Experimental Mathematics and Optimization

1
Experimental Mathematics and Optimization

• David H Bailey
• Lawrence Berkeley National Laboratory
• http//crd.lbl.gov/dhbailey

2
Outline

• Introduction
• Application the uncertainty principle.
• Integer relation detection
• The PSLQ algorithm.
• Applications bifurcation constants, sculpture.
• Sequence limit extrapolation
• Application the Quinn-Rand-Strogatz constant.
• Infinite series summation
• The Euler-Maclaurin formula.
• Eulers transformation.
• Applications Euler sums, Apery-like series,
  Ramanujan-like series.
• Numerical integration
• Gaussian quadrature.
• Tanh-sinh quadrature.
• Applications tetrahedral integral, box
  integrals, Ising integrals, spin integrals.
• Summary

3
Methods Used in Experimental and
Computer-Assisted Mathematics

• Computer-based symbolic computation (e.g.,
  Mathematica, Maple, Magma).
• High-precision (hundreds or thousands of digits)
  numerical computation.
• Integer relation detection algorithms (notably
  the PSLQ algorithm).
• Computer-based sequence and constant recognition
  facilities.
• Minimization and maximization schemes.
• Prime verification algorithms and other
  computational number theory methods.
• The Wilf-Zeilberger algorithm for proving certain
  infinite series identities.
• Formal verification methods for instance, the
  effort to verify Thomas Hales recent proof of
  the Kepler conjecture using symbolic logic.

4
The Uncertainty Principle
Uncertainty principle from quantum mechanics
The uncertainty in position times the uncertainty
in momentum must be at least some minimum
value. Uncertainty principle from signal
processing A signal cannot be both time-limited
and band-limited -- the dispersion of the signal
times the dispersion of the Fourier transform
must be at least some minimum value. The
mathematical formulations of these two principles
are identical
where
5
A Numerical Minimization Approach to the
Uncertainty Principle
A formal proof of the uncertainty principle (UP)
is not very difficult, but it is hardly intuitive
at first reading. Here is a more computationally
and intuitively appealing approach 1. Start with
a simple tent function. 2. Evaluate the UP
product for the all perturbations of the current
function on a grid with the current
resolution. 3. Select the function from step 2
that has the smallest UP product. 4. Refine the
grid and go to 2. 5. End when grid is
sufficiently fine.
Minimizing function Gaussian function Minimum
UP product 0.25
Jonathan M. Borwein and David H. Bailey,
Mathematics by Experiment, AK Peters, 2004, pg.
183-188.
6
The Integer Relation Problem

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

(or at least within epsilon of zero). This can
be viewed as an integer programming problem
find a set of integers (ai) that minimizes
subject to the constraint
for some M. The input (xi) values must be
specified to at least dn-digit precision, where
d is the size (in digits) of the largest of the
(ai), and the integer relation detection
algorithm must be performed using at least
dn-digit precision.
7
The PSLQ Integer Relation Algorithm

• 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.
• A multi-pair variant of PSLQ is known that is
  well-suited for parallel computation.
• PSLQ constructs a sequence of integer-valued
  matrices Bn that reduces the vector y x Bn,
  until either the relation is found (as one of the
  columns of Bn), or else precision is exhausted.
• When a relation is found, the size of smallest
  entry of the vector y suddenly drops to roughly
  epsilon (i.e. 10-p, where p is the number of
  digits of precision).
• The size of this drop can be viewed as a
  confidence level that the relation is real and
  not merely a numerical artifact -- a drop of 20
  orders of magnitude almost always indicates a
  real relation.
• 1. H. R. P. Ferguson, D. H. Bailey and S. Arno,
  Analysis of PSLQ, An Integer Relation Finding
  Algorithm, Mathematics of Computation, vol. 68,
  no. 225 (Jan 1999), pg. 351-369.
• 2. D. H. Bailey and D. J. Broadhurst, Parallel
  Integer Relation Detection Techniques and
  Applications, Mathematics of Computation, vol.
  70, no. 236 (Oct 2000), pg. 1719-1736.

