Computing as the Third Mode of Scientific andMathematical Discovery
David H Bailey
Lawrence Berkeley National Laboratory
http://crd.lbl.gov/~dhbailey1Laplace: Father of Modern Scientific
Computing?
Computer simulations can be seen as the modern realization of a centuriesold dream known as the “clockwork universe.” This was stated mostclearly by Pierre Simon Laplace in 1773:
 “An intelligence knowing all the forces acting in nature at a given instant, as wellas the momentary positions of all things in the universe, would be able tocomprehend in one single formula the motions of the largest bodies as well as ofthe lightest atoms in the world, provided that its intellect were sufficiently powerful
to subject all data to analysis; to it nothing would be uncertain, the future as wellas the past would be present to its eyes.”

We now know that this dream, taken to its logical extreme, is unrealistic: Chaos theory teaches us that even simple physical systems exhibit chaos: slightchanges to the present state are exponentially magnified in future states.
 Quantum theory teaches us that it is fundamentally impossible to know both
positions and velocities at any instant to arbitrarily high accuracy.

But many physical systems are amenable to computer simulation, and evenchaotic systems can be studied computationally.

2Computing:
The Third Mode of Scientific Discovery

Experiment

TheoryComputing3

Progress of Scientific Supercomputers:
Data from the Top500 List

4

New, Multi-Petaflfops Systems
 Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA: Currently
installing “Hopper” system, 1 Pflop/s peak.
 National Center for Supercomputer Applications (NCSA), University of
Illinois: Plans to install IBM “Blue Waters” system, 10 Pflop/s peak, in2012.
 Lawrence Livermore National Laboratory (LLNL), Livermore, CA: Plans
to install “Sequoia” system, 20 Pflop/s peak in 2011.
 Japan’s Institute for Physical and Chemical Research (RIKEN), Kobe,
Japan: Plans to install Fujitsu system, 10 Pflop/s peak in 2011.

5

Do We Really Need a Such PowerfulComputers?
In 1998, just prior to the time when the first 1 Tflop/s (109 flop/sec) computer
was installed, a U.S. Congressional Budget Officer asked me, “won’t that
finally satisfy the needs of you scientists?”
Needless to say, the answer was a resounding “no.”In 2010, just after passing the 1 Pflop/s (1015 flop/sec) level, the answer is thesame: For many existing applications, scientists need much finer 3-D grids, or
require more sophisticated physical/mathematical models.
 Many new applications require huge amounts of computing and/or storage– chemistry, biology, environmental science, engineering.
 New, larger systems are typically fully utilized within weeks of installation.
Scientists already have sketched applications for 1 Eflop/s (1018 flop/sec)
systems (probably available in 2020). After that, 1 Zflop/s (1021 flop/sec)...
6

Computing: A Multi-Disciplinary Symphonyof 21st Century Science and Engineering












Basic physical laws.
Mathematical formulations of these laws.
Numerical algorithms to solve the mathematics.
Computational techniques to implement algorithms.
Grid generation.
Multidimensional optimization techniques to explore parameter variations.
Parallel computing methods.
Scientific visualization.
Performance monitoring and analysis.
Computer system software.
Computer system hardware.

7

Basic Physical Laws
Many large scientific computations are
merely repeated applications of simple
physical laws, e.g., Newtonʼs laws, e.g.: ##force = mass x acceleration##gravity= G M1 M2 / R2.#

200

200

400

600

800

1000

1200

�200

�400

Example: Suppose initial velocity = 100 m/s at a 45 deg angle (i.e. 70.71 m/s horizontal and
70.71 m/s vertical vel), and is subject to gravity.
 Then after 0.1 sec, horiz vel = 70.71 m/s, vert vel = 70.71 - 0.1 x 9.8 = 69.73 m/s.
 Thus object will have risen 0.1 x 69.73 = 6.973 m, and moved forward 7.071 m.
 Repeated applications yield the above curve. More accurate results can beobtained by using a smaller time interval, e.g. 0.01 sec or 0.001 sec.
 This is just for one object. Large simulations do similar computations for millions
or billions of objects, on a large 3-D grid of spatial points.
81400

Mathematical Formulationsof Physical Laws
All basic laws of physics can be encapsulated into mathematical equations.
For example, Maxwell’s equations governing light and electromagnetic
radiation can be written as:

