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Material Type: Notes; Class: INTRO NUM ANALYS & COMPUT; Subject: Mathematics; University: San Diego State University; Term: Fall 2009;
Typology: Study notes
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Approximation Theory Pad´e Approximation
Lecture Notes #
— Approximation Theory —
Rational Function Approximation
Peter Blomgren,
Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182-
http://terminus.sdsu.edu/
Fall 2009
Approximation Theory Pad´e Approximation
Outline
(^1) Approximation Theory
Pros and Cons of Polynomial Approximation
New Bag-of-Tricks: Rational Approximation
Pad´e Approximation: Example #
2 Pad´e Approximation
Example #
Finding the Optimal Pad´e Approximation
Approximation Theory Pad´e Approximation
Pros and Cons of Polynomial Approximation New Bag-of-Tricks: Rational Approximation Pad´e Approximation: Example #
Moving Beyond Polynomials: Rational Approximation
We are going to use rational functions, r (x), of the form
r (x) =
p(x)
q(x)
∑^ n
i=
pi x
i
∑^ m
j=
qi x
i
and say that the degree of such a function is N = n + m.
Since this is a richer class of functions than polynomials — rational
functions with q(x) ≡ 1 are polynomials, we expect that rational
approximation of degree N gives results that are at least as
good as polynomial approximation of degree N.
Approximation Theory Pad´e Approximation
Pros and Cons of Polynomial Approximation New Bag-of-Tricks: Rational Approximation Pad´e Approximation: Example #
Pad´e Approximation
Extension of Taylor expansion to rational functions; selecting the
pi ’s and qi ’s so that r
(k) (x 0 ) = f
(k) (x 0 ) ∀k = 0, 1 ,... , N.
f (x) − r (x) = f (x) −
p(x)
q(x)
f (x)q(x) − p(x)
q(x)
Now, use the Taylor expansion f (x) ∼
i=
ai (x − x 0 )
i , for
simplicity x 0 = 0:
f (x) − r (x) =
i= 0
aix
i
∑m
i= 0
qix
i −
∑^ n
i= 0
pix
i
q(x)
Next, we choose p 0 , p 1 ,... , pn and q 1 , q 2 ,... , qm so that the numerator
has no terms of degree ≤ N.
Approximation Theory Pad´e Approximation
Pros and Cons of Polynomial Approximation New Bag-of-Tricks: Rational Approximation Pad´e Approximation: Example #
Pad´e Approximation: Abstract Example 1 of 2
Find the Pad´e approximation of f (x) of degree 5, where
f (x) ∼ a 0 + a 1 x +... a 5 x
5 is the Taylor expansion of f (x) about
the point x 0 = 0.
The corresponding equations are:
x
0 a 0 − p 0 = 0
x^1 a 0 q 1 + a 1 − p 1 = 0
x
2 a 0 q 2 + a 1 q 1 + a 2 − p 2 = 0
x
3 a 0 q 3 + a 1 q 2 + a 2 q 1 + a 3 − p 3 = 0
x
4 a 0 q 4 + a 1 q 3 + a 2 q 2 + a 3 q 1 + a 4 − p 4 = 0
x
5 a 0 q 5 + a 1 q 4 + a 2 q 3 + a 3 q 2 + a 4 q 1 + a 5 − p 5 = 0
Note: p 0 = a 0 !!! (This reduces the number of unknowns and
equations by one (1).)
Approximation Theory Pad´e Approximation
Pros and Cons of Polynomial Approximation New Bag-of-Tricks: Rational Approximation Pad´e Approximation: Example #
Pad´e Approximation: Abstract Example 2 of 2
We get a linear system for p 1 , p 2 ,... , pN and q 1 , q 2 ,... , qN :
a 0
a 1 a 0
a 2 a 1 a 0
a 3 a 2 a 1 a 0
a 4 a 3 a 2 a 1 a 0
q 1
q 2
q 3
q 4
q 5
p 1
p 2
p 3
p 4
p 5
a 1
a 2
a 3
a 4
a 5
If we want n = 3, m = 2:
a 0 0 − 1
a 1 a 0 0 − 1
a 2 a 1 0 0 − 1
a 3 a 2 0 0 0
a 4 a 3 0 0 0
q 1
q 2
p 1
p 2
p 3
a 1
a 2
a 3
a 4
a 5
Approximation Theory Pad´e Approximation
Example # Finding the Optimal Pad´e Approximation
Pad´e Approximation: Concrete Example, e
−x 2 of 3
All the possible Pad´e approximations of degree 5 are:
r 5 , 0 (x) = 1 − x +
1 2
x^2 −
1 6
x^3 +
1 24
x^4 −
1 120
x^5
r 4 , 1 (x) =
1 − 45 x+ 103 x^2 − 151 x^3 + 1201 x^4 1+ 15 x
r 3 , 2 (x) =
1 − 35 x+ 203 x^2 − 601 x^3 1+ 25 x+ 201 x^2
r 2 , 3 (x) =
1 − 25 x+ 201 x^2 1+ 35 x+ 203 x^2 + 601 x^3
r 1 , 4 (x) =
1 − 15 x 1+ 45 x+ 103 x^2 + 151 x^3 + 1201 x^4
r 0 , 5 (x) =
1 1+x+ 12 x^2 + 16 x^3 + 241 x^4 + 1201 x^5
Note: r 5 , 0 (x) is the Taylor polynomial of degree 5.
Approximation Theory Pad´e Approximation
Example # Finding the Optimal Pad´e Approximation
Pad´e Approximation: Concrete Example, e
−x 3 of 3
The Absolute Error.
