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 O N E 4. Asymptotic Approximations http://aofa.cs.princeton.edu
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 O N E 4. Asymptotic Approximations •Standard scale •Manipulating expansions •Asymptotics of finite sums OF •Bivariate asymptotics http://aofa.cs.princeton.edu 4a.Asympt.scale
Asymptotic approximations Goal: Develop accurate and concise estimates of quantities of interest � (log � ) ✗ not accurate � � � � = ✗ not concise � � ≤ � ≤ � ln � + γ + � ( � ✓ � ) Informal definition of concise : ‟ easy to compute with constants and standard functions” 3
Notation (revisited) “Big-Oh” notation for upper bounds � ( � ) = � ( � ( � )) ��� | � ( � ) / � ( � ) | ������������������������ � → ∞ “Little-oh” notation for lower bounds � ( � ) = � ( � ( � )) ��� � ( � ) / � ( � ) → � �� � → ∞ “Tilde” notation for asymptotic equivalence � ( � ) ∼ � ( � ) ��� � ( � ) / � ( � ) → � �� � → ∞ 4
Notation for approximations “Big-Oh” approximation � ( � ) = � ( � ) + � ( � ( � )) Error will be at most within a constant factor of h ( N ) as N increases. “Little-oh” approximation � ( � ) = � ( � ) + � ( � ( � )) Error will decrease relative to h ( N ) as N increases. “Tilde” approximation � ( � ) ∼ � ( � ) Weakest nontrivial o-approximation. 5
Standard asymptotic scale Definition. A decreasing series g k ( N ) with g k +1 ( N ) = o( g k ( N )) is called an asymptotic scale. The series � ( � ) ∼ � � � � ( � ) + � � � � ( � ) + � � � � ( � ) + . . . is called an asymptotic expansion of f . The expansion represents the collection of formulae � ( � ) = � ( � � ( � )) � ( � ) = � � � � ( � ) + � ( � � ( � )) � ( � ) = � � � � ( � ) + � � � � ( � ) + � ( � � ( � )) � ( � ) = � � � � ( � ) + � � � � ( � ) + � � � � ( � ) + � ( � � ( � )) � � � The standard scale is products of powers of N, log N, iterated logs and exponentials. Typically, we : • use only 2, 3, or 4 terms (continuing until unused terms are extremely small) • use ~-notation to drop information on unused terms. • use O-notation or o-notation to specify information on unused terms. Methods extend in principle to any desired precision. 6
Example: Asymptotics of linear recurrences Theorem. Assume that a rational GF f ( z )/ g ( z ) with f ( z ) and g ( z ) relatively prime and g (0)=0 has a unique pole 1/ β of smallest modulus and that the multiplicity of β is ν . Then [ � � ] � ( � ) � = ν ( − β ) ν � ( � / β ) � ( � ) ∼ � β � � ν − � ����� � ( ν ) ( � / β ) � � + � � � � Proof sketch � Ex. 7 2187 275 � � � � � β � � � � � � β � � �� � � β � � � � � + � + . . . + � 8 6561 17811 � ≤ � < � � � ≤ � < � � � ≤ � < � � 9 19683 20195 Largest term dominates. 10 59049 60073 11 177147 179195 Notes : Example 1. ✓ • Pole of smallest modulus usually dominates. 3 N + 2 N ~ 3 N • Easy to extend to cover multiple poles in neighborhood Example 2. of pole of smallest modulus. ✗ 2 N + 1.99999 N ~ 2 N 7
Asymptotics of linear recurrences Theorem. Assume that a rational GF f ( z )/ g ( z ) with f ( z ) and g ( z ) relatively prime and g (0)=0 has a unique pole 1/ β of smallest modulus and that the multiplicity of β is ν . Then [ � � ] � ( � ) � = ν ( − β ) ν � ( � / β ) � ( � ) ∼ � β � � ν − � ����� � ( ν ) ( � / β ) Example from earlier lectures. � � = � � � − � − � � � − � ��� � ≥ � ���� � � = � ��� � � = � � � = � � � − � − � � � − � + δ � � Make recurrence valid for all n. � ( � ) = � �� ( � ) − � � � � ( � ) + � Multiply by z n and sum on n. � � ( � ) = Solve. � − � � + � � � � = � ( − � )( � / � ) � � ∼ � � Smallest root of denominator is 1/3. − � + �� / � = � 8
Fundamental asymptotic expansions are immediate from Taylor’s theorem. � � = � + � + � � � + � � exponential � + � ( � � ) ln( � + � ) = � − � � � + � � logarithmic � + � ( � � ) � � � � � � ( � + � ) � = � + �� + � � + � � + � ( � � ) binomial � � � � − � = � + � + � � + � � + � ( � � ) geometric as x ➛ 0. 9
