/*** A. LANGUASCO & A. ZACCAGNINI ***/ /*** Implementation of Pintz-Ruzsa's method paper ***/ /*** (as described in their paper paper on Acta Arith. 109,(2003)) ***/ /*** Improved by K. BELABAS using the gexpo function ***/ /*** Improved by K. BELABAS using a more clever matrices generation ***/ /*** Improved dyadic search in the PintzRuzsa_psiapprox function - it allows to work with smaller inputs ***/ global(L, L1, majpsiU, minpsiU, majpsiL, minpsiL, momentmajU, momentminU, momentmajL, momentminL); global(SIN, COS); \\ Goal function and its first derivative : gam e gamprime \\ gamma is internally used for the Euler Gamam function gam(i,lambda) = exp(lambda*COS[i+1]); gamprime(i,lambda) = { local(g = gam(i,lambda)); [g, -lambda*SIN[i+1]*g]; } \\ gexpo is a libpari function; returns the binary exponent of x or \\ the maximal binary exponent of the coefficients of x. \\ lg is a libpari function; returns the length of x install(gexpo, lG); \\ main function: it realizes the approximation via the matrix method \\ c is the level \\ k is the degree of the polynomials \\ lambda is the point in which we evaluate psi \\ numdigits is the precision for the final result {Ruzsa(c, k, lambda, numdigits) = \\ local variables local(log2 = log(2)); local(c1 = c*log2); local(eU, eL, expU, expL, eps2, K2, g1, g2); local(sigmaupper, sigmalower, sigma0upper,sigma0lower, rhoupper, delta, rholower, Uupper, Ulower, Bupper, Blower, K, j, l, eps1, i, i1,i2, m, j1, j2,alphaupp, alphalow, betalow,betaupp); \\ Building the coefficients of the polynomials \\print("Building the coefficients of the polynomials"); K = k+1; K2 = 2*K; sigma0upper = 0; sigmaupper=vector(k); rhoupper=vector(k); sigma0lower = 0; sigmalower=vector(k); rholower=vector(k); for(i=1,K, g1 = gamprime(2*i-2, lambda); sigma0upper += g1[1]; g2 = gamprime(2*i-1, lambda); sigma0lower += g2[1]; for (j=1, k, sigmaupper[j] += g1[1] * COS[(j*(2*i-2)) % K2 + 1]; sigmalower[j] += g2[1] * COS[(j*(2*i-1)) % K2 + 1]; rhoupper[j] += g1[2] * SIN[(j*(2*i-2)) % K2 + 1]; rholower[j] += g2[2] * SIN[(j*(2*i-1)) % K2 + 1]; ); ); sigma0upper /= K; sigmaupper /= K; rhoupper /= K; sigma0lower /= K; sigmalower /= K; rholower /= K; \\ print(sigma0upper); \\ print(sigma0lower); \\ print(sigmaupper); \\ print(sigmalower); \\ print(rhoupper); \\ print(rholower); Bupper=vector(k+1); Blower=vector(k+1); Bupper[1]=sigma0upper; Blower[1]=sigma0lower; for (i=1, k, Bupper[i+1]= (2/K) * ((K-i)*sigmaupper[i]-rhoupper[i]); Blower[i+1]= (2/K) * ((K-i)*sigmalower[i]-rholower[i]); ); \\ printp("Bupper =", Bupper); \\ printp("Blower =", Blower); \\ Building the matricies U \\print("Building the matricies U"); m=k-1; Uupper = matrix(m+1,m+1); Ulower = matrix(m+1,m+1); \\ print("k= ", k); \\ This definition follows from line -6 and -5 at p.178 pf Pintz-Ruzsa paper. for (j=1,m+1, forstep (n = (j-1)%2, m, 2, j1=(j-1+n)\2; j2= abs(j-1-n)\2; Uupper[n+1,j1+1] += Bupper[j]/2; Uupper[n+1,j2+1] += Bupper[j]/2; Ulower[n+1,j1+1] += Blower[j]/2; Ulower[n+1,j2+1] += Blower[j]/2; ) ); \\ printp("Uupper =", Uupper); \\ printp("Ulower =", Ulower); \\ eps1 is the desired precision eps1= 10^(-(numdigits)); \\ print(eps1); L = 0; \\ expU and expL are used to keep track of the exponents used expU = 0; expL = 0; eps2 = eps1^3; \\print("starting the repeated squaring step"); until (delta>= eU; expU += eU); eL = gexpo(Ulower); if (eL, Ulower >>= eL; expL += eL); \\ second normalization step: the entry in the lower right corner \\ decreases too quickly comparing to the others if (abs(Uupper[m+1,m+1]) < eps2, Uupper[m+1,m+1] = 0); if (abs(Ulower[m+1,m+1]) < eps2, Ulower[m+1,m+1] = 0); \\ squaring out the matrices and keeping track of the exponents Uupper = Uupper^2; expU *= 2; Ulower = Ulower^2; expL *= 2; \\ keeping track of the number of squares L++; \\ print ("L= ", L); \\ approximations, see Lemma 7, p. 179, of Pintz-Ruzsa paper betaupp = sum(i=1,m+1, Uupper[1,i]); alphaupp = Uupper[1,1]; \\ computing the precision delta = log(betaupp / alphaupp) / (lambda*2^L); \\ print("delta= ", delta); ); alphalow = Ulower[1,1]; betalow = sum(i=1,m+1, Ulower[1,i]); \\ computing the moments and the approximations \\print("computing the moments and the approximations"); L1 = 2^L; momentmajU = (log(betaupp) + expU * log2)/L1; momentminU = (log(alphaupp)+ expU * log2)/L1; momentmajL = (log(betalow) + expL * log2)/L1; momentminL = (log(alphalow)+ expL * log2)/L1; \\ computing the approximations for the function psi of Pintz-Ruzsa paper \\print("computing the approximations for the function psi of Pintz-Ruzsa paper"); majpsiU= (momentmajU + c1)/lambda; minpsiU= (momentminU + c1)/lambda; majpsiL= (momentmajL + c1)/lambda; minpsiL= (momentminL + c1)/lambda; \\ if the degree of the polynomials is too small we are not able to reach \\ the fixed precision \\ print("test on the degree of the polynomials"); if (abs(majpsiU-minpsiL) > eps1, error("increase the degree of the polynomials") ); \\ return the approximated value of psi(lambda) return(majpsiU); } \\ Auxiliary functions needed to generate the sin and cos values used \\ to build the approximation matrices /* COS[i] = cos((i-1)*Pi/K), SIN[i] = sin((i-1)*Pi/K) */ initSINCOS(K) = { COS = vector(2*K); SIN = vector(2*K); COS[1] = 1; COS[2] = cos(Pi/K); SIN[1] = 0; SIN[2] = sin(Pi/K); for (i = 3, K, /* could use 3M/Karatsuba to gain 1 mul*/ COS[i] = COS[i-1]*COS[2] - SIN[i-1]*SIN[2]; SIN[i] = COS[i-1]*SIN[2] + SIN[i-1]*COS[2]; ); for (i = K+1, 2*K, COS[i] = -COS[i-K]; SIN[i] = -SIN[i-K]; ); } \\ This function realizes a dyadic approximation using the previous function \\ it starts using the approximated values of psi in 1-T, 1, 1+T \\ and it moves to a new set of three points to look for a saddle point \\ which is approximated using a precision 10^(-numdigits) \\ c is the level \\ k is the degree of the polynomials \\ numdigits is the precision for the final result {PintzRuzsa_psiapprox(c,k,numdigits) = local(iteration = 0,eps0,r0,r1,r2,l0,T,ltrial); numdigits += 2; default(realprecision,numdigits+10); \\ set working precision; print("The expected number of correct decimal digits is = ", numdigits-2); eps0=10^(-numdigits); print("Approximation for the minimal lambda is = 10^(-", numdigits-2,")"); r0=0; r1=0; r2=0; T=1/10; l0=1; initSINCOS(k+1); print("start of the dyadic step"); print("Initial T = ",T*1.0); until (T0; new T = ",T)); while((ltrial-2*T) <=0, T/=10); l0 -= 2*T; r2 = r1; r1 = r0; \\ print("l0 = ", l0); r0 = Ruzsa(c,k,l0,numdigits); \\ print ("r0 = ", r0); ); until (r2>r1, iteration++; \\ print("moving to the right"); l0 += 2*T; r0 = r1; r1 = r2; \\ print("l0 = ", l0); r2 = Ruzsa(c,k,l0,numdigits); \\ print ("r2 = ", r2); ); T /= 10; ); \\ output of the final results print("Final T = ",T*1.0); print("end of the dyadic step"); print ("Needed matrix exponent for this precision is = 2^", L); print ("Number of iterations in the dyadic procedure = ", iteration); print ("The approximated final values are: "); print ("max-upper-matrix = ", majpsiU); print ("min-upper-matrix = ", minpsiU); print ("max-lower-matrix = ", majpsiL); print ("min-lower-matrix = ", minpsiL); print ("The approximated values for the moments are: "); print ("moment-max-upper-matrix = ", momentmajU); print ("moment-min-upper-matrix = ", momentminU); print ("moment-max-lower-matrix = ", momentmajL); print ("moment-min-lower-matrix = ", momentminL); print ("The minimal lambda is in [", (l0-T)*1.0, ",",(l0+T)*1.0,"]"); print ("Final result (in the centre of the interval): d = ", r1); } /*********** - with gp2c (faster) to be run as 1) > gp2c-run -pmy_ -g -W PRMethod-KB-2.gp 2) (10:01) gp > PintzRuzsa_psiapprox(1/2,14,12) - with gp (slowler) to be run as 1) gp 2) \r /path/PRMethod-KB-2.gp 3) (10:01) gp > PintzRuzsa_psiapprox(1/2,14,12) **************** piu' veloce del 25% circa rispetto alla mia. I miglioramenti sono: - la precomputazione dei valori di seno e coseno da usare - la riorganizzazione in un unico loop dei calcoli dei coefficienti **************** Inserito un test per controllare che il punto piu' a sinistra dello step diadico non diventi negativo o nullo. ****************/ /*********** Results: ----------------------------------------------------- 4/5 on a double quad core pc ----------------------------------------------------- [languasc@labsrv0 ~]$ nice /usr/local/Gruppi/PariGP/bin/gp2c-run -pmy_ -g -W PRmethod-KB.gp PRmethod-KB.gp.c: In function `my_PintzRuzsa': PRmethod-KB.gp.c:107: warning: unused variable `my_j' PRmethod-KB.gp.c:107: warning: unused variable `my_l' PRmethod-KB.gp.c:107: warning: unused variable `my_i' PRmethod-KB.gp.c:107: warning: unused variable `my_i1' PRmethod-KB.gp.c:107: warning: unused variable `my_i2' GP/PARI CALCULATOR Version 2.3.2 (released) amd64 running linux (x86-64/GMP-4.2.2 kernel) 64-bit version compiled: Nov 30 2007, gcc-3.4.3 20041212 (Red Hat 3.4.3-9.EL4) (readline v4.3 enabled, extended help available) Copyright (C) 2000-2006 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. ----------------------------------------------------- 10 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(4/5+10^(-20),13,10) The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) Needed matrix exponent for this precision is = 2^39 Number of iterations in the dyadic procedure = 162 The approximated final values are: max-upper-matrix = 0.9123781030527322323638 min-upper-matrix = 0.9123781030521349899098 max-lower-matrix = 0.9123781030527020894117 min-lower-matrix = 0.9123781030521048469576 The approximated values for the moments are: moment-max-upper-matrix = 0.6181690752736390706245 moment-min-upper-matrix = 0.6181690752728714301565 moment-max-lower-matrix = 0.6181690752736003276488 moment-min-lower-matrix = 0.6181690752728326871808 The minimal lambda is in [1.285307939545900000000,1.285307939546100000000] Final result (in the centre of the interval): d = 0.9123781030527322323638 time = 3,459 ms. ----------------------------------------------------- 20 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(4/5+10^(-20),20,20) The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) Needed matrix exponent for this precision is = 2^72 Number of iterations in the dyadic procedure = 286 The approximated final values are: max-upper-matrix = 0.91237810305275834972134056121053 min-upper-matrix = 0.91237810305275834972127103303804 max-lower-matrix = 0.91237810305275834972133828300078 min-lower-matrix = 0.91237810305275834972126875482828 The approximated values for the moments are: moment-max-upper-matrix = 0.61816907526618862724769127210477 moment-min-upper-matrix = 0.61816907526618862724760190699264 moment-max-lower-matrix = 0.61816907526618862724768834390369 moment-min-lower-matrix = 0.61816907526618862724759897879155 The minimal lambda is in [1.2853079395377972469979900000000,1.2853079395377972469980100000000] Final result (in the centre of the interval): d = 0.91237810305275834972134056121053 time = 40,779 ms. ----------------------------------------------------- 30 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(4/5+10^(-20),27,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 427 The approximated final values are: max-upper-matrix = 0.912378103052758349721358590459910416545890 min-upper-matrix = 0.912378103052758349721358590459902322401313 max-lower-matrix = 0.912378103052758349721358590459910399454390 min-lower-matrix = 0.912378103052758349721358590459902305309813 The approximated values for the moments are: moment-max-upper-matrix = 