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Short of Proof: How Many Digits Are Nonetheless Correct? Short of Proof: How Many Digits are Nonetheless Correct? Folkmar Bornemann Warm-Up Quiz >> exp(pi*sqrt(67)/6) - sqrt(5280) ans = 6.12120487630818e-08 How many digits are correct? No tools allowed . ICMS 2018, NOTRE DAME,JULY 26 FOLKMAR BORNEMANN 1 CAVEAT At its highest level, numerical analysis is a mixture of science, art and bar-room brawl. —T.W. Körner: The Pleasures of Counting ICMS 2018, NOTRE DAME,JULY 26 FOLKMAR BORNEMANN 2 THE TITLE-QUESTION IS A WEE BIT OLDFASHIONED, ISN’TIT? ultimately, numerical analysis is about efficient algorithms/software . so, answering the title-question is left to the discretion of the educated user? prevalent attitudes towards the accuracy of numerical output level 0: scared! (errors all over the place: roundoff, approximation, . ) level 1: far more accurate than ever needed in practice (lots of plots) level 2: as good as it gets in the realm of noisy data (a.k.a. backward stability) level 3: adaptivity steered by accuracy requirement (“user tolerance”) in short 0: pessimistic; 1: optimistic; 2: blame the data; 3: gimme steering wheel ICMS 2018, NOTRE DAME,JULY 26 FOLKMAR BORNEMANN 3 Prologue Setting the Stage The mathematical proposition has been given the stamp of incontestability: “dispute other things; this is immovable.” —L. Wittgenstein: On Certainty ICMS 2018, NOTRE DAME,JULY 26 FOLKMAR BORNEMANN 4 POINT OF ENTRY I: THE SIAM 100-DIGIT CHALLENGE prehistory: Problem Solving Squad @ Stanford George Pólya George Forsythe Bob Floyd Don Knuth Nick Trefethen ! ! ! ! the challenge (Nick Trefethen, SIAM News & Science, January ’02) 10 problems, the answer to each is a real number sought to 10 sig. digits • the rules teams w/ six people maximum • any ideas and advice from friends and literature far and wide admissible • three months • the award $100 for the team w/ most accurate answers • They’re hard! If anyone gets 50 digits in total, I will be impressed. — Nick Trefethen ICMS 2018, NOTRE DAME,JULY 26 FOLKMAR BORNEMANN 5 THE OUTCOME contestants: 94 teams from 25 countries, 20 teams w/ perfect score among the winners (A, D, CH, GB, NL, AUS, CAN, USA, ZA) D. Lichtblau/M. Trott (team Mathematica); R. Israel/ G. Gonnet (of Maple fame) • the book Folkmar Bornemann (D) Dirk Laurie (ZA) Stan Wagon (USA) Jörg Waldvogel (CH) interviews • a wealth of different methods • 9 problems to 10,000 digits • 6 problems w/ proven digits SIAM 2004 Springer 2006 • ICMS 2018, NOTRE DAME,JULY 26 FOLKMAR BORNEMANN 6 HOW DO YOU KNOW YOUR DIGITS ARE CORRECT? letter to the editor of SIAM News by Joe Keller (Dec. ’02) I found it surprising that no proof of the correctness of the answers was given. Omitting such proofs is the accepted procedure in scientific computing. However, in a contest for calculating precise digits, one might have hoped for more. answers ranged from psychological, to sociological, to pragmatic proofs of correctness of digits are rote & dull (would have killed fun/contest) • different algorithms/programs/computers/countries, still same result • tools of the trade provide “incontestability” which is short of proof • ICMS 2018, NOTRE DAME,JULY 26 FOLKMAR BORNEMANN 7 POINT OF ENTRY II: DISTRIBUTIONS IN RANDOM MATRIX THEORY experimental evidence for universal fluctuations Takeuchi/Sano/Sasamoto/Spohn ’11 (Nature) 15 6 (b) (a) (b) 1.2 (a) 1000 0.8 int 500 g int int 2/3 2 0 g – C (t) ) 2 s ζ t B./Ferrari/Prähofer ‘08 1 10 0 slope -1/3 Γ ) d -500 0.6 ζ '( ) / ( 0.8 s -1000 l ( Circular, Airy C t (s) -1000 -500 0 500 1000 s -1 2 (µm) 10 C ∞ 0 0 1 2 ∫ (d) 0.4 (c) 10 10 10 ≡ 0.6 ≡ 2000 [email protected] ) int ζ 1500 s ( ' C s 0.2 1000 Flat, 0.4 C 500 Airy 1 int 0 g 0 1000 2000 3000 1 (µm) 0 0.2 0 1 2 3 0 20 40 60 Fig. 3 Growing DSM2 cluster with a circular (a,b) and flat (c,d) interface. (a,c) Raw images. Indicated below each image is the elapsed time after the emission of laser pulses. (b,d) Snapshots of the interfaces at t = 3,8,...,28 s for the circular case (b) and -2/3 t = 10,20,...,60 s forgrowth the flat case (d).in The gray turbulent dashed lines indicate liquid the mean radius crystal (height) of all the circular (flat) interfaces t (s) recorded at t = 28 s (t = 60 s). See also Supplementary Movies of Ref. [96]. ζ ≡ (Al/2)(Γt) 1/3 h(t, x) lt + Gt c Fig. 10 Spatial correlationtwo-point function C covariance(l;t). (a) Rescaled of c correlation function C (z;t) C (l;t)/(G t)2/3 against rescaled length recent study showed that overhangs are irrelevant' for the scaling of the interfaces [79], here, for the sake of s s0 s simplicity and direct