Experimental Mathematics & Computer Algebra(high precision numerical approx., power series truncated to high order, large number of terms in a sequence,…) 2. Guess/conjecture a general

Experimental Mathematics
& Computer Algebra

Bruno Salvy

Saint-Flour, 
June 21-25, 2018

Computer Algebra

Effective mathematics: what can we compute exactly? And complexity: how fast? (also, how big is the result?)

50+ years of algorithmic progress

Systems with several million users

in computational mathematics!

Experimental Mathematics

A 3-step process:

1. Compute a high order approximation

(high precision numerical approx., 
power series truncated to high order, 
large number of terms in a sequence,…)

2. Guess/conjecture a general formula

3. Prove it

(with the help of a computer)

(using computer-algebra algorithms)

Examples

From the SIAM 100 Digits ChallengeFrom the SIAM 100-Digit Challenge1/4

1/4

1/4-ε 1/4+ε

Chapter 6

Biasing for a Fair ReturnFolkmar Bornemann

It was often claimed that [direct and “exact” numeri-cal solution of the equations of physics] would make thespecial functions redundant. ... The persistence of spe-cial functions is puzzling as well as surprising. Whatare they, other than just names for mathematical objectsthat are useful only in situations of contrived simplicity?Why are we so pleased when a complicated calculation“comes out” as a Bessel function, or a Laguerre polyno-mial? What determines which functions are “special”?

— Sir Michael Berry [Ber01]

People who like this sort of thing will find this the sortof thing they like.

— Barry Hughes, quoting Abraham Lincoln at thebeginning of an appendix on “Special Functions for Ran-dom Walk Problems” [Hug95, p. 569]

Problem 6

A flea starts at (0, 0) on the infinite two-dimensional integer lattice andexecutes a biased random walk: At each step it hops north or south withprobability 1/4, east with probability 1/4 + �, and west with probability1/4 � �. The probability that the flea returns to (0, 0) sometime duringits wanderings is 1/2. What is �?

Asking for the � that gives a certain probability p of return yields a problem hardlyany more di�cult than calculating the probability for a given �: it just adds theneed to use a numerical root-finder. But the problem looks more interesting the wayit is stated. In §6.1 we give a short argument, why the problem is solvable.

We will discuss several methods for calculating the probability of return. In§6.2, using virtually no probability theory, we transform the problem to one of lin-ear algebra. Solving a sparse linear system of dimension 25 920 gives us 15 correct

123

I Computer algebra conjectures and proves

p(✏) = 1�r

A

2· 2F1

0

@12 ,

12

1

������2p1� 16✏2A

1

A�1

, with A = 1 + 8✏2 +p

1� 16✏2.

Gessel’s ConjectureGessel’s conjecture

• Gessel walks: walks in N2 using only steps in S = {%,., ,!}

• g(i, j, n) = number of walks from (0, 0) to (i, j) with n steps in S

Question: Nature of the generating function

G(x, y, t) =1X

i,j,n=0

g(i, j, n) xiyjtn 2 Q[[x, y, t]]

I Computer algebra conjectures and proves:Theorem [B. & Kauers 2010] G(x, y, t) is an algebraic function† and

G(1, 1, t) =1

2t· 2F1

✓�1/12 1/42/3

���� �64t(4t + 1)2

(4t� 1)4

◆� 1

2t.

I A simpler variant as an exercise tomorrow.†Minimal polynomial P (x, y, t, G(x, y, t)) = 0 has > 1011 monomials; ⇡30Gb (!)

Bailey-Borwein-Plouffe Formula (1997)

⇡ =1X

i=0

1

16i

✓4

8i+ 1� 2

8i+ 4� 1

8i+ 5� 1

8i+ 6

◆

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

We found the identity by a combination of inspired guessing and extensive searching using […] integer relation algorithm.

