Computer discovery of large Poisson polynomials with
51,000-digit computations
David H. Bailey
http://www.davidhbailey.com
Lawrence Berkeley National Lab (retired)
University of California, Davis, Dept. of Computer Science
Collaborators:
Jonathan M. Borwein and Jason Kimberley
CARMA, University of Newcastle, Australia

December 16, 2015
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The PSLQ integer relation algorithm
Let X = (xk ) be an (m + 1)-long real or complex vector. An integer relation algorithm
such as PSLQ finds a nontrivial integer vector A = (ak ) such thata0 x0 + a1 x1 + · · · + am xm = 0.
I The multipair PSLQ algorithm is a more efficient and moderately parallelizablevariant of PSLQ, the most widely used integter relation algorithm (although someuse a variant of LLL).
I Integer relation detection (by any algorithm) requires very high precision: at least(m + 1) · maxk log10 |ak | digits, both in the input data and the algorithm.
I Multipair PSLQ is extremely efficient with precision — it can usually detect a
relation when the precision is only a few percent higher than this minimum level.

I D. H. Bailey and D. J. Broadhurst, “Parallel integer relation detection: Techniques andapplications,” Mathematics of Computation, vol. 70, no. 236 (Oct 2000), 1719–1736.
2 / 21

Application of multipair PSLQ
One simple but important application of multipair PSLQ is to recognize a computed
numerical value as the root of an integer polynomial of degree m.
Example: The following constant is suspected to be an algebraic number:
α = 1.232688913061443445331472869611255647068988824547930576057634684778 . . .
What is its minimal polynomial?
Method: Compute the vector (1, α, α2 , · · · , αm ) for m = 30, then input this vector tomultipair PSLQ.
Answer (using 250-digit arithmetic):
0 = 6971 − 1440α − 20520α2 − 98280α3 − 102060α4 − 1458α5 + 80α6 − 43920α7+ 538380α8 − 336420α9 + 1215α10 − 80α12 − 56160α13 − 135540α14 − 540α15+ 40α18 − 7380α19 + 135α20 − 10α24 − 18α25 + α30
3 / 21

The Poisson potential function problem
In a 2013 study, we numerically discovered, and then proved the intriguing fact that for
rational (x , y ), the 2-D Poisson potential function φ2 (x , y ) satisfies
φ2 (x , y ) =1 X cos(mπx ) cos(nπy )
1
=
log α
2
2
2
π m,n oddm +nπwhere α is algebraic, i.e., the root of a some integer polynomial of degree m.
By computing high-precision numerical values of φ2 (x , y ) for various specific rational x
and y , and applying a multipair PSLQ program, we were able to produce the explicit
minimal polynomials for α in several specific cases.
This study was hampered by the huge complexity and computational costs involved.
I D. H. Bailey, J. M. Borwein, R. E. Crandall and J. Zucker, “Lattice sums arising from the Poisson

equation,” Journal of Physics A: Mathematical and Theoretical, vol. 46 (2013), 115201.
4 / 21Samples of minimal polynomials found by multipair PSLQ
s
5
6
7
8
9

10

Minimal polynomial corresponding to x = y = 1/s:
1 + 52α − 26α2 − 12α3 + α4
1 − 28α + 6α2 − 28α3 + α4
−1 − 196α + 1302α2 − 14756α3 + 15673α4 + 42168α5 − 111916α6 + 82264α7−35231α8 + 19852α9 − 2954α10 − 308α11 + 7α12
1 − 88α + 92α2 − 872α3 + 1990α4 − 872α5 + 92α6 − 88α7 + α8
−1 − 534α + 10923α2 − 342864α3 + 2304684α4 − 7820712α5 + 13729068α6−22321584α7 + 39775986α8 − 44431044α9 + 19899882α10 + 3546576α11−8458020α12 + 4009176α13 − 273348α14 + 121392α15−11385α16 − 342α17 + 3α18
1 − 216α + 860α2 − 744α3 + 454α4 − 744α5 + 860α6 − 216α7 + α8These computations are very expensive. The case x = y = 1/32, for instance, required
10,000-digit arithmetic and ran for 45 hours. Other runs, using even higher precision,
ultimately failed, evidently due to subtle program bugs. Help!