∇·E
∇×E
∇·B
∇×B

=

4πρ
1 ∂B
= −
c ∂t
= 0
4π
1 ∂E
=
J+
c
c ∂t

9

Numerical Algorithms
 Advanced numerical algorithms are used to solve the underlying
mathematical formulations of physical laws.
 These algorithms dramatically lower the amount of computation normallyrequired – modern scientific computing could not be done without them.
 Examples:











Dense linear algebra.
Sparse linear algebra.
Spectral methods (i.e., fast Fourier transforms).
N-body methods.
Structured grid methods for partial differential equations.
Unstructured grid methods for partial differential equations.
Distribute-reduce schemes.
Sorting, searching.
Optimization, maximization, minimization.
Many others.
10Computational Techniques
Because of the realities of modern computer architectures, efficient
implementations of basic algorithms require considerable sophistication:
 Data locality is very important:The “hierarchical” design of processors (Level-1 cache, Level-2 cache,
Level-3 cache, main memory, disk memory) means that computations must
be carefully structured for good performance.
It is often more efficient to recompute a value, rather than store it to memory
and later fetch it.

 Changing computer designs means that computer programs that once
were “efficient” now must be revised.


Floating-point multiply operations once were much more expensive than add/
subtract operations, so codes were changed to exploit this. Now there is nodifference.
Computation (add, subtract, multiply, divide) was once much more expensive
than fetching and storing data to memory. Now the opposite is true.

11

Grid Generation
 In many computational simulations, a key aspect of the computer
implementation is the construction of a grid of points or polygons thatencompass the physical object under study.
 Techniques for generating efficient and effective grids are an active fieldof research.

12

Optimization Techniques After a computational simulation program has been successfully
developed, say to simulate the operation of a jet engine, then researchers
run many instances of this simulation with variations of the input
parameters, in an attempt to find an optimal configuration.
 Advanced optimization techniques permit these searches to be done
significantly more efficiently than by exhaustive search.
 The resulting discipline of multidimensional optimization now is an
essential part of high-performance computing.
 On some large supercomputers, almost all jobs perform multidimensional
optimization.

13

Parallel Computing Parallel computing (using multiple processors for a single computation)
until recently was the exclusive province of large government laboratories.
 Now, even consumer personal computers have more two, four, eight ormore “cores,” and it is essential to take advantage of these cores to fully
utilize the power of the system.
 Scientists who do not convert their programs to utilize parallel computerswill soon be left behind.
 Constructs must be inserted into computer programs to control:







Data layout.
Broadcast of data from the control node to other nodes.
Synchronization between nodes.
Communication between nodes.
Collection of data from all nodes back to a single node.
Parallel input and output of data to external disk drives.

14

Amdahl’s Law
A simple principle first enunciated by Gene Amdahl in the 1960s places limits
on speedup from parallel processing: If some fraction P of a computation is amenable to parallelization withspeedup S, then the maximum possible speedup of the entire calculation
is no more than 1 / ((1-P) + P/S).
For instance, if 99.9% of a computation can be effectively run in parallel on
10,000 processors, and the rest must be done serially, then the maximum
possible speedup is only 909 (out of 10,000).
So far scientists working on large-scale computers have been able to keep
Amdahl’s Law “at bay.” But how long will our luck hold out?
 In some arenas, such as climate modeling, computer programs are
already pressing the limits of available concurrency.
Present-day leading-edge computations typically must exhibit 108-way
concurrency at EVERY step of the computation.
15

Scientific Visualization
 With the enormous volumes of data now involved in a large scientificsimulation, it is no longer possible for a scientist to examine bits of data
one by one.
 Instead, sophisticated scientific visualization software must be employed.
 Finding better ways to generate displays, and finding newer approaches
to visualization, are active areas of research.

16Performance Monitoringand Analysis
 Scientists are often disappointed in the performance (computational
speed) of their programs – often only a few percent of the peak theoretical
performance is achieved.
 Numerous sophisticated software tools are now available to findbottlenecks and improve performance.
 The development of automatic performance tuning tools for scientificcomputer programs is an active area of research.

17System Software
Underlying all of scientific computing is an enormous body of system
software, without which scientific computing would not be possible: Operating system – Linux is the most widely used, although BSD Unix
and IBM’s AIX Unix are also used.
 Compilers – before any program is executed, it must be “compiled”, i.e.,translated to machine instructions, using sophisticated techniques toobtain the best performance.
 Support for parallel computing.
 Support for large-scale data storage.
 Support for networking.