Rooted Planar Eulerian OrientationsFrom a recent arXiv paper (March 2018)


by Mireille Bousquet-Mélou and Andrew Elvey Price60 N. Bonichon et al. / European Journal of Combinatorics 65 (2017) 59–91

Fig. 1. A rooted Eulerian map and a rooted Eulerian orientation.

degree) with n edges is [54]:

mn =3 · 2n�1

(n + 1)(n + 2)

✓2nn

◆. (1)

A bijective explanation involving plane trees can be found in [15]. The associated generatingfunction M(t) = P

n�0 mntn is known to be algebraic, that is, to satisfy a polynomial equation. More

precisely:

t2 + 11t � 1 � (8t2 + 12t � 1)M(t) + 16t2M(t)2 = 0.

Beyond their enumerative implications, bijections involving maps have been applied to encode,sample and draw maps efficiently [11,21,31,50]. More recently, they have played a key role in thestudy of large random planar maps, culminating with the existence of a universal scaling limit knownas the Brownian map [42].

Planar maps equipped with an additional structure (e.g. a spanning tree [43], a proper colouring[56,57], an Ising or Potts configuration [4,12,13,16,18,22,25,39]. . . ) are also much studied, both incombinatorics and in theoretical physics, where maps are considered as a model for two-dimensionalquantumgravity [23]. However, formany of these structures, we are still in the early days of the study,as even their enumeration remains elusive, not to mention bijections and asymptotic properties.

Recent progresses in this direction include the enumeration of planarmapsweighted by their Tuttepolynomial, or equivalently, maps equipped with a Potts configuration. The associated generatingfunction P(t) is known to be differentially algebraic. That is, there exists a polynomial equation relatingP(t) and its derivatives [6,7]. The Tutte polynomial hasmany interesting specializations (in particular,it counts all structures cited above, like spanning trees and colourings) and several special cases hadbeen solved earlier. One key tool in the solution is that the Tutte polynomial of amap can be computedinductively, by deleting and contracting edges.

Another solved example, which does not seem to belong to the Tutte/Potts realm, consists of maps(in fact, triangulations) equipped with certain orientations called Schnyder orientations. The resultsobtained there have analogies with those obtained for another class of orientations, called bipolar,(which do belong to the Tutte realm). Indeed, for both classes of oriented maps:

• orientedmaps are counted by simple numbers, which are also known to count other combinatorialobjects (various lattice paths and permutations, among others);

• there exist nice bijections explaining these equi-enumeration results [9,10,28,33];• for a fixed map M , the set of Schnyder/bipolar orientations of M has a lattice structure [51,27,45].

The above bijections, once specialized to maps equipped with their (unique) minimal orientation,coincide with attractive bijections designed earlier for (unoriented) maps [5,10];

• specializing the bijections further to maps that have only one Schnyder/bipolar orientation alsoyields interesting combinatorial results [5,10].

These observations led us to wonder about another natural class of orientations, namely those inwhich every vertex has equal in- and out-degree, known as Eulerian orientations (Fig. 1). Clearly, amap needs to be Eulerian to admit an Eulerian orientation. The condition is in fact sufficient (suchmaps even admit an Eulerian circuit [37]). One analogy with the above two classes is that the set ofEulerian orientations of a given planarmap can be equippedwith a lattice structure [51,27].Moreover,Eulerian maps (equivalently, Eulerian maps equipped with their minimal Eulerian orientation) haverich combinatorial properties: not only are they counted by simple numbers (see (1)), but they are

Wanted:

G(t) :=X

n�0gnt

n

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

nb of rooted planar orientations with n edges

Result:

Found by a guess-and-prove approach!

