PARI/GP C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Oliver Laboratoire A2X, U.M.R du C.N.R.S Université Bordeaus I, 351 Cours de la Libérati

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1 PARI/GP C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Oliver Laboratoire A2X, U.M.R du C.N.R.S Université Bordeaus I, 351 Cours de la Libération TALENCE Cedex, FRANCE Home Page: Primary ftp site: ftp://pari.math.u-bordeaux.fr/pub/pari/ last updated September 17, 2002 (this document distributed with version 2.2.8) Translation: TOMINAGA Daisuke Computational Biology Research Center, National Institute of Advanced Industrial Science and Technology Aomi , Koto, Tokyo, , JAPAN February 24, 2004

2 Copyright c The PARI Group Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permissionis granted to copy and distribute modified version, or translations, of this manual under the conditions for verbatim copying, provided also that the entire resulting derived work is distributed under the terms of a permission notice indentical to this one. PARI/GP is Copyright c The PARI Group PARI/GP is free software; you can redistribute it and/or modify it under the terms of the GNU General Public Licence as published by the Free Software Fundation. It is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY WHATSOEVER. GNU General Public Licence

3 GP GP 1 - Greetings! gp Enter Macintosh Retern 1 GP? gp > GP Return Newline Macintosh Enter Macsyma Maple Return ; GP GP 27 * 37 * 27*37 GP GP GP quit exit system GP quit \q q q 2 GP 1/7 GP 1 / 7 GP 1/7 Q 1/7 2/14 PARI GP 1 Return 2 ver (development CHANGES-1.907) q 3

4 GP 1./ 7 1 / 7. 1./ 7. 1 / GP GP 12 GP 64bit integer DEC alpha 28 exp(1) e 28 log(exp(1)) 1 pi Pi GP pi stupid garbage GP stupidgarbage 27 * 37 GP GP??Pi?pi??Pi GP readline enabled readline exp(pi * sqrt(163)) 4

5 \p 50 exp(pi * sqrt(163)) 1 9 PARI sqr(log(%) / Pi) % %1 %2... GP sqr sqr(x) = x * x sqrt 163 exp(π n) n GP GP 1) PARI ver xx ell default(compatible, 3) break next return whatnow(function) 2) * * 1.4 3) 100! 28 4) 28 \p 28 exp(24 * Pi) GP 28 exp(24 π) 33 GP 33 5

6 5) %n n GP exp(24 * Pi) ) 28 default(format, "e0.50") exp(24 * Pi) log(exp(1)) default(format,) \p 7) default GP default(realprecision) default(realprecision, 38) \p(38) default format realprecision default 8)default GP 2 - Warming up 1/0 28 floor(exp(24 *Pi)) floor truncate floor(-3.4) 4 truncate(-3.4) 3 4 exp(24 * Pi) TEX gp 6

7 log(x) GP log(x) 5 log?log sqrt(-1) GP log(-2) exp(i*pi) I^I I i I^2 i^2 6*zeta(2) / Pi^2 GP log(x) log exp(x) exp(x) GP x = seriesprecision \psn default(seriesprecision, n) n O(x^16) 6 GP Big-O x = 0 0 log(x) log(1+x) 0 x x a x= 2 log log(x+2) serieslength \ps cos(x) cos(x)^2 + sin(x)^2 exp(cos(x)) gamma(1+x) exp(exp(x) - 1) 1 / tan(x) (1 + x)^3 O(x) 7 (1 + x)^(-3) -3 1 / 7 1/7 PARI 1/7 (1 + x)^(-3) + O(x^16) (1 + O(x^16)) * (1 + x)^(-3) (1 + x + O(x^16))^(-3),... seriesprecision 5 log is not analytic at 0. 6 O(x^17) 7 7

8 Ser( (1 + x)^(-3) ) (1 + x)^(1/2) PARI PARI (1 + x)^pi (1 + x)^(-3.) -3. PARI taylor((1 + x)^(-3), x) taylor PARI 1) \y 0 n = 3 + I - I 3 type(n) 8 (1+x)^n 1 + 3*x + 3*x^2 + x^3 9 eulerphi arithmetic functions want integer x - x 3 x 3 = 3x 0 10 simplify GP \y 2) x=0 x=a x z + a subst p = 1 / (x^4 + 3*x^3 + 5*x^2-6*x + 7) subst(p, x, z+a) x z + a subst p = 1 + x + x^2 + O(x^10) subst(p, x, z+1) 8 t INT t COMPLEX 9 10 type(3 + x - x) t POL 8

9 3 - The Remaining PARI Types PARI p = x * exp(-x) 16 pr = serreverse(p) x 1 subst(p, x, pr) subst(pr, x, p) x + O(x^17) 11 pr x^n n! PARI serlaplace ps = serlaplace(pr) x n n n 1 pr = n 1 n n 1 X n n! PARI [1,2,3,4] [1,2,3,4]~ ~ \b \b m = [a,b,c; d,e,f] ; \a default(output, 1) default(output, 0) m[1,2] m[1,] m[,2] m[j,k] j k 1 0 m[1,2] = 5; m m[1,] = [15,-17,8] m[,2] = [j,k] m[,2] m[,2] = [j,k]~ h = mathilbert(20) i j (i + j 1) 1 h ; h = mathildert(20); PARI precomputed 11 p 17 +O(x^18) x + O(x^18) default seriesprecision = 16 significant terms 9

