Numerical
LinearMatters
Algebra in the UK:
Research
From Cayley to Exascale Computing
February 25, 2009
Nick
Higham
Nick
Higham
SchoolofofResearch
Mathematics
Director
The University of Manchester
School of Mathematics
higham@ma.man.ac.uk
http://www.ma.man.ac.uk/~higham/
ENUMATH Conference 2011
September 5–9, 2011
Leicester
1/6
Outline
MIMS
1
Matrices
2
Applications
3
History
4
Machines and Computation
5
Towards Exascale
Nick Higham
Linear Algebra in the UK
2 / 53
What is a Matrix?
Matrix = array = table of numbers. E.g.

−4 −11
3 −6
 −17
12
2 22


1
12 −2 −1
3
0
7
1


.

Penguin Dictionary of Mathematics (4th ed., 2008):
A set of quantities arranged in a rectangular
array, with certain rules governing their
combination.
Term “matrix” coined in 1850
by James Joseph Sylvester,
FRS (1814–1897).
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Linear Algebra in the UK
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Correlation Matrix
An n × n matrix A for which aij is the correlation between
variables i and j. E.g.


1.00 −0.28
0.75 −0.09
 −0.28
1.00
0.25 −0.53 

.
 0.75
0.25
1.00 −0.08 
−0.09 −0.53 −0.08
1.00
Some properties:
symmetric,
1s on the diagonal,
off-diagonal elements between −1 and 1.
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Linear Algebra in the UK
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Correlation Matrix
An n × n matrix A for which aij is the correlation between
variables i and j. E.g.


1.00 −0.28
0.75 −0.09
 −0.28
1.00
0.25 −0.53 

.
 0.75
0.25
1.00 −0.08 
−0.09 −0.53 −0.08
1.00
Some properties:
symmetric,
1s on the diagonal,
off-diagonal elements between −1 and 1.
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Linear Algebra in the UK
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Leslie Matrix
Model for growth of female portion of an animal population;
P. H. Leslie (1945).
Model with 4 age classes:


0
9
12 6
 1/3
0
0
0
.
L=
 0
1/2
0
0
0
0
1/4 0
Row 1: average births per age class.
Subdiagonal: survival rates from one age class to next.
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Magic Square
Dürer’s Melencolia I (1514)
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Linear Algebra in the UK
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Magic Square
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Matrix (1)
Consider set of all Web pages on the internet. Define
gij = 1 if page i links to page j:
1 2 3 4

1 0 1 1 0

2
 1 0 1 0 .
3 0 1 1 1
4 0 0 1 0

Then scale rows so they sum to 1 (stochastic matrix):
1
2
3
1
0
1/2 1/2

2  1/2
0
1/2

3
0
1/3 1/3
4
0
0
1

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Nick Higham
4

0
0 
.
1/3 
0
Linear Algebra in the UK
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Matrix (2)
Google matrix for http://www.manchester.ac.uk:
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Popularity
Number of hits from Google search on exact phrase:
Correlation Matrix
Magic Square
Leslie Matrix
Google Matrix
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Nick Higham
2011
2007
1,030,000 702,000
915,000 418,000
35,900
37,600
46,000
928
Linear Algebra in the UK
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Outline
MIMS
1
Matrices
2
Applications
3
History
4
Machines and Computation
5
Towards Exascale
Nick Higham
Linear Algebra in the UK
10 / 53
“Matrices offer some
of the most powerful
techniques in modern
mathematics. In the
social sciences they
provide fresh insights
into an astonishing
variety of topics.”
Penguin, 1986.
Chapter 4:
Matrices and Matrimony in Tribal Societies.
Alan Turing Building Panorama
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Linear Algebra in the UK
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Alan Turing Building Panorama
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Alan Turing Building Panorama
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Alan Turing Building Panorama
Nonlinear least squares, Levenberg–Marquardt:
(J T J + λD)d = J T r ,
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Nick Higham
J ∈ R3200×32 for 8 images.
Linear Algebra in the UK
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Jpeg Image Format
Jpeg compression first converts from RGB to YCb Cr colour
space where Y = luminance, Cb , Cr = blue, red
chrominances, by
  