G(t) =t� 2t2 �R(t)

4t2AAACw3icbVFNaxRBEO0dv+L4tdGjCI1hIQpZZlbBXISgiN6Mkk0CO+vS01O902x3z9BdY7IMc/Lu//Cqf8SrJ/+KPTsJmo0FDY9XVVS/99JSSYdR9KsXXLl67fqNjZvhrdt37t7rb94/dEVlOYx5oQp7nDIHShoYo0QFx6UFplMFR+nidds/+gzWycIc4LKEqWZzI4XkDD016z96u41PXibCMl7jzgg/jXY+eqapn3vYzPpb0TBaFb0M4jOwtbf7+yclhOzPNntfk6zglQaDXDHnJnFU4rRmFiVX0ISDxIH/pJljXicIp3giM8yb+lnEdRMmlYOS8QWbw8RDwzS4aX26ktnQgacyKgrrn0G6Yv9dqZl2bqlTP6kZ5m6915L/600qFLvTWpqyQjC8OyQqRbGgrWc0kxY4qqUHjFvppVCeM28ZemfDxIKBE15ozUz2NHEiA8EqhXVVlM16VzAt1fJ84u+wPzqg742/IQXFHGgbaqez6IhzW6l07cdSoI4ZRx1YKS6oOYindbvYSgl9gvF6XpfB4WgYR8P4g4/yFelqgzwkj8k2ickLskfekX0yJpx8Id/Id/IjeBMsAhtgNxr0znYekAsVNH8A1F3fEA==AAACw3icbVFda9RAFJ2NXzV+bfVBRITBslCFLskq2BehKKJvVum2hU1cJpObzbAzkzBzY7uEPPnu//BV/4iv+uKrP8PJpkW79cLA4dx7uXPOSUopLAbBj5534eKly1fWrvrXrt+4eau/fnvfFpXhMOaFLMxhwixIoWGMAiUclgaYSiQcJPOXbf/gIxgrCr2HixJixWZaZIIzdNS0/+D1Jj56HmWG8Rq3RvhhtPXeMU391MFm2t8IhsGy6HkQnoCNne1f3+/+/H1vd7re+xylBa8UaOSSWTsJgxLjmhkUXELjDyIL7pN6hnkdIRzjkUgxb+onAVeNH1UWSsbnbAYTBzVTYOP6eCmzoQNHpTQrjHsa6ZL9d6VmytqFStykYpjb1V5L/q83qTDbjmuhywpB8+5QVkmKBW09o6kwwFEuHGDcCCeF8pw5y9A560cGNBzxQimm08eRzVLIWCWxroqyWe1mTAm5OJ34O+yODuhb7W6IjGIOtA2101l0xKmtVNj2YwlQy7SlFozIzqjZC+O6XWyl+C7BcDWv82B/NAyDYfjORfmCdLVG7pOHZJOE5BnZIW/ILhkTTj6RL+Qr+ea98uae8bAb9XonO3fImfKaP5ve4OI=AAACw3icbVFda9RAFJ2NXzV+bfVBRITBslCFLskq2BehKKJvVum2hU1cJpObzbAzkzBzY7uEPPnu//BV/4iv+uKrP8PJpkW79cLA4dx7uXPOSUopLAbBj5534eKly1fWrvrXrt+4eau/fnvfFpXhMOaFLMxhwixIoWGMAiUclgaYSiQcJPOXbf/gIxgrCr2HixJixWZaZIIzdNS0/+D1Jj56HmWG8Rq3RvhhtPXeMU391MFm2t8IhsGy6HkQnoCNne1f3+/+/H1vd7re+xylBa8UaOSSWTsJgxLjmhkUXELjDyIL7pN6hnkdIRzjkUgxb+onAVeNH1UWSsbnbAYTBzVTYOP6eCmzoQNHpTQrjHsa6ZL9d6VmytqFStykYpjb1V5L/q83qTDbjmuhywpB8+5QVkmKBW09o6kwwFEuHGDcCCeF8pw5y9A560cGNBzxQimm08eRzVLIWCWxroqyWe1mTAm5OJ34O+yODuhb7W6IjGIOtA2101l0xKmtVNj2YwlQy7SlFozIzqjZC+O6XWyl+C7BcDWv82B/NAyDYfjORfmCdLVG7pOHZJOE5BnZIW/ILhkTTj6RL+Qr+ea98uae8bAb9XonO3fImfKaP5ve4OI=AAACw3icbVFdaxNBFJ2sX3X9aKqPIgyGQBUadqNgX4SiiL5ZpWkL2RhmZ+9kh8zMLjN324Zln3z3f/iqv8Z/42ySok29MHA4917unHPSUkmHUfS7E9y4eev2na274b37Dx5ud3ceHbuishxGvFCFPU2ZAyUNjFCigtPSAtOpgpN0/q7tn5yBdbIwR7goYaLZzEghOUNPTbtPP+zi8zeJsIzXuDfEr8O9L55p6lceNtNuLxpEy6LXQbwGPbKuw+lO53uSFbzSYJAr5tw4jkqc1Myi5AqasJ848J80M8zrBOECz2WGeVO/jLhuwqRyUDI+ZzMYe2iYBjepL5YyG9r3VEZFYf0zSJfsvys1084tdOonNcPcbfZa8n+9cYVif1JLU1YIhq8OiUpRLGjrGc2kBY5q4QHjVnoplOfMW4be2TCxYOCcF1ozk71InMhAsEphXRVls9kVTEu1uJz4O+yP9ukn429IQTEH2oa60lmsiEtbqXTtx1KgjhlHHVgprqg5iid1u9hKCX2C8WZe18HxcBBHg/hz1Dt4u85yizwhz8guiclrckA+kkMyIpx8Iz/IT/IreB/MAxvgajTorHcekysVNH8AEhjcyQ==