8
Statement of Basic PSLQ Algorithm

• Initialize
• 1. Set the n x n matrices A and B to the
  identity.
• 2. For k 1 to n do set s(k) sqrtsum_(k
  ltjltn) x(j)2 enddo set t 1 / s(1) for k
  1 to n do set y(k) t x(k) s(k) t s(k)
  enddo.
• 3. Compute the initial n x (n-1) matrix H as
  H(I,j) 0 if i lt j, H(j,j) s(j1) / s(j), and
  H(I,j) - y(i) y(j) / (s(j) s(j1)) if i gt j.
• 4. Reduce H for i 2 to n do for j i - 1
  to 1 step -1 do set t nint (H(i,j) / H(j,j))
  and y(j) y(j) t y(i) for k 1 to j do
  set H(i,k) H(i,k) - t H(j,k) enddo for k
  1 to n do set A(i,k) A(i,k) - t A(j,k) and
  B(k,j) B(k,j) t B(k,i) enddo enddo enddo.
• Iterate the following until an entry of y is
  within a reasonable tolerance of zero, or
  precision has been exhausted
• 1. Select m such that gammai H(i,i) is maximal
  when i m.
• 2. Exchange the entries of y indexed m and m 1,
  the corresponding rows of A and H, and the
  corresponding columns of B.
• 3. Remove the corner on H diagonal If m lt n - 2
  then set t(0) sqrtH(m,m)2 H(m,m1)2),
  t(1) H(m,m) / t(0) and t(2) H(m,m1) /
  t(0) for i m to n do set t(3) H(i,m),
  t(4) H(i,m1), H(i,m) t(1) t(3) t(2) t(4)
  and H(i,m1) - t(2) t(3) t(1) t(4) enddo
  endif.
• 4. Reduce H for i m 1 to n do for j
  min(i - 1, m 1) to 1 step -1 do set t nint
  (H(i,j) / H(j,j)) and y(j) y(j) t y(i) for
  k 1 to j do set H(i,k) H(i,k) - t H(j,k)
  enddo for k 1 to n do set A(i,k) A(i,k) -
  t A(j,k) and B(k,j) B(k,j) t B(k,i) enddo
  enddo enddo.
• 5. Norm bound Compute M 1 / \max_j H(j,j).
  Then there can exist no relation vector whose
  Euclidean norm is less than M.
• Upon completion, the desired relation is found in
  the column of B corresponding to the zero entry
  of y.

9
Decrease of Smallest Entry of the Vector y x
Bn in a Typical PSLQ Run
x axis n (iteration number) y axis
log10 (mink y(k))
10
Typical Scenario for Using PSLQ in Experimental
Mathematics

• Compute the numerical value of some constant
  alpha to high precision (typically several
  hundred digits).
• Examples of alpha limit of sequence, sum of
  infinite series, value of definite integral,
  result of minimization or maximization, etc.
• Conjecture, based on experience with related
  problems, the terms ti of an analytic formula
  that this constant might satisfy (with unknown
  integer or rational coefficients).
• Compute the value of each of these terms to the
  same precision as alpha.
• Form the vector X (alpha, t1, t2, , tn) as
  input to an integer relation finding algorithm.
• If the algorithm finds an integer relation for X,
  then a formula for alpha can be found by simply
  solving the relation for alpha.
• Even if no relation is found, PSLQ produces
  useful bounds on the size of any possible
  relation involving the given terms.

11
LBNLs High-Precision Software (ARPREC and QD)

• 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 (gt64 digits)
  available.
• Special routines for extra-high precision (gt1000
  dig).
• High-precision integer, real and complex
  datatypes.
• Includes many common functions sqrt, cos, exp,
  gamma, etc.
• PSLQ, root finding, numerical integration.
• An interactive Experimental Mathematicians
  Toolkit is also available.
• Available at http//www.experimentalmath.info
• This software is being used by physicists,
  climate modelers, chemists and engineers, in
  addition to mathematicians.
• Authors Xiaoye Li, Yozo Hida, Brandon Thompson
  and DHB