 
Y
0.299
0.587
0.114
R
 Cb  =  −0.1687 −0.3313
0.5   G  .
Cr
0.5
−0.4187 −0.0813
B
Vision has poor response to spatial detail in coloured areas
of same luminance ⇒ Cb , Cr can take greater compression.
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Linear Algebra in the UK
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Outline
MIMS
1
Matrices
2
Applications
3
History
4
Machines and Computation
5
Towards Exascale
Nick Higham
Linear Algebra in the UK
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Linear System
Jiu Zhang Suanshu
(Nine Chapters of the Mathematical Art), around 1 AD.
T /M/L = sheaves of rice stalks from top/medium/low
grade paddies.
Find yield (in dou) for each quality of sheaf, given overall
yields as follows:
3T + 2M + L = 39,
2T + 3M + L = 34,
T + 2M + 3L = 26.
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Linear Algebra in the UK
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Cayley and Sylvester
Term “matrix” coined in 1850
by James Joseph Sylvester,
FRS (1814–1897).
Matrix algebra developed by
Arthur Cayley, FRS (1821–
1895).
Memoir on the Theory of Matrices (1858).
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Linear Algebra in the UK
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Cayley
Sylvester
Enter Cambridge
University
Wrangler in Tripos
examinations
Work in London
Trinity College,
1838
Senior Wrangler,
1842
Pupil barrister from
1846; called to the
Bar in 1849
St. John’s College,
1831
Second wrangler, 1837
Elected Fellow of the
Royal Society
President of the
London Mathematical
Society
Awarded Royal
Society Copley Medal
Awarded LMS De
Morgan Medal
British Assoc. for the
Advancement of
Science
1852
Actuary from 1844;
pupil barrister from
1846; called to the Bar
in 1850
1839
1868–1869
1866–1867
1882
1880
1884
1887
President, 1883
Vice President,
1863–1865; President
of Section A, 1869
Academic Positions
Cayley
Sylvester
• Sadleirian Chair,
Cambridge 1863
• UCL, 1838
• U Virginia, 1841
• Royal Military Academy, Woolwich,
London, 1855
• Johns Hopkins University 1876
• Savilian Chair of Geometry, Oxford,
1883
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Biographies
Tony Crilly, Arthur Cayley: Mathematician Laureate of the Victorian Age,
2006.
Karen Hunger Parshall, James Joseph
Sylvester. Jewish Mathematician in a
Victorian World, 2006.
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Matrices in Applied Mathematics
Frazer, Duncan & Collar, Aerodynamics Division of
NPL: aircraft flutter, matrix structural analysis.
Elementary Matrices & Some Applications to
Dynamics and Differential Equations, 1938.
Emphasizes importance of eA .
Arthur Roderick Collar, FRS
(1908–1986): “First book to treat
matrices as a branch of applied
mathematics”.
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Linear Algebra in the UK
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History of Gaussian Elimination
Chinese used variant of GE in Nine Chapters of the
Mathematical Art.
Gauss developed GE for his work in linear regression
theory. GE first appears in Theoria Motus (1809).
Variants of GE went by various names in the first half of
20th century:
the bordering method,
the escalator method (for matrix inversion),
the square root method (Cholesky factorization),
pivotal condensation,
Doolittle’s method and Crout’s method.
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Outline
MIMS
1
Matrices
2
Applications
3
History
4
Machines and Computation
5
Towards Exascale
Nick Higham
Linear Algebra in the UK
24 / 53
William Thomson (Lord Kelvin, 1824–1907)
On a Machine for the Solution of
Simultaneous Equations,
Proc Roy Soc, 1878.
Proposed a system involving tilting plates, cords,
pulleys, for 8–10 unknowns.
Suggested iterative refinement: “There is, of course, no
limit to the accuracy thus obtainable by successive
approximations.”
Actual system for 9 unknowns built by Wilbur (1936) at
MIT. Tapes 60ft long. For 3 sig figs, about 3 times
faster than human with desk calculator.
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Linear Algebra in the UK
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R. R. M. Mallock’s Machine (1933)
Experimental analogue m/c (variable coil transformers) for solving 6
lin eqns built & tested in 1931.
M/c for 10 equations built by Cambridge Instrument Co.
Accurate to ≈ 1% of largest component. Cost ≈ £2000.