with R defined by

t =X

n�0

1

n+ 1

✓2n

n

◆2R(t)n+1

AAAC2nicbVFdaxNBFJ2sH63xo6k++jJYAlUh7EahvghBX/TJKk1byKZhdvZuMnQ+lpm7tmHYF99EEASf/DWCT/oP/DfOJi3a1AsDh3Pu5c65JyulcBjHv1vRlavXrq+t32jfvHX7zkZn8+6+M5XlMORGGnuYMQdSaBiiQAmHpQWmMgkH2fHLRj94D9YJo/dwXsJYsakWheAMAzXpPMXnqavUxOt0CnHt08Iynnj9OKnTTGijfF/XXtdH/Xfb+PBoIdSTzlbcixdFL4PkDGwNXv/4MiCE7E42W5/T3PBKgUYumXOjJC5x7JlFwSXU7W7qIPxdT3HmU4RTPBE5zmr/JOaqbqeVg5LxYzaFUYCaKXBjf7pwX9NuoHJaGBueRrpg/x3xTDk3V1noVAxnblVryP9powqLZ2MvdFkhaL5cVFSSoqHNKWkuLHCU8wAYtyJYoXzGwv0wHLydWtBwwo1STOePUlfkULBKoq9MWa+qBVNCzs87/jaHpV36RocdoqA4A9pkvfRplsT5WalwzccyoI5pRx1YUVxws5eMfTPYWGmHBJPVvC6D/X4viXvJ2xDlC7KsdXKfPCDbJCE7ZEBekV0yJJx8I9/JT/IrSqMP0cfo07I1ap3N3CMXKvr6Bx0M6eA=AAAC2nicbVFdaxNBFJ2sXzV+NNVHXwZLoCqE3VTQF6Uogj5ZpWkL2TTMzt5Nhs7HMnPXNgz7oG8iCIJP/hrBJ/0H/htnkxZt6oWBwzn3cufck5VSOIzj363owsVLl6+sXG1fu37j5mpn7dauM5XlMOBGGrufMQdSaBigQAn7pQWmMgl72eHzRt97B9YJo3dwVsJIsYkWheAMAzXuPMQnqavU2Ot0AnHt08Iynnj9IKnTTGijfF/XXtcH/bcbeO9gLtTjznrci+dFz4PkBKxvvfrx5en7F6vb47XW5zQ3vFKgkUvm3DCJSxx5ZlFwCXW7mzoIf9cTnPoU4RiPRI7T2m/GXNXttHJQMn7IJjAMUDMFbuSP5+5r2g1UTgtjw9NI5+y/I54p52YqC52K4dQtaw35P21YYfF45IUuKwTNF4uKSlI0tDklzYUFjnIWAONWBCuUT1m4H4aDt1MLGo64UYrp/H7qihwKVkn0lSnrZbVgSsjZacff5rC0S1/rsEMUFKdAm6wXPs2COD0rFa75WAbUMe2oAyuKM252kpFvBhsr7ZBgspzXebDb7yVxL3kTonxGFrVC7pC7ZIMk5BHZIi/JNhkQTr6R7+Qn+RWl0YfoY/Rp0Rq1TmZukzMVff0D0zLqsw==AAAC2nicbVFdaxNBFJ2sXzV+NNVHXwZLoCqE3VTQF6Uogj5ZpWkL2TTMzt5Nhs7HMnPXNgz7oG8iCIJP/hrBJ/0H/htnkxZt6oWBwzn3cufck5VSOIzj363owsVLl6+sXG1fu37j5mpn7dauM5XlMOBGGrufMQdSaBigQAn7pQWmMgl72eHzRt97B9YJo3dwVsJIsYkWheAMAzXuPMQnqavU2Ot0AnHt08Iynnj9IKnTTGijfF/XXtcH/bcbeO9gLtTjznrci+dFz4PkBKxvvfrx5en7F6vb47XW5zQ3vFKgkUvm3DCJSxx5ZlFwCXW7mzoIf9cTnPoU4RiPRI7T2m/GXNXttHJQMn7IJjAMUDMFbuSP5+5r2g1UTgtjw9NI5+y/I54p52YqC52K4dQtaw35P21YYfF45IUuKwTNF4uKSlI0tDklzYUFjnIWAONWBCuUT1m4H4aDt1MLGo64UYrp/H7qihwKVkn0lSnrZbVgSsjZacff5rC0S1/rsEMUFKdAm6wXPs2COD0rFa75WAbUMe2oAyuKM252kpFvBhsr7ZBgspzXebDb7yVxL3kTonxGFrVC7pC7ZIMk5BHZIi/JNhkQTr6R7+Qn+RWl0YfoY/Rp0Rq1TmZukzMVff0D0zLqsw==AAAC2nicbVFNbxMxEHWWrxK+UjhysYgiFZCi3YAEF6QKLtwoqGkrxWnk9c5mrdrelT1LG1l74YaQOHHi13CFf8C/wZukgqaMZOnpvRmN37y0UtJhHP/uRFeuXrt+Y+tm99btO3fv9bbvH7iytgLGolSlPUq5AyUNjFGigqPKAtepgsP05E2rH34E62Rp9nFRwVTzuZG5FBwDNes9x1fM1XrmDZtD3HiWWy4Sb54mDUulKbUfmcab5nj0YQcfHy+FZtbrx8N4WfQySNagT9a1N9vufGVZKWoNBoXizk2SuMKp5xalUNB0B8xB+LuZY+EZwhmeygyLxj+LhW66rHZQcXHC5zAJ0HANburPlu4bOghURvPShmeQLtl/RzzXzi10Gjo1x8Jtai35P21SY/5y6qWpagQjVovyWlEsaXtKmkkLAtUiAC6sDFaoKHi4H4aDd5kFA6ei1Jqb7AlzeQY5rxX6uqyaTTXnWqrFecff5rB0QN+ZsEPmFAugbdYrn+WKOD8rla79WArUceOoAyvzC272k6lvB1sr3ZBgspnXZXAwGibxMHkf93dfr7PcIg/JI7JDEvKC7JK3ZI+MiSDfyQ/yk/yKWPQp+hx9WbVGnfXMA3Khom9/ALJu58M=

Asymptotic behaviour


follows

gn ⇠(4⇡)n

16n2 log2 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

American Mathematical Monthly Problems with a Combinatorial Flavor

I Last one as an exercise tomorrow.

1971] ELEMENTARY PROBLEMS AND SOLUTIONS 543

E 2296. Proposed by Erwin Just and Norman Schaumberger, Bronx Com- munity College

A nonconstant polynomial f with integral coefficients has the property that for each prime pi, there exists a prime qi and an integer m such that f(pi) =q';. Prove that the polynomials contained in { xn,n =1, 2, . , are the only poly- nomials which possess this property. (This generalizes E 1632 [1964, 795].)