12
Bifurcation Points in Chaos Theory
Let B3 be the smallest r such that the logistic
iteration
exhibits 8-way periodicity instead of 4-way
periodicity. By means of a sequential
approximation scheme, one can obtain the
numerical value B3 3.54409035955 to any
desired precision.
13
PSLQ and Logistic Iteration Bifurcation Points
By computing the vector (1, t, t2, t3, , t12),
where t B3, to 200-digit precision, then
applying PSLQ, one can find that B3 is a root of
Similarly define B4 as the smallest r such that
the logistic iteration exhibits 16-way
periodicity instead of 8-way periodicity. One
can compute B4 3.564407266095 to arbitrarily
high precision as before. Recently B4(2-B4) was
identified as the root of a 128-degree polynomial
by a much more challenging computation. Thus B4
is a root of a 256-degree polynomial. These
results have subsequently been proven
formally. David H. Bailey, Jonathan M. Borwein,
Vishal Kapoor and Eric Weisstein, "Ten Problems
in Experimental Mathematics," American
Mathematical Monthly, vol. 113, no. 6 (Jun 2006),
pg. 481-409.
14
PSLQ and Sculpture
The complement of the figure-eight knot, when
viewed in hyperbolic space, has finite volume
that can be computed to be V 2.02988321281930725
0042 Recently David Broadhurst found that
This was done by computing V/sqrt(3) and the six
terms Tk (k 0 to 5)
to high precision, then applying PSLQ.
Jonathan M. Borwein and David H. Bailey,
Mathematics by Experiment, AK Peters, 2004, pg.
53-56.
15
Some Supercomputer-Class PSLQ Computations

• Identification of B4, the fourth bifurcation
  point of the logistic iteration
• Integer relation of size 121 10,000-digit
  arithmetic.
• Identification of Apery sums
• 15 integer relation problems, with size up to
  118, requiring up to 5,000-digit arithmetic.
• Discovery of recursions in the rows of Ising
  Integrals
• 2660 individual PSLQ runs, each of size 36
  1200-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 a relation involving root of a Lehmer
  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.
• Papers by DHB, Jonathan Borwein, David Bradley,
  David Broadhurst, Richard Crandall and Roland
  Girgensohn.

16
BBP-Type Formulas

• Until recently, it was widely believed that it is
  impossible to compute digits of various
  well-known mathematical constants beginning at an
  arbitrary position n, except by computing all
  digits up through position n.
• In 1996, Peter Borwein and Simon Plouffe observed
  that a well-known formula permitted this to be
  done for the binary digits of log(2)

They immediately began to seek other constants
with formulas of the type
where p(n) and q(n) are integer polynomials, with
q nonzero at nonnegative integer arguments. Any
rational-linear combination of such constants
also has this property.
17
PSLQ and the BBP Formula for Pi

• Simon Plouffe used DHBs PSLQ program, with pi,
  log(2) and about 25 other BBP-type constants as
  input (all computed to 200-digit precision), in
  an attempt to see if pi satisfies a BBP-type
  formula.
• After several attempts, the program discovered
  this formula

Indeed, this formula permits one to compute
binary (or hexadecimal) digits of pi beginning at
an arbitrary starting position. The largest such
computation to date (by Colin Percival of Simon
Fraser University) found approximately 100 binary
digits of pi beginning at the quadrillionth
(1015) position. David H. Bailey, Peter B.
Borwein and Simon Plouffe, On the Rapid
Computation of Various Polylogarithmic
Constants, Mathematics of Computation, vol. 66,
no. 218 (Apr 1997), pg. 903-913.
18
Some Other BBP-Type Identities
Papers by DHB, Peter B. Borwein, Simon Plouffe,
David Broadhurst and Richard Crandall.
19
Is There a Base-10 Formula for Pi?

• For some constants, both a base-2 and a base-3
  formula are known.
• Question Is there any base-n BBP-type formula
  for pi for nonbinary n?
• Answer A 2004 paper by Borwein, Galway and
  Borwein ruled out the existence of any degree-1
  BBP formula for pi in a non-binary base.
• This does not rule out some completely different
  scheme for computing non-binary digits of pi
  beginning at an arbitrary starting point.
• J. M. Borwein, W. F. Galway and D. Borwein,
  Finding and Excluding b-ary Machin-Type BBP
  Formulae, Canadian Journal of Mathematics, vol.
  56 (2004), pg 1339-1342.