Aware of conditioning issue: “if the equations are
ill-conditioned, these errors may be serious”.
Used equilibration and iterative refinement.
“The machine could not adequately deal with ill
conditioned equations, letting out a very sharp whistle
when equilibrium could not be reached” (Croarken).
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Linear Algebra in the UK
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Lord Vivian Bowden (1910–1989)
Many years ago we made out of half a dozen transformers
a simple and rather inaccurate machine for
solving simultaneous equations—the solutions being
represented as flux in the cores of the transformers.
During the course of our experiments we
set the machine to solve the equations—
X +Y +Z =1
X +Y +Z =2
X +Y +Z =3
The machine reacted sharply—
it blew the main fuse and put all the lights out.
— B. V. BOWDEN,
The Organization of a Typical Machine (1953)
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Linear Algebra in the UK
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Lewis Fry Richardson (1881–1953)
Num soln of PDEs (1910): finitedifference methods, Richardson extrapolation (“deferred approach to
the limit”). Richardson’s method:
xk +1 = xk + αk (Axk − b).
Met. office, 1913–1916; Paisley College, 1929–1940.
First to apply mathematics, in particular the method of
finite differences, to weather prediction:
Weather Prediction by Numerical Process, 1922
(2ed, 2007).
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Linear Algebra in the UK
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Richardson on Weather Forecasting
“The detailed example of Ch. IX was worked out in
France in the intervals of transporting wounded in
1916–1918. During the battle of Champagne in April
1917 the working copy was sent to the rear, where it
became lost, to be re-discovered some months later
under a heap of coal.”
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Linear Algebra in the UK
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Richardson on Weather Forecasting
“The detailed example of Ch. IX was worked out in
France in the intervals of transporting wounded in
1916–1918. During the battle of Champagne in April
1917 the working copy was sent to the rear, where it
became lost, to be re-discovered some months later
under a heap of coal.”
“Perhaps some day in the dim future it will be possible
to advance the computations faster than the weather
advances . . . But that is a dream.”
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Linear Algebra in the UK
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Richardson on Weather Forecasting
“The detailed example of Ch. IX was worked out in
France in the intervals of transporting wounded in
1916–1918. During the battle of Champagne in April
1917 the working copy was sent to the rear, where it
became lost, to be re-discovered some months later
under a heap of coal.”
“Perhaps some day in the dim future it will be possible
to advance the computations faster than the weather
advances . . . But that is a dream.”
“Imagine a large hall like a theatre. . . the walls of this
chamber are painted to form a map of the globe.. . . A
myriad computers are at work upon the weather of the
part of the map where each sits, but each computer
attends only to one equation or part of an equation.”
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Linear Algebra in the UK
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Forecast Factory
Artist’s impression: Francois Schuiten.
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Linear Algebra in the UK
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Richard Vynne Southwell (1888–1970)
Relaxation method for Ax = b.
Examine patterns & relative magnitudes of residuals, identify best way to
reduce them.
“Like a game of chess” (Fox).
Cgce acceleration: terms overrelaxation, underrelaxation coined.
Multigrid ideas used.
Relaxation Methods in Engineering Science, 1940.
Relaxation Methods in Theoretical Physics, 1946.
“Any attempt to mechanize relaxation methods would
be a waste of time” (quoted by Young, 1990).
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Linear Algebra in the UK
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2D Flow Round Aerofoil
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Linear Algebra in the UK
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Leslie Fox (1918–1992)