E 2297. Proposed by Richard Stanley, Harvard University

Let L(n) be the total nu'mber of distinct monomials appearing in the ex- pansion of the determinant of an n Xn symmetric matrix A = (a,). For instance, L(3) = 5. Show that

00

Z L(n)xn/n ! = (1 - x)f"2 exp(.!x + Ix2), n=O

where I x I < 1, and where we define L(O) -1.

E 2298. Proposed by Anders Bager, Hjorring, Denmark

Prove that in every triangle

B-C C-A A-B cos + cos + cos - 2 2 2

A B C\ < (cos A + cos B + cos C) + (sin - + sin -+ sin -) 3,

with equality if and only if A = B = C.

E 2299. Proposed by Anders Bager, Hjorring, Denmark

It is given thal the roots of a certain cubic equation

ax3 + bx2 + CX + d = O (a - O)

are tan('A), tan(IB), and tan(C), where A, B, C are the angles of a triangle. Prove that a+b = c+d.

E 2300. Proposed by T. C. Brown, Simon Fraser University, British Columbia

Let S be a semigroup in which, for some fixed k _ 1, xk+1 = x and XykX = yXky for all x, y in S. Show that S is commutative.

SOLUTIONS OF ELEMENTARY PROBLEMS

The Fermat Relation as a Matrix Equation

E 2030 [1967, 1133; 1968, 1123]. Proposed by J. L. Brenner, University of Arizona, and Bernard Jacobson, Franklin and Marshall College.

In the article, Solutions of x4+y4 Z4 in 2 X 2 integral matrices (this MONTHLY,

This content downloaded from 188.1.230.74 on Thu, 17 May 2018 13:30:55 UTCAll use subject to http://about.jstor.org/terms

Live Example Session

Guess and Prove an explicit formula for

⇣px2 � 1

⌘(n)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

Exercise for the afternoon

Number of domino tilings of the Aztec diamond of size n:

size 8

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

,

1

1� w � z � wz =X

i,j

mi,jwizj

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

Mn = (mi,j)0i,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

where

Guess & Prove a general formula for det Mn

Delannoy numbers

Basic Tools of Computer Algebra

What Cannot be Done

Thm. [Richardson-Matiyasevich] In the class of expressions obtained from a variable X and the constant 1 par the operations +,-,x and composition with functions sin and absolute value, testing for equivalence to 0 is undecidable.

Mathematical Objects

• Main objects

– integers Z– polynomials K[x]– rational functions K(x)– power series K[[x]]– matrices Mr(K)– linear recurrences with constant, or polynomial, coe�cients K[n]hSni– linear di↵erential equations with polynomial coe�cients K[x]h@xiwhere K is a field (generally supposed of characteristic 0 or large)

• Secondary/auxiliary objects

– polynomial matrices Mr(K[x])– power series matrices Mr(K[[x]])

Classes where 0 is recognized

Complexity yardsticks

Important features:

• addition is easy: naive algorithm already optimal

• multiplication is the most basic (non-trivial) problem

• almost all problems can be reduced to multiplication

Are there quasi-optimal algorithms for:

• integer/polynomial/power series multiplication? Yes!

• matrix multiplication? Big open problem!

Complexity Issues

Complexity yardsticks

M(n) = complexity of multiplication in K[x]

What can be computed in 1 min?

Integersproduct of 2 integers of 1,000,000,000 digits factorial of 40,000,000 (approx 280,000,000 digits)factorisation of an integer of 72 digits.

