mat 4725 numerical analysis section 8.2 orthogonal polynomials and least squares approximations...
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MAT 4725Numerical Analysis
Section 8.2
Orthogonal Polynomials and Least Squares
Approximations (Part II)
http://myhome.spu.edu/lauw
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Inner Product Spaces Gram-Schmidt Process
A Different Technique for Least Squares Approximation
Computationally Efficient Once Pn(x) is known, it is easy to
determine Pn+1(x)
Recall (Linear Algebra)
General Inner Product Spaces
Inner Product
Example 0
Let f,gC[a,b]. Show that
is an inner product on C[a,b]
,b
a
f g f x g x dx
Norm, Distance,…
Orthonormal Bases
A basis S for an inner product space V is orthonormal if
1. For u,v S, <u,v>=0.
2. For u S, u is a unit vector.
Gram-Schmidt Process
1 2
1 2
1 2
, , , Basis
, , , Orthogonal Basis
, , , Orthonormal Basis
n
n
n
v v v
w w w
u u u
Gram-Schmidt Process
Gram-Schmidt Process
The component in v2 that is “parallel” to w1 is removed to get w2.
So w1 is “perpendicular” to w2.
Simple Example
1 2v i
2v i j
Specific Inner Product Space
Definition 8.1
0 1
0
, , , is said to be linearly independent
on [ , ] if ,
whenever ( ) 0, [ , ],
we have 0 0,1,..., .
(Otherwse, linearly dependent.)
n
n
j jj
k
a b
c x x a b
c j n
Theorem 8.2
0 1
If ( ) is a polynomial of degree ,
then , , , is linearly independent
on any interval [ , ].
j
n
x j
a b
Idea
Definition
the set of polynomials of degree nn
Theorem 8.3
0 1 n
n
0
If ( ), ( ), , ( ) is linearly independent in
Then Q , unique such that
( ) ( )
n
j
n
j jj
x x x
c
Q x c x
Example 1
0
1
22
20 1 2
0 1 2
( ) 2
( ) 3
( ) 2 7
Express ( ) as a
linear combination of ( ), ( ), and ( ).
x
x x
x x x
Q x a a x a x
x x x
Definition (Skip it for the rest)
Weight function ( ) on an interval :
( ) integrable
( ) ( ) 0
( ) ( ) 0 on any subinterval of
w x I
a
b w x x
c w x I
Weight Functions
to assign varying degree of importance to certain portion of the interval
1
Modification of the Least Squares Approximation
Recall from part I
Least Squares Approximation of Functions
0
Given [ , ],
approximate ( ) by
( )n
n kk
k
f C a b
f x
P xx a
a b
( )f x
( )nP x
Least Squares Approximation of Functions
a b
( )f x
( )nP x
2
Find such that
( ) ( )
is minimized
k
b
n
a
a
E f x P x dx
Normal Equations
0
( ) , 0,1, ,
Solve for
b bnk j j
kk a a
k
a x dx f x x dx j n
a
Modification of the Least Squares Approximation
0
Given [ , ],
approximate ( ) by
( )n
kk
k
f C a b
f x
P x a x
0
Given [ , ],
approximate ( ) by
( )n
n kk
k
f C a b
f x
P xx a
Modification of the Least Squares Approximation
2
Find such that
( ) ( )
is minimized
k
b
n
a
a
E f x P x dx 2
Find such that
( ) ( )
is minimized
k
b
a
a
E f x P x dx
Modification of the Least Squares Approximation
0
0
For 0,1, , , solve for
( )
( ) ( ) ( ) ( )
k
b bnk j j
kk a a
b bn
k k j jk a a
j n a
a x dx f x x dx
a x x dx f x x dx
Where are the Improvements?
0
( ) ( ) ( ) ( )b bn
k k j jk a a
a x x dx f x x dx
Where are the Improvements?
Find such that
0( ) ( )
0
Then......
k
b
k jja
j kx x dx
j k
0
( ) ( ) ( ) ( )b bn
k k j jk a a
a x x dx f x x dx
Definition 8.5
0 1, , , is said to be an orthogonal
set of functions on [ , ] with respect to if
0 ( ) ( )
0
(Orthonormal if all =1)
n
b
k jja
j
a b w
j kx x dx
j k
Theorem 8.6
2
( ) ( )
( )
1( ) ( )
b
k
ak b
k
a
b
kk a
f x x dx
a
x dx
f x x dx
ak are easier to solve
ak are “reusable”
0
( ) ( )n
k kk
P x a x
Where to find Orthogonal Poly.?
the Gram-Schmidt Process
Gram-Schmidt Process
0
2
0
1 1 12
0
1 2
2
1 1 2
2 2
1 2
( ) 1
( )
( ) where
( )
For 2, ( ) ( ) ( )
( ) ( ) ( )
where and
( ) ( )
b
ab
a
k k k k k
b b
k k k
a ak kb b
k k
a a
x
x x dx
x x B B
x dx
k x x B x C x
x x dx x x x dx
B C
x dx x dx
( ) 1w x
Legendre Polynomials
0
12
0
11 1 1 1
2
0
1
22 2 1 2 0
1 12
1 1 0
1 12 21 1
2 2
1 0
1 1
( ) 1
( )
( ) where 0
( )
1( ) ( ) ( )
3
( ) ( ) ( )1
where 0 and 3
( ) ( )
P x
x P x dx
P x x B x B
P x dx
P x x B P x C P x x
x P x dx xP x P x dx
B C
P x dx P x dx
[-1,1]
Legendre Polynomials
0
1
22
33
( ) 1
( )
1( )
3
3( )
5
P x
P x x
P x x
P x x x
( ) 1, [-1,1]w x
Example 2
Find the least squares approx. of
f(x)=sin(x)
on [-1,1] by the Legendre Polynomials.
Example 2
1 1
0
1 10 1 1
2 2
0
1 1
( ) ( ) sin 1
0
( ) 1
f x P x dx x dx
a
P x dx dx
2
( ) ( )
( )
b
k
ak b
k
a
f x x dx
a
x dx
Example 2
0
1 1
1
1 11 1 1
2 2
1
1 1
0
( ) ( ) sin( )3
( )
a
f x P x dx x xdx
a
P x dx x dx
2
( ) ( )
( )
b
k
ak b
k
a
f x x dx
a
x dx
Example 2
0 1
12
12 21
2
1
30,
1sin( )
30
13
a a
x x dx
a
x dx
2
( ) ( )
( )
b
k
ak b
k
a
f x x dx
a
x dx
Example 2
0 1 2
13
2
13 2 31
3
1
30, , 0
3sin( )
35 155
235
a a a
x x x dx
a
x x dx
2
( ) ( )
( )
b
k
ak b
k
a
f x x dx
a
x dx
Example 2
2
0 1 2 3 3
0 0 1 1 2 2 3 3
35 1530, , 0,
2( ) ( ) ( ) ( ) ( )
a a a a
P x a P x a P x a P x a P x
Example 2
Homework
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