0.618169075266188626931299358124429477729334 moment-min-upper-matrix = 0.618169075266188626931299358124419074261045 moment-max-lower-matrix = 0.618169075266188626931299358124429455761493 moment-min-lower-matrix = 0.618169075266188626931299358124419052293204 The minimal lambda is in [1.28530793953779724665119741228005900000000,1.28530793953779724665119741228006100000000] Final result (in the centre of the interval): d = 0.912378103052758349721358590459910416545890 time = 3mn, 48,573 ms. ----------------------------------------------------- 50 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(4/5+10^(-20),39,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) Needed matrix exponent for this precision is = 2^172 Number of iterations in the dyadic procedure = 642 The approximated final values are: max-upper-matrix = 0.91237810305275834972135859045990929414085615244675930311308271 min-upper-matrix = 0.91237810305275834972135859045990929414085615244675924826502561 max-lower-matrix = 0.91237810305275834972135859045990929414085615244675930308170857 min-lower-matrix = 0.91237810305275834972135859045990929414085615244675924823365148 The approximated values for the moments are: moment-max-upper-matrix = 0.61816907526618862693129935808253892939620562407405246163756394 moment-min-upper-matrix = 0.61816907526618862693129935808253892939620562407405239114092069 moment-max-lower-matrix = 0.61816907526618862693129935808253892939620562407405246159723852 moment-min-lower-matrix = 0.61816907526618862693129935808253892939620562407405239110059527 The minimal lambda is in [1.2853079395377972466511974122341479975562011781514459900000000,1.2853079395377972466511974122341479975562011781514460100000000] Final result (in the centre of the interval): d = 0.91237810305275834972135859045990929414085615244675930311308271 ? ## *** last result computed in 29mn, 1,394 ms. ----------------------------------------------------- ----------------------------------------------------- 2/3 on a double quad core pc ----------------------------------------------------- [languasc@labsrv0 ~]$ nice /usr/local/Gruppi/PariGP/bin/gp2c-run -pmy_ -g -W PRmethod-KB.gp PRmethod-KB.gp.c: In function `my_PintzRuzsa': PRmethod-KB.gp.c:107: warning: unused variable `my_j' PRmethod-KB.gp.c:107: warning: unused variable `my_l' PRmethod-KB.gp.c:107: warning: unused variable `my_i' PRmethod-KB.gp.c:107: warning: unused variable `my_i1' PRmethod-KB.gp.c:107: warning: unused variable `my_i2' GP/PARI CALCULATOR Version 2.3.2 (released) amd64 running linux (x86-64/GMP-4.2.2 kernel) 64-bit version compiled: Nov 30 2007, gcc-3.4.3 20041212 (Red Hat 3.4.3-9.EL4) (readline v4.3 enabled, extended help available) Copyright (C) 2000-2006 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. ----------------------------------------------------- 10 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(2/3+10^(-20),12,10) The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) Needed matrix exponent for this precision is = 2^39 Number of iterations in the dyadic procedure = 160 The approximated final values are: max-upper-matrix = 0.8337213168426473838898 min-upper-matrix = 0.8337213168420918764748 max-lower-matrix = 0.8337213168425383672048 min-lower-matrix = 0.8337213168419828597898 The approximated values for the moments are: moment-max-upper-matrix = 0.4413015839592436256564 moment-min-upper-matrix = 0.4413015839586416916237 moment-max-lower-matrix = 0.4413015839591254978852 moment-min-lower-matrix = 0.4413015839585235638525 The minimal lambda is in [1.083575154049900000000,1.083575154050100000000] Final result (in the centre of the interval): d = 0.8337213168426473838898 time = 2,774 ms. ----------------------------------------------------- 20 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(2/3+10^(-20),19,20) The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) Needed matrix exponent for this precision is = 2^72 Number of iterations in the dyadic procedure = 281 The approximated final values are: max-upper-matrix = 0.83372131684338485515459227435255 min-upper-matrix = 0.83372131684338485515452760477806 max-lower-matrix = 0.83372131684338485515458851235650 min-lower-matrix = 0.83372131684338485515452384278202 The approximated values for the moments are: moment-max-upper-matrix = 0.44130158394251203320300447723651 moment-min-upper-matrix = 0.44130158394251203320293440289238 moment-max-lower-matrix = 0.44130158394251203320300040083107 moment-min-lower-matrix = 0.44130158394251203320293032648693 The minimal lambda is in [1.0835751540289729521859900000000,1.0835751540289729521860100000000] Final result (in the centre of the interval): d = 0.83372131684338485515459227435255 time = 34,920 ms. ----------------------------------------------------- 30 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(2/3+10^(-20),26,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 419 The approximated final values are: max-upper-matrix = 0.833721316843384855154592152588205415890590 min-upper-matrix = 0.833721316843384855154592152588197887361263 max-lower-matrix = 0.833721316843384855154592152588205404464508 min-lower-matrix = 0.833721316843384855154592152588197875935182 The approximated values for the moments are: moment-max-upper-matrix = 0.441301583942512033013203743647633606807313 moment-min-upper-matrix = 0.441301583942512033013203743647625449079988 moment-max-lower-matrix = 0.441301583942512033013203743647633594426295 moment-min-lower-matrix = 0.441301583942512033013203743647625436698970 The minimal lambda is in [1.08357515402897295195834526955999900000000,1.08357515402897295195834526956000100000000] Final result (in the centre of the interval): d = 0.833721316843384855154592152588205415890590 time = 3mn, 24,120 ms. ----------------------------------------------------- 50 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(2/3+10^(-20),37,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) Needed matrix exponent for this precision is = 2^172 Number of iterations in the dyadic procedure = 632 The approximated final values are: max-upper-matrix = 0.83372131684338485515459215258820372013640387733149037455160858 min-upper-matrix = 0.83372131684338485515459215258820372013640387733149032353630951 max-lower-matrix = 0.83372131684338485515459215258820372013640387733149037427262730 min-lower-matrix = 0.83372131684338485515459215258820372013640387733149032325732823 The approximated values for the moments are: moment-max-upper-matrix = 0.44130158394251203301320374362026095967315416832525801753662405 moment-min-upper-matrix = 0.44130158394251203301320374362026095967315416832525796225771350 moment-max-lower-matrix = 0.44130158394251203301320374362026095967315416832525801723432687 moment-min-lower-matrix = 0.44130158394251203301320374362026095967315416832525796195541632 The minimal lambda is in [1.0835751540289729519583452695271703132640461989595857900000000,1.0835751540289729519583452695271703132640461989595858100000000] Final result (in the centre of the interval): d = 0.83372131684338485515459215258820372013640387733149037455160858 time = 24mn, 37,985 ms. ----------------------------------------------------- 1/2 sul double quad core pc ----------------------------------------------------- ? PintzRuzsa_psiapprox(1/2,40,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) Needed matrix exponent for this precision is = 2^171 Number of iterations in the dyadic procedure = 603 The approximated final values are: max-upper-matrix = 0.71634354447695461010278515973159601502134200920356411936380717 min-upper-matrix = 0.71634354447695461010278515973159601502134200920356402930428895 max-lower-matrix = 0.71634354447695461010278515973159601502134200920356411936380717 min-lower-matrix = 0.71634354447695461010278515973159601502134200920356402930428895 The approximated values for the moments are: moment-max-upper-matrix = 0.29605059650077903586483304592589958334239193569045999914586253 moment-min-upper-matrix = 0.29605059650077903586483304592589958334239193569045991835442509 moment-max-lower-matrix = 0.29605059650077903586483304592589958334239193569045999914586253 moment-min-lower-matrix = 0.29605059650077903586483304592589958334239193569045991835442508 The minimal lambda is in [0.89708938083607657209433754033922238166044084406199979000000000,0.89708938083607657209433754033922238166044084406199981000000000] Final result (in the centre of the interval): d = 0.71634354447695461010278515973159601502134200920356411936380717 time = 29mn, 59,986 ms. ----------------------------------------------------- 3/5 sul double quad core pc ----------------------------------------------------- ? PintzRuzsa_psiapprox(3/5+10^(-20),12,10) The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) Needed matrix exponent for this precision is = 2^39 Number of iterations in the dyadic procedure = 154 The approximated final values are: max-upper-matrix = 0.7894008449783857472384 min-upper-matrix = 0.7894008449778546494365 max-lower-matrix = 0.7894008449783376500635 min-lower-matrix = 0.7894008449778065522616 The approximated values for the moments are: moment-max-upper-matrix = 0.3767998579988440381267 moment-min-upper-matrix = 0.3767998579983107286612 moment-max-lower-matrix = 0.3767998579987957406595 moment-min-lower-matrix = 0.3767998579982624311940 The minimal lambda is in [1.004164324597900000000,1.004164324598100000000] Final result (in the centre of the interval): d = 0.7894008449783857472384 ? ## *** last result computed in 2,680 ms. ----------------------------------------------------- ? PintzRuzsa_psiapprox(3/5+10^(-20),19,20) The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) Needed matrix exponent for this precision is = 2^72 Number of iterations in the dyadic procedure = 266 The approximated final values are: max-upper-matrix = 0.78940084497861113672781602661469 min-upper-matrix = 0.78940084497861113672775419869316 max-lower-matrix = 0.78940084497861113672781504957826 min-lower-matrix = 0.78940084497861113672775322165673 The approximated values for the moments are: moment-max-upper-matrix = 0.37679985799100143983167940968316 moment-min-upper-matrix = 0.37679985799100143983161732429009 moment-max-lower-matrix = 0.37679985799100143983167842857803 moment-min-lower-matrix = 0.37679985799100143983161634318496 The minimal lambda is in [1.0041643245877784168451900000000,1.0041643245877784168452100000000] Final result (in the centre of the interval): d = 0.78940084497861113672781602661469 ? ## *** last result computed in 29,929 ms. ----------------------------------------------------- ? PintzRuzsa_psiapprox(3/5+10^(-20),27,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 413 The approximated final values are: max-upper-matrix = 0.789400844978611136727801410347773567829976 min-upper-matrix = 0.789400844978611136727801410347766370112593 max-lower-matrix = 0.789400844978611136727801410347773567798719 min-lower-matrix = 0.789400844978611136727801410347766370081336 The approximated values for the moments are: moment-max-upper-matrix = 0.376799857991001439681626008895641571260780 moment-min-upper-matrix = 0.376799857991001439681626008895634343569765 moment-max-lower-matrix = 0.376799857991001439681626008895641571229392 moment-min-lower-matrix = 0.376799857991001439681626008895634343538378 The minimal lambda is in [1.00416432458777841665513341392003900000000,1.00416432458777841665513341392004100000000] Final result (in the centre of the interval): d = 0.789400844978611136727801410347773567829976 