(in Maple on a recent laptop)

Polynomials in K[x]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

(in )K[x, y]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

product of 2 polynomials of degree 2,000,000 (1,100) gcd of 2 polynomials of degree 100,000 factorisation of a polynomial of degree 2,000 (600)

Matricesdet of a 5,500x5,500 matrix with coeffs inK

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

char. poly of a 1,700x1,700 matrix with coeffs 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

det of a 650x650 matrix with 32 bit integer entries

K = 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

p = 67108879 (28 bits)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

Where to Learn More 
Computer Algebra

Computer algebra books

�x

�y

�z

Algorithmes E�cacesen Calcul Formel

Alin BostanFrédéric Chyzak

Marc GiustiRomain LebretonGrégoire Lecerf

Bruno SalvyÉric Schost

Free from our homepages,


legally!

When CA Systems Need Help with High-Precision: Newton’s Iteration

1671

Numerical Newton Iteration

Quadratic convergence

�(y) = 1 + zy2 � yTo solve , iterate 

with

�(y) = 0

y[n+1] = y[n] + u[n],

�(y[n]) + �0(y[n])u[n] = 0.

z = 0.23 y[0] = 0,

y[1] = 1.0000000000000000,

y[2] ' 1.4259259259259259,y[3] ' 1.5471933181836303,y[4] ' 1.5589256602748822,y[5] ' 1.5590375713926592,y[6] ' 1.5590375815769151

y[n+1] = N (y[n]) = y[n] + 1 + zy[n]2 � y[n]

1� 2zy[n]

Newton Iteration for Power SeriesSame

Newton Iteration

Expand(N) = { 
 res=Expand(N/2); 
 a=ɸ(res); b=ɸ’(res); 
 u=Solve(a+bx,x); 
 return res+u; }

Cost(N) ≤ ct ⨉ Cost(last step)

y[0] = 0

y[1] = 1

y[2] = 1 + z + 2z2 + 4z3 + 8z4 + 16z5 + 32z6 + 64z7 + · · ·y[3] = 1 + z + 2z2 + 5z3 + 14z4 + 42z5 + 132z6 + 428z7 + · · ·

y[n+1] = N (y[n]) = y[n] + 1 + zy[n]2 � y[n]

1� 2zy[n]

On power series: y � y[1] = O(zm) =) N (y)� y[1] = O(z2m(+1))

Example: Newton Iteration for Inverses

To solve , iterate 

with

�(y) = 0

y[n+1] = y[n] + u[n],

�(y[n]) + �0(y[n])u[n] = 0.

�(y) = a� 1/y) 1/�0(y) = y2

) y[n+1] = y[n] � y[n](ay[n] � 1).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

Cost: a small number of multiplications. Works for:

numerical inversion; reciprocal of power series; inversion of matrices.

Basis for many more efficient operations

Tools for Conjectures

Online Encyclopedia of Integer SequencesA treasure for combinatorists

Exercise for the afternoon

Also accessible from within Sage

Stanley’s problem E2297

http://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.orghttp://www.oeis.org

Hermite-Padé ApproximantsDefinition

Definition: Given a column vector F = (f1, . . . , fn)T 2 K[[x]]n and an n-tupled = (d1, . . . , dn) 2 Nn, a Hermite-Padé approximant of type d for F is a rowvector P = (P1, . . . , Pn) 2 K[x]n, (P 6= 0), such that:

(1) P · F = P1f1 + · · · + Pnfn = O(x�) with � =P

i(di + 1) � 1,

(2) deg(Pi) di for all i.

� is called the order of the approximant P.

I Very useful concept in number theory (transcendence theory):

• [Hermite, 1873]: e is transcendent.

• [Lindemann, 1882]: ⇡ is transcendent, and so does e↵ for any ↵ 2 Q \ {0}.

• [Beukers, 1981]: reformulate Apéry’s proof that ⇣(3) =P

n1n3 is irrational.

• [Rivoal, 2000]: there exist an infinite number of k such that ⇣(2k + 1) /2 Q.

Definition

Definition: Given a column vector F = (f1, . . . , fn)T 2 K[[x]]n and an n-tupled = (d1, . . . , dn) 2 Nn, a Hermite-Padé approximant of type d for F is a rowvector P = (P1, . . . , Pn) 2 K[x]n, (P 6= 0), such that:

(1) P · F = P1f1 + · · · + Pnfn = O(x�) with � =P

i(di + 1) � 1,

(2) deg(Pi) di for all i.