5 / 21

Kimberley’s formula for the degree of the polynomial
Based on our preliminary results, Jason Kimberley of the University of Newcastle,
Australia observed that the degree m(s) of the minimal polynomial associated with thecase x = y = 1/s appears to be given by the following:
Set m(2) = 1/2. Otherwise for primes p congruent to 1 mod 4, set m(p) = int2 (p/2),
where int denotes greatest integer, and for primes p congruent to 3 mod 4, setm(p) = int (p/2)(int (p/2) + 1). Then for any other positive integer s whose prime
factorization is s = p1e1 p2e2 · · · prer ,?

r −1m(s) = 4rY
2(ei −1)pim(pi ).

i=1

Does Kimberley’s formula hold for larger s? Why?
What is the true mathematical connection between the pair of rationals (x , y ) and thealgebraic number α?
6 / 21

Three improvements to the Poisson polynomial computation program

1. MPFUN2015: A new thread-safe multiprecision package.
I

Speedup: 3X

2. A new 3-level multipair PSLQ program.
I

Speedup: 4.2X

3. Parallel implementation on a 16-core system.
I

Speedup: 12.2X

Overall speedup: 156X

7 / 21

Thread safety: A major challenge for high-precision software
I Because of greatly magnified run times (often 1000X or more) of high-precisioncomputations, highly parallel implementations are in order. However, mosthigh-precision packages are not thread-safe.
I Many packages employ global read/write variables, e.g., for transcendental
function evaluation, which ruin thread safety.
I The working precision level, a global variable that is changed frequently within the
package itself and often by users also, is particularly troublesome.
One thread-safe package: The MPFR C++ package, which calls the lower-level MPFR
package, is thread-safe (provided MPFR is compiled with the thread-safe build option).
MPFR is a very well-designed, features correct rounding and is the fastest low-levelpackage currently available, but is only a low-level package.
There is no thread-safe high-level Fortran package.
8 / 21MPFUN2015: A new thread-safe arbitrary precision package
DHB has written a new package (approx. 50,000 lines of code) for arbitrary precision
floating-point computation. It is available in two versions:I MPFUN-Fort: Completely self-contained, all-Fortran version. Compilation is a
simple one-line command, which completes in a few seconds.
I MPFUN-MPFR: Calls the MPFR package for lower-level operations. Installation
is significantly more complicated (since GMP and MPFR must first be installed),
but performance is roughly 3X faster.
Both versions include a high-level language interface, using custom datatypes andoperator overloading. For most codes, only minor changes are required.
A C++ version is in the works.
The software is available at http://www.davidhbailey.com/dhbsoftware.
Full details of design, algorithms, installation and usage are given in
I D. H. Bailey, “MPFUN2015: A thread-safe arbitrary precision computation package,”manuscript, 1 Oct 2015. http://www.davidhbailey.com/dhbpapers/mpfun2015.pdf.
9 / 21

New three-level multipair PSLQ program
Employs three levels of numeric precision:
I Ordinary double precision.
I Medium precision, typically 100–2000 digits.
I Full precision, typically many thousands of digits.

When an entry of the double precision reduced vector is smaller than 10−14 , or whenan entry of one of the integer-valued double precision arrays exceeds
253 ≈ 9.007 · 1015 , the medium precision arrays are updated by matrix multiplication.
Similarly, when an entry of the medium precision reduced vector is smaller than themedium precision “epsilon,” the full-precision arrays are updated.
Substantial care must be taken to manage this three-level hierarchy, and to correctly
handle numerous atypical scenarios.

10 / 21

A fast algorithm to compute the Poisson potential function φ2 (x , y )φ2 (x , y ) =1
θ2 (z, q)θ4 (z, q)log,
2π
θ1 (z, q)θ3 (z, q)

where q = e −π and z = π2 (y + ix ). Compute the four theta functions using thefollowing rapidly convergent formulas involving complex variables:θ1 (z, q) = 2
θ2 (z, q) = 2∞
X2(−1)k−1 q (2k−1) /4 sin((2k − 1)z),

k=1
∞
X

2

q (2k−1) /4 cos((2k − 1)z),k=1

θ3 (z, q) = 1 + 2
θ4 (z, q) = 1 + 2

∞
X
k=1
∞
X

2

q k cos(2kz),
2(−1)k q k cos(2kz).