time = 3mn, 43,752 ms. ? ----------------------------------------------------- ? PintzRuzsa_psiapprox(3/5+10^(-20),37,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) Needed matrix exponent for this precision is = 2^171 Number of iterations in the dyadic procedure = 606 The approximated final values are: max-upper-matrix = 0.78940084497861113672780141034777146847626707023970566622571072 min-upper-matrix = 0.78940084497861113672780141034777146847626707023970556867845042 max-lower-matrix = 0.78940084497861113672780141034777146847626707023970566620725108 min-lower-matrix = 0.78940084497861113672780141034777146847626707023970556865999079 The approximated values for the moments are: moment-max-upper-matrix = 0.37679985799100143968162600887215587533323934891943983851757242 moment-min-upper-matrix = 0.37679985799100143968162600887215587533323934891943974056409367 moment-max-lower-matrix = 0.37679985799100143968162600887215587533323934891943983849903591 moment-min-lower-matrix = 0.37679985799100143968162600887215587533323934891943974054555717 The minimal lambda is in [1.0041643245877784166551334138902913773822002000066039900000000,1.0041643245877784166551334138902913773822002000066040100000000] Final result (in the centre of the interval): d = 0.78940084497861113672780141034777146847626707023970566622571072 ? ## *** last result computed in 23mn, 39,518 ms. 0:7894008449792309 ----------------------------------------------------- ----------------------------------------------------- 7/8 on a MacBook PRO ----------------------------------------------------- macbook:programmi languasc$ gp2c-run -pmy_ -g -W PRMethod-KB-2.gp PRMethod-KB-2.gp.c: In function Ômy_RuzsaÕ: PRMethod-KB-2.gp.c:105: warning: unused variable Ômy_i2Õ PRMethod-KB-2.gp.c:105: warning: unused variable Ômy_i1Õ PRMethod-KB-2.gp.c:105: warning: unused variable Ômy_iÕ PRMethod-KB-2.gp.c:105: warning: unused variable Ômy_lÕ PRMethod-KB-2.gp.c:105: warning: unused variable Ômy_jÕ GP/PARI CALCULATOR Version 2.3.5 (released) i386 running darwin (x86-64/GMP-5.0.1 kernel) 64-bit version compiled: May 27 2010, gcc-4.2.1 (Apple Inc. build 5659) (readline v6.1 enabled, extended help available) Copyright (C) 2000-2006 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 8000000, primelimit = 500000 ----------------------------------------------------- 20 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(7/8+10^(-20),22,20) The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^72 Number of iterations in the dyadic procedure = 290 The approximated final values are: max-upper-matrix = 0.95050874997654961769559757815260 min-upper-matrix = 0.95050874997654961769552604696525 max-lower-matrix = 0.95050874997654961769559755469396 min-lower-matrix = 0.95050874997654961769552602350662 The approximated values for the moments are: moment-max-upper-matrix = 0.77828634236773010541149111211392 moment-min-upper-matrix = 0.77828634236773010541138689878407 moment-max-lower-matrix = 0.77828634236773010541149107793718 moment-min-lower-matrix = 0.77828634236773010541138686460733 The minimal lambda is in [1.4568936113337693770005900000000,1.4568936113337693770006100000000] Final result (in the centre of the interval): d = 0.95050874997654961769559757815260 ? ## *** last result computed in 44,500 ms. ----------------------------------------------------- 30 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(7/8+10^(-20),28,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) start of the dyadic step Initial T = 0.100000000000000000000000000000000000000000 Final T = 1.00000000000000000000000000000000000000000 E-33 end of the dyadic step Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 439 The approximated final values are: max-upper-matrix = 0.950508749976549617695590442433410458611151 min-upper-matrix = 0.950508749976549617695590442433402131284956 max-lower-matrix = 0.950508749976549617695590442433410448478203 min-lower-matrix = 0.950508749976549617695590442433402121152008 The approximated values for the moments are: moment-max-upper-matrix = 0.778286342367730105512886995973533908449269 moment-min-upper-matrix = 0.778286342367730105512886995973521776420936 moment-max-lower-matrix = 0.778286342367730105512886995973533893686643 moment-min-lower-matrix = 0.778286342367730105512886995973521761658309 The minimal lambda is in [1.45689361133376937710728631913783900000000,1.45689361133376937710728631913784100000000] Final result (in the centre of the interval): d = 0.950508749976549617695590442433410458611151 time = 4mn, 6,027 ms. ----------------------------------------------------- 50 digits ----------------------------------------------------- ? PintzRuzsa_psiapprox(7/8+10^(-20),40,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) start of the dyadic step Initial T = 0.10000000000000000000000000000000000000000000000000000000000000 Final T = 1.0000000000000000000000000000000000000000000000000000000000000 E-53 end of the dyadic step Needed matrix exponent for this precision is = 2^172 Number of iterations in the dyadic procedure = 645 The approximated final values are: max-upper-matrix = 0.95050874997654961769559044243340976733785259695974679604327740 min-upper-matrix = 0.95050874997654961769559044243340976733785259695974673961512020 max-lower-matrix = 0.95050874997654961769559044243340976733785259695974679598425146 min-lower-matrix = 0.95050874997654961769559044243340976733785259695974673955609426 The approximated values for the moments are: moment-max-upper-matrix = 0.77828634236773010551288699596045007909072749419383465020891552 moment-min-upper-matrix = 0.77828634236773010551288699596045007909072749419383456799909379 moment-max-lower-matrix = 0.77828634236773010551288699596045007909072749419383465012292100 moment-min-lower-matrix = 0.77828634236773010551288699596045007909072749419383456791309928 The minimal lambda is in [1.4568936113337693771072863191240759791562018045798121900000000,1.4568936113337693771072863191240759791562018045798122100000000] Final result (in the centre of the interval): d = 0.95050874997654961769559044243340976733785259695974679604327740 time = 29mn, 36,593 ms. ? ----------------------------------------------------- ----------------------------------------------------- 7/8 sul double quad core pc ----------------------------------------------------- ? PintzRuzsa_psiapprox(7/8+10^(-20),14,10) The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) Needed matrix exponent for this precision is = 2^39 Number of iterations in the dyadic procedure = 165 The approximated final values are: max-upper-matrix = 0.9505087499765653257047 min-upper-matrix = 0.9505087499759508774842 max-lower-matrix = 0.9505087499765593071068 min-lower-matrix = 0.9505087499759448588863 The approximated values for the moments are: moment-max-upper-matrix = 0.7782863423736752519884 moment-min-upper-matrix = 0.7782863423727800663013 moment-max-lower-matrix = 0.7782863423736664835316 moment-min-lower-matrix = 0.7782863423727712978445 The minimal lambda is in [1.456893611339900000000,1.456893611340100000000] Final result (in the centre of the interval): d = 0.9505087499765653257047 ? ## *** last result computed in 4,405 ms. ? PintzRuzsa_psiapprox(7/8+10^(-20),21,20) The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) Needed matrix exponent for this precision is = 2^72 Number of iterations in the dyadic procedure = 289 The approximated final values are: max-upper-matrix = 0.95050874997654961769559263750997 min-upper-matrix = 0.95050874997654961769552110632263 max-lower-matrix = 0.95050874997654961769559189499786 min-lower-matrix = 0.95050874997654961769552036381052 The approximated values for the moments are: moment-max-upper-matrix = 0.77828634236773010581830165911321 moment-min-upper-matrix = 0.77828634236773010581819744578336 moment-max-lower-matrix = 0.77828634236773010581830057735205 moment-min-lower-matrix = 0.77828634236773010581819636402220 The minimal lambda is in [1.4568936113337693774285900000000,1.4568936113337693774286100000000] Final result (in the centre of the interval): d = 0.95050874997654961769559263750997 ? ## *** last result computed in 43,106 ms. ? PintzRuzsa_psiapprox(7/8+10^(-20),28,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) ## Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 439 The approximated final values are: max-upper-matrix = 0.950508749976549617695590442433410458611151 min-upper-matrix = 0.950508749976549617695590442433402131284956 max-lower-matrix = 0.950508749976549617695590442433410448478203 min-lower-matrix = 0.950508749976549617695590442433402121152008 The approximated values for the moments are: moment-max-upper-matrix = 0.778286342367730105512886995973533908449269 moment-min-upper-matrix = 0.778286342367730105512886995973521776420936 moment-max-lower-matrix = 0.778286342367730105512886995973533893686643 moment-min-lower-matrix = 0.778286342367730105512886995973521761658309 The minimal lambda is in [1.45689361133376937710728631913783900000000,1.45689361133376937710728631913784100000000] Final result (in the centre of the interval): d = 0.950508749976549617695590442433410458611151 ? ## *** last result computed in 4mn, 21,867 ms. ? ? PintzRuzsa_psiapprox(7/8+10^(-20),40,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) ## Needed matrix exponent for this precision is = 2^172 Number of iterations in the dyadic procedure = 645 The approximated final values are: max-upper-matrix = 0.95050874997654961769559044243340976733785259695974679604327740 min-upper-matrix = 0.95050874997654961769559044243340976733785259695974673961512020 max-lower-matrix = 0.95050874997654961769559044243340976733785259695974679598425146 min-lower-matrix = 0.95050874997654961769559044243340976733785259695974673955609426 The approximated values for the moments are: moment-max-upper-matrix = 0.77828634236773010551288699596045007909072749419383465020891552 moment-min-upper-matrix = 0.77828634236773010551288699596045007909072749419383456799909379 moment-max-lower-matrix = 0.77828634236773010551288699596045007909072749419383465012292100 moment-min-lower-matrix = 0.77828634236773010551288699596045007909072749419383456791309928 The minimal lambda is in [1.4568936113337693771072863191240759791562018045798121900000000,1.4568936113337693771072863191240759791562018045798122100000000] Final result (in the centre of the interval): d = 0.95050874997654961769559044243340976733785259695974679604327740 ? ## *** last result computed in 31mn, 28,007 ms. ----------------------------------------------------- 19/24 sul double quad core pc ----------------------------------------------------- ? PintzRuzsa_psiapprox(19/24+10^(-20),30,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 418 The approximated final values are: max-upper-matrix = 0.907856803841950190921307382791751979081108 min-upper-matrix = 0.907856803841950190921307382791743915716763 max-lower-matrix = 0.907856803841950190921307382791751979080989 min-lower-matrix = 0.907856803841950190921307382791743915716644 The approximated values for the moments are: moment-max-upper-matrix = 0.604172659650528896412938023712629896682698 moment-min-upper-matrix = 0.604172659650528896412938023712619656778086 moment-max-lower-matrix = 0.604172659650528896412938023712629896682546 moment-min-lower-matrix = 0.604172659650528896412938023712619656777934 The minimal lambda is in [1.26992954474187215006177213402417900000000,1.26992954474187215006177213402418100000000] Final result (in the centre of the interval): d = 0.907856803841950190921307382791751979081108 ? ## *** last result computed in 5mn, 5,072 ms. ? PintzRuzsa_psiapprox(19/24+10^(-20),39,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) Needed matrix exponent for this precision is = 2^172 Number of iterations in the dyadic procedure = 630 The approximated final values are: max-upper-matrix = 0.90785680384195019092130738279175068034507513702259955923219504 min-upper-matrix = 0.90785680384195019092130738279175068034507513702259950459271291 max-lower-matrix = 0.90785680384195019092130738279175068034507513702259955921217317 min-lower-matrix = 0.90785680384195019092130738279175068034507513702259950457269104 The approximated values for the moments are: moment-max-upper-matrix = 0.60417265965052889641293802357568261761464830407162130395211294 moment-min-upper-matrix = 0.60417265965052889641293802357568261761464830407162123456382028 moment-max-lower-matrix = 0.60417265965052889641293802357568261761464830407162130392668658 moment-min-lower-matrix = 0.60417265965052889641293802357568261761464830407162123453839392 The minimal lambda is in [1.2699295447418721500617721338733350328363996382438599900000000,1.2699295447418721500617721338733350328363996382438600100000000] Final result (in the centre of the interval): d = 0.90785680384195019092130738279175068034507513702259955923219504 ? ## *** last result computed in 28mn, 28,075 ms. calcolo delle potenze di due necessarie: ? 