� is called the order of the approximant P.

I Very useful concept in number theory (transcendence theory):

• [Hermite, 1873]: e is transcendent.

• [Lindemann, 1882]: ⇡ is transcendent, and so does e↵ for any ↵ 2 Q \ {0}.

• [Beukers, 1981]: reformulate Apéry’s proof that ⇣(3) =P

n1n3 is irrational.

• [Rivoal, 2000]: there exist an infinite number of k such that ⇣(2k + 1) /2 Q.

Worked example

Let us compute a Hermite-Padé approximant of type (1, 1, 1) for (1, C, C2),

where C(x) = 1 + x + 2x2 + 5x3 + 14x4 + 42x5 + O(x6).

This boils down to finding ↵0, ↵1, �0, �1, �0, �1 such that

↵0+↵1x+(�0+�1x)(1+x+2x2+5x3+14x4)+(�0+�1x)(1+2x+5x

2+14x3+42x4) = O(x5).

By identifying coe�cients, this is equivalent to a homogeneous linear system:

2

666666664

1 0 1 0 1 0

0 1 1 1 2 1

0 0 2 1 5 2

0 0 5 2 14 5

0 0 14 5 42 14

3

777777775

⇥

2

666666666664

↵0

↵1

�0

�1

�0

�1

3

777777777775

= 0 ()

2

666666664

1 0 1 0 1

0 1 1 1 2

0 0 2 1 5

0 0 5 2 14

0 0 14 5 42

3

777777775

⇥

2

666666664

↵0

↵1

�0

�1

�0

3

777777775

= ��1

2

666666664

0

1

2

5

14

3

777777775

.

By homogeneity, one can choose �1 = 1. Then, the violet minor shows that

one can take (�0, �1, �0) = (�1, 0, 0). The other values are ↵0 = 1, ↵1 = 0.Thus the approximant is (1, �1, x), which corresponds to P = 1 � y + xy2

such that P (x, C(x)) = 0 mod x5.

Worked Example

Algebraic and di↵erential approximation = guessing

• Hermite-Padé approximants of n = 2 power series are related to Padéapproximants, i.e. to approximation of series by rational functions

• algebraic approximants = Hermite-Padé approximants for f` = A`�1,where A 2 K[[x]] seriestoalgeq, listtoalgeq

• di↵erential approximants = Hermite-Padé approximants for f` = A(`�1),where A 2 K[[x]] seriestodi↵eq, listtodi↵eq

> listtoalgeq([1,1,2,5,14,42,132,429],y(x));

2

[1 - y(x) + x y(x) , ogf]

> listtodiffeq([1,1,2,5,14,42,132,429],y(x));

/ 2 \

/d \ |d |

[{-2 y(x) + (2 - 4 x) |-- y(x)| + x |--- y(x)|, y(0) = 1, D(y)(0) = 1}, egf]

\dx / | 2 |

\dx /

Algebraic and Differential Approximants = Guessing

LLL and Linear Relations Between Constants

Def. Euclidean lattice: integer linear combinations of vectors in ℝn.

Shortest Vector Problem is NP-hard 
Input: a basis in ℤn 
Output: the shortest nonzero vector in the lattice.

LLL (Lenstra, Lenstra, Lovász 1982). A polynomial time 
approximation algorithm up to a factor 2(n-1)/2.

Application.

take C large and find short vectors in

(↵1, . . . ,↵n) 2 Rn,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

0

BBBBBB@

[C↵1] [C↵2] . . . [C↵n]1 0 0

0 1...

.... . .

0 0 · · · 1

1

CCCCCCA

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 for the afternoon: BPP formula.

Summary of the Exercises
for this Afternoon

1. Identities for nice constants 2. Stanley’s Problem E 2297 on the

determinant of symmetric matrices 3. Number of domino tilings of the

Aztec diamond

Tomorrow: Tools for Proofs

1. Resultants

2. D-finite Sequences & Series

3. Creative Telescoping