k=1
11 / 21

High-level computational algorithm
1. Given rationals x = p/s and y = q/s, select a conjectured minimal polynomial
degree m(s) (using Kimberley’s formula) and other parameters for the run.
2. Calculate φ2 (x , y ) to P2 -digit precision using the formulas from two viewgraphs
above. When done, calculate α = exp(8πφ2 (x , y )) and generate the (m + 1)-longvector X = (1, α, α2 , · · · , αm ), to P2 -digit precision.
3. Apply the three-level multipair PSLQ algorithm to X . For larger problems, employ
a parallel version of the three-level multipair PSLQ code, using the OpenMP
construct DO PARALLEL to perform certain time-intensive loops in parallel.
4. If no numerically significant relation is found, try again with a larger degree m or
higher precision P2 . If a relation is found, employ the polynomial factorization
facilities in Mathematica and Maple to ensure that the polynomial is irreducible.

12 / 21

Application program and libraries for the Poisson calculations
Description
Poisson polynomial program∗
MPFUN-MPFR package
MPFR package
GMP package
Total

Language
FortranFortranCC

Lines of code
2,000
12,00093,00083,000190,000∗ This includes the computation of φ (x , y ) and the 3-level multipair PSLQ program.
2

13 / 21

192-degree minimal polynomial found by multipair PSLQ for x = y = 1/35
−1−127936α1−119729568α2+86259615424α3−63414593420752α4−23761356230294720α5−21521006149475920864α6+611939502140287079872α7−368241444434585386649704α8+34534137997535622208800832α9+190487328667825240861480928α10+70731866698900460686569086656α11−30639321248717091789168538879600α12+1922209242084456061370070376570688α13−92225937246234731111918315818791008α14+5237085972458845348795423777776516544α15−235187168740813424983476512954481139732α16+7429223399159195923652315325576416851776α17−183888805409443551177519325744878574722592α18+3271185405201792963719366162166966763587520α19−30299858585212592953207274837044089721625936α20−359992131877582010695076712447360628122541504α21+21102025339314733846086441916632636325491195040α22−476414747523697562008419279360098519444810134848α23+7169682338305684610208768903330082567075749930088α24−77147038334570163492295981291240215533152479746240α25+576164594448741453064385715717563185909190291963744α26−2736845608824484767533372475869861068967680432337984α27+14259726273826198867906144815986191950918103600120720α28−418898418536517746458185180042147268305073686192184768α29+11576727630588055309375415098495616370526328495842993184α30−215864800528660839585002162871715414654518484882221347136α31+2990075428515164844805618172618994337159659201108555443326α32−32103405822691698883721869657536188329165809065032597536320α33+266563559238616545234045835785848715794979563412588549450912α34−1588318769762074127932309766905276050093368042694667151097024α35+4203983024865006644081516487096952711041894782888827508128720α36+45741795572596990194281792252600295928845217432176402329342656α37−888295322846459251495779433538199608179522746791003115766467360α38+9318868074863097497435367536227145321575550333899160360356887616α39−75360568030730663716613169159158032198635749984420852664531432312α40+511904470165868318936871843389229757000581029769096726546021748160α41−3023546278817700158319248501909046229060364677626955289951128897760α42+15762439699463291788277556458704672972787957041647009977980115608384α43−72736254087237112123433235647299134668422519990232924303557659578512α44+294655977143150502457165877012542972911987116693855121083332478262976α45−1023418666617067140855492177628022043641503497647927192890895515496096α46+2872077837131433597858520147550093715024860106120898508958083768560704α47−5244656757384067630568130031166724732476128944651080774800327815250116α48−4375176099824059357126327245955265067024929918868045662842672259365696α49+106846263928061290720995820179791755195031418846948198250527939644422688α50−723639781157123885224054475278025279616831219096501715148796353565656000α51+3859388734169523941501510320228124865152568559427283813598867448036672464α52−18252524309522593867333335673661626604394606878814374964432136569382704704α53+77801314121940709085266334072671847312436136451041737155601244216316832608α54−295924487894698517430363594146438746974018698578472622597771256623091267264α55+990883094386701201740101120896676716687418747972363663196338999871139087736α56−2890647897108739627601089028825831836951153432467718962671967596342224331584α57+7329280206578701233923536747523603867159738577214393279268225554697992024224α58−16529926761499923756105727653427646107756611125773189972311741253023019544512α59+36621401257350141587688985844214102107297318891842289523223570545646432107248α60−97806177157646206090474078202543089426709440278475346399840243186928760712