Jonathan Borwein, FRSC www.cs.dal.ca/~jborwein Canada Research Chair in Collaborative Technology

“The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. 'Mathematizing' may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalisation.” Hermann Weyl

Revised 18/07/2007

MAA Summer Seminar on Experimental Math in Action (Carleton College July 15-

20, 2007)

Jonathan Borwein, FRSC www.cs.dal.ca/~jborwein Canada Research Chair in Collaborative Technology “Elsewhere Kronecker said ``In mathematics, I recognize true scientific value only in

concrete mathematical truths, or to put it more pointedly, only in mathematical formulas." ... I would rather say ``computations" than ``formulas", but my view is

essentially the same.”

Harold Edwards, Essays in Constructive Mathematics, 2004

L2-3 Computer-assisted Discovery & Proof

Chapters Two and Three

Computer-assisted Discovery and Proof

Gfun, Coupon Collecting, Formulae for Pi, and Generating Functions for Riemann’s Zeta

Jonathan M. BorweinDalhousie D-Drive

David H BaileyLawrence Berkeley National Lab

“All truths are easy to understand once they are discovered; the point is to discover them.” – Galileo Galilei

Details in Experimental Mathematics in Action,Bailey, Borwein et al, A.K. Peters, 2007.

Ex 1. Generating Functions in Maple

Sums of 2, 3, 4 squares: what we can tell the easy way

Ex 2. Generating Functions in ‘gfun’

Identifying and confirming in Maple

And what Sloane tells us …

And what Sloane tells us …

• What a wonderful resource!

•The more technical the result, the less we will learn from Sloane and the more from Salvy-Zimmerman

Ex 3. Coupon Collecting and ConvexityB. The origin of the problem.

This arose as the objective function in a 1999 PhD on coupon collection. Ian Affleck wished to show pN was convex on the positive orthant. I hoped not!

Coupon Collecting and ConvexityB. Doing What is Easy.

Coupon Collecting and ConvexityB. A Very Convex Integrand. (Is there a direct proof?)

A year later, Omar Hijab suggested re-expressing pN as the joint expectation of Poisson distributions. This leads to:

Now yi xi yi and standard techniques show 1/pN is concave, as the integrand is. [We can now ignore probability if we wish!] Q.“inclusion-exclusion” convexity: OK for 1/g(x) > 0, g concave.

Algorithms Used in Experimental Mathematics

Symbolic computation for algebraic and calculus manipulations (as in Ex. 1, 2 & 3).

Integer-relation methods, especially the “PSLQ” algorithm. High-precision integer and floating-point arithmetic. High-precision evaluation of integrals and infinite series

summations. The Wilf-Zeilberger algorithm for proving summation

identities. Iterative approximations to continuous functions. Identification of functions based on graph characteristics. Graphics and visualization methods targeted to

mathematical objects.

Goals for Today

First lecture: fast arithmetic, Newton’s method, PSLQ and numerical quadrature

Second lecture: more quadrature, Wilf-Zeilberger and applications to physics, binomial series and pi.

“As you know Mahlburg proved that for every prime p > 3 there are infinitely many pairs (A,B) such that M(r, p, A*n + B) = 0 (mod p) for  r=0,1,...p-1. Here M(r,p,n)= number of partitions of n with crank congruent to r mod p. Actually he proved more. Recently, I have proved the rank-analog. I found a couple of nontrivial examples:

(1) N(r, 11, 5^4*11*19^4*n + 4322599) = 0 (mod 11)

(2) N(r, 11, 11^2*19^4*n + 172904)  = 0  (mod 11)  (r=0,1,..10)

Here N(r,p,n)= number of partitions of n with rank congruent to r mod p. Anyway, I have verified (2) for n=0 using Fortran (Maple has no hope of doing such a computation). I have not even verified (1) for n=0 (I hope it is correct). I hope to find examples for higher primes, next case is p=13. Frank Garvan”

Question #1.

Continued fraction is = [1,2,3,4,5,….] “google” arithmetic continued fraction (or “sloane” ) http://

mathworld.wolfram.com/ContinuedFractionConstant.html tells us that = ratio of Bessel functions indeed:

“The general infinite continued fraction with partial quotients that are in arithmetic progression is given by

(Schroeppel 1972).”

