MINIMIZING THE INFINITY NORM
Runge’s Phenomenon and
Chebyshev Polynomials in
Approximation
CONTENTS
1.
Introduction
2.
Weierstrass approximation theorem
3.
Minimax polynomial
4.
Oscillation theorem
5.
Examples
6.
Existence and uniqueness
7.
Runge’s phenomenon
8.
Chebyshev polynomials
9.
Remez algorithm
INTRODUCTION
Want to approximate a function
Possible solution: interpolation
Is it the best approximation?
Can we find the “best” approximation?
WEIERSTRASS APPROXIMATION THEOREM
We can!
Proof by construction: Bernstein polynomials
Linear, monotonic, positive
Essentially convergence
MINIMAX POLYNOMIAL
OSCILLATION THEOREM
EXAMPLE
EXAMPLE
EXISTENCE OF MINIMAX POLYNOMIAL
UNIQUENESS OF MINIMAX POLYNOMIAL
ERROR
-
Looks like a Taylor Series
Same error we know
RUNGE’S PHENOMENON
Red - f(x)
Blue - P5
Green - P9
-
A higher-order polynomial does not
necessarily interpolate better.
-
Weierstrauss asserts the existence of
a polynomial with uniform convergence;
not every polynomial has this property
Uniform Nodes
n=4
Uniform Nodes
n=8
Uniform Nodes
n=16
Uniform Nodes
n=32
Chebyshev Nodes
n=4
Chebyshev Nodes
n=8
Chebyshev Nodes
n=16
Chebyshev Nodes
n=32
THE CHEBYSHEV POLYNOMIAL
CHEBYSHEV
RECURRENCE RELATION
CHEBYSHEV NODES
•Chebyshev nodes are the roots of the Chebyshev polynomials of the 1st kind
•We can easily find the roots:
𝑇𝑛 𝑥 = 0 → cos 𝜃 = 0 → 𝑛𝜃 = 2𝑗 − 1
𝜋
2
, where 𝑗 is an integer
𝑇𝑛 𝑥 = 0 → cos 𝜃𝑗 , where 𝜃𝑗 =
𝑥𝑗 =
(2𝑗−1)𝜋
2𝑛
,1≤𝑗≤𝑛
2𝑗−1 𝜋
2𝑛
EXTREME POINTS
•𝑇𝑛 𝑥 assumes its absolute extrema at:
𝑥′𝑗 = cos
𝑗𝜋
𝑛
, for 𝑗 = 1, 2, . . . , 𝑛
•Extreme points are bounded by 1
•A maximum occurs at each even value of 𝑗 and a minimum for each odd
•See oscillation theorem
MONIC POLYNOMIALS
•Polynomials with a leading coefficient of 1
are monic
•For any 𝑇𝑛 𝑥 we can make it monic by
1
multiplying by 𝑛−1
2
•Denote 𝑇𝑛 𝑥 the set of monic Chebyshev
polynomials
UNIQUE PROPERTY OF CHEBYSHEV POLYNOMIALS
•Denote Π𝑛 as the set of all monic polynomials of degree 𝑛
1
2𝑛−1
= max 𝑇𝑛 𝑥
𝑥∈[−1,1]
•Equality can only occur if 𝑃𝑛 ≡ 𝑇𝑛
≤ max 𝑃𝑛 𝑥
𝑥∈[−1,1]
, for all 𝑃𝑛 (𝑥) ∈ Π𝑛
MINIMAX POLYNOMIAL
𝑓 𝑥 −𝑃 𝑥 =
𝑓
𝑛+1
𝜉 𝑥
𝑛+1 !
𝑥 − 𝑥0 𝑥 − 𝑥1 … (𝑥 − 𝑥𝑛 )
•We are want to minimize
𝑥 − 𝑥0 𝑥 − 𝑥1 … (𝑥 − 𝑥𝑛 )
by choosing our nodes 𝑥1 , … , 𝑥𝑛+1 as the roots of the 𝑇𝑛+1 polynomial
REMEZ ALGORITHM
Iterative algorithm
Approximations in Chebyshev space
Best in the uniform norm L∞ sense
POLYNOMIAL OF BEST APPROXIMATION
Minimize absolute minimum error
Difference between the function and the polynomial.
PROCEDURE
The Remez algorithm starts with the function f to be approximated and a set X of
n+2 sample points x1, x2,…….xn+2 in the approximation interval, usually the
Chebyshev nodes linearly mapped to the interval, the steps are:
STEPS
CASES
SIX POSSIBILITIES
ITERATIONS
EXAMPLE
CONCLUSION
Oscillation Minimax
Chebyshev polynomials and roots as “near-minimizer”
Interpolation and approximation