768α61+342578593578669077931375617617876157438332517966938868387799492392630391951328α62−1318494126999138496197980784859753020673610723076044612133782854663456341738176α63+4770648039063802379141989053619421884321933507524681278243277292208295535890193α64−15472388710953109453763060955215809455238864792962040591046439233238251454944640α65+44993126133327412718394996675533962755303579158603905782013384675151509701621696α66−118993678023270686331414029749780909390537125214711043931901528442137124714048640α67+291344068959030996949609862561213601556661749401236700666696596438800158224020448α68−672232544304160020140857460151229781783982323755372371953704546852887742475693952α69+1483756546102154371991621621590988762258290163435879224071206411473630010421819712α70−3160698702072288781771809020419553277725945343183442108817290538424291765729105536α71+6501409535318122589047227869235783252522832570490016457500770280864513874652673648α72−12819483008841743167047686498254296089188806357669061135254105501043843161878338944α73+23940036962388573943673383046497610566473660443396196928270109685668366770350448320α74−41783013070841149166516454269070382975961508352312427230333483156459522180289158272α75+67368007706334464267463678880196649459723255934263643619439168617608299192825564576α76−99642265418236291841000783588241107943576928164973329187609122090423042369846900608α77+135684188062925983721829145899247186881926375911313411751554577363782314702542167104α78−175027359690635875118927274192015444148378662148930449526142186378361018115616958080α79+23061047037097602217994243304224041474746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41879784121657187449503319934776774553449947624311536879983067197120α137+2046994405582919610833914133280166599368341899820908062960382811295175875424α138−475319952147035704834648908024038830280982173827054926876842801020099391040α139+101156139413765995787243855087479738205336119604403640417144416030882137040α140−19536944183496787665897200701914355830359619924729440672204518600726485952α141+3365653573540320310956912862417397989082549665698949749718751351520029216α142−500297582644104844305032452251740515847130656744730715102464280225674048α143+59315729762438733681847579101255419553245420619721207125629069846855484α144−4088968402271371424704252106469344820514085294011337888699549616483264α145−410467411100906673223801525015092827907382852275953565568989408591520α146+244373061239573525170364306606773136690590544856331578133540062357184α147−67137173574208581929099251980225280832815704879993996789961828208784α148+15109453726979262114404719422671376896934991181254480301068783496000α149−3041110741874930638456938010940447684323532404585513230577319188704α150+543474006539496836413675511283771054220572953580018501137140774336α151−82818210860123209370721479679382853454718857015790579069608342392α152+10208909053669116781213384172905303479740054958245095883752727616α153−950607878263673381203873433978784840581837837196127719161731872α154+61537081222427663468738382440653874375219870006734261147376320α155−3614460685557548015719619196028506524834155796764598110372912α156+735146337727640447184879802116259476500819273333226229041984α157−195982957610545601975537049907586449965158403571666750972768α158+35410924134603994553881822516846688340985279903764142254528α159−4586047122379261183418802995335871699189992574849634423682α160+461731528371721725878159689755046846967144056271095432896α161−39548982563052670609179355793571460666409257926095752160α162+3249771246267792941295436011847663684436436914874939968α163−278663520870823657308884684973631236730880421051628656α164+23298555433192575099637973614016576299882162116142016α165−1572480211244215449798675303409379604821388457521312α166+64631732188205970561818215170229051834055602691904α167−359811432336185820581657539275402497742419221400α168−81408350985197555231086181640295965142859955520α169−2670526194482509900914512598125562096574018400α170+824346510881081434287176385644420962607200832α171−40421524575366405147127382177082176049208656α172+859850260877732200189610333732782641323968α173−21746942237477507610278314875487367745056α174+1166204976072113679141136387334878097216α175−16255593447525276321168587153278001172α176−1031047179999272558707644762054830656α177+57315125130135149120405695909830560α178−2190911023714361858443237127579840α179+42156348386945234230569584900496α180−400797347138034992145950828864α181−3239264180104926121034061856α182+57518923520963556655145024α183−302326293499892000458856α184−648961315190292320832α185+4804670128512328736α186+97238301302930240α187−106616047418832α188−478181605696α189+208298592α190+25664α191−α192This distinctive semi-elliptical pattern is strong evidence that the result is correct.