0 0.5 1 1.5 2
1e-
R{5,0}(x) R{4,1}(x) R{3,2}(x) R{2,3}(x) R{1,4}(x) R{0,5}(x)
Approximation Theory Pad´e Approximation
Example # Finding the Optimal Pad´e Approximation
Optimal Pad´e Approximation?
One Point Optimal Points
Polynomials Taylor Chebyshev
Rational Functions Pad´e ???
From the example e
−x we can see that Pad´e approximations suffer
from the same problem as Taylor polynomials – they are very
accurate near one point, but away from that point the
approximation degrades.
“Chebyshev-placement” of interpolating points for polynomials
gave us an optimal (uniform) error bound over the interval.
Can we do something similar for rational approximations???
Approximation Theory Pad´e Approximation
Example # Finding the Optimal Pad´e Approximation
Chebyshev Basis for the Pad´e Approximation!
We use the same idea — instead of expanding in terms of the
basis functions x
k , we will use the Chebyshev polynomials,
Tk (x), as our basis, i.e.
rn,m(x) =
∑n
k=0 pk^ Tk^ (x) ∑m
k=0 qk^ Tk^ (x)
where N = n + m, and q 0 = 1.
We also need to expand f (x) in a series of Chebyshev polynomials:
f (x) =
k=
ak Tk (x),
so that
f (x) − rn,m(x) =
∞ k=0 ak^ Tk^ (x)^
m k=0 qk^ Tk^ (x)^ −^
n
∑ k=0^ pk^ Tk^ (x) m k=0 qk^ Tk^ (x)^
Approximation Theory Pad´e Approximation
Example # Finding the Optimal Pad´e Approximation
Example: Revisiting e
−x with Chebyshev-Pad´e Approximation 1/
The 8
th order Chebyshev-expansion (All Praise Maple) for e
−x is
PCT 8 (x) = 1. 266065878 T 0 (x) − 1. 130318208 T 1 (x) + 0. 2714953396 T 2 (x) − 0. 04433684985 T 3 (x) + 0. 005474240442 T 4 (x) − 0. 0005429263119 T 5 (x) + 0. 00004497732296 T 6 (x) − 0. 000003198436462 T 7 (x) + 0. 0000001992124807 T 8 (x)
and using the same strategy — building a matrix and
right-hand-side utilizing the coefficients in this expansion, we can
solve for the Chebyshev-Pad´e polynomials of degree (n + 2m) ≤ 8:
Next slide shows the matrix set-up for the r
CP 3 , 2 (x) approximation.
Note: Due to the “folding”, Ti (x)Tj (x) =
1 2
Ti+j (x) + T|i−j|(x)
we need n + 2m Chebyshev-expansion coefficients. (Burden-
Faires do not mention this, but it is “obvious” from algo-
rithm 8.2; Example 2 (p. 519) is broken, – it needs ˜P 7 (x).)
Approximation Theory Pad´e Approximation
Example # Finding the Optimal Pad´e Approximation
Example: Revisiting e
−x with Chebyshev-Pad´e Approximation 2/
T 0 (x) :
1 2
[
a 1 q 1 + a 2 q 2 − 2 p 0 = 2 a 0
]
T 1 (x) :
1 2
[
(2a 0 + a 2 )q 1 + (a 1 + a 3 )q 2 − 2 p 1 = 2 a 1
]
T 2 (x) :
1 2
[
(a 1 + a 3 )q 1 + (2a 0 + a 4 )q 2 − 2 p 2 = 2 a 2
]
T 3 (x) :
1 2
[
(a 2 + a 4 )q 1 + (a 1 + a 5 )q 2 − 2 p 3 = 2 a 3
]
T 4 (x) :
1 2
[
(a 3 + a 5 )q 1 + (a 2 + a 6 )q 2 − 0 = 2 a 4
]
T 5 (x) :
1 2
[
(a 4 + a 6 )q 1 + (a 3 + a 7 )q 2 − 0 = 2 a 5
]
Approximation Theory Pad´e Approximation
Example # Finding the Optimal Pad´e Approximation
Example: Revisiting e
−x with Chebyshev-Pad´e Approximation 4/
−2 −1 −0.5 0 0.5 1
−1.
−
−0.
0
0.
1
1.
2 x 10
−5 Error for Chebyshev−Pade−4−1 Approximation
−2 −1 −0.5 0 0.5 1
−1.
−
−0.
0
0.
1
1.
2 x 10
−5 Error for Chebyshev−Pade−3−2 Approximation
−2 −1 −0.5 0 0.5 1
−1.
−
−0.
0
0.
1
1.
2 x 10
−5 (^) Error for Chebyshev−Pade−2−3 Approximation
−2 −1 −0.5 0 0.5 1
−1.
−
−0.
0
0.
1
1.
2 x 10
−5 (^) Error for Chebyshev−Pade−1−4 Approximation
Approximation Theory Pad´e Approximation
Example # Finding the Optimal Pad´e Approximation
Example: Revisiting e
−x with Chebyshev-Pad´e Approximation 5/
−5 −1 −0.5 0 0.5 1
−4.
−
−3.
−
−2.
−
−1.
−
−0.
0
x 10−4^ Error Comparison for 4−1 Approximations
Chebyshev−Pade Pade −1 −0.5 0 0.5 1
0
0.
1
1.
2
2.
3
3.5 x 10
−4 (^) Error Comparison for 3−2 Approximations Chebyshev−Pade Pade
−5 −1 −0.5 0 0.5 1
−4.
−
−3.
−
−2.
−
−1.
−
−0.
0
x 10−4^ Error Comparison for 2−3 Approximations
Chebyshev−Pade Pade −1 −0.5 0 0.5 1
0
5
10
15 x 10
−4 (^) Error Comparison for 1−4 Approximations Chebyshev−Pade Pade