20
A Connection Between BBP Formulas and Normality

• A real number x is said to be b-normal (or normal
  base b) if every m-long string of base-b digits
  appears, in the limit, with frequency b-m.
• Let denote fractional part. Consider the
  sequence defined by x0 0,

Result log(2) is 2-normal if and only if this
sequence is equidistributed in the unit
interval. In a similar vein, consider the
sequence x0 0, and
Result pi is 16-normal if and only if this
sequence is equidistributed in the unit interval.
A similar result holds for any constant with a
BBP-type formula. David H. Bailey and Richard E.
Crandall, "On the Random Character of Fundamental
Constant Expansions," Experimental Mathematics,
vol. 10, no. 2 (Jun 2001), pg. 175-190.
21
Richardsons Algorithm for Sequence Limit
Extrapolation

• Richardsons algorithm is one of the most widely
  used techniques for extrapolating the limit of a
  sequence to high precision.
• Algorithm Pick r (we typically set r 2 or r
  4). Given a sequence (xm), for each m gt 0 set
  Am,1 xm, and then for k 2 to k m,
  successively set

This recursive scheme generates a triangular
matrix A. The best estimates for the limit of xm
are the diagonal values Am,m. Avram Sidi,
Practical Extrapolation Methods Theory and
Applications, Cambridge University Press, 2002,
pg. 21-41.
22
Application The Quinn-Rand-Strogatz Constant of
Nonlinear Physics

• Quinn, Rand, and Strogatz recently described a
  nonlinear Winfree-oscillator mean-field system,
  by means of the formula

For large N, s 1 c / N, where c
0.605443657... What is this constant? By means
of a Richardson extrapolation scheme, implemented
on 64-CPUs of a highly parallel computer system,
we computed c 0.60544365719673274947892284244720
74752208996 This led to the conclusion that c is
the root of
where the Hurwicz zeta function is defined as
D. H. Bailey, J. M. Borwein and R. E. Crandall,
Resolution of the Quinn-Rand-Strogatz Constant
of Nonlinear Physics, manuscript, Jun 2007,
http//crd.lbl.gov/dhbailey/dhbpapers/QRS.pdf.
23
The Euler-Maclaurin Formula and Infinite Series
Summation

• The Euler-Maclaurin summation formula
  approximates a finite sum as an integral with
  high-order corrections

Here h (b - a)/n and xj a j h. Dm f(x)
means m-th derivative of f. One can use the E-M
to compute a high-precision sum for an infinite
series Explicitly compute, to high precision,
the sum of the first N terms of the series, where
N 10p (we typically set p 8, so that N
100,000,000). Then use the E-M formula to
calculate a high-precision value for the tail.
Each term of the E-M formula adds roughly p more
correct digits. A related technique, Boole
summation, can be used for alternating series.
24
Eulers Transformation for Summing Alternating
Infinite Series

• For example, Catalans constant

can be computed to 500-digit precision by setting
n 1000, then evaluating 400 terms of the second
series (a total of 1400 function
evaluations). William H. Press, et al, Numerical
Recipes, Cambridge University Press, 1966, pg.
133-134.
25
Converting All-Positive Series to Alternating
Series

• Given an all-positive series (xn), one can
  construct an alternating series (yn) with the
  same sum as follows Set y0 x0, then for n gt 0

Each of these individual summations converges
quite rapidly, so only a modest number of terms
typically need to be computed. Eulers
transformation can then be applied to find the sum
This method works fairly well, but is many times
more costly than the alternating series case. Is
there an efficient, general-purpose, numerically
robust scheme for finding high-precision values
for infinite series sums?
26
Application Euler Sum Identities
and many other specific and general
formulas. These were found by computing
high-precision values of the left-hand side (via
the E-M formula), speculating what terms might be
involved in the right-hand side, then applying
PSLQ to find the coefficients. David H. Bailey,
Jonathan M. Borwein and Roland Girgensohn,
"Experimental Evaluation of Euler Sums,"
Experimental Mathematics, vol. 3, no. 1 (1994),
pg. 17-30.
27
Apery-Like Sum Identities
These were found using PSLQ and straightforward
numerical summation
D. H. Bailey, J. M. Borwein and D. M. Bradley,
Experimental Determination of Apery-Like
Identities for Zeta(2n2), Experimental
Mathematics, vol. 15 (2006), pg. 281-289 .
28
General Apery-Like Sum Identities Discovered
and Proven by Computer

• Based on specific results obtained via a
  bootstrap process, we found the following
  general formulas and then proved them using the
  Wilf-Zeilberger algorithm

PSLQ Great for discovering new identities, but
not helpful for proof. Wilf-Zeilberger Great
for proving identities, but not helpful for
discovery. Together they are a great combination!
29
Ramanujan-Like Identities

