8. Saddle-Point Asymptotics http://ac.cs.princeton.edu Analytic - PowerPoint PPT Presentation

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 8. Saddle-Point Asymptotics http://ac.cs.princeton.edu

⬅ Analytic combinatorics overview specification A. SYMBOLIC METHOD 1. OGFs 2. EGFs GF equation 3. MGFs SYMBOLIC METHOD B. COMPLEX ASYMPTOTICS asymptotic 4. Rational & Meromorphic estimate 5. Applications of R&M COMPLEX ASYMPTOTICS 6. Singularity Analysis desired 7. Applications of SA result ! 8. Saddle point 2

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 8. Saddle-Point Asymptotics Analytic •Modulus surfaces Combinatorics •Saddle point bounds •Saddle point asymptotics Philippe Flajolet and Robert Sedgewick OF •Applications CAMBRIDGE http://ac.cs.princeton.edu II.8a.Saddle.Surfaces

Warmup: 2D absolute value plots Consider 2D plots of functions: all points ( x , | f ( x )| ) in a Cartesian plot. � � | � � | � / � | � / � | sin � � ( � − � ) � | � ( � − � ) � | � − � � | � − � � | | sin � | 4

Welcome to absolute-value-land! Consider 3D versions of our plots of analytic functions. A modulus surface is a plot of ( x , y , | f ( z )| ) where z = x + yi . 3D version 2D version � + � � � pole Example: � − � � � ordinary point saddle point zero Q. Can a modulus surface assume any shape ? A. No. A. (A surprise.) Only four types of points. 5

Modulus surface points type I: zeros θ , | f ( z )| = 2 r Ex. f ( z ) = 2 z = 2 re i A zero is a point where f ( z ) = 0 and f ' ( z ) ≠ 0. Key point: All zeros have the same local behavior. � ( � ) = � ( � � ) + � � ( � � )( � − � � ) + � �� ( � � ) ( � − � � ) � + . . . ∼ � � ( � � )( � − � � ) � ! same for all θ 2 r � − � � � − � � � − � � r ∼ � ( � + � ) ∼ − � ( � − � ) A zero of order p is a point where f ( k ) ( z ) = 0 for 0 ≤ k < p and f ( p ) ( z ) ≠ 0. � � � � + � � zero of order 3 zero (order 1) zero of order 2 6

Modulus surface points type II: poles � � � ( � ) ∼ � − � same for all θ A pole is a point z 0 where � − � � By definition, all poles have the same local behavior. � A pole of order p is a point z 0 where � ( � ) ∼ ( � − � � ) � 7

Quick in-class exercise Q. What function is this? pole � A. � − � zero 8

Modulus surface points type III: ordinary points An ordinary point is a point where f ( z ) ≠ 0 and f ' ( z ) ≠ 0. All ordinary points have the same local behavior. � ( � ) = � ( � � ) + � � ( � � )( � − � � ) + � �� ( � � ) ( � − � � ) � + . . . ∼ � � ! 9

Modulus surface points type III: saddle points A saddle point is a point where f ( z ) ≠ 0 and f ' ( z ) = 0. � ( � − � )( � − � ) All saddle points have the same local behavior. � ( � ) = � ( � � ) + � � ( � � )( � − � � ) + � �� ( � � ) ( � − � � ) � + . . . ∼ � ( � − � � ) � � ! Basic characteristic • Downwards-oriented parabola at one angle • Upwards-oriented parabola at perpendicular angle 10

Modulus surface points: summary poles at 0 + i /2 and 0 − i /2 f(z) f ' ( z ) local behavior � − � � � simple zero 0 not 0 ~ c ( z − z 0 ) � + � � � saddle point at 0 + 0 i zero of order p > 1 0 0 ~ c ( z − z 0 ) p saddle point not 0 0 ~ c ( z − z 0 ) 2 ordinary point not 0 not 0 ~ c simple pole ~ c / ( z − z 0 ) simple zeros at 1/2 + 0 i and − 1/2 + 0 i � � − � � � � � = − � � ( � + � � � ) − ( � − � � � ) � � � + � � � ( � + � � � ) � Maximum modulus principle : There are no other possibilities (!) �� � = − ( � − � � � ) � Example: No local maxima 11

Quick in-class exercise Q. Where are the saddle points? zeros ( − 1, − i , + i ) saddle points � + � + � � + � � bottom view √ � = − � � ± � � � + � � + � � � = � A. Where , or � 12

Modulus surface plots for familiar AC GFs � − � − � � / � − � � / � − � � / � − � � / � � − � � + � + � � + � � + � � � − � − � � − � � − � � − � � � � − � � 13