PhD (1942) with Southwell. “Outstanding exponent of relaxation method”.
Mathematics Division, NPL, 1945–1956.
Set up Oxford Univ Computing Lab., 1957.
1950s papers on A−1 , Ax = b.
An Introduction to Numerical Linear
Algebra, 1964.
Early textbook treatment of computational aspects.
First textbook to describe Wilkinson’s backward error analysis.
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Linear Algebra in the UK
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James Hardy Wilkinson (1919–1986)
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Enter Cambridge
University
Second World War
National Physical
Laboratory
Elected Fellow of
the Royal Society
Turing
Wilkinson
Kings College,
1931
Bletchley Park;
breaks the Enigma
code.
1945: Senior
Scientific Officer
Trinity College, 1936
Proposal for
Development . . . of
ACE
1948–51: Head of Pilot
ACE group.
1951–56: Works on
exploitation of Pilot ACE
for solving scientific
problems
1969
1951
Ordnance Board of
Ministry of Supply
1946: Working half time
each with Turing and
Desk Computing Group
Gaussian Elimination at NPL
1946 Fox, Goodwin, Turing & Wilkinson solve 18 × 18
system on desk calculator in 2 weeks. Obtained
small residual.
1948 Fox, Huskey & Wilkinson give empirical evidence
in support of GE, even for ill conditioned matrices.
1948 Wilkinson’s confidential NPL report on the Automatic Computing Engine (ACE) gives program
implementing GE with partial pivoting and iterative refinement.
1963 Wilkinson’s backward error analysis: Rounding
Errors in Algebraic Processes.
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Turing’s Paper
Rounding-Off Errors in Matrix
Processes, Quart. J. Mech. and
Applied Math., 1948.
Proves ∃ce of A = LU; shows GE computes it.
Introduces term “condition number”.
Uses term “preconditioning”.
Describes iterative refinement for linear systems.
Exploits backward error ideas.
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Pilot Ace
1950 The Pilot ACE at the National Physical Laboratory
runs for the first time.
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Linear Algebra in the UK
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Linear Equation Solvers on the Pilot ACE
An interesting feature of the codes is
that they made a very intensive use of subroutines;
the addition of two vectors,
multiplication of a vector by a scalar,
inner products, etc.,
were all coded in this way
— J. H. Wilkinson
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Daily Mirror, 1952
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Mathematicians on Flutter
Olga Taussky, in Frazer’s group at NPL,
1940s. 6 × 6 QEPs from flutter in
supersonic aircraft.
Used Gershgorin.
Peter Lancaster, English Electric Co.,
1950s. QEPs, 2 ≤ n ≤ 20.
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Linear Algebra in the UK
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Outline
MIMS
1
Matrices
2
Applications
3
History
4
Machines and Computation
5
Towards Exascale
Nick Higham
Linear Algebra in the UK
45 / 53
TOP500, http://www.top500.org
Ranks world’s fastest computers by their performance on
the LINPACK benchmark.
Performance measured in flops (floating point
operations) per second.
Updated twice a year: SC (Nov, USA) and Germany
(June).
User can tune provided code (C and MPI).
Must obtain correct result (small residual).
Must not use Strassen’s method.
Tera: 1012 , Peta: 1015 , Exa: 1018 .
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Linear Algebra in the UK
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Rank
Site
Computer
Country
Cores
Rmax
[Pflops]
% of
Peak
1
RIKEN Advanced Inst
for Comp Sci
K Computer Fujitsu SPARC64
VIIIfx + custom
Japan
548,352
8.16
93
9.9
82
2
Nat. SuperComputer
Center in Tianjin
Tianhe-1A, NUDT
Intel + Nvidia GPU + custom
China
186,368
2.57
55
4.04
63
3
DOE / OS
Oak Ridge Nat Lab
Jaguar, Cray
AMD + custom
USA
224,162
1.76
75
7.0
25
4
Nat. Supercomputer
Center in Shenzhen
Nebulea, Dawning
Intel + Nvidia GPU + IB
China
120,640
1.27
43
2.58
49
5
GSIC Center, Tokyo
Institute of Technology
Tusbame 2.0, HP
Intel + Nvidia GPU + IB
Japan
73,278
1.19
52
1.40
85
6
DOE / NNSA
LANL & SNL
Cielo, Cray
AMD + custom
USA
142,272
1.11
81
3.98
27
NASA Ames Research
Center/NAS
DOE / OS
Lawrence Berkeley Nat
Lab
Commissariat a
l'Energie Atomique
(CEA)
Plelades SGI Altix ICE
8200EX/8400EX + IB
USA