comparison to theoretical predictions, we take the mean of all the detected heights at a 2/3 ⌘ given coordinate x to define a single-valued function h(x,t) for each interface. The spatial profilez h(xAl,t) is/ sta-2)(G t)− . The symbols indicate the experimental data for the circular and flat interfaces (top and bottom pairs of tistically equivalent at any point x because of the isotropic and homogeneous growth of thesymbols, interfaces,⌘ which, respectively), obtained at t = 10 s and 30 s for the former and t = 20 s and 60 s for the latter (from bottom to top). together with the large numbers of the realizations, provides accurate statistics for the interface fluctuations analyzedKPZ below. equation & universality The dashed and dashed-dotted lines indicate the correlation function for the Airy2 and Airy1 processes, respectively, estimated Before presenting the results of the analysis, it is worth noting different characters of the “system size” int • L, or the total lateral length, of the circular and flat interfaces in general. While the systemnumerically size of the flat by Bornemann et al. [9, 10]. (b) Integral of the rescaled correlation function Cs (t) 0 Cs0(z;t)dz for the circular interfaces is chosen a priori and fixed2 during the evolution, that of the circular interfaces is(blue the circumference circles) and flat (red diamonds) interfaces. The dashed and dashed-dotted lines indicate the⌘ values for the Airy and Airy whichh grows= linearlyh with+ timeh and is therefore• not+ independentspace-time of dynamics. This white matters most noise when one Kardar/Parisi/Zhang ’86 (4990 citations) 2 1 t xx x int • int int R takes an average of a stochastic variable,− e.g., the interface height. For the flat interfaces,processes, the spatial and respectively, gi 0 gi(z)dz. The inset shows the difference g Cs (t) for the circular interfaces, with a guide ensemble averages are equivalent provided that the system size L is much larger than the correlation length ⌘ 2 − l t1/z. In contrast, for the circular interfaces, the two averages make a significant difference,for the because eyes the indicatingrigorous the slope solution1/3. concept: M. Hairer ’13 system⇤ ⇠ size is inevitably finite and the influence of finite-size effects varies in time. To avoid this complication, R − we take below the ensemble average denoted by unless otherwise stipulated, which turns out to be the right choice when one measures characteristic quantitiesh···i such as the growth exponent b. ICMS 2018, NOTRE DAME,JULY 26 FOLKMAR BORNEMANN 8 3 Experimental results the KPZ-class interfaces. We note that this actually implies qualitative difference between the circular and flat 3.1 Scaling exponents 2 cases; it is theoretically known that the Airy2 correlation for the circular interfaces decreases as g2(z) z − First we test the Family-Vicsek scaling (2) and (4) and measure the roughness exponent a and the growth ⇠ exponent b. Figure 4 shows the interface width w(l,t) and the square root of the height-differencefor large correlationz, while the Airy1 correlation g1(z) for the flat interfaces decays faster than exponentially [10]. 1/2 function Ch(l,t) measured at different times t, for both circular and flat interfaces [Fig. 4(a,b) and (c,d), respectively]. They grow algebraically for short lengths l l and converge to time-dependent constants for ⇤ ⌧ a 1/2 a large l, in agreement with the Family-Vicsek scaling (2) and (4). Fitting w l and Ch l in the power- law regime of the data at the latest time in Fig. 4, we estimate a = 0.48(5)⇠and 0.43(6) for⇠ the circular and 3.6 Temporal correlation function In contrast to the spatial correlation, correlation in time axis is a statistical property that has not been solved yet by analytical means. It is characterized by the temporal correlation function C (t,t ) h(x,t)h(x,t ) h(x,t) h(x,t ) . (18) t 0 ⌘h 0 ih ih 0 i The temporal correlation should be measured along the directions in which fluctuations propagate in space- time, called the characteristic lines [20, 25]. In our experiment, these are simply perpendicular to the mean spatial profile of the interfaces, namely the upward and radial directions for the flat and circular interfaces, respecitvely, which are represented in both cases by the fixed x in Eq. (18). Figure 11 displays the experimental results for Ct(t,t0), obtained with different t0 for the flat (a) and circular (b) interfaces. Here, again, we find different functional forms for the two cases; with increasing t, the temporal correlation function Ct(t,t0) decays toward zero for the flat interfaces [Fig.
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