18Computer Hardware
A large-scale scientific computer is much more than just a collection of chips– the large-scale system architecture is also important.
 Until about 2000, vector supercomputers were the most common platform
for large-scale scientific computing.
 Now most scientific computing is done on large clusters of units, each ofwhich is often an off-the-shelf personal computer system.
 The interprocessor network is very important – without a very high-speed
network, many scientific computations would be mired in network
congestion.

19

Moore’s Law: 45 Years of Sustained
Exponential Growth in Computer PowerIn 1965 Gordon Moore (cofounder of Intel with Andy
Grove and Robert Noyce)
predicted that the transistor
density of semiconductor
chips would double every 18
months or so. After 45 years,
no end is yet in sight!

2X transistors/chip every 1.5 years
Called “Moore’s Law”
20

LBNL’s National Energy Research
Scientific Computing Center (NERSC)NERSC serves a large population of users:~3000 users, ~400 projects, ~500 codes
Allocations managed by the Department of Energy:
 10% INCITE awards:
 Open to all of science, not just DOE-funded projects.
 Large allocations, extra service.
 70% production (ERCAP) awards:
 From 10,000 CPU-hours to 5,000,000 CPU-hours.
 10% each NERSC and DOE reserve for special needs.

21

NERSC 2009 Configuration
Current flagship system: “Franklin”• Cray XT4#
• 9,740 nodes; 19,480 cores• 78 Tbyte main memory• 355 Tflops/s theoretical peak• ~25 Tflops/s sustained on real scientificwork
New upgrade (3Q 2010): “Hopper”• Cray XT-5, 6,400 nodes, 153,600 cores• 1.9+ GHz AMD Opteron chips• 1.17 Pflop/s peak performance (1.17 x 1015)• Expect 100 Tflop/s sustained performance• 217 TB DDR3 memory total• Gemini Interconnect• 2 Pbyte disk, 80 Gbyte/s bandwidth• Liquid cooled22

Global Cloud System Resolving Models
Surface Altitude (feet)#

200km	  Typical	  resolu5on	
  of	
  
IPCC	
  AR4	  models	
  

25km	
  
Upper	
  limit	
  of	
  climate	
  models	
  with	  cloud	  parameteriza5ons	
  
23

1km	
  
Cloud	
  system	
  resolving	
  models	
  
are	
  a	
  transforma5onal	
  change	  Climate Modeling with CCSM
 Climate Change Simulations with CCSM: Moderate and High
Resolution Studies.
 Principal investigator: Warren Washington, NCAR.
•

Science Results:– 2000-2100 simulation onpreserving polar bear
habitat by reducing nonCO2 emissions.
– Separating human andnatural forcings inclimate change.
– Sulfate and carbon
sulfate impact isolated.
Granted 12,000,000 CPUhours in 2009.
Typical runs utilize
5,000-6,000 cores.
24

Material Modeling for Geoscience Calculation: Simulation of seismic waves through silicates, which make up
80% of the Earth’s mantle; important for understanding structures in oil welldrilling, carbon sequestration, earthquakes, etc.
 PI: John Wilkins, Ohio State University

Science Result:– Seismic analysis shows jumps in wave velocity due to structuralchanges in silicates under pressureFirst use of Quantum
Monte Carlo (QMC)
for computing elastic
constants.
Typical runs utilize
8,000 cores.

25

Low-Swirl Combustion Burner Simulation
 Computation: Numerical simulation of a lean premixed hydrogen
flame in a laboratory-scale low-swirl burner (LMC code). Uses a low
Mach number formulation, adaptive mesh refinement (AMR) anddetailed chemistry and transport.
 PI: John Bell, LBNL
Science Result:• Simulations capture cellular structure of leanhydrogen flames and provide a quantitative
characterization of enhanced local burning
structure.
• LMC dramatically reduces time and memory.
• Scales to 4K cores, typically run at 2K• Used 9,600,000 CPU-hours in 2008; allocated5,500,000 CPU-hours in 2009.
J B Bell, R K Cheng, M S Day, V E Beckner and M J Lijewski, Journal of Physics: Conference Series
125 (2008) 012027.
26