Fundamental asymptotic expansions are immediate from Taylor’s theorem. Substitute x = 1/N to get expansions as N ➛ ∞ . � � / � = � + � � � � � + � ( � � exponential � + � � � + � � ) ln( � + � � ) = � � � � � + � ( � � logarithmic � � � + � � ) � − � � � � ( � + � � ) � = � + � � � � � � � + � ( � binomial � + � � + � � ) � � � − � = � � � + � � � + � � � + � ( � � � ) geometric as N ➛ ∞ . 10
Inclass exercise Develop the following asymptotic approximations ln( � + � � ) + ln( � − � � ( � / � � ) �� � ) = � � � � + � ( � � � � ) − � � � � + � ( � � � � ) � − � − = − � � � + � ( � � � ) ln( � + � � ) − ln( � − � � ( � / � � ) �� � ) = � � � � + � ( � � � � ) + � � � � + � ( � � � + � � ) � − = � � + � ( � � � ) 11
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 O N E 4. Asymptotic Approximations •Standard scale •Manipulating expansions •Asymptotics of finite sums OF •Bivariate asymptotics http://aofa.cs.princeton.edu 4a.Asympt.scale
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 O N E 4. Asymptotic Approximations •Standard scale •Manipulating expansions •Asymptotics of finite sums OF •Bivariate asymptotics http://aofa.cs.princeton.edu 4b.Asympt.manip
Manipulating asymptotic expansions Goal. Develop expansion on the standard scale for any given expression. � � � � � � � − � � �� � � � ( � � ) � � � � + � � ln( � + � ) � Techniques. Why? Facilitate comparisons of different quantities. simplification Simplify computations. substitution � � � � � factoring Ex. � � � N = 10 6 ?? multiplication division composition exp/log 14
Manipulating asymptotic expansions Simplification. An asymptotic series is only as good as its O-term. Discard smaller terms. ln � + γ + � ( � ) ✗ ✓ ln � + � ( � ) Substitution. Change variables in a known expansion. ln( � + � ) = � − � � � + � � � + � ( � � ) �� � → � Taylor series ln( � + � � ) = � � � � � + � ( � � �� � → ∞ � � � + � � )) Substitute x = 1/ N � − 15
Manipulating asymptotic expansions Factoring. Estimate the leading term, factor it out, expand the rest. � � � + � = � � Factor out 1/ N 2 . � � � + � / � = � � − � � + � ( � � � � � ) Expand the rest. � � = � � � − � � � + � ( � � � � ) Distribute. 16
Manipulating asymptotic expansions Multiplication. Do term-by-term multiplication, simplify, collect terms. ln � + γ + � ( � ln � + γ + � ( � ( � � ) � = � �� � Ex. � ) � ) (ln � ) � + γ ln � + � (log � � � = ) Term-by-term � multiplication. γ ln � + γ � + � ( � � � + � ) � (log � ) + � ( � � ) + � ( � � � + � � ) � slight improvement big improvement in precision in precision = (ln � ) � + � γ ln � + γ � + � (log � Collect terms. ) � (ln � ) � + � γ ln � ( � � ) � + γ � 1000 56.032 47.717 55.692 56.025 10000 95.797 84.830 95.463 95.796 May need trial-and-error to get desired precision. 146.172 ✓ 100000 146.172 132.547 145.838 17
Manipulating asymptotic expansions Division. Expand, factor denominator, expand 1/(1 − x ), multiply. � � Ex. ln( � + � ) ln � + γ + � ( � � ) Expand. = ln � + � ( � � ) ln � + � ( � γ � + � ) OK to simplify by replacing Factor denominator. = O( 1 /N log N ) by O(1/ N ) � + � ( � � ) ln � + � ( � � + � ( � γ � + � �� � Expand 1/(1 − x ). = � ) � ) ln � + � ( � γ = � + Multiply. � ) 18
Manipulating asymptotic expansions Composition. Substitute an expansion. �� � = � ln � + γ + � ( � / � ) Substitute H N expansion. Lemma. � � ( � / � ) = � + � ( � = �� γ � � ( � / � ) � ) = �� γ ( � + � ( � � ( � � ) � � � ) + � Expand e x . � = �� γ ( � + � ( � � Simplify. � ) �� � � �� γ = �� γ + � ( � ) Distribute. 1000 1782 1781 10000 17812 17811 big improvement 100000 178108 178107 in precision ✓ 1000000 1781073 1781072 19
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