Here I is the modified Bessel function of the first kind

Typical Scheme for High-Precision Floating-Point Arithmetic

A high-precision number is represented as a string of n + 4 integers (or a string of n + 4 floating-point numbers with integer values):

First word contains sign and n, the number of words. Second word contains p, the exponent (power of 2b). Words three through n + 2 contain mantissas m1 through mn. Words n + 3 and n + 4 are for convenience in arithmetic. The value is then given by:

•For basic arithmetic operations, conventional schemes suffice up to about 1000 digits. Beyond that level, Karatsuba’s algorithm (next slide) or FFTs can be used for significantly faster multiply performance.

•Division and square roots can be performed by Newton iterations (next slide).

•For transcendental functions, Taylor’s series or (for higher precision) quadratically convergent elliptic algorithms can be used.

Multiplication Karatsuba multiplication (200 digits +) or Fast Fourier Transform

(FFT) … in ranges from 100 to 1,000,000,000,000 digits

• The other operations via Newton’s method

• Elementary and special functions via Elliptic integrals and Gauss AGM

For example:

Karatsuba replaces one

‘times’ by many ‘plus’

FFT multiplication of multi-billion digit numbers reduces centuries to minutes. Trillions must be done with Karatsuba!

….all based on Fast Arithmetic (Complexity Reduction in Action)

1. Doubles precision at each stepNewton is self correcting and quadratically convergent

Newton’s Methodfor

Elementary Operations and Functions

3. For the logarithm we approximate by elliptic integrals (AGM) which admit quadratic transformations: near zero

4. We use Newton to obtain the complex exponential

So So allall elementary functionselementary functions are fast computable are fast computable

Newton’s arcsin

Now multiply by A

2. Consequences for work needed:

Initial guess

DHB’s Arbitrary Precision Computation (ARPREC) Package

Low-level routines written in C++. C++ and F-90 translation modules permit use with existing

programs with only minor code changes. Double-double (32 digits), quad-double, (64 digits) and arbitrary

precision (>64 digits) available. Special routines for extra-high precision (>1000 dig). Includes common math functions: sqrt, cos, exp, etc. PSLQ, root finding, numerical integration. An interactive “Experimental Mathematician’s Toolkit”

employing this software is also available. Available at: http://www.experimentalmath.info

Also recommended: GMP/MPFR package, available at

http://www.mpfr.org

A Matrix Example

The key discovery:

… the discovery

Log Convexity A Full Case Study

The PSLQ Integer Relation Algorithm

Let (xn) be a vector of real numbers. An integer relation algorithm finds (or excludes) integers (an) such that

At the present time, the PSLQ algorithm of mathematician-sculptor Helaman Ferguson is the best-known integer relation algorithm.

PSLQ was named one of ten “algorithms of the century” by Computing in Science and Engineering.

High precision arithmetic software is required: at least d £ n digits, where d is the size (in digits) of the largest of the integers ak. [APPENDIX II on PSLQ]

Ferguson’s Sculpture

Time for a movie?

Decrease of error = minj |Aj x| in PSLQ

Number of iterates

Application of PSLQ: Bifurcation Points in Chaos Theory

In other words, B3 is the smallest r such that the iteration exhibits 8-way periodicity instead of 4-way periodicity. In 1990, a predecessor to PSLQ found that B3 is a root of

B3 = 3.54409035955… is third bifurcation point of the logistic iteration of chaos theory:

Recently B4 was identified as the root of a 240-degree polynomial by a much more challenging computation. These results have subsequently been proven formally (by Groebner basis methods).

•An iterative approximation scheme to calculate high-precision values of these constants is described in EMA.

Evaluation of Ten Constants from Quantum Field Theory

where

PSLQ and Sculpture

The complement of the figure-eight knot, when viewed in hyperbolic space, has finite volume

2.029883212819307250042…

David Broadhurst found, using PSLQ, that this constant is given by the “BBP” formula:

Some Supercomputer-Class PSLQ Solutions

Identification of B4, the fourth bifurcation point of the logistic iteration. (earlier) Integer relation of size 121; 10,000 digit arithmetic.

Identification of Apery sums (later). 15 integer relation problems, with size up to 118,

requiring up to 5,000 digit arithmetic. Identification of Euler-zeta sums.