• Guillera recently found some 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?
30
Relations Found by PSLQ(in addition to
Guilleras three relations)
David H. Bailey and Jonathan M. Borwein,
"Computer-Assisted Discovery and Proof," Feb
2007, http//crd.lbl.gov/dhbailey/dhbpapers/comp-
disc-proof.pdf.
31
Gaussian Quadrature
Gaussian quadrature approximates the integral of
a function on -1, 1 as
Here the abscissas xj are the roots of the
Legendre polynomial Pn(x) and the weights wj are
given by
Values of Pn(x) can be computed using the
recurrence
The abscissas and weights can be
pre-computed. Gaussian quadrature is the best
integration scheme for continuous, well-behaved
functions at moderate levels of precision (lt 1000
digits).
32
Numerical Integration and the Euler-Maclaurin
Formula
Here h (b - a)/n and xj a j h. Dm f(x)
means m-th derivative of f. Note when f(t) and
all of its derivatives are zero at a and b (as in
a bell-shaped curve), the error E(h) of a simple
trapezoidal approximation to the integral goes to
zero more rapidly than any power of h. Kendall
Atkinson, An Introduction to Numerical Analysis,
John Wiley, 1989, pg. 289.
33
Trapezoidal Approximation to a Bell-Shaped
Function
34
Tanh-Sinh Quadrature
Given f(x) defined on (-1,1), define g(t) tanh
(pi/2 sinh t). Then setting x g(t) yields
where xj g(hj) and wj g(hj). Since g(t)
goes to zero very rapidly for large t, then even
if f(x) has a vertical derivative or blow-up
singularity at an endpoint, the product f(g(t))
g(t) typically is a nice bell-shaped function
for which the E-M formula applies. Reducing h by
half typically doubles the number of correct
digits. Tanh-sinh quadrature is the best
integration scheme for functions with vertical
derivatives or blow-up singularities at
endpoints, or for any function at very high
precision (gt 1000 digits). 1. David H. Bailey,
Xiaoye S. Li and Karthik Jeyabalan, A Comparison
of Three High-Precision Quadrature Schemes,
Experimental Mathematics, vol. 14 (2005), no. 3,
pg. 317-329. 2. H. Takahasi and M. Mori, Double
Exponential Formulas for Numerical Integration,
Publications of RIMS, Kyoto University, vol. 9
(1974), pg. 721741.
35
Original and Transformed Integrand Functions
Original function on -1,1 with singularities at
endpoints
Transformed function on real line using tanh-sinh
substitution
36
A Tetrahedral Integral Identity
This conjectured identity arises from analysis of
volumes of ideal tetrahedra in hyperbolic space.
We have verified this numerically to 20,000
digits (using highly parallel tanh-sinh
quadrature), but no formal proof is known. D.H.
Bailey, J.M. Borwein, V. Kapoor and E. Weisstein,
Ten Problems in Experimental Mathematics,
American Mathematical Monthly, vol. 113, no. 6
(Jun 2006), pg. 481-409 .
37
Parallel Run Times for the 20,000-Digit
Tetrahedral Integral Computation
1-CPU timings are sums of timings from a 64-CPU
run, where barrier waits and communication were
not timed. We believe this is the largest single
numerical integration ever done, parallel or
serial.
38
Box Integrals

• In 2006, Luis Goddyn of SFU suggested we examine
  integrals of the form

which can be thought of as the average distance
from the origin to the sides of an n-dimensional
hypercube. The following evaluations are now
known
where
D. H. Bailey, J. M. Borwein and R. E. Crandall,
Box Integrals, Journal of Computational and
Applied Mathematics, vol. 206 (2007), pg.
196-208.
39
Ising Integrals

• We recently applied our methods to study three
  classes of integrals that arise in the Ising
  theory of mathematical physics

David H. Bailey, Jonathan M. Borwein and Richard
E. Crandall, Integrals of the Ising Class,
Journal of Physics A Mathematical and General,
vol. 39 (2006), pg. 12271-12302.
40
Computing and Evaluating Cn
Richard Crandall showed that the
multi-dimensional Cn integrals can be transformed
to 1-D integrals
where K0 is the modified Bessel function. We
used this formula to compute 1000-digit numerical
values of various Cn, from which the following
results and others were found, then proven
41
Limiting Value of Cn