New Method for Electronic Structure Calculations
 A new linear scaling three dimensional
fragment (LS3DF) method forelectronic structure calculations now
makes possible the simulation ofnanostructures with the sameaccuracy as a direct ab initio method.
 The LS3DF method is based on theobservation that the total energy of agiven system can be broken down into
two parts:
 Long-range electrostatic energy
 Short-range quantum mechanicalenergySimulation of CdSe nanorods. The green atoms are Cd,
yellow atoms are Se, and white atoms are surface
 LBNL researchers have used a divide
hydrogen. (top) the potential on the central axis;
and conquer approach to study the(bottom) the electron state (red) and hole state (green)
total dipole moments of CdSe
isosurfaces with an isovalue of 0.0002 e/Bohr3.
quantum dots.
J. Meza, L.-W. Wang, Z. Zhao, LBNL, Nanoscience-Math27

LS3DF Code Wins 2008 ACM Gordon
Bell Prize for “Algorithm Innovation”• Run on Franklin (Cray XT4) at LBNL’s NERSC supercomputer center.
• Subsequent runs on Intrepid (IBM BlueGene/P) at Argonne National Lab and lateron Jaguar (Cray XT5) at Oak Ridge National Lab.
LS3DF

450

XT5: 442 Tflops

400• NCCS Jaguar• 36,864 cores• 33% of peak.

350

TFlop/s

Performance [ Tflop/s]#

500

JaguarIntrepidFranklin300

BG/P: 224 Tflops• ALCF Intrepid• 163,840 cores• 40% of peak

250
200

XT4: 139 TFlops

150• NERSC Franklin• 147,146 cores• 40% of peak

100
50
0
0

50,000100,000

NumberCoresof cores#28

150,000

200,000NERSC User George Smootwins 2006 Nobel Prize in Physics

COBE Experiment showedanisotropy of CMB#

Cosmic Microwave
Background Radiation(CMB): an image of theuniverse at 400,000 years.#29

Cosmic Microwave Background
Computations
Calculation: Planck full focal plane# 1 year simulation of CMB, detector
noise and foregrounds.#
 74 detectors at 9 frequencies. # 750 billion observations.#
 54,000 files, 3 TBbyte data.#
 Principal investigator: J. Borrill, LBNL.#

Science Result: #• Unprecedented 9-frequency map withentire simulated Planck data set.#

Scaling Result:#• 9-frequency problem ran for < 1 hour on
16,384 processor cores.#30

Supernovae Simulations
 Researchers at LBNL have developed
a new mathematical model based on
low Mach number analysis.
 The new method allows
astrophysicists to study the ignition
process in Type Ia supernovae, which
are used as “standard candles” incosmological studies.
 The low Mach number approach has
been validated by comparison withother approaches and promises
significant computational savings.
Vortical Structure induced by local heating in awhile dwarf. The simulation was performed
with the MAESTRO code that is based on the
low Mach number analysis developed at LBNL.
A. Almgren, J. Bell, LBNL, Base Math Program31

Quantum Effects of Photosynthesis

Sunlight absorbed bybacteriochlorophyll (green)
within the FMO protein (gray)generates a wavelike motionof excitation energy.
Quantum mechanicalproperties can be mapped
through the use of twodimensional electronic
spectroscopy.
(Image courtesy of G. Engel)Spectrum modeling and quantum dynamics simulations: Oscillations correlate with the quantum coherence in thesystem.
 Energy transfer pathways shown inside network ofphotosynthetic pigment-protein complexes.
32Single Molecule Electrical Junctions

Electron flow#

H2#
Pd#

Pd#

Shown at left is a single hydrogen molecule (in white) bridging
palladium point contacts. At right, a density plot of the dominanttransmitting electronic state reveals a significant reflection of charge
at the left Pd contact, leading to a high resistance, consistent withrecent experiments. (Red is high electronic density in the plot, blue
is low.)

33

Steven Louie,
Marvin Cohen,
UC Berkeley
Jeff Neaton, LBNL
Molecular Foundry

Experimental Math: Discovering New
Mathematical Results by Computer Compute various mathematical entities (limits, infinite series sums,
definite integrals) to high precision, typically 100-1000 digits.
 Use algorithms such as PSLQ to recognize these entities in terms of wellknown mathematical constants.
 When results are found experimentally, seek to find formal mathematical
proofs of the discovered relations.
Many results have recently been found using this methodology, both in puremathematics and in mathematical physics.
“If mathematics describes an objective world just like physics, there is no
reason why inductive methods should not be applied in mathematics justthe same as in physics.” – Kurt Godel

Mathematics

Computerscience

Scientificcomputing34

Mathematics

The PSLQ Integer Relation Algorithm
Let (xn) be a given vector of real numbers given to high precision. An
integer relation algorithm finds integers (an) such thata1 x1 + a2 x2 + · · · + an xn= 0(or within some “epsilon” of zero).

