require the derivative at the endpoints
-
7/28/2019 97nach4
41/83
41
Natural SplineFrom
n-1
-
7/28/2019 97nach4
42/83
42
Complete SplineFrom
n+1
-
7/28/2019 97nach4
43/83
43
Example 4.7
-
7/28/2019 97nach4
44/83
44
-
7/28/2019 97nach4
45/83
45
-
7/28/2019 97nach4
46/83
46
-
7/28/2019 97nach4
47/83
47
-
7/28/2019 97nach4
48/83
48
Example 4.8
-
7/28/2019 97nach4
49/83
49
-
7/28/2019 97nach4
50/83
50
-
7/28/2019 97nach4
51/83
51
-
7/28/2019 97nach4
52/83
-
7/28/2019 97nach4
53/83
4 9 A li ti S l ti f
-
7/28/2019 97nach4
54/83
54
4.9 Application: Solution of
Boundary Value Exercises Consider the two-point boundary value problem:
We construct the uniform grid of points:
We now look for our approximation in the form of a cubic splinedefine on this grid.
Consider the function:
-
7/28/2019 97nach4
55/83
55
The advantage of this approach is we can get a continuoussmooth function.
Because we know the values of and its derivatives ateach of the nodes, we can easily reduce this to the systemof equations: (n+1 equations in n+3 unknown)
where
-
7/28/2019 97nach4
56/83
56
We can eliminate the two extra unknowns by imposing
the boundary conditions on the approximation:
Substitute these into the first and last equations of therectangular system, we get
-
7/28/2019 97nach4
57/83
57
We are then left with the square system:
where
-
7/28/2019 97nach4
58/83
58
Example 4.9
h
-
7/28/2019 97nach4
59/83
59
where
The solution we get
-
7/28/2019 97nach4
60/83
60
4 10 Least Squares Concepts in
-
7/28/2019 97nach4
61/83
61
4.10 Least Squares Concepts in
Approximation
4.10.1 An introduction to data fitting
-
7/28/2019 97nach4
62/83
62
Least Square Data Fitting
-
7/28/2019 97nach4
63/83
63
-
7/28/2019 97nach4
64/83
64
-
7/28/2019 97nach4
65/83
65
Example 4.10
-
7/28/2019 97nach4
66/83
66
-
7/28/2019 97nach4
67/83
67
Example 4.11
-
7/28/2019 97nach4
68/83
68
-
7/28/2019 97nach4
69/83
69
-
7/28/2019 97nach4
70/83
-
7/28/2019 97nach4
71/83
71
4 10 2 Least Squares Approximation
-
7/28/2019 97nach4
72/83
72
4.10.2 Least Squares Approximation
and Orthogonal Polynomials
Let , we can seek such thatis minimized.
-
7/28/2019 97nach4
73/83
73
Inner Productions
Inner productions of functions:
Inner product on real vector spaces:
-
7/28/2019 97nach4
74/83
74
Inner Productions
-
7/28/2019 97nach4
75/83
75
The definition of inner product will allow us to apply anumber of ideas from linear algebra to the construction
of approximations.
-
7/28/2019 97nach4
76/83
76
The system can be organized along matrix-vector linesas
If our basis function satisfy the orthogonality condition
the special basis functions that satisfy this equation arecalled orthogonal polynomials.
Then the above matrix is a diagonal matrix, and we veryeasily have
-
7/28/2019 97nach4
77/83
77
-
7/28/2019 97nach4
78/83
78
Orthogonal Polynomials Legendre polynomials:
Example 4 12
-
7/28/2019 97nach4
79/83
79
Example 4.12
-
7/28/2019 97nach4
80/83
80
-
7/28/2019 97nach4
81/83
81
-
7/28/2019 97nach4
82/83
82
Example4.13
4.11 Advanced Topics in
-
7/28/2019 97nach4
83/83
4.11 Advanced Topics in
Interpolation Error
You can read it by yourselves.