• The Cn numerical values approach a limit

What is this limit? We copied the first 50
digits of this numerical value into the online
Inverse Symbolic Calculator tool, available
at http//oldweb.cecm.sfu.ca/projects/ISC/ISCmain.
html The result was
where gamma denotes Eulers constant. Note An
improved and updated version of the ISC (2.0) is
now available at http//ddrive.cs.dal.ca/isc
42
Other Ising Integral Evaluations
43
The Ising Integral E5
We were able to reduce E5, which is a 5-D
integral, to an extremely complicated 3-D
integral (see below). We computed this integral
to 250-digit precision, using a highly parallel,
high-precision 3-D quadrature program. Then we
used a PSLQ program to discover the evaluation
given on the previous page.
44
Recursions in Ising Integrals

• Consider the 2-parameter class of Ising integrals

After computing 1000-digit numerical values for
all n up to 36 and all k up to 75 (a total of
2660 individual quadrature calculations,
performed on a highly parallel computer system),
we discovered (using PSLQ) linear relations in
the rows of this array. For example, when n 3
These recursions have been proven for n 1, 2,
3, 4. Similar, but more complicated, recursions
have been found for larger n (see next
page). David H. Bailey, David Borwein, Jonathan
M. Borwein and Richard Crandall, Hypergeometric
Forms for Ising-Class Integrals, Experimental
Mathematics, to appear, http//crd.lbl.gov/dhbail
ey/dhbpapers/meijer/pdf.
45
Experimental Recursion for n 24
46
General Recursion Formulas

• We were then able to find general recursion
  formulas for each n up to 36

New Result Jonathan Borwein and Bruno Salvy
have given an explicit form for these recursions,
together with code to compute any desired case.
Jonathan M. Borwein and Bruno Salvy, A Proof of
a Recursion for Bessel Moments, manuscript,
2007, http//users.cs.dal.ca/jborwein/recursion.p
df.
47
Heisenberg Spin Integrals

• In another application of experimental
  high-precision integration to mathematical
  physics, we recently investigated some integrals
  first studied by Boos and Korepin

Here C denotes the contour x - i/2, x on real
line), and
D. H. Bailey, J. M. Borwein, R. E. Crandall, D.
Manna, New Representations for Spin Integrals,
manuscript, 2007 (work in progress).
48
Transformation of Spin Integrals

• We were able to transform this expression to the
  following more manageable form over a finite
  n-dimensional interval

By evaluating this n-dimensional integral
numerically, we have verified some analytic
evaluations given by Boos and Korepin, and hope
to extend their results.
49
Evaluations of P(n)Derived Analytically,
Confirmed Numerically
50
Evaluation of P(6)
where
51
Computational Cost of P(n)

• n Digits Processors Run Time
• 2 120 1 10 seconds
• 3 120 8 55 minutes
• 4 60 64 27 minutes
• 5 30 256 39 minutes
• 6 6 256 59 hours

52
Cautionary Example
These constants agree to 42 decimal digit
accuracy, but are NOT equal
Computing this integral is nontrivial, due to
difficulty in evaluating the integrand function
to high precision. However, examples like this
are extremely rare -- in almost all other cases
where we have found a high-precision confirmation
of this sort, the identity was later established
formally (or remains an open question).
53
Two Additional Cautionary Examples
Also
holds if and only if N does not exceed 40248. R.
Baillie, D. Borwein and J. M. Borwein,
manuscript, http//users.cs.dal.ca/jborwein/sinc-
sums.pdf.
54
Summary

• With the rise of experimental and
  computer-assisted mathematics, the field of
  mathematical research is now taking advantage of
  the relentless exponential march of Moores Law.
• High-precision computations and integer relation
  searches are at the heart of numerous new results
  in mathematics and physics.
• Efficient, robust schemes are known for
  high-precision 1-D quadrature.
• Schemes are also known for multi-D quadrature,
  sequence extrapolation and infinite series
  summation, but they are not as efficient or
  robust as those available for 1-D quadrature.
• Questions
• Can integer programming schemes or other
  optimization methods be applied to the problem of
  integer relation detection?
• Are there integer programming schemes or other
  optimization methods that can search for more
  general formulas (not just linear relations) in
  experimental math problems?
• Are there techniques in the optimization field
  that can lead to improved schemes for multi-D
  integrals, sequence extrapolation or infinite
  series summation?
• Are there applications of experimental math in
  the optimization field?