Hundreds of integer relation problems, each of size 145 and requiring 5,000 digit arithmetic.

Run on IBM SP parallel system. Finding relation for root of Lehmer’s polynomial.

Integer relation of size 125; 50,000 digit arithmetic. Utilizes 3-level, multi-pair parallel PSLQ program. Run on IBM SP using ARPEC; 16 hours on 64 CPUs.

Numerical Integration and PSLQ

where

is a primitive Dirichlet series modulo three. [Note homogeneity of evaluation.]

Numerical Integration: Example 2

This arises in mathematical physics, from analysis of the volumes of ideal tetrahedra in hyperbolic space.

This “identity” (one of 998) has now been verified numerically to 20,000 digits, but no proof is known.

Note that the integrand function has a nasty singularity.

Numerical Integration: Example 3 (Jan 2006)

The following integrals arise in Ising theory of mathematical physics:

where K0 is a modified Bessel function. We then computed 400-digit numerical values, from which we found these results (now proven):

We first showed that this can be transformed to a 1-D integral:

Identifying the Limit Using the Inverse Symbolic Calculator

We discovered the limit result as follows: We first calculated:

We then used the Inverse Symbolic Calculator, anor online numerical constant recognition facility available at:

http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html

or http://ddrive.cs.dal.ca/~isc/ (ISC2.0)

Output: Mixed constants, 2 with elementary transforms. 6304735033743867 = sr(2)^2/exp(gamma)^2 In other words,

For full details see “An Integral of the Ising Class,” (J. Phys. A, 2007) available at

http://crd.lbl.gov/~dhbailey/dhbpapers/IsingBBC.pdf

New Ramanujan-Like Identities

Guillera has recently found Ramanujan-like identities, including:

where

Guillera proved the first two of these using the Wilf-Zeilberger algorithm. He ascribed the third to Gourevich, who found it using integer relation methods.

Are there any higher-order analogues?

Not as far as we can tell

Part II

JM Borwein and DH Bailey

“Anyone who is not shocked by quantum theory has not understood a single word." - Niels Bohr

Algorithms Used in Experimental Mathematics (again)

Symbolic computation for algebraic and calculus manipulations.

Integer-relation methods, especially the “PSLQ” algorithm. High-precision integer and floating-point arithmetic. High-precision evaluation of integrals and infinite series

summations. The Wilf-Zeilberger algorithm for proving summation

identities. Iterative approximations to continuous functions. Identification of functions based on graph characteristics. Graphics and visualization methods targeted to

mathematical objects.

The Wilf-Zeilberger Algorithmfor Proving Identities

A slick, computer-assisted proof scheme to prove certain types of (holonomic) identities

Provides a nice complement to PSLQ

PSLQ and the like permit one to discover new identities but do not constitute rigorous proof

W-Z methods permit one to prove certain types of identities but do not suggest any means to discover the identity

Example Usage of W-Z

Recall these experimentally-discovered identities (from Part I):

Guillera cunningly started by defining

He then used the EKHAD software package to obtain the companion

Example Usage of W-Z, II

When we define

Zeilberger's theorem yields the identity

which when written out is

A limit argument completes the proof of Guillera’s identities.

History of Numerical Quadrature

1670: Newton devises Newton-Coates integration. 1740: Thomas Simpson develops Simpson's rule. 1820: Gauss develops Gaussian quadrature. 1950-1980: Adaptive quadrature, Romberg integration,

Clenshaw-Curtis integration, others. 1985-1990: Maple and Mathematica feature built-in numerical

quadrature facilities. 2000: Very high-precision (exp-exp) quadrature (1000+ digits).

With these extreme-precision values, we can now use PSLQ to obtain analytical evaluations of integrals (not just sums).

Goals for Today

First lecture: numerical quadrature more quadrature, Wilf-Zeilberger and applications to physics, binomial series and pi

Second lecture: Continued and exploring strange functions

15million popsicle sticks

The Euler-Maclaurin Formula

[Here h = (b - a)/n and xj = a + j h. Dm f(x) means m-th derivative of f(x).]

Note when f(t) and all of its derivatives are zero at a and b, the error E(h) of a simple block-function approximation to the integral goes to zero more rapidly than any power of h.

• See also Borwein-Calkin-Manna, Euler-Boole summation revisited, preprint, July 2007.