111,104
1.09
83
4.10
26
Hopper, Cray
AMD + custom
USA
153,408
1.054
82
2.91
36
Tera-10, Bull
Intel + IB
France
138,368
1.050
84
4.59
22
DOE / NNSA
Los Alamos Nat Lab
Roadrunner, IBM
AMD + Cell GPU + IB
USA
122,400
1.04
76
2.35
44
7
8
9
10
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Linear Algebra in the UK
Power GFlo
[MW] Wa
47 / 53
28 Supercomputers in the UK
Rank
24
65
69
70
93
154
160
186
190
191
211
212
213
228
233
234
278
279
339
351
365
404
405
415
416
482
484
MIMS
Site
Computer
University of Edinburgh
Cray XE6 12-core 2.1 GHz
Atomic Weapons Establishment Bullx B500 Cluster, Xeon X56xx 2.8Ghz, QDR Infiniband
ECMWF
Power 575, p6 4.7 GHz, Infiniband
ECMWF
Power 575, p6 4.7 GHz, Infiniband
University of Edinburgh
Cray XT4, 2.3 GHz
University of Southampton iDataPlex, Xeon QC 2.26 GHz, Ifband, Windows HPC2008 R2
IT Service Provider
Cluster Platform 4000 BL685c G7, Opteron 12C 2.2 Ghz, GigE
IT Service Provider
Cluster Platform 3000 BL460c G7, Xeon X5670 2.93 Ghz, GigE
Computacenter (UK) LTD
Cluster Platform 3000 BL460c G1, Xeon L5420 2.5 GHz, GigE
Classified
xSeries x3650 Cluster Xeon QC GT 2.66 GHz, Infiniband
Classified
BladeCenter HS22 Cluster, WM Xeon 6-core 2.66Ghz, Ifband
Classified
BladeCenter HS22 Cluster, WM Xeon 6-core 2.66Ghz, Ifband
Classified
BladeCenter HS22 Cluster, WM Xeon 6-core 2.66Ghz, Ifband
IT Service Provider
Cluster Platform 4000 BL685c G7, Opteron 12C 2.1 Ghz, GigE
Financial Institution
iDataPlex, Xeon X56xx 6C 2.66 GHz, GigE
Financial Institution
iDataPlex, Xeon X56xx 6C 2.66 GHz, GigE
UK Meteorological Office
Power 575, p6 4.7 GHz, Infiniband
UK Meteorological Office
Power 575, p6 4.7 GHz, Infiniband
Cluster Platform 3000 BL460c, Xeon 54xx 3.0GHz,
Computacenter (UK) LTD
GigEthernet
Asda Stores
BladeCenter HS22 Cluster, WM Xeon 6-core 2.93Ghz, GigE
Financial Services
xSeries x3650M2 Cluster, Xeon QC E55xx 2.53 Ghz, GigE
Financial Institution
BladeCenter HS22 Cluster, Xeon QC GT 2.53 GHz, GigEthernet
Financial Institution
BladeCenter HS22 Cluster, Xeon QC GT 2.53 GHz, GigEthernet
Bank
xSeries x3650M3, Xeon X56xx 2.93 GHz, GigE
Bank
xSeries x3650M3, Xeon X56xx 2.93 GHz, GigE
IT Service Provider
Cluster Platform 3000 BL460c G6, Xeon L5520 2.26 GHz, GigE
IT Service Provider
Cluster Platform 3000 BL460c G6, Xeon X5670 2.93 GHz, 10G
Nick Higham
Linear Algebra in the UK
Cores
44376
12936
8320
8320
12288
8000
14556
9768
11280
6368
5880
5880
5880
12552
9480
9480
3520
3520
Rmax
Tflop/s
279
124
115
115
95
66
65
59
58
58
55
55
55
54
53
53
51
51
7560
8352
8096
7872
7872
7728
7728
8568
4392
47
47
46
44
44
43
43
40
40
48 / 53
59 PFlop/s 100 Pflop/s
100000000
10 Pflop/s
8.2 PFlop/s 10000000
1 Pflop/s
1000000
100 Tflop/s
SUM 100000
41 TFlop/s 10 Tflop/s
10000
1 Tflop/s
1000
N=1 1.17 TFlop/s 6-8 years
100 Gflop/s
100
N=500 59.7 GFlop/s 10 Gflop/s
10
My Laptop (6 Gflop/s)
1 Gflop/s
1
100 Mflop/s
0.1
MIMS
My iPad2 (620 Mflop/s)
400 MFlop/s 1993
1995
1997
Nick Higham
1999
2001
2003
2005
2007
Linear Algebra in the UK
2009
2011
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Need for Software Redesign
Fast ascent terascale →
petascale → exascale.
Extreme parallelism & hybrid
design.
Limits on power/clock speed.
Reducing communication
essential.
Fault tolerance required.
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Issues for Exascale
Synchronization-reducing algorithms.
Communication-reducing algorithms.
Fault resilient algorithms.
Mixed precision methods.
Reproducibility of results.
Rethink our approach to Ax = b.
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Asynchronous Jacobi
EPSRC project Novel Asynchronous Algorithms and
Software for Large Sparse Systems, 2010–2014
(Manchester, Edinburgh, Hull, Leeds & Strathclyde).
Avge iterations
250
200
150
100
50
0
2
4
6
8
10
12
14
16
18
20
22
24
16
18
20
22
24
Time (secs)
8000
7000
6000
5000
4000
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2
4
Nick Higham
6
8
10
12
14
Linear Algebra in the UK
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References I
D. J. Albers and G. L. Alexanderson, editors.
Mathematical People: Profiles and Interviews.
Birkhäuser, Boston, MA, USA, 1985.
B. V. Bowden.
The organization of a typical machine.
In B. V. Bowden, editor, Faster than Thought: A
Symposium on Digital Computing Machines, pages
67–77. Pitman, London, 1953.
M. Brown and D. G. Lowe.
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