3+(log(1.516492)-log(21616))/log(0.9078568038419501909213073827917506803450751370225995) %4 = 101.944098936576218601378923634008590253158015879904790692138 ----------------------------------------------------- 3/4 sul double quad core pc ----------------------------------------------------- languasc@labsrv0:~$ nice /usr/local/Gruppi/Languasco/bin/gp PRMethod-KB-2.gp GP/PARI CALCULATOR Version 2.5.0 (released) amd64 running linux (x86-64/GMP-5.0.2 kernel) 64-bit version compiled: Jun 23 2011, gcc-4.4.3 (Ubuntu 4.4.3-4ubuntu5) (readline v5.2 enabled, extended help enabled) Copyright (C) 2000-2011 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 8000000, primelimit = 500509 ? PintzRuzsa_psiapprox(3/4+10^(-20),30,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) start of the dyadic step Initial T = 0.100000000000000000000000000000000000000000 Final T = 1.00000000000000000000000000000000000000000 E-33 end of the dyadic step Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 439 The approximated final values are: max-upper-matrix = 0.884447213166270618266830682874112263722197 min-upper-matrix = 0.884447213166270618266830682874104364604498 max-lower-matrix = 0.884447213166270618266830682874112263722173 min-lower-matrix = 0.884447213166270618266830682874104364604475 The approximated values for the moments are: moment-max-upper-matrix = 0.541418030802965850285418783154371639345198 moment-min-upper-matrix = 0.541418030802965850285418783154362160924106 moment-max-lower-matrix = 0.541418030802965850285418783154371639345170 moment-min-lower-matrix = 0.541418030802965850285418783154362160924077 The minimal lambda is in [1.19993415143862411505624754152643900000000,1.19993415143862411505624754152644100000000] Final result (in the centre of the interval): d = 0.884447213166270618266830682874112263722197 ? ## *** last result computed in 5min, 20,870 ms. parisize = 8000000, primelimit = 500509 ? PintzRuzsa_psiapprox(3/4+10^(-20),39,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) start of the dyadic step Initial T = 0.10000000000000000000000000000000000000000000000000000000000000 Final T = 1.0000000000000000000000000000000000000000000000000000000000000 E-53 end of the dyadic step Needed matrix exponent for this precision is = 2^172 Number of iterations in the dyadic procedure = 628 The approximated final values are: max-upper-matrix = 0.88444721316627061826683068287411076854971436259485375655803261 min-upper-matrix = 0.88444721316627061826683068287411076854971436259485370303152906 max-lower-matrix = 0.88444721316627061826683068287411076854971436259485375655562630 min-lower-matrix = 0.88444721316627061826683068287411076854971436259485370302912274 The approximated values for the moments are: moment-max-upper-matrix = 0.54141803080296585028541878314714891602759377296533843283324999 moment-min-upper-matrix = 0.54141803080296585028541878314714891602759377296533836860497037 moment-max-lower-matrix = 0.54141803080296585028541878314714891602759377296533843283036258 moment-min-lower-matrix = 0.54141803080296585028541878314714891602759377296533836860208295 The minimal lambda is in [1.1999341514386241150562475415182756583619838663980797900000000,1.1999341514386241150562475415182756583619838663980798100000000] Final result (in the centre of the interval): d = 0.88444721316627061826683068287411076854971436259485375655803261 ? ## *** last result computed in 28min, 22,890 ms. ----------------------------------------------------- Articolo di Hongze Li quattro quadrati 3/4 sul macbookpro ----------------------------------------------------- macbook:programmi languasc$ gp2c-run -pmy_ -g -W PRMethod-KB-2.gp PRMethod-KB-2.gp.c: In function Ômy_RuzsaÕ: PRMethod-KB-2.gp.c:106: warning: unused variable Ômy_i2Õ PRMethod-KB-2.gp.c:106: warning: unused variable Ômy_i1Õ PRMethod-KB-2.gp.c:106: warning: unused variable Ômy_iÕ PRMethod-KB-2.gp.c:106: warning: unused variable Ômy_lÕ PRMethod-KB-2.gp.c:106: warning: unused variable Ômy_jÕ GP/PARI CALCULATOR Version 2.5.0 (released) i386 running darwin (x86-64/GMP-5.0.2 kernel) 64-bit version compiled: Jul 5 2011, gcc-4.2.1 (Apple Inc. build 5666) (dot 3) (readline v6.2 enabled, extended help enabled) Copyright (C) 2000-2011 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 8000000, primelimit = 500509 ? PintzRuzsa_psiapprox(3/4+10^(-20),22,20) The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^72 Number of iterations in the dyadic procedure = 288 The approximated final values are: max-upper-matrix = 0.88444721316627061826684352081761 min-upper-matrix = 0.88444721316627061826677566791325 max-lower-matrix = 0.88444721316627061826684352036181 min-lower-matrix = 0.88444721316627061826677566745745 The approximated values for the moments are: moment-max-upper-matrix = 0.54141803080296585043415616112521 moment-min-upper-matrix = 0.54141803080296585043407474210799 moment-max-lower-matrix = 0.54141803080296585043415616057827 moment-min-lower-matrix = 0.54141803080296585043407474156105 The minimal lambda is in [1.1999341514386241152243900000000,1.1999341514386241152244100000000] Final result (in the centre of the interval): d = 0.88444721316627061826684352081761 ? ## *** last result computed in 42,486 ms. ? ? PintzRuzsa_psiapprox(3/4+10^(-20),28,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) start of the dyadic step Initial T = 0.100000000000000000000000000000000000000000 ## Final T = 1.00000000000000000000000000000000000000000 E-33 end of the dyadic step Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 431 The approximated final values are: max-upper-matrix = 0.884447213166270618266830682874112262612883 min-upper-matrix = 0.884447213166270618266830682874104363495184 max-lower-matrix = 0.884447213166270618266830682874112262551169 min-lower-matrix = 0.884447213166270618266830682874104363433471 The approximated values for the moments are: moment-max-upper-matrix = 0.541418030802965850285418783204767440220309 moment-min-upper-matrix = 0.541418030802965850285418783204757961799216 moment-max-lower-matrix = 0.541418030802965850285418783204767440146257 moment-min-lower-matrix = 0.541418030802965850285418783204757961725164 The minimal lambda is in [1.19993415143862411505624754158341900000000,1.19993415143862411505624754158342100000000] Final result (in the centre of the interval): d = 0.884447213166270618266830682874112262612883 ? ## *** last result computed in 3min, 54,556 ms. ? PintzRuzsa_psiapprox(3/4+10^(-20),27,30) The expected number of correct decimal digits is = 30 Approximation for the minimal lambda is = 10^(-30) start of the dyadic step Initial T = 0.100000000000000000000000000000000000000000 ## Final T = 1.00000000000000000000000000000000000000000 E-33 end of the dyadic step Needed matrix exponent for this precision is = 2^105 Number of iterations in the dyadic procedure = 431 The approximated final values are: max-upper-matrix = 0.884447213166270618266830682874112242471026 min-upper-matrix = 0.884447213166270618266830682874104343353328 max-lower-matrix = 0.884447213166270618266830682874112239485099 min-lower-matrix = 0.884447213166270618266830682874104340367401 The approximated values for the moments are: moment-max-upper-matrix = 0.541418030802965850285418783210799346045201 moment-min-upper-matrix = 0.541418030802965850285418783210789867624109 moment-max-lower-matrix = 0.541418030802965850285418783210799342462285 moment-min-lower-matrix = 0.541418030802965850285418783210789864041193 The minimal lambda is in [1.19993415143862411505624754159023900000000,1.19993415143862411505624754159024100000000] Final result (in the centre of the interval): d = 0.884447213166270618266830682874112242471026 ? ## *** last result computed in 3min, 30,461 ms. ? PintzRuzsa_psiapprox(3/4+10^(-20),39,50) The expected number of correct decimal digits is = 50 Approximation for the minimal lambda is = 10^(-50) start of the dyadic step Initial T = 0.10000000000000000000000000000000000000000000000000000000000000 ## Final T = 1.0000000000000000000000000000000000000000000000000000000000000 E-53 end of the dyadic step Needed matrix exponent for this precision is = 2^172 Number of iterations in the dyadic procedure = 628 The approximated final values are: max-upper-matrix = 0.88444721316627061826683068287411076854971436259485375655803261 min-upper-matrix = 0.88444721316627061826683068287411076854971436259485370303152906 max-lower-matrix = 0.88444721316627061826683068287411076854971436259485375655562630 min-lower-matrix = 0.88444721316627061826683068287411076854971436259485370302912274 The approximated values for the moments are: moment-max-upper-matrix = 0.54141803080296585028541878314714891602759377296533843283324999 moment-min-upper-matrix = 0.54141803080296585028541878314714891602759377296533836860497037 moment-max-lower-matrix = 0.54141803080296585028541878314714891602759377296533843283036258 moment-min-lower-matrix = 0.54141803080296585028541878314714891602759377296533836860208295 The minimal lambda is in [1.1999341514386241150562475415182756583619838663980797900000000,1.1999341514386241150562475415182756583619838663980798100000000] Final result (in the centre of the interval): d = 0.88444721316627061826683068287411076854971436259485375655803261 ? ## *** last result computed in 26min, 28,021 ms. calcolo delle potenze di due necessarie: ? c=(32^4*101*1.620767/3+(8*(log(2))^2)/Pi^2) %1 = 57216412.10843787777109934792 ? 