Block-Function Approximation to the Integral of a Bell-Shaped Function

Quadrature and the Euler-Maclaurin Formula

Given f(x) defined on (-1,1), employ a function g(t) such that g(t) goes from -1 to 1 over the real line, with g’(t) going to zero for large |t|. Then substituting x = g(t) yields

[Here xj = g(hj) and wj = g’(hj).]

If g’(t) goes to zero rapidly enough for large t, then even if f(x) has an infinite derivative or blow-up singularity at an endpoint, f(g(t)) g’(t) often is a nice bell-shaped function for which the E-M formula applies.

Change of perspective: change the function not the rule

Three Suitable ‘g’ Functions

The third & fourth are known as “tanh-sinh” quadrature.

Original and Transformed Integrand Function

Original function (on [-1,1]):

Transformed function using g(t) = erf t:

Tanh-Sinh Quadrature Example 1

Let

Then PSLQ yields

Several general results have now been proven, including

Example 2 (again)

where

is the primitive Dirichlet series modulo three.

Example 3 (again)

This arises in mathematical physics, from analysis of the volumes of ideal tetrahedra in hyperbolic space.

This “identity” has now been verified numerically to 20,000 digits, but no proof is known.

Note that the integrand function has a nasty singularity.

Expected and unexpected scientific spinoffs

• 1986-1996. Cray used quartic-Pi to check machines in factory• 1986. Complex FFT sped up by factor of two• 2002. Kanada used hex-pi (20hrs not 300hrs to check computation)• 2005. Virginia Tech (this integral pushed the limits)• 1995- Math Resources (another lecture)

Ш. The integral was split at the nasty interior singularityШ. The sum was `easy’.Ш. All fast arithmetic & function evaluation ideas used

Extreme Quadrature … 20,000 Digits (50 Certified)

on 1024 CPUs

Run-times and speedup ratios on the Virginia Tech G5 Cluster

Perko knots 10162 and 10163 agree: a dynamic proof

Example 4

Define

Then

This has been verified to over 1000 digits. The interval in J23 includes the singularity.

Numerical Integration: Example 5 (Jan 2006)

As we saw, the following integrals arise in Ising models

where K0 is a modified Bessel function. We then computed 400-digit numerical values, from which we found these results (now proven):

We first showed that this can be transformed to a 1-D integral:

An Ising Susceptibility Integral (bis)

As well Bailey, Crandall and I recently studied

The first few values are known: D1=2, D2= 2/3, while

Computer Algebra Systems can (with help) find the first 3

D_4 is a remarkable 1977 result due to McCoy--Tracy--Wu

An Extreme Ising Quadrature

Recently Tracy asked for help ‘experimentally’ evaluating D5

Using `PSLQ` this entails being able to evaluate a five dimensional integral to at least 50 or 100 places so that one can search for combinations of 6 to10 constants

Monte Carlo methods can certainly not do this We are able to reduce D5 to a horrifying several-page-long 3-D symbolic integral !A 256 cpu `3D-tanh-sinh’ computation at LBNL provided 500 digits in 18.2 hours on ``Bassi", an IBM Power5 system: 0.00248460576234031547995050915390974963506067764248751615870769216182213785691543575379268994872451201870687211063925205118620699449975422656562646708538284124500116682230004545703268769738489615198247961303552525851510715438638113696174922429855780762804289477702787109211981116063406312541360385984019828078640186930726810988548230378878848758305835125785523641996948691463140911273630946052409340088716283870643642186120450902997335663411372761220240883454631501711354084419784092245668504608184468...

A FIRST

Error Estimation in Tanh-Sinh Quadrature

Let F(t) be the desired integrand function, and then define f(t) = F(g(t)) g'(t), where g(t) = tanh (sinh t) (or one of the other g functions above). Then an estimate of the error of the quadrature result, with interval h, is:

• First order (m = 1) estimates are remarkably accurate (at roughly 10% cost)

• Higher-order estimates (m > 1) can be used to obtain “certificates” on the accuracy of a tanh-sinh quadrature result

For convergence analysis see:

JMB and Peter Ye, “Quadratic Convergence of ‘tanh-sinh’ Quadrature,” manuscript, available at

http://users.cs.dal.ca/~jborwein/tanh-sinh.pdf

Example of Error Estimates

Results for tanh-sinh quadrature to integrate the function

DHB and JMB, “Effective Error Estimates in Euler-Maclaurin Based Quadrature Schemes,” available at http://crd.lbl.gov/~dhbailey/dhbpapers/em-error.pdf

Cautionary Example

The two constants below agree to 42 decimal digits accuracy, but are NOT equal:

Computing this integral is nontrivial, due largely to difficulty in evaluating the integrand function to high precision. We’ll see this example again in the last lecture.

