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Title: Polynomial interpolation
Description: Polynomial interpolation is a method of estimating values between known data points. When graphical data contains a gap, but data is available on either side of the gap or at a few specific points within the gap, an estimate of values within the gap can be made by interpolation.
Description: Polynomial interpolation is a method of estimating values between known data points. When graphical data contains a gap, but data is available on either side of the gap or at a few specific points within the gap, an estimate of values within the gap can be made by interpolation.
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This chapter is about
working with data
...
If we want to know the
population of the US in
year 1965 or year
2010, we have to fit a
function through the
given data
...
Definition: The process of
fitting a function through
given data is called
interpolation
...
So we fit a
certain class of functions
...
We will see why
polynomials are fitted through data when we
don’t know f(x)
...
Polynomials are often used because they have
the property of approximating any continuous
function
...
This fact is guaranteed by
the following Theorem:
Theorem: (Weierstrass
Approximation Theorem):
Suppose that f(x) is defined
and continuous on [a,b]
...
f(x) in red approximated
by P(x) in blue within ε
...
, fn
where fi = f(xi) for i=0,…,n
...
We have n+1
conditions given
...
a1x a0
n
When n=2, the problem becomes: Given:
x0 x1
f0 f1
Find a polynomial of degree one such that
P(x0)=f0
P(x1)=f1
f1 f 0
x x0
y f0
( x x0 ) f 0 ( f1 f 0 )
x1 x0
x1 x0
x x0
x x1
f0
f1
x0 x1
x1 x0
Denote by L0(x) and L1(x) the two first degree
polynomials:
L0 ( x)
x x1
x0 x1
L1 ( x )
x x0
x1 x0
These polynomials have the following
properties:
L0(x0)=1
L0(x1)=0
L1(x0)=0
L1(x1)=1
So we can rewrite the polynomial that fits the
data in the form:
P(x) = f0L0(x) + f1L1(x)
P(x0)=f0L0(x0)+f1L1(x1)= f0
P(x1)=f0L0(x1)+f1L1(x1) = f1
L0(x0)=1, L0(x1)=0
L1(x0)=0, L1(x1)=1
Note: P(x) is the unique linear polynomial
passing through the points (x0,f0) and (x1,f1)
Now we generalize the approach to n+1
points
...
,fn
First we construct the special polynomials
Lk(x) so that they have the properties:
◦ Lk(xi)=0 if i≠k
◦ Lk(xk)=1
These are polynomials that are zero at all
points except one
...
Thus,
( x x0 )
...
( x xn )
Lk ( x)
( xk x0 )
...
( xk xn )
Lk(x) is called basic Lagrange polynomial of
degree n
...
Definition: P(x) is called the nth Lagrange
interpolating polynomial
...
5
2
1
3
1
...
The basic Lagrange
polynomials are:
( x 0
...
5)
(0 0
...
5)
x ( x 1)( x 1
...
5(0
...
5 1
...
5)( x 1
...
5)(1 1
...
5)( x 1)
=L0(x)+2L1(x)+3L2(x)+4L3(x)
L3 ( x )
1
...
5 0
...
5 1)
L0 ( x )
If we replace the function f(x) with the
polynomial P(x), we would like to know what
error we are making
...
In the
interpolation points the error is zero but it is
non-zero in other points
...
This
maximum is called error bound
...
3: Suppose x0,x1,…,xn are n+1
distinct points in the interval [a,b] and f(x)
has n+1 continuous derivatives
...
( x xn )
(n 1)!
( n 1)
where P(x) is the nth Lagrange interpolating
polynomial
...
( x xn )
(n 1)!
Note:
◦ If the interval [a,b] is not given, we take
[a,b]=[x0,xn]
...
Example: For the function
f(x)=cos(x)
let x0=0, x1=0
...
9
...
45)
...
45
...
3 to find the error bound for the
error
...
6)( x 0
...
6)(0 0
...
9)
L1 ( x )
0
...
6 0
...
6)
L2 ( x )
0
...
9 0
...
6)L1(x) +cos(0
...
P(0
...
898100747
cos(0
...
9004471024
Actual error at x=0
...
45)-P(0
...
0023
Error bound:
f ' ' ' ( ( x))
E2 ( x, f )
x( x 0
...
9)
3!
We want to compute
maxx|E2(x,f)|
Plot of the error
...
4
However, we will not be able to compute this
best bound because we don’t know the
location of ξ
...
6)( x 0
...
9]
...
9) |
a x b
Computing
max | x( x 0
...
9) |
a x b
g(x)=x(x-0
...
9)
g ' ( x) 3( x x 0
...
2354248689
p2=0
...
05704
|g(p2)|=0
...
6)( x 0
...
9 |
0
...
0074468
6
Construct the Lagrange interpolating
polynomial of degree 2 for
f(x)=sin(lnx)
on the interval [2,2
...
0 x1=2
...
6
...
Thus, the following table is given:
x
f(x)
2
sin(ln2)
2
...
4)
2
...
6)
We construct the basic
Lagrange polynomials:
( x 2
...
6)
( 2 2
...
6)
( x 2)( x 2
...
4 2)( 2
...
6)
( x 2)( x 2
...
6 2)( 2
...
4)
L0 ( x )
P(x)=sin(ln2)L0(x)+sin(ln2
...
6)L2(x)
The error is given by
f ' ' ' ( ( x))
E2 ( x, f )
( x 2)( x 2
...
6)
3!
f(x)=sin(lnx))
3 sin(ln x) cos(ln x)
f ' ' ' ( x)
3
x
Plot of the error f(x)-P(x)
Bounding the third
derivative:
|f’’’(x)| is plotted on the
right
...
The maximum is at 2:
max | f ' ' ' ( x) || f ' ' ' (2) |
2 x 2
...
335765
8
f’’’(x) plotted over the
interval [2,2
...
Next we bound |g(x)|:
g(x)=(x-2)(x-2
...
6)
To find the maximum of
|g(x)|, we need to take
the derivative:
g ' ( x) 3x 2 14 x 16
...
157
p2=2
...
157)|=0
...
5)|=0
...
Obtaining the error bound:
| f ' ' ' ( ( x)) |
| E2 ( x, f ) |
| ( x 2)( x 2
...
6) |
3!
0
...
0169 9
Title: Polynomial interpolation
Description: Polynomial interpolation is a method of estimating values between known data points. When graphical data contains a gap, but data is available on either side of the gap or at a few specific points within the gap, an estimate of values within the gap can be made by interpolation.
Description: Polynomial interpolation is a method of estimating values between known data points. When graphical data contains a gap, but data is available on either side of the gap or at a few specific points within the gap, an estimate of values within the gap can be made by interpolation.