4+(log(4.99456)-log(3*c))/(log(0.884447213166270618266830682874112263722197)) %2 = 145.3166259795537255526834907 ? (-1)/log(0.884447213166270618266830682874112263722197) %1 = 8.1438233724053744320317160690928078409 -------------------------------------------- Per valori molto piccoli sul double quad core pc -------------------------------------------- [languasc@labsrv0 ~]$ nice /usr/local/Gruppi/PariGP/bin/gp2c-run -pmy_ -g -W PRMethod-KB-2.gp PRMethod-KB-2.gp.c: In function `my_Ruzsa': PRMethod-KB-2.gp.c:105: warning: unused variable `my_j' PRMethod-KB-2.gp.c:105: warning: unused variable `my_l' PRMethod-KB-2.gp.c:105: warning: unused variable `my_i' PRMethod-KB-2.gp.c:105: warning: unused variable `my_i1' PRMethod-KB-2.gp.c:105: warning: unused variable `my_i2' GP/PARI CALCULATOR Version 2.3.2 (released) amd64 running linux (x86-64/GMP-4.2.2 kernel) 64-bit version compiled: Nov 30 2007, gcc-3.4.3 20041212 (Red Hat 3.4.3-9.EL4) (readline v4.3 enabled, extended help available) Copyright (C) 2000-2006 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 8000000, primelimit = 500000 ? # timer = 1 (on) ? for(i=2,50,print("******************");print(1/i);PintzRuzsa_psiapprox(1/i,15,10);print("*****************")) ****************** 1/2 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^38 Number of iterations in the dyadic procedure = 160 The approximated final values are: max-upper-matrix = 0.7163435444773103410122 min-upper-matrix = 0.7163435444763296796994 max-lower-matrix = 0.7163435444773103406372 min-lower-matrix = 0.7163435444763296793244 The approximated values for the moments are: moment-max-upper-matrix = 0.2960505965024759934496 moment-min-upper-matrix = 0.2960505965015962525997 moment-max-lower-matrix = 0.2960505965024759931132 moment-min-lower-matrix = 0.2960505965015962522633 The minimal lambda is in [0.8970893808379000000000,0.8970893808381000000000] Final result (in the centre of the interval): d = 0.7163435444773103410122 ***************** ****************** 1/3 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^38 Number of iterations in the dyadic procedure = 160 The approximated final values are: max-upper-matrix = 0.5736915641439311888986 min-upper-matrix = 0.5736915641431139407080 max-lower-matrix = 0.5736915641439311888792 min-lower-matrix = 0.5736915641431139406886 The approximated values for the moments are: moment-max-upper-matrix = 0.1867056939310429693181 moment-min-upper-matrix = 0.1867056939304478597884 moment-max-lower-matrix = 0.1867056939310429693040 moment-min-lower-matrix = 0.1867056939304478597743 The minimal lambda is in [0.7281870263179000000000,0.7281870263181000000000] Final result (in the centre of the interval): d = 0.5736915641439311888986 ***************** ****************** 1/4 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^38 Number of iterations in the dyadic procedure = 165 The approximated final values are: max-upper-matrix = 0.4891089146781871136905 min-upper-matrix = 0.4891089146774707999437 max-lower-matrix = 0.4891089146781871136877 min-lower-matrix = 0.4891089146774707999408 The approximated values for the moments are: moment-max-upper-matrix = 0.1388526915174259398958 moment-min-upper-matrix = 0.1388526915169688028482 moment-max-lower-matrix = 0.1388526915174259398940 moment-min-lower-matrix = 0.1388526915169688028464 The minimal lambda is in [0.6381799171719000000000,0.6381799171721000000000] Final result (in the centre of the interval): d = 0.4891089146781871136905 ***************** ****************** 1/5 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^38 Number of iterations in the dyadic procedure = 151 The approximated final values are: max-upper-matrix = 0.4321376222390310691089 min-upper-matrix = 0.4321376222383851497899 max-lower-matrix = 0.4321376222390310691082 min-lower-matrix = 0.4321376222383851497892 The approximated values for the moments are: moment-max-upper-matrix = 0.1113469894029675290140 moment-min-upper-matrix = 0.1113469894025938873910 moment-max-lower-matrix = 0.1113469894029675290136 moment-min-lower-matrix = 0.1113469894025938873906 The minimal lambda is in [0.5784648515899000000000,0.5784648515901000000000] Final result (in the centre of the interval): d = 0.4321376222390310691089 ***************** ****************** 1/6 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^38 Number of iterations in the dyadic procedure = 176 The approximated final values are: max-upper-matrix = 0.3906251842737457953796 min-upper-matrix = 0.3906251842731526530301 max-lower-matrix = 0.3906251842737457953794 min-lower-matrix = 0.3906251842731526530299 The approximated values for the moments are: moment-max-upper-matrix = 0.09330139269255368777787 moment-min-upper-matrix = 0.09330139269223659737135 moment-max-lower-matrix = 0.09330139269255368777776 moment-min-lower-matrix = 0.09330139269223659737123 The minimal lambda is in [0.5345941101419000000000,0.5345941101421000000000] Final result (in the centre of the interval): d = 0.3906251842737457953796 ***************** ****************** 1/7 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^38 Number of iterations in the dyadic procedure = 171 The approximated final values are: max-upper-matrix = 0.3587362310467993263489 min-upper-matrix = 0.3587362310462476910205 max-lower-matrix = 0.3587362310467993263489 min-lower-matrix = 0.3587362310462476910204 The approximated values for the moments are: moment-max-upper-matrix = 0.08048006318311469470414 moment-min-upper-matrix = 0.08048006318283867256418 moment-max-lower-matrix = 0.08048006318311469470410 moment-min-lower-matrix = 0.08048006318283867256414 The minimal lambda is in [0.5003706719379000000000,0.5003706719381000000000] Final result (in the centre of the interval): d = 0.3587362310467993263489 ***************** ****************** 1/8 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^38 Number of iterations in the dyadic procedure = 154 The approximated final values are: max-upper-matrix = 0.3332950945506240000517 min-upper-matrix = 0.3332950945501061382160 max-lower-matrix = 0.3332950945506240000516 min-lower-matrix = 0.3332950945501061382160 The approximated values for the moments are: moment-max-upper-matrix = 0.07086806107028648433605 moment-min-upper-matrix = 0.07086806107004174874156 moment-max-lower-matrix = 0.07086806107028648433603 moment-min-lower-matrix = 0.07086806107004174874154 The minimal lambda is in [0.4725885895579000000000,0.4725885895581000000000] Final result (in the centre of the interval): d = 0.3332950945506240000517 ***************** ****************** 1/9 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 158 The approximated final values are: max-upper-matrix = 0.3124130173091621412904 min-upper-matrix = 0.3124130173081827977336 max-lower-matrix = 0.3124130173091621412904 min-lower-matrix = 0.3124130173081827977336 The approximated values for the moments are: moment-max-upper-matrix = 0.06337710485123463604805 moment-min-upper-matrix = 0.06337710485079453457619 moment-max-lower-matrix = 0.06337710485123463604804 moment-min-lower-matrix = 0.06337710485079453457618 The minimal lambda is in [0.4493841500459000000000,0.4493841500461000000000] Final result (in the centre of the interval): d = 0.3124130173091621412904 ***************** ****************** 1/10 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 170 The approximated final values are: max-upper-matrix = 0.2948898084619342474545 min-upper-matrix = 0.2948898084610029049187 max-lower-matrix = 0.2948898084619342474545 min-lower-matrix = 0.2948898084610029049187 The approximated values for the moments are: moment-max-upper-matrix = 0.05736502346585214642724 moment-min-upper-matrix = 0.05736502346545205721320 moment-max-lower-matrix = 0.05736502346585214642723 moment-min-lower-matrix = 0.05736502346545205721319 The minimal lambda is in [0.4295833151459000000000,0.4295833151461000000000] Final result (in the centre of the interval): d = 0.2948898084619342474545 ***************** ****************** 1/11 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 168 The approximated final values are: max-upper-matrix = 0.2799227684758584903490 min-upper-matrix = 0.2799227684749686682670 max-lower-matrix = 0.2799227684758584903490 min-lower-matrix = 0.2799227684749686682670 The approximated values for the moments are: moment-max-upper-matrix = 0.05242721568348307339113 moment-min-upper-matrix = 0.05242721568311610934669 moment-max-lower-matrix = 0.05242721568348307339112 moment-min-lower-matrix = 0.05242721568311610934669 The minimal lambda is in [0.4124015933499000000000,0.4124015933501000000000] Final result (in the centre of the interval): d = 0.2799227684758584903490 ***************** ****************** 1/12 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 175 The approximated final values are: max-upper-matrix = 0.2669526977037857297291 min-upper-matrix = 0.2669526977029322919203 max-lower-matrix = 0.2669526977037857297291 min-lower-matrix = 0.2669526977029322919203 The approximated values for the moments are: moment-max-upper-matrix = 0.04829551399090675522762 moment-min-upper-matrix = 0.04829551399056769246732 moment-max-lower-matrix = 0.04829551399090675522762 moment-min-lower-matrix = 0.04829551399056769246732 The minimal lambda is in [0.3972905310559000000000,0.3972905310561000000000] Final result (in the centre of the interval): d = 0.2669526977037857297291 ***************** ****************** 1/13 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 148 The approximated final values are: max-upper-matrix = 0.2555768538198036540002 min-upper-matrix = 0.2555768538189824469535 max-lower-matrix = 0.2555768538198036540002 min-lower-matrix = 0.2555768538189824469535 The approximated values for the moments are: moment-max-upper-matrix = 0.04478484885156687260720 moment-min-upper-matrix = 0.04478484885125165007363 moment-max-lower-matrix = 0.04478484885156687260720 moment-min-lower-matrix = 0.04478484885125165007363 The minimal lambda is in [0.3838526896099000000000,0.3838526896101000000000] Final result (in the centre of the interval): d = 0.2555768538198036540002 ***************** ****************** 1/14 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 174 The approximated final values are: max-upper-matrix = 0.2454970436038778173708 min-upper-matrix = 0.2454970436030854249966 max-lower-matrix = 0.2454970436038778173708 min-lower-matrix = 0.2454970436030854249966 The approximated values for the moments are: moment-max-upper-matrix = 0.04176321022469438757425 moment-min-upper-matrix = 0.04176321022439978279570 moment-max-lower-matrix = 0.04176321022469438757425 moment-min-lower-matrix = 0.04176321022439978279570 The minimal lambda is in [0.3717915368019000000000,0.3717915368021000000000] Final result (in the centre of the interval): d = 0.2454970436038778173708 ***************** ****************** 1/15 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 156 The approximated final values are: max-upper-matrix = 0.2364872654665903826139 min-upper-matrix = 0.2364872654658239546151 max-lower-matrix = 0.2364872654665903826139 min-lower-matrix = 0.2364872654658239546151 The approximated values for the moments are: moment-max-upper-matrix = 0.03913380779936523868603 moment-min-upper-matrix = 0.03913380779908864984039 moment-max-lower-matrix = 0.03913380779936523868603 moment-min-lower-matrix = 0.03913380779908864984039 The minimal lambda is in [0.3608804037219000000000,0.3608804037221000000000] Final result (in the centre of the interval): d = 0.2364872654665903826139 ***************** ****************** 1/16 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 159 The approximated final values are: max-upper-matrix = 0.2283727738405002200675 min-upper-matrix = 0.2283727738397573485675 max-lower-matrix = 0.2283727738405002200675 min-lower-matrix = 0.2283727738397573485675 The approximated values for the moments are: moment-max-upper-matrix = 0.03682400127603958564102 moment-min-upper-matrix = 0.03682400127577888049907 moment-max-lower-matrix = 0.03682400127603958564102 moment-min-lower-matrix = 0.03682400127577888049907 