New Yorker - circa 1995

Apery-Like Summations

The following formulas for (n) have been known for many decades:

These results have led many to speculate that

might be some nice rational or algebraic value.

Sadly, PSLQ calculations have established that if Q5 satisfies a polynomial with degree at most 25, then at least one coefficient has 380 digits.

Apery-Like Relations Found Using Integer Relation Methods

Formulas for 7 and 11 were found by JMB and David Bradley; 5 and 9 by Kocher 25 years ago, as part of the general formula:

Newer (2005) Results

Using bootstrapping and the “Pade/pade” function JMB and Dave Bradley then found the following remarkable result (1996):

Following an analogous – but more deliberate – experimental-based procedure, we have obtained a similar general formula for (2n+2) that is pleasingly parallel to above:

Note that this gives an Apery-like formula for (2n), since the LHS equals

• We will sketch our experimental discovery of (1) in the next few slides.

The Experimental Scheme

1. We first supposed that (2n+2) is a rational combination of terms of the form:

where r + a1 + a2 + ... + aN = n + 1 and the ai are listed increasingly.

2. We could then write:

where (m) denotes the additive partitions of m.

3. We may then deduce that

where Pk(x) are polynomials whose general form we hope to discover:

The Bootstrap Process

We can predict many of the coefficients in the next case

Coefficients Obtained

Resulting (ugly) Polynomials

After Using “Padé” Function in Mathematica or Maple

which immediately suggests the general form:

Henri Padé (1863-1953)

= Z(x)Zeta(x) =

Several Confirmations of Z(2n+2)=Zeta(2n+2) Formula

We symbolically computed the power series coefficients of the LHS and the RHS , and verified that they agree up to the term with x100.

We verified that Z(1/6), Z(1/2), Z(1/3), Z(1/4) give numerically correct values (analytic values are known).

We then affirmed that the formula gives numerically correct results for 100 pseudo-randomly chosen arguments.

We subsequently proved this formula two different ways, including using the Wilf-Zeilberger method….a summary follows.

3. was easily computer proven (Wilf-Zeilberger) (MAA?)

2

Riemann (1826-66)

Euler (1707-73)

3

1

2005 Bailey, Bradley & JMB discovered and

proved - in 3Ms - three equivalent

binomial identities

1. via PSLQ to 5,000 digits

(70 terms)

2. reduced as hoped

Wilf-Zeilberger Algorithm

is a form of automated telescoping:

AMS Steele Research Prize winner. In Maple 9.5 set:

Automating the Steps?

1. HUMAN CONJECTURE “There is a generating function for (2n+2) in terms of ”

3. PATTERN DETECTION3. PATTERN DETECTION

4. STRUCTURE DETERMINATION via Maple/Mathematica - INFINITE IDENTITY

5. ANALYTIC CONTINUATION via Gosper - FINITE IDENTITY I

6. HUMAN PURIFICATION - FINITE IDENTITY II

7. WILF-ZEILBERGER PROOF

2. DATA COLLECTION via PSLQ and Maple or Mathematica

Nothing New under the Sun

The case a=0 above is Apery’s formula for (3) !

Summary

New techniques now permit integrals, infinite series sums and other entities to be evaluated to high precision (hundreds or thousands of digits), thus permitting PSLQ-based schemes to discover new identities.

These methods typically do not suggest proofs, but often it is much easier to find a proof when one “knows” the answer is right.

Full details are in Excursions in Experimental Mathematics, or in one of the two slightly older books by Jonathan M. Borwein, David H. Bailey and (for vol 2) Roland Girgensohn. A “Reader’s Digest” version of these two books is available at http://www.experimentalmath.info

"The plural of 'anecdote' is not 'evidence'." - Alan L. Leshner, Science's publisher