14 / 21

Timings for the case x = y = 1/35
Multiprecision software
ARPRECMPFUN-MPFR
MPFUN-MPFR

PSLQ code
2-level2-level3-level

Cores
1
1
1
2
4
8
16

Run time
1.599 · 106
5.249 · 105
1.240 · 105
7.585 · 104
4.121 · 104
2.476 · 104
1.021 · 104

Speedup
1.00
3.05
12.90
21.08
38.80
64.58
156.61

The run times are wall-clock run times (in seconds), measured on a 16-core 2.4 GHz
MacPro, in a typically busy environment with similar jobs running on other cores.

15 / 21

Selected runs (degrees, precision, timings, etc.) for x = y = 1/s
s
20
24
30
32
34
36
37
38
39
40
42
44
45
46
48
50

m
32
64
64
128
128
144
324
180
288
128
192
240
288
264
256
200

log10 (D)-463.84-1883.78-1868.01-7577.07-7574.93-9570.86-48431.32-14951.64-38330.14-7580.00-16993.99-26604.14-38315.08-32036.34-30248.55-18421.18

P1
160
320
350
650
650
750
1650
900
1450
650
1000
1200
1450
1350
1350
1000

P2
700
2200
2300
8200
8200
10300
51000
16000
40000
8200
18000
28000
40000
34000
32000
20000

N
1967
9297
9064
45893
45914
62282
93125412098466715345655
150364323762660001476902415316168947

M
0.81
11.33
11.33
168.20168.20267.106579.66642.984124.24
168.20829.41
2003.33
4124.24
2921.57
2586.39974.44

C
1
1
1
1
1
1
16
1
16
1
8
8
16
16
16
8

T (sec.)2.19 · 100
7.73 · 101
1.02 · 102
5.13 · 103
5.16 · 103
9.54 · 103
4.84 · 105
3.88 · 104
2.68 · 105
5.02 · 103
1.57 · 104
7.43 · 104
2.09 · 105
1.06 · 105
8.98 · 104
2.12 · 104

C · T (sec.)
2.19 · 100
7.73 · 101
1.02 · 102
5.13 · 103
5.16 · 103
9.54 · 103
7.74 · 106
3.88 · 104
4.29 · 106
5.02 · 103
1.26 · 105
5.94 · 105
3.35 · 106
1.70 · 106
1.44 · 106
1.69 · 105

s = denominator; m = degree; D = detection level; P1 = medium precision; P2 = full precision; N =number of iterations; M = Mbytes; C = cores; T = wall clock time; C · T = total core-seconds.
16 / 21

Selected runs (degrees, precision, timings, etc.) for x = 1/s, y = q/s
s
20
24
30
32
34
36
37
38
39
40
42
44
45
46
48
50

q
3
5
7
3
3
5
2
3
2
3
5
3
2
3
5
3

m
32
32
64
128
128
144
324
180
288
128
192
240
288
264
256
200

log10 (D)-359.23-336.46-1291.21-5578.06-5411.70-6831.83-35412.09-11011.83-27943.70-5674.61-12183.50-19581.93-27857.00-23318.37-21480.15-13409.44

P1
160
320
350
650
650
750
1650
900
1450
650
1000
1200
1450
1350
1350
1000

P2
700
2200
2300
8200
8200
10300
51000
16000
40000
8200
18000
28000
40000
34000
32000
20000

N
1637
1613
6867
33829
32842
45559
69127789722
45823834273
106770232713482959346987292974122468

M
0.81
11.33
11.33
168.20168.20267.106579.66642.984124.24
168.20829.41
2003.33
4124.24
2921.57
2586.39974.44

C
1
1
1
1
1
1
16
1
16
1
8
8
16
16
16
8

T (sec.)3.06 · 100
2.83 · 100
1.01 · 102
4.16 · 103
2.53 · 103
6.93 · 103
3.42 · 105
7.42 · 103
1.84 · 105
4.13 · 103
1.16 · 104
6.18 · 104
1.47 · 105
7.28 · 104
5.93 · 104
1.63 · 104