The minimal lambda is in [0.3509424469179000000000,0.3509424469181000000000] Final result (in the centre of the interval): d = 0.2283727738405002200675 ***************** ****************** 1/17 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 173 The approximated final values are: max-upper-matrix = 0.2210160965302377995948 min-upper-matrix = 0.2210160965295164283892 max-lower-matrix = 0.2210160965302377995948 min-lower-matrix = 0.2210160965295164283892 The approximated values for the moments are: moment-max-upper-matrix = 0.03477817363003893236287 moment-min-upper-matrix = 0.03477817362979234080539 moment-max-lower-matrix = 0.03477817363003893236287 moment-min-lower-matrix = 0.03477817362979234080539 The minimal lambda is in [0.3418372615319000000000,0.3418372615321000000000] Final result (in the centre of the interval): d = 0.2210160965302377995948 ***************** ****************** 1/18 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 168 The approximated final values are: max-upper-matrix = 0.2143074363023662311868 min-upper-matrix = 0.2143074363016645876443 max-lower-matrix = 0.2143074363023662311868 min-lower-matrix = 0.2143074363016645876443 The approximated values for the moments are: moment-max-upper-matrix = 0.03295299630814487341045 moment-min-upper-matrix = 0.03295299630791090919788 moment-max-lower-matrix = 0.03295299630814487341045 moment-min-lower-matrix = 0.03295299630791090919788 The minimal lambda is in [0.3334516722279000000000,0.3334516722281000000000] Final result (in the centre of the interval): d = 0.2143074363023662311868 ***************** ****************** 1/19 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 166 The approximated final values are: max-upper-matrix = 0.2081579226110904825997 min-upper-matrix = 0.2081579226104070256577 max-lower-matrix = 0.2081579226110904825997 min-lower-matrix = 0.2081579226104070256577 The approximated values for the moments are: moment-max-upper-matrix = 0.03131419690758230327044 moment-min-upper-matrix = 0.03131419690735970596716 moment-max-lower-matrix = 0.03131419690758230327044 moment-min-lower-matrix = 0.03131419690735970596716 The minimal lambda is in [0.3256932362359000000000,0.3256932362361000000000] Final result (in the centre of the interval): d = 0.2081579226110904825997 ***************** ****************** 1/20 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 159 The approximated final values are: max-upper-matrix = 0.2024947662680409379394 min-upper-matrix = 0.2024947662673743177706 max-lower-matrix = 0.2024947662680409379394 min-lower-matrix = 0.2024947662673743177706 The approximated values for the moments are: moment-max-upper-matrix = 0.02983429960786909993140 moment-min-upper-matrix = 0.02983429960765679103494 moment-max-lower-matrix = 0.02983429960786909993140 moment-min-lower-matrix = 0.02983429960765679103494 The minimal lambda is in [0.3184855580439000000000,0.3184855580441000000000] Final result (in the centre of the interval): d = 0.2024947662680409379394 ***************** ****************** 1/21 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 157 The approximated final values are: max-upper-matrix = 0.1972577149205193226624 min-upper-matrix = 0.1972577149198683489592 max-lower-matrix = 0.1972577149205193226624 min-lower-matrix = 0.1972577149198683489592 The approximated values for the moments are: moment-max-upper-matrix = 0.02849101224498835935747 moment-min-upper-matrix = 0.02849101224478540864204 moment-max-lower-matrix = 0.02849101224498835935747 moment-min-lower-matrix = 0.02849101224478540864204 The minimal lambda is in [0.3117648446239000000000,0.3117648446241000000000] Final result (in the centre of the interval): d = 0.1972577149205193226624 ***************** ****************** 1/22 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 164 The approximated final values are: max-upper-matrix = 0.1923964165462284319735 min-upper-matrix = 0.1923964165455920486990 max-lower-matrix = 0.1923964165462284319735 min-lower-matrix = 0.1923964165455920486990 The approximated values for the moments are: moment-max-upper-matrix = 0.02726605341641789223364 moment-min-upper-matrix = 0.02726605341622349157070 moment-max-lower-matrix = 0.02726605341641789223364 moment-min-lower-matrix = 0.02726605341622349157070 The minimal lambda is in [0.3054773290319000000000,0.3054773290321000000000] Final result (in the centre of the interval): d = 0.1923964165462284319735 ***************** ****************** 1/23 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 183 The approximated final values are: max-upper-matrix = 0.1878684287704976131021 min-upper-matrix = 0.1878684287698748781642 max-lower-matrix = 0.1878684287704976131021 min-lower-matrix = 0.1878684287698748781642 The approximated values for the moments are: moment-max-upper-matrix = 0.02614428530450927998636 moment-min-upper-matrix = 0.02614428530432272272644 moment-max-lower-matrix = 0.02614428530450927998636 moment-min-lower-matrix = 0.02614428530432272272644 The minimal lambda is in [0.2995773138159000000000,0.2995773138161000000000] Final result (in the centre of the interval): d = 0.1878684287704976131021 ***************** ****************** 1/24 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 164 The approximated final values are: max-upper-matrix = 0.1836376953386838176829 min-upper-matrix = 0.1836376953380738864059 max-lower-matrix = 0.1836376953386838176829 min-lower-matrix = 0.1836376953380738864059 The approximated values for the moments are: moment-max-upper-matrix = 0.02511306288615771842953 moment-min-upper-matrix = 0.02511306288597838298046 moment-max-lower-matrix = 0.02511306288615771842953 moment-min-lower-matrix = 0.02511306288597838298046 The minimal lambda is in [0.2940256645559000000000,0.2940256645561000000000] Final result (in the centre of the interval): d = 0.1836376953386838176829 ***************** ****************** 1/25 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 160 The approximated final values are: max-upper-matrix = 0.1796733657063105033072 min-upper-matrix = 0.1796733657057126148669 max-lower-matrix = 0.1796733657063105033072 min-lower-matrix = 0.1796733657057126148669 The approximated values for the moments are: moment-max-upper-matrix = 0.02416173890244767950956 moment-min-upper-matrix = 0.02416173890227501612268 moment-max-lower-matrix = 0.02416173890244767950956 moment-min-lower-matrix = 0.02416173890227501612268 The minimal lambda is in [0.2887886355379000000000,0.2887886355381000000000] Final result (in the centre of the interval): d = 0.1796733657063105033072 ***************** ****************** 1/26 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 148 The approximated final values are: max-upper-matrix = 0.1759488701694863148097 min-upper-matrix = 0.1759488701688997810076 max-lower-matrix = 0.1759488701694863148097 min-lower-matrix = 0.1759488701688997810076 The approximated values for the moments are: moment-max-upper-matrix = 0.02328128265504523752484 moment-min-upper-matrix = 0.02328128265487875756291 moment-max-lower-matrix = 0.02328128265504523752484 moment-min-lower-matrix = 0.02328128265487875756291 The minimal lambda is in [0.2838369439459000000000,0.2838369439461000000000] Final result (in the centre of the interval): d = 0.1759488701694863148097 ***************** ****************** 1/27 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 156 The approximated final values are: max-upper-matrix = 0.1724411877425392323376 min-upper-matrix = 0.1724411877419634282378 max-lower-matrix = 0.1724411877425392323376 min-lower-matrix = 0.1724411877419634282378 The approximated values for the moments are: moment-max-upper-matrix = 0.02246398314924628093942 moment-min-upper-matrix = 0.02246398314908554808530 moment-max-lower-matrix = 0.02246398314924628093942 moment-min-lower-matrix = 0.02246398314908554808530 The minimal lambda is in [0.2791450324479000000000,0.2791450324481000000000] Final result (in the centre of the interval): d = 0.1724411877425392323376 ***************** ****************** 1/28 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 144 The approximated final values are: max-upper-matrix = 0.1691302611221708158300 min-upper-matrix = 0.1691302611216051718954 max-lower-matrix = 0.1691302611221708158300 min-lower-matrix = 0.1691302611216051718954 The approximated values for the moments are: moment-max-upper-matrix = 0.02170321553548473803173 moment-min-upper-matrix = 0.02170321553532936102992 moment-max-lower-matrix = 0.02170321553548473803173 moment-min-lower-matrix = 0.02170321553532936102992 The minimal lambda is in [0.2746904762979000000000,0.2746904762981000000000] Final result (in the centre of the interval): d = 0.1691302611221708158300 ***************** ****************** 1/29 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 151 The approximated final values are: max-upper-matrix = 0.1659985251035098964411 min-upper-matrix = 0.1659985251029538918842 max-lower-matrix = 0.1659985251035098964411 min-lower-matrix = 0.1659985251029538918842 The approximated values for the moments are: moment-max-upper-matrix = 0.02099325560357028474966 moment-min-upper-matrix = 0.02099325560341991136985 moment-max-lower-matrix = 0.02099325560357028474966 moment-min-lower-matrix = 0.02099325560341991136985 The minimal lambda is in [0.2704535024719000000000,0.2704535024721000000000] Final result (in the centre of the interval): d = 0.1659985251035098964411 ***************** ****************** 1/30 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 146 The approximated final values are: max-upper-matrix = 0.1630305233751840705282 min-upper-matrix = 0.1630305233746372276551 max-lower-matrix = 0.1630305233751840705282 min-lower-matrix = 0.1630305233746372276551 The approximated values for the moments are: moment-max-upper-matrix = 0.02032913115434722038641 moment-min-upper-matrix = 0.02032913115420153236929 moment-max-lower-matrix = 0.02032913115434722038641 moment-min-lower-matrix = 0.02032913115420153236929 The minimal lambda is in [0.2664165965599000000000,0.2664165965601000000000] Final result (in the centre of the interval): d = 0.1630305233751840705282 ***************** ****************** 1/31 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 151 The approximated final values are: max-upper-matrix = 0.1602125947958325625142 min-upper-matrix = 0.1602125947952944418862 max-lower-matrix = 0.1602125947958325625142 min-lower-matrix = 0.1602125947952944418862 The approximated values for the moments are: moment-max-upper-matrix = 0.01970650194899264463470 moment-min-upper-matrix = 0.01970650194885135343381 moment-max-lower-matrix = 0.01970650194899264463470 moment-min-lower-matrix = 0.01970650194885135343381 The minimal lambda is in [0.2625641790039000000000,0.2625641790041000000000] Final result (in the centre of the interval): d = 0.1602125947958325625142 ***************** ****************** 1/32 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 143 The approximated final values are: max-upper-matrix = 0.1575326147625790256632 min-upper-matrix = 0.1575326147620492219360 max-lower-matrix = 0.1575326147625790256632 min-lower-matrix = 0.1575326147620492219360 The approximated values for the moments are: moment-max-upper-matrix = 0.01912156201109647378180 moment-min-upper-matrix = 0.01912156201095931695495 moment-max-lower-matrix = 0.01912156201109647378180 moment-min-lower-matrix = 0.01912156201095931695495 The minimal lambda is in [0.2588823366199000000000,0.2588823366201000000000] Final result (in the centre of the interval): d = 0.1575326147625790256632 ***************** ****************** 1/33 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 156 The approximated final values are: max-upper-matrix = 0.1549797806100425194469 min-upper-matrix = 0.1549797806095206577741 max-lower-matrix = 0.1549797806100425194469 min-lower-matrix = 0.1549797806095206577741 The approximated values for the moments are: moment-max-upper-matrix = 0.01857095955947936740791 moment-min-upper-matrix = 0.01857095955934610554254 moment-max-lower-matrix = 0.01857095955947936740791 moment-min-lower-matrix = 0.01857095955934610554254 The minimal lambda is in [0.2553585985259000000000,0.2553585985261000000000] Final result (in the centre of the interval): d = 0.1549797806100425194469 ***************** ****************** 1/34 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 152 The approximated final values are: max-upper-matrix = 0.1525444324611224946499 min-upper-matrix = 0.1525444324606082275612 max-lower-matrix = 0.1525444324611224946499 min-lower-matrix = 0.1525444324606082275612 The approximated values for the moments are: moment-max-upper-matrix = 0.01805173096475868123062 moment-min-upper-matrix = 0.01805173096462909531064 moment-max-lower-matrix = 0.01805173096475868123062 moment-min-lower-matrix = 0.01805173096462909531064 The minimal lambda is in [0.2519817480439000000000,0.2519817480441000000000] Final result (in the centre of the interval): d = 0.1525444324611224946499 ***************** ****************** 1/35 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 137 The approximated final values are: max-upper-matrix = 0.1502179028209084055075 min-upper-matrix = 0.1502179028204014101856 max-lower-matrix = 0.1502179028209084055075 min-lower-matrix = 0.1502179028204014101856 The approximated values for the moments are: moment-max-upper-matrix = 0.01756124593127827153045 moment-min-upper-matrix = 0.01756124593115216067052 moment-max-lower-matrix = 0.01756124593127827153045 moment-min-lower-matrix = 0.01756124593115216067052 The minimal lambda is in [0.2487416638659000000000,0.2487416638661000000000] Final result (in the centre of the interval): d = 0.1502179028209084055075 ***************** ****************** 1/36 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^37 Number of iterations in the dyadic procedure = 149 The approximated final values are: max-upper-matrix = 0.1479923896266495610283 min-upper-matrix = 0.1479923896261495369256 max-lower-matrix = 0.1479923896266495610283 min-lower-matrix = 0.1479923896261495369256 The approximated values for the moments are: moment-max-upper-matrix = 0.01709716174452279402094 moment-min-upper-matrix = 0.01709716174439997350796 moment-max-lower-matrix = 0.01709716174452279402094 moment-min-lower-matrix = 0.01709716174439997350796 The minimal lambda is in [0.2456291852919000000000,0.2456291852921000000000] Final result (in the centre of the interval): d = 0.1479923896266495610283 ***************** ****************** 1/37 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 152 The approximated final values are: max-upper-matrix = 0.1458608485570385640973 min-upper-matrix = 0.1458608485560518975834 max-lower-matrix = 0.1458608485570385640973 min-lower-matrix = 0.1458608485560518975834 The approximated values for the moments are: moment-max-upper-matrix = 0.01665738486987598959587 moment-min-upper-matrix = 0.01665738486963658878229 moment-max-lower-matrix = 0.01665738486987598959587 moment-min-lower-matrix = 0.01665738486963658878229 The minimal lambda is in [0.2426359972719000000000,0.2426359972721000000000] Final result (in the centre of the interval): d = 0.1458608485570385640973 ***************** ****************** 1/38 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 163 The approximated final values are: max-upper-matrix = 0.1438169012433973776557 min-upper-matrix = 0.1438169012424235687387 max-lower-matrix = 0.1438169012433973776557 min-lower-matrix = 0.1438169012424235687387 The approximated values for the moments are: moment-max-upper-matrix = 0.01624003856247757425912 moment-min-upper-matrix = 0.01624003856224409915805 moment-max-lower-matrix = 0.01624003856247757425912 moment-min-lower-matrix = 0.01624003856224409915805 The minimal lambda is in [0.2397545319239000000000,0.2397545319241000000000] Final result (in the centre of the interval): d = 0.1438169012433973776557 ***************** ****************** 1/39 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 150 The approximated final values are: max-upper-matrix = 0.1418547566902292109130 min-upper-matrix = 0.1418547566892677688791 max-lower-matrix = 0.1418547566902292109130 min-lower-matrix = 0.1418547566892677688791 The approximated values for the moments are: moment-max-upper-matrix = 0.01584343541398399944421 moment-min-upper-matrix = 0.01584343541375615894564 moment-max-lower-matrix = 0.01584343541398399944421 moment-min-lower-matrix = 0.01584343541375615894564 The minimal lambda is in [0.2369778837739000000000,0.2369778837741000000000] Final result (in the centre of the interval): d = 0.1418547566902292109130 ***************** ****************** 1/40 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 154 The approximated final values are: max-upper-matrix = 0.1399691437120459178435 min-upper-matrix = 0.1399691437110963824541 max-lower-matrix = 0.1399691437120459178435 min-lower-matrix = 0.1399691437110963824541 The approximated values for the moments are: moment-max-upper-matrix = 0.01546605398146312343969 moment-min-upper-matrix = 0.01546605398124064754812 moment-max-lower-matrix = 0.01546605398146312343969 moment-min-lower-matrix = 0.01546605398124064754812 The minimal lambda is in [0.2342997365399000000000,0.2342997365401000000000] Final result (in the centre of the interval): d = 0.1399691437120459178435 ***************** ****************** 1/41 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 155 The approximated final values are: max-upper-matrix = 0.1381552526145130402257 min-upper-matrix = 0.1381552526135749791297 max-lower-matrix = 0.1381552526145130402257 min-lower-matrix = 0.1381552526135749791297 The approximated values for the moments are: moment-max-upper-matrix = 0.01510651880010639783806 moment-min-upper-matrix = 0.01510651879988903566823 moment-max-lower-matrix = 0.01510651880010639783806 moment-min-lower-matrix = 0.01510651879988903566823 The minimal lambda is in [0.2317142995899000000000,0.2317142995901000000000] Final result (in the centre of the interval): d = 0.1381552526145130402257 ***************** ****************** 1/42 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 148 The approximated final values are: max-upper-matrix = 0.1364086846616697331408 min-upper-matrix = 0.1364086846607427395629 max-lower-matrix = 0.1364086846616697331408 min-lower-matrix = 0.1364086846607427395629 The approximated values for the moments are: moment-max-upper-matrix = 0.01476358322583461347700 moment-min-upper-matrix = 0.01476358322562213148285 moment-max-lower-matrix = 0.01476358322583461347700 moment-min-lower-matrix = 0.01476358322562213148285 The minimal lambda is in [0.2292162526339000000000,0.2292162526341000000000] Final result (in the centre of the interval): d = 0.1364086846616697331408 ***************** ****************** 1/43 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 135 The approximated final values are: max-upper-matrix = 0.1347254081319653550094 min-upper-matrix = 0.1347254081310490456790 max-lower-matrix = 0.1347254081319653550094 min-lower-matrix = 0.1347254081310490456790 The approximated values for the moments are: moment-max-upper-matrix = 0.01443611464942007679483 moment-min-upper-matrix = 0.01443611464921225719965 moment-max-lower-matrix = 0.01443611464942007679483 moment-min-lower-matrix = 0.01443611464921225719965 The minimal lambda is in [0.2268006974079000000000,0.2268006974081000000000] Final result (in the centre of the interval): d = 0.1347254081319653550094 ***************** ****************** 1/44 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 152 The approximated final values are: max-upper-matrix = 0.1331017199728307958578 min-upper-matrix = 0.1331017199719248091490 max-lower-matrix = 0.1331017199728307958578 min-lower-matrix = 0.1331017199719248091490 The approximated values for the moments are: moment-max-upper-matrix = 0.01412308170363139515864 moment-min-upper-matrix = 0.01412308170342803455957 moment-max-lower-matrix = 0.01412308170363139515864 moment-min-lower-matrix = 0.01412308170342803455957 The minimal lambda is in [0.2244631152959000000000,0.2244631152961000000000] Final result (in the centre of the interval): d = 0.1331017199728307958578 ***************** ****************** 1/45 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 153 The approximated final values are: max-upper-matrix = 0.1315342122311592903081 min-upper-matrix = 0.1315342122302632845651 max-lower-matrix = 0.1315342122311592903081 min-lower-matrix = 0.1315342122302632845651 The approximated values for the moments are: moment-max-upper-matrix = 0.01382354316178037749233 moment-min-upper-matrix = 0.01382354316158128561648 moment-max-lower-matrix = 0.01382354316178037749233 moment-min-lower-matrix = 0.01382354316158128561648 The minimal lambda is in [0.2221993300839000000000,0.2221993300841000000000] Final result (in the centre of the interval): d = 0.1315342122311592903081 ***************** ****************** 1/46 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 152 The approximated final values are: max-upper-matrix = 0.1300197425732985237940 min-upper-matrix = 0.1300197425724121758189 max-lower-matrix = 0.1300197425732985237940 min-lower-matrix = 0.1300197425724121758189 The approximated values for the moments are: moment-max-upper-matrix = 0.01353663826748361711263 moment-min-upper-matrix = 0.01353663826728861570529 moment-max-lower-matrix = 0.01353663826748361711263 moment-min-lower-matrix = 0.01353663826728861570529 The minimal lambda is in [0.2200054750919000000000,0.2200054750921000000000] Final result (in the centre of the interval): d = 0.1300197425732985237940 ***************** ****************** 1/47 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 155 The approximated final values are: max-upper-matrix = 0.1285554083193913529470 min-upper-matrix = 0.1285554083185143566328 max-lower-matrix = 0.1285554083193913529470 min-lower-matrix = 0.1285554083185143566328 The approximated values for the moments are: moment-max-upper-matrix = 0.01326157828813370434522 moment-min-upper-matrix = 0.01326157828794262617374 moment-max-lower-matrix = 0.01326157828813370434522 moment-min-lower-matrix = 0.01326157828794262617374 The minimal lambda is in [0.2178779641139000000000,0.2178779641141000000000] Final result (in the centre of the interval): d = 0.1285554083193913529470 ***************** ****************** 1/48 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 157 The approximated final values are: max-upper-matrix = 0.1271385235081585657688 min-upper-matrix = 0.1271385235072906308587 max-lower-matrix = 0.1271385235081585657688 min-lower-matrix = 0.1271385235072906308587 The approximated values for the moments are: moment-max-upper-matrix = 0.01299763910942290495546 moment-min-upper-matrix = 0.01299763910923559291459 moment-max-lower-matrix = 0.01299763910942290495546 moment-min-lower-matrix = 0.01299763910923559291459 The minimal lambda is