higher arithmetic with PARI/GP
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1 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences MCS 507 Lecture 34 Mathematical, Statistical and Scientific Software Jan Verschelde, 14 November 2012 Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
2 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
3 PARI/GP PARI stems from Pascal ARIthmetic (although switched to C), and GP means the Great Programmable calculator, distributed under the GNU General Public License. PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves...), but also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers, etc., and a lot of transcendental functions. Originally ( ) developed at Université Bordeaux I by a team led by Henri Cohen, PARI/GP is maintained since 1995 by Karim Belabas (Université Bordeaux I) with the help of many volunteer contributors. Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
4 PARI/GP in Sage Although originally intended as a debugging device for the PARI system library, gp became a powerful user-friendly stand-alone programmable calculator. GP is the name of gp s scripting language. The compiler gp2c translates GP code to C which makes it run 3 to 10 times faster (from the User s Guide to PARI/GP). We run gp explicitly in Sage via 1 the class Gp, do help(gp); or 2 opening a Terminal Session with gp, type gp_console(); in a Sage terminal. Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
5 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
6 approximating π To approximate π we could use π 4 = ( 1) k 2k + 1 k=0 but this series converges very slowly. = Because of slow convergence, we could conclude that π cannot be computed as above, but gp proves us wrong. Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
7 the command sumalt? \p 100 realprecision = 115 significant digits \ (100 digits displayed)? pi100 = 4*sumalt(k=0, (-1)^k/(2*k+1) ) %1 = \ \ ? Pi %2 = \ \ ? pi100 - Pi %3 = E-115 sumalt accelerates the convergence of alternating series with algorithms of Cohen, Villegas, and Zagier. Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
8 in a Sage session sage: gp("default(realprecision,100)"); sage: pi100 = gp("4*sumalt(k=0, (-1)^k/(2*k+1))"); sage: pi \ \ sage: type(pi100) <class sage.interfaces.gp.gpelement > sage: RF = RealField(340); sage: sagepi100 = RF(pi100) sage: sagepi \ \ sage: type(sagepi100) <type sage.rings.real_mpfr.realnumber > Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
9 checking the slow sum? \p 10 realprecision = 19 significant digits \ (10 digits displayed)? sum4pi = 4*sum(k=0,1000,(-1.0)^k/(2*k+1)) %4 = ? sum4pi - Pi %5 = ? sum4pi = 4*sum(k=0,100000,(-1.0)^k/(2*k+1)) %6 = ? sum4pi - Pi %7 = E-6? sum4pi = 4*sum(k=0, ,(-1.0)^k/(2*k+1)) %8 = ? sum4pi - Pi %9 = E-8 Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
10 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
11 rational and real numbers? \p 10 realprecision = 19 significant digits \ (10 digits displayed)? a = 3/4 %10 = 3/4? type(a) %11 = "t_frac"? b = 3/ %12 = ? type(b) %13 = "t_real" Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
12 best rational approximation? bestappr(pi) %14 = / \ ? \p 100 realprecision = 115 significant digits \ (100 digits displayed)? bestappr(pi) %15 = \ \ /553\ \ \ ? %16 - Pi %17 = 0.E-114 Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
13 the function bestappr The second optional argument of bestappr is the upper bound on the size of the denominator.? bestappr(sqrt(2),10) %18 = 7/5? bestappr(sqrt(2),10^4) %19 = 8119/5741? bestappr(sqrt(2),10^8) %20 = / For approximations of increasing precision:? apply(k->bestappr(sqrt(2),10^k),[1,2,3,4]) %21 = [7/5, 99/70, 1393/985, 8119/5741] Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
14 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
15 functions and formulas? f(x) = x*(x^2-1) %1 = (x)->x*(x^2-1)? f(2) %2 = 6? f(z) %3 = z^3 - z? dfz = deriv(f(z),z) %4 = 3*z^2-1? g(x) = substpol(dfz,z,x) %5 = (x)->substpol(dfz,z,x)? g(2) %6 = 11? g(z) %7 = 3*z^2-1 Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
16 the Cauchy integral formula The number of roots of f(z) in a disk C a,r in the complex plane centered at z = a and with radius r is 1 2πi C a,r f (z) f(z) dx We compute circular integrals with intcirc.? \p 5? f(z) %8 = z^3 - z? g(z) %9 = 3*z^2-1? intcirc(z=0,0.5,g(z)/f(z)) %10 = E-21*I? intcirc(z=0,1.5,g(z)/f(z)) %11 = E-20*I? intcirc(z=0.9,0.3,g(z)/f(z)) %13 = E-21*I Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
17 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
18 binary and other expansions? b = binary(142) %1 = [1, 0, 0, 0, 1, 1, 1, 0]? c = O(4^6) %2 = 2 + 3*4 + 2*4^3 + O(4^6)? c = O(4^8) %3 = 2 + 3*4 + 2*4^3 + O(4^8)? d = truncate(c) %4 = 142? e = O(7^6) %5 = 2 + 6*7 + 2*7^2 + O(7^6) The same syntax applies for power series:? exp(z)*sin(z) + O(z^6) %6 = z + z^2 + 1/3*z^3-1/30*z^5 + O(z^6) Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
19 continued fractions? L = contfrac(sqrt(21),19) %7 = [4, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, The continued fraction expansion of a rational number is computed by repeated greatest common divisor computations of numerator and denominator = We denote the periodicity in 21 by [4, 1, 1, 2, 1, 1, 8]. Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