At the present time the “PSLQ” algorithm of mathematician-sculptor
Helaman Ferguson is the most widely used integer relation algorithm,
although the “LLL” algorithm is also used. PSLQ was named one of ten“algorithms of the century” by Computing in Science and Engineering.

1. H. R. P. Ferguson, DHB and S. Arno, “Analysis of PSLQ, an integer relation finding algorithm,”
Mathematics of Computation, vol. 68, no. 225 (Jan 1999), pg. 351-369.
2. DHB and D. J. Broadhurst, “Parallel integer relation detection: Techniques and applications,” Mathematics
of Computation, vol. 70, no. 236 (Oct 2000), pg. 1719-1736.

35

Ferguson’s PSLQ Algorithm Solves
Problem Inspired by Ferguson’s Sculpture
The complement of the figure-eight knot,
when viewed in hyperbolic space, has finite
volume#2.029883212819307250042… #Recently physicist David Broadhurst found,using PSLQ, that this constant is given by
the formula:#

J. M. Borwein and D. H. Bailey, Mathematics by Experiment,
A.K. Peters, 2004, pg. 53.#
36

PSLQ Discovery:
The BBP Formula for Pi
In 1996, this new formula for π was found using a PSLQ program:

π

=

�
�
∞
�
1
4
2
1
1
−
−
−
k
16
8k + 1 8k + 4 8k + 5 8k + 6

k=0This formula permits one to compute binary (or hexadecimal) digits of π
beginning at an arbitrary starting position, using a very simple scheme thatcan run on any system, using only standard 64-bit or 128-bit arithmetic.
Recently it was proven that no base-n formulas of this type exist for π,
except n = 2m.

1. DHB, P. B. Borwein and S. Plouffe, “On the rapid computation of various polylogarithmic constants,”
Mathematics of Computation, vol. 66, no. 218 (Apr 1997), pg. 903-913.
2. 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.
37

Some Other New BBP-Type Formulas
Discovered Using PSLQ
π2

=

π2

=

ζ(3) =

∞

1� 1
8
64k

�

144
216
72
54
9
−
−
−
+
(6k + 1)2
(6k + 2)2
(6k + 3)2
(6k + 4)2
(6k + 5)2
k=0
�
∞
2 � 1
243
405
81
27
−
−
−
27
729k (12k + 1)2(12k + 2)2(12k + 4)2(27k + 5)2
k=0
�
72
9
9
5
1
−
−
−
−
+
(12k + 6)2(12k + 7)2(12k + 8)2(12k + 10)2
(12k + 11)2
�
∞
1 � 1
6144
43008
24576
30720
−
+
+
1792
212k (24k + 1)3(24k + 2)3(24k + 3)3(24k + 4)3

�

k=0

1536
3072
768
3072
2688
+
+
−
−
(24k + 5)3(24k + 6)3(24k + 7)3(24k + 9)3(24k + 10)3
192
1536
96
672
384
−
−
−
−
−
(24k + 11)3
(24k + 12)3
(24k + 13)3
(24k + 14)3
(24k + 15)3
24
48
12
120
48
+
+
−
+
+
(24k + 17)3
(24k + 18)3
(24k + 19)3
(24k + 20)3
(24k + 21)3
�
42
3
−
+
(24k + 22)3
(24k + 23)3−

where ζ is the Riemann zeta function.

DHB, “A compendium of BBP-type formulas,” 2010, http://crd.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf.
38

A Connection Between BBP Formulasand Digit Randomness
Let {} denote�fractional part. Consider the sequence defined by x0 = 0, �xn=

120n2 − 89n + 16
16xn−1 +512n4 − 1024n3 + 712n2 − 206n + 21

We showed that π is 16-normal (“random” base-16 digits in a certain sense)
if and only if this sequence is equidistributed in the unit interval.
Further, we proved that the following mathematical constant is 2-normal:

α2,3

∞
�

=

1
n 23n
3
n=1

=

0.041883680831502985071252898624571682426096 . . .10=

0.0ab8e38f 684bda12f 684bf 35ba781948b0f cd6e9e0 . . .16

1. D. H. Bailey and R. E. Crandall, "On the Random Character of Fundamental Constant Expansions,"
Experimental Mathematics, vol. 10, no. 2 (Jun 2001), pg. 175-190.
2. D. H. Bailey and R. E. Crandall, “Random Generators and Normal Numbers,” Experimental Mathematics,
vol. 11, no. 4 (2002), pg. 527-546.#
3. D. H. Bailey, "A Pseudo-Random Number Generator Based on Normal Numbers," manuscript, Dec 2004,
http://crd.lbl.gov/~dhbailey/dhbpapers/normal-random.pdf.#
39