in [0.2158134656119000000000,0.2158134656121000000000] Final result (in the centre of the interval): d = 0.1271385235081585657688 ***************** ****************** 1/49 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 141 The approximated final values are: max-upper-matrix = 0.1257665985834185924779 min-upper-matrix = 0.1257665985825594434394 max-lower-matrix = 0.1257665985834185924779 min-lower-matrix = 0.1257665985825594434394 The approximated values for the moments are: moment-max-upper-matrix = 0.01274415473212607660005 moment-min-upper-matrix = 0.01274415473194238290654 moment-max-lower-matrix = 0.01274415473212607660005 moment-min-lower-matrix = 0.01274415473194238290654 The minimal lambda is in [0.2138088798039000000000,0.2138088798041000000000] Final result (in the centre of the interval): d = 0.1257665985834185924779 ***************** ****************** 1/50 The expected number of correct decimal digits is = 10 Approximation for the minimal lambda is = 10^(-10) start of the dyadic step Initial T = 0.1000000000000000000000 Final T = 1.000000000000000000000 E-13 end of the dyadic step Needed matrix exponent for this precision is = 2^36 Number of iterations in the dyadic procedure = 158 The approximated final values are: max-upper-matrix = 0.1244373223558763737019 min-upper-matrix = 0.1244373223550257487007 max-lower-matrix = 0.1244373223558763737019 min-lower-matrix = 0.1244373223550257487007 The approximated values for the moments are: moment-max-upper-matrix = 0.01250051153639539186037 moment-min-upper-matrix = 0.01250051153621517732634 moment-max-lower-matrix = 0.01250051153639539186037 moment-min-lower-matrix = 0.01250051153621517732634 The minimal lambda is in [0.2118613181999000000000,0.2118613182001000000000] Final result (in the centre of the interval): d = 0.1244373223558763737019 ***************** time = 3mn, 47,011 ms. ? for(i=1,10,print("******************");print(10^(-i));PintzRuzsa_psiapprox(10^(-i),20,20);print("*****************")) ****************** 1/10 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^71 Number of iterations in the dyadic procedure = 280 The approximated final values are: max-upper-matrix = 0.29488980846125294572124490305267 min-upper-matrix = 0.29488980846125294572119069178216 max-lower-matrix = 0.29488980846125294572124490305267 min-lower-matrix = 0.29488980846125294572119069178216 The approximated values for the moments are: moment-max-upper-matrix = 0.057365023464857569365701584398947 moment-min-upper-matrix = 0.057365023464857569365678296141641 moment-max-lower-matrix = 0.057365023464857569365701584398946 moment-min-lower-matrix = 0.057365023464857569365678296141641 The minimal lambda is in [0.42958331514361978480019000000000,0.42958331514361978480021000000000] Final result (in the centre of the interval): d = 0.29488980846125294572124490305267 ***************** ****************** 1/100 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^69 Number of iterations in the dyadic procedure = 254 The approximated final values are: max-upper-matrix = 0.086641223890764507332381598589666 min-upper-matrix = 0.086641223890764507332311342908960 max-lower-matrix = 0.086641223890764507332381598589666 min-lower-matrix = 0.086641223890764507332311342908960 The approximated values for the moments are: moment-max-upper-matrix = 0.0064231138618413486541840115218075 moment-min-upper-matrix = 0.0064231138618413486541731825484334 moment-max-lower-matrix = 0.0064231138618413486541840115218075 moment-min-lower-matrix = 0.0064231138618413486541731825484334 The minimal lambda is in [0.15413662304999282680199000000000,0.15413662304999282680201000000000] Final result (in the centre of the interval): d = 0.086641223890764507332381598589666 ***************** ****************** 1/1000 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^67 Number of iterations in the dyadic procedure = 289 The approximated final values are: max-upper-matrix = 0.026671901455027692778215205958212 min-upper-matrix = 0.026671901455027692778126041198534 max-lower-matrix = 0.026671901455027692778215205958212 min-lower-matrix = 0.026671901455027692778126041198534 The approximated values for the moments are: moment-max-upper-matrix = 0.00067560796799354770185504071504572 moment-min-upper-matrix = 0.00067560796799354770185046493614656 moment-max-lower-matrix = 0.00067560796799354770185504071504572 moment-min-lower-matrix = 0.00067560796799354770185046493614656 The minimal lambda is in [0.051318244065253197244190000000000,0.051318244065253197244210000000000] Final result (in the centre of the interval): d = 0.026671901455027692778215205958212 ***************** ****************** 1/10000 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^66 Number of iterations in the dyadic procedure = 278 The approximated final values are: max-upper-matrix = 0.0083601305363790781677139820591352 min-upper-matrix = 0.0083601305363790781676575683732661 max-lower-matrix = 0.0083601305363790781677139820591352 min-lower-matrix = 0.0083601305363790781676575683732661 The approximated values for the moments are: moment-max-upper-matrix = 0.000068744807102128842197410932008347 moment-min-upper-matrix = 0.000068744807102128842196479314176590 moment-max-lower-matrix = 0.000068744807102128842197410932008347 moment-min-lower-matrix = 0.000068744807102128842196479314176590 The minimal lambda is in [0.016514039410910851516590000000000,0.016514039410910851516610000000000] Final result (in the centre of the interval): d = 0.0083601305363790781677139820591352 ***************** ****************** 1/100000 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^64 Number of iterations in the dyadic procedure = 274 The approximated final values are: max-upper-matrix = 0.0026362322945150477383610179294311 min-upper-matrix = 0.0026362322945150477382896568945637 max-lower-matrix = 0.0026362322945150477383610179294311 min-lower-matrix = 0.0026362322945150477382896568945637 The approximated values for the moments are: moment-max-upper-matrix = 0.0000069132947999626581673382929571782 moment-min-upper-matrix = 0.0000069132947999626581669635244382862 moment-max-lower-matrix = 0.0000069132947999626581673382929571782 moment-min-lower-matrix = 0.0000069132947999626581669635244382862 The minimal lambda is in [0.0052517248325830660899900000000000,0.0052517248325830660900100000000000] Final result (in the centre of the interval): d = 0.0026362322945150477383610179294311 ***************** ****************** 1/1000000 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^62 Number of iterations in the dyadic procedure = 274 The approximated final values are: max-upper-matrix = 0.00083290111253256113053386609514297 min-upper-matrix = 0.00083290111253256113044360038237823 max-lower-matrix = 0.00083290111253256113053386609514297 min-lower-matrix = 0.00083290111253256113044360038237823 The approximated values for the moments are: moment-max-upper-matrix = 0.00000069257081800820692309269361401516 moment-min-upper-matrix = 0.00000069257081800820692294251633492455 moment-max-lower-matrix = 0.00000069257081800820692309269361401516 moment-min-lower-matrix = 0.00000069257081800820692294251633492455 The minimal lambda is in [0.0016637245138917730541900000000000,0.0016637245138917730542100000000000] Final result (in the centre of the interval): d = 0.00083290111253256113053386609514297 ***************** ****************** 1/10000000 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^61 Number of iterations in the dyadic procedure = 277 The approximated final values are: max-upper-matrix = 0.00026331153985052285695169305930381 min-upper-matrix = 0.00026331153985052285689460398768856 max-lower-matrix = 0.00026331153985052285695169305930381 min-lower-matrix = 0.00026331153985052285689460398768856 The approximated values for the moments are: moment-max-upper-matrix = 0.000000069296476298644333770428813139509 moment-min-upper-matrix = 0.000000069296476298644333740376258641979 moment-max-lower-matrix = 0.000000069296476298644333770428813139509 moment-min-lower-matrix = 0.000000069296476298644333740376258641979 The minimal lambda is in [0.00052641519028496017839000000000000,0.00052641519028496017841000000000000] Final result (in the centre of the interval): d = 0.00026331153985052285695169305930381 ***************** ****************** 1/100000000 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^59 Number of iterations in the dyadic procedure = 256 The approximated final values are: max-upper-matrix = 0.000083258926779529207901524176437391 min-upper-matrix = 0.000083258926779529207829311575299783 max-lower-matrix = 0.000083258926779529207901524176437391 min-lower-matrix = 0.000083258926779529207829311575299783 The approximated values for the moments are: moment-max-upper-matrix = 0.0000000069308947947825192064402466011144 moment-min-upper-matrix = 0.0000000069308947947825191944170607535783 moment-max-lower-matrix = 0.0000000069308947947825192064402466011144 moment-min-lower-matrix = 0.0000000069308947947825191944170607535783 The minimal lambda is in [0.00016649706087480218419000000000000,0.00016649706087480218421000000000000] Final result (in the centre of the interval): d = 0.000083258926779529207901524176437391 ***************** ****************** 1/1000000000 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^57 Number of iterations in the dyadic procedure = 287 The approximated final values are: max-upper-matrix = 0.000026328035048650673021525832250336 min-upper-matrix = 0.000026328035048650672930183313750208 max-lower-matrix = 0.000026328035048650673021525832250336 min-lower-matrix = 0.000026328035048650672930183313750208 The approximated values for the moments are: moment-max-upper-matrix = 0.00000000069312893231757505501666310895619 moment-min-upper-matrix = 0.00000000069312893231757505020711498837627 moment-max-lower-matrix = 0.00000000069312893231757505501666310895619 moment-min-lower-matrix = 0.00000000069312893231757505020711498837627 The minimal lambda is in [0.000052653990710505674990000000000000,0.000052653990710505675010000000000000] Final result (in the centre of the interval): d = 0.000026328035048650673021525832250336 ***************** ****************** 1/10000000000 The expected number of correct decimal digits is = 20 Approximation for the minimal lambda is = 10^(-20) start of the dyadic step Initial T = 0.10000000000000000000000000000000 Final T = 1.0000000000000000000000000000000 E-23 end of the dyadic step Needed matrix exponent for this precision is = 2^56 Number of iterations in the dyadic procedure = 277 The approximated final values are: max-upper-matrix = 0.0000083255807688638694576403365488115 min-upper-matrix = 0.0000083255807688638693998702553909941 max-lower-matrix = 0.0000083255807688638694576403365488115 min-lower-matrix = 0.0000083255807688638693998702553909941 The approximated values for the moments are: moment-max-upper-matrix = 6.9314140980319905539089707296330 E-11 moment-min-upper-matrix = 6.9314140980319904577162766740520 E-11 moment-max-lower-matrix = 6.9314140980319905539089707296330 E-11 moment-min-lower-matrix = 6.9314140980319904577162766740520 E-11 The minimal lambda is in [0.000016650953595304810590000000000000,0.000016650953595304810610000000000000] Final result (in the centre of the interval): d = 0.0000083255807688638694576403365488115 ***************** time = 4mn, 41,348 ms. ? ********************************************/ 1,348 ms. ? ********************************************/