20 consecutive convergents? A = contfrac(sqrt(21),9) %8 = [4, 1, 1, 2, 1, 1, 8, 2]? B = contfracpnqn(a) %9 = [ ] [ ]? 999^2-21*(218^2) %10 = -3? 472^2-21*(103^2) %11 = -5 This implies = 3 or 21 = , so 21 = Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
21 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
22 integer relation detection We have log(15) = log(5 3) = log(5) + log(3).? \p 10 realprecision = 19 significant digits \ (10 digits displayed)? u = [log(15.0), log(5.0), log(3.0)] %12 = [ , , ]? lindep(u) %13 = [-1, 1, 1]~ We find ( 1) log(15) + (+1) log(5) + (+1) log(3). Calling lindep(u,-3) uses the PSLQ Algorithm. Note: the type of %13 is "t_col" (column vector). Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
23 lattice basis reduction A lattice is similar to a vector space. Like a vector space, a lattice is spanned (or generated) by a finite number of basis vectors. Unlike a vector space, the basis vectors have only integer entries and only integer linear combinations of the basis vectors are allowed. Given any basis of a lattice, the LLL-algorithm finds a reduced basis, consisting of short vectors. A short vector has small length and its entries are small numbers, like the entries in the vectors of the subset sum problem are either zero or one. Consider the following weights: w = (366, 385, 392, 401, 422, 437) and sum s = This leads to the lattice basis (1, 0, 0, 0, 0, 0, 366), (0, 1, 0, 0, 0, 0, 385), (0, 0, 1, 0, 0, 0, 392), (0, 0, 0, 1, 0, 0, 401), (0, 0, 0, 0, 1, 0, 422), (0, 0, 0, 0, 0, 1, 437), and (0, 0, 0, 0, 0, 0, 1215). The output of the LLL-algorithm gives a reduced basis, which may contain a solution to the subset sum problem s = w x. Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
24 solving a knapsack problem? w = [366, 385, 392, 401, 422, 437] %1 = [366, 385, 392, 401, 422, 437]? type(w) %2 = "t_vec"? L = List(w) %3 = List([366, 385, 392, 401, 422, 437])? s = 1215 %4 = 1215? listinsert(l,-s,1) %5 = -1215? L %6 = List([-1215, 366, 385, 392, 401, 422, 437])? v = Vec(L) %7 = [-1215, 366, 385, 392, 401, 422, 437]? x = lindep(v) %8 = [1, 0, 0, 1, 1, 1, 0]~ Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
25 algebraic numbers For a floating point number x, we can ask to compute the polynomial that has x as its root. algdep(x, k) is a call to lindep([1, x,...,x k ])? \p 10? x = 3.0^(1/6) %14 = ? a = algdep(x,10) %15 = x^8-3*x^2? a = algdep(x,10,-3) %16 = x^10 - x^7-3*x^4 + 3*x Using PSLQ with the flag -3 does not work well. Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
26 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
27 linear algebra? A = [5, -4; -1, 1] %17 = [5-4] [-1 1]? x = [80; 100] %18 = [80] [100]? b = A*x %19 = [0] [20] Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
28 the Hermite normal form The Hermite normal form is computed by mathnf:? L = mathnf(a,1) %20 = [[1, 0; 0, 1], [1, 4; 1, 5]]? U = L[2] %21 = [1 4] [1 5]? U*A %22 = [1 0] [0 1] Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
29 solving linear systems Transforming a right hand side vector:? b = [0; 20] %23 = [0] [20]? U*b %24 = [80] [100] To solve the linear system directly:? y = matsolve(a,b) %25 = [80] [100] Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
30 INTMOD objects? a = Mod(19,5) %23 = Mod(4, 5)? b = Mod(23,5) %24 = Mod(3, 5)? a + b %25 = Mod(2, 5)? ( ) % 5 %26 = 2? 19 % % 5 %27 = 7? type(a) %28 = "t_intmod"? c = lift(a) %29 = 4? type(c) %30 = "t_int" Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
31 modular arithmetic? inva = 1/a %31 = Mod(4, 5)? a*inva %32 = Mod(1, 5)? cba = a^(1/2)j %33 = Mod(3, 5)? cba^2 %34 = Mod(4, 5) Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
32 higher arithmetic with PARI/GP 1 PARI/GP in Sage the computer algebra system PARI/GP acceleration of alternating series rational approximations functions and formulas 2 Arithmetic Functions binary expansions and continued fractions integer relation detection modular arithmetic linear systems of congruences Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
33 linear systems of congruences Consider the input data: M = [m ij ] is any integral matrix; D a column vector of moduli; and B an integer column vector. We look for x, a small integer solution to m ij x j b i mod d i for i running over all rows of M. j Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
34 Chinese remaindering? A = [1,1; 1,2] %1 = [1 1] [1 2]? m = [3,7]~ %2 = [3, 7]~? y = [0,5]~ %3 = [0, 5]~? x = matsolvemod(a,m,y) %4 = [1, 2]~? A*x %5 = [3, 5]~? Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
35 Summary + Exercises Read A Tutorial for PARI/GP (available online). 1 Redo the experiments of lecture one with lindep. 2 Explore the computation of the Smith normal form of a matrix and compare with the implementation in GAP. Homework due Friday 16 November at 10AM: exercises 1 and 3 of lecture 21; exercise 1 of lecture 22; exercise 2 of lecture 23; exercises 2 and 4 of lecture 24; exercises 1 and 2 of lecture 25; exercises 1 and 3 of lecture 26. Scientific Software (MCS 507) higher arithmetic with PARI/GP 14 November / 35
higher arithmetic with PARI/GP
higher arithmetic with PARI/GP 1 2 MCS 507 Lecture 38 Mathematical, Statistical and Scientific Software Jan Verschelde, 18 November 2011 higher arithmetic with PARI/GP 1 2 PARI/GP PARI stems from Pascal
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