Ising Integrals from Mathematical
Physics
We recently applied our methods to study three classes of integrals thatarise in the Ising theory of mathematical physics – Dn and two others:

Cn

:=

Dn

:=

En

=

� ∞

� ∞

du1
dun
·
·
·
�2
��
u1
un
n
0
0
(u
+
1/u
)
j
j=1 j
�2
�
�
ui −uj� ∞� ∞i<j ui +uj
du1
4
dun
···
···
�2
��
n! 0
u1
un
n
0
(u
+
1/u
)
j
j=1 j
2

� 1
� 1
� uk − uj dt2 dt3 · · · dtn
2···
uk + uj
0
0
4
n!

···

1

1≤j<k≤nwhere in the last line uk = t1 t2 … tk.

DHB, J. M. Borwein and R. E. Crandall, “Integrals of the Ising class,” Journal of Physics A: Mathematicaland General, vol. 39 (2006), pg. 12271-12302.
40

Limiting Value of Cn:
What Is This Number?
The Cn numerical values appear to approach a limit. For instance,

C1024 = 0.63047350337438679612204019271087890435458707871273234 . . .
What is this limit? We copied the first 50 digits of this numerical value into
the online Inverse Symbolic Calculator (ISC):
http://ddrive.cs.dal.ca/~isc or http://carma-lx1.newcastle.edu.au:8087/

The result was:lim Cn

n→∞

=

2e−2γ

where gamma denotes Euler’s constant. Finding this limit led us to theasymptotic expansion and made it clear that the integral representation of
Cn is fundamental.

41

Other Ising Integral Evaluations Found
Using PSLQ Calculations
D2

=

1/3

D3

=

D4

=

8 + 4π 2 /3 − 27 L−3 (2)

E2

=

E3

=

E4

=

E5

?

=

4π 2 /9 − 1/6 − 7ζ(3)/2
6 − 8 log 2

10 − 2π 2 − 8 log 2 + 32 log2 2

22 − 82ζ(3) − 24 log 2 + 176 log2 2 − 256(log3 2)/3+16π 2 log 2 − 22π 2 /3

42 − 1984 Li4 (1/2) + 189π 4 /10 − 74ζ(3) − 1272ζ(3) log 2+40π 2 log2 2 − 62π 2 /3 + 40(π 2 log 2)/3 + 88 log4 2+464 log2 2 − 40 log 2

where ζ is the Riemann zeta function and Lin(x) is the polylog function. D2,
D3 and D4 were originally provided to us by mathematical physicist Craig
Tracy, who hoped that our tools could help identify D5.
42Discovering AND Proving New
Mathematical Formulas by Computer
For certain types of mathematical formulas, we can discover them using
PSLQ, then prove them using the Wilf-Zeilberger algorithm.
Here is one example of a new mathematical result that was both discovered
and proven by computer:
∞
�

n=0

ζ(2n + 2)x2n∞�
1 − πx cot(πx)
1
=
=
2
2x
k 2 − x2
k=1
�
�
k−1
∞
2
2
�
�
1 − 4 x /m
1
= 3
�2k�
2 /m2
2
2 /k 2 )
1
−
x
k
(1
−
x
k
m=1
k=1

DHB, J. M. Borwein and D. M. Bradley, “Experimental Determination of Apery-Like Identities for Zeta(2n+2),”
Experimental Mathematics, vol. 15 (2006), pg. 281-289.
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Summary
 Scientific computing has vastly expanded in sophistication and powerover the past 40 years, and is now widely regarded as a third mode ofscientific discovery, after theory and experiment.
 Modern scientific computing is a multidisciplinary “symphony” involving
scientists and engineers from many fields.
 The power of the leading-edge systems has closely followed Moore’s
Law in an exponentially upward path, and no end is yet in sight.
 Computers have recently been applied into mathematical research,
discovering many new previously unknown results, including a new
formula for pi.
Major challenges:• In order to utilize future systems efficiently, computer programs mustpossess and exhibit enormous parallelism (1010-way or more).
• Exotic architectures (e.g., hybrid systems employing game processors)
will present difficult programming and software challenges.
• Daunting electric power requirements projected for future systems will
require innovation in hardware, software and applications.
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