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 8. Saddle-Point Asymptotics Analytic •Modulus surfaces Combinatorics •Saddle point bounds •Saddle point asymptotics Philippe Flajolet and Robert Sedgewick OF •Applications CAMBRIDGE http://ac.cs.princeton.edu II.8a.Saddle.Surfaces

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 8. Saddle-Point Asymptotics Analytic •Modulus surfaces Combinatorics •Saddle point bounds •Saddle point asymptotics Philippe Flajolet and Robert Sedgewick OF •Applications CAMBRIDGE http://ac.cs.princeton.edu II.8b.Saddle.Bounds

Saddle-point bound for GFs: basic idea � � / � � Example: Cauchy coefficient formula � ( � ) �� � � [ � � ] � ( � ) = � �� � � + � � Saddle point bound: ζ "zeta" • Saddle point at ζ • Use circle of radius ζ • Integrand is ≤ G ( ζ )/ ζ N +1 everywhere on circle Note: ζ is the solution to � � ( � ) � � = � � � + � � � ( � ) � � + � − ( � + � ) � ( � ) � � + � = � �� � ( � ) / � ( � ) = � + � "saddle point equation" 16

Saddle-point bounds for GFs Theorem. Saddle point bounds for GFs . Let G ( z ), not a polynomial, be analytic at the origin with finite radius of convergence R . [ � � ] � ( � ) ≤ � ( ζ ) / ζ � If G has nonnegative coefficients, then where ζ is the saddle point ζ � � ( ζ ) / � ( ζ ) = � + � closest to the origin, the unique real root of the saddle point equation . [ � � ] � � Example: � � / � � Proof (sketch). By Cauchy coefficient formula � ( � ) = � � [ � � ] � ( � ) = � � ( � ) �� � � � ( � ) = � � � �� � � + � � � � � ζ = � Take C to be a � ( � ) � θ = ζ circle of radius ζ � � + � � � and change to � � ! ≤ � � [ � � ] � � = � polar coordinates ≤ � ( ζ ) � � G ( z ) ≤ G ( ζ )/ ζ N +1 on C ζ � ≐ .008333 ≐ .009498 17

Saddle point GF bound example I: factorial/exponential � � ! = [ � � ] � � Goal. Estimate � � / � � � ( � ) = � � Saddle point equation �� � �� � ( � ) � � = � + � � ( � ) = � + � Saddle point equation ζ = � + � Saddle point Saddle point bound � � + � [ � � ] � � = � Saddle point bound � ! ≤ [ � � ] � ( � ) ≤ � ( ζ ) / ζ � ( � + � ) � → � � � + � � � → � � � � � � � � √ � �� Bound is too high by only a factor of , since � � √ � ! ∼ � �� 18

Saddle point GF bound example II: Catalan/central binomial � � � � = [ � � ]( � + � ) � � Goal. Estimate � ( � + � ) �� / � � Saddle point equation � ( � ) = ( � + � ) � � �� � ( � ) � ( � ) = � + � � �� = ( � + � )( � + � ) Saddle point equation � � � ( � + � ) � � − � = � + � ( � + � ) � � ζ = � + � Saddle point � − � � � � � � � � � � � Saddle point bound � � � � � � � − � Saddle point bound [ � � ] � ( � ) ≤ � ( ζ ) / ζ � ≤ � � + � � � = � � − � � � − � → � � � � � � � � √ �� Bound is too high by only a factor of , since √ ∼ � �� 19

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 8. Saddle-Point Asymptotics Analytic •Modulus surfaces Combinatorics •Saddle point bounds •Saddle point asymptotics Philippe Flajolet and Robert Sedgewick OF •Applications CAMBRIDGE http://ac.cs.princeton.edu II.8b.Saddle.Bounds

A N A L Y T I C C O M B I N A T O R I C S P A R T T W O 8. Saddle-Point Asymptotics Analytic •Modulus surfaces Combinatorics •Saddle point bounds •Saddle point asymptotics Philippe Flajolet and Robert Sedgewick OF •Applications CAMBRIDGE http://ac.cs.princeton.edu II.8c.Saddle.Method

Saddle-point method for GFs: basic idea � � / � � ( � + � ) �� / � � Cauchy coefficient formula � ( � ) �� � � [ � � ] � ( � ) = � �� � � + � � Saddle point bound: • Saddle point at ζ • Use circle of radius ζ • Integrand is ≤ G ( ζ )/ ζ N +1 everywhere on circle Saddle point method : • Focus on path near saddle point • Bound “tail” contribution • Use Laplace’s method 22