C · T (sec.)
3.06 · 100
2.83 · 100
1.01 · 102
4.16 · 103
2.53 · 103
6.93 · 103
5.47 · 106
7.42 · 103
2.94 · 106
4.13 · 103
9.27 · 104
4.95 · 105
2.35 · 106
1.17 · 106
9.50 · 105
1.30 · 105

s = denominator; m = degree; D = detection level; P1 = medium precision; P2 = full precision; N =number of iterations; M = Mbytes; C = cores; T = wall clock time; C · T = total core-seconds.
17 / 21

Analysis
From our results, in the case (1/s, 1/s) where s is even, the resulting polynomial isalways palindromic (ak = am−k ). For instance, when s = 16, it is:1 − 1376x 1 − 12560x 2 − 3550496x 3 + 81241720x 4 − 169589984x 5+ 1334964944x 6 − 24307725984x 7 + 238934926108x 8 − 1043027124704x 9+ 2328675366384x 10 − 3219896325280x 11 + 4238551472456x 12− 10247414430048x 13 + 28552105805904x 14 − 55832851687968x 15+ 70020268309062x 16− 55832851687968x 17 + 28552105805904x 18 − 10247414430048x 19+ 4238551472456x 20 − 3219896325280x 21 + 2328675366384x 22− 1043027124704x 23 + 238934926108x 24 − 24307725984x 25 + 1334964944x 26− 169589984x 27 + 81241720x 28 − 3550496x 29 − 12560x 30 − 1376x 31 + x 32

The eight real roots of this polynomial are symmetric (if α is root, so is 1/α):7.21047139876944319282 . . . × 10−4
5.70748499825567769711 . . . × 10−2
3.77402362524897556853 . . . × 10−1
8.98134799939575125663 . . . × 10−1

1.38687187660249572981 . . . × 103
1.75208520093459751406 . . . × 101
2.64969194498359597353 . . . × 100
1.11341860939725103576 . . . × 100

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Jason Kimberley’s new observations for the case (1/s, 1/s)
By examining some of the polynomials produced by our program, Jason Kimberleyobserved the following:
I The algebraic number αs is the largest real root of the associated polynomial.
I The polynomial has ϕ(s) real roots, where ϕ is the Euler totient function.
√I−αs is the largest purely imaginary root of ψs (x , 1), where ψ is a sequence ofpolynomials defined in a 2010 paper by Savin and Quarfoot of the Univ. of Utah.
Kimberley found the Savin-Quarfoot paper by doing an Internet search for“387221579866,” which is a coefficient of the polynomial for the case (1/11, 1/11).
He found another relevant paper by Zudilin by searching for “221753896032,” which isa coefficient for the case (1/13, 1/13).
Latest news: Kimberley reports that he has found a recursion that generates thepolynomials in the (1/s, 1/s) case. Details will follow.
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Conclusions
I As far as we are aware, these computations, which employed up to 51,000-digit

precision, producing polynomials with degrees up to 324 and integer coefficients
up to 10145 , constitute the largest successful integer relation computations
performed to date.
I Kimberley’s formula was affirmed in every case x = y = 1/s, for s up to 52(except for s = 41, 43, 47, 49, 51, which are still too costly to test).
I The resulting polynomial coefficients have yielded clues as to why Kimberley’s
formula holds.
I Additional research is needed to understand many other combinations, e.g.,
x = p/s and y = q/s, for different values of p, q and s, which will require evenmore extreme computation.
I While a 12X parallel speedup, with a 156X overall speedup, are certainly welcome,a scheme to efficiently employ hundreds or thousands of processor cores is needed.
A fundamentally new integer relation algorithm may well be required.
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Thanks!
I This talk is available at

http://www.davidhbailey.com/dhbtalks/dhb-wcnt2015.pdf.
I A technical report with more details, results and analysis is available at

http://www.davidhbailey.com/dhbpapers/poisson-res.pdf.
I The MPFUN-Fort and MPFUN-MPFR software packages are available at

http://www.davidhbailey.com/dhbsoftware.
I A technical report giving full details of MPFUN-Fort and MPFUN-MPFR isavailable at http://www.davidhbailey.com/dhbpapers/mpfun2015.pdf.

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