10 PARI d = matdet(h) * d sizedigit(1/d) hr = 1. * h; dr = matdet(hr) * d 2 norml2(1/h - 1/hr) norml2 L 2 1/h matdet(h) /hr i i 2 10 vector(10, i, i^2) matrix(5,5,i,j, 1/(i+j-1)) mathilbert i j I 1 vector(10, I, I^2) GP I Pi Euler i pi euler if() if if matrix(8,8, i,j, if ((i-j)%2, x, 0)) x :1 mathilbert(50) 2 :30 10

11 for (i = 1, 5, v[i] = 1/i) v v = vector(5,j,0) PARI n = 10^ PARI factor factor(n, ) factor n PARI factor n n a a n 1 1 (mod n) a = 2 2 n a = Mod(2,n) R = Z/nZ 2 R n R a n 1 =Mod(1,n) a^(n-1) n Mod(1,n) Mod(1,n) n fa = factor(n) factor isprime(fa[,1]) isprime fa 1 p p p n p O(p^n) n= 1 O(p) n = O(7^8) s = sqrt(n) n s^2-n l = log(n) e = exp(l) p e n (n/e)^6 e 11

12 n 1 p 1 p n 569 lift(n) truncate(n) Q quadgen d w (d + a)/2 a d 0 1 quadgen w w w w = quadgen(d) w1 w = quadgen(-163) charpoly(w) w x charpoly(w, y) subst w 1.*w Q( 163) w^10 norm(3 + 4*w) 1 / (4+w) a = Mod(1,23) * w b = a^ lift(b) p = x^2 + w*x + 5*w + 7 norm(p) Q(w) p Q wr = sqrt(w^2) w sqrt algdep(wr,2) w w algdep(sqrt(3*w + 5), 4) algdep Q p Z/pZ w = quadgen(-163) w = Mod(x, x^2 - x + 41) w^10 norm(3 + 4*w) 1 / (4+w) a = Mod(1,23)*w b = a^264 lift(...) x^2 - x + 41 w 1.*w sqrt(w) 2 C w w n n 12

13 a = Mod(x, x^3 - x -1) b = a^5 a b b modreverse(b) b a 4 - Elementary Arithmetic Functions PARI factor factor factor(x^15-1) p factormod factorff GCD gcd GCD bezout chinese isprime ispseudoprime precprime nextprime ispseudoprime isprime µ moebius φ eulerphi ω Ω omega bigomega k σ k sigma \p 1000 contfrac(exp(1)) GP isprime issquare isfundamental 1 forprime fordiv sumdiv n p=1; fordiv(n,d, p *= d); p p *= d p = p * d p forprime(p=3,1000, if (znprimroot(p) == 2, print(p))) znprimroot forprime(p=3,1000, if (znorder(mod(2,p)) == p-1, print(p))) Z/pZ 13

14 5 - Performing Linear Algebra Z lattice PARI PARI p Z Using Transcendental Functions K polylogarithm GP e = exp(1) e = \p 50 e exp(1) e 50 e length(e) ln(2 32 )/ ln(10) \p 28 exp(1.) 28 f = exp( ^(-30)) ^(-30) PARI ^(-30) 59 = f cos(10.^(-15)) length(%) words = 96 bits = 12 bytes 14

15 PARI x 1 cos(x) 1 x 2 / = 58 PARI cos(0) GP 1 GP gamma(10) 9! gamma(100) 99! gamma(100) 156 \p 170 gamma(100) \p 28 lngamma(100) log(99!) gamma(1/2 + 10*I) t = 1000; z= gamma(1 + I*t) * t^(-1/2) * exp(pi/2*t)/sqrt(2*pi) norm(z) norm(z) 1 # zeta(2) Pi^2/6 zeta(3) zeta(3.) 14 n PARI zeta(n) zeta(1 + I) zeta 1/2 + i t t zeta.gp GP \r zeta.gp { t1 = 1/2 + 14*I; t2 = 1/2 + 15*I; eps = 10^(-50); z1 = zeta(t1); until (norm(z2) < eps, z2 = zeta(t2); if (norm(z2) < norm(z1), 14 Apple /PowerBook G4(867MHz 640MB) p 500 zeta(3.)

16 } ) t3 = t1; t1 = t2; t2 = t3; z1 = z2 ); t2 = (t1+t2) / 2.; print(t1 ": " z1) GP \ GP.gp GP 15 GP 25 p a = exp(7 + O(7^10)) log(a) b = log(5 + O(7^10)) exp(b) exp(b) * teichmuller(5 + O(7^10)) teichmuller(x) Teichmüller x p p x p 1 agm(1,sqrt(2)) 1 2 arithmeticgeometric mean AGM Pi/2 / intnum(t=0,pi/2, 1 / sqrt(1 + sin(t)^2)) x = sin(t) 1 0 dx 1 x 4 = π 2agm(1, 2) 2 * agm(1,i) / (1+I) agm(1, O(7^10)) p p p p p = 2 a/b agm(a,b)?3 p 7 - Using Numerical Tools PARI 15 zeta.gp zeta 16

17 π = 4( ) atan(x) π \p * sumalt(k=0, (-1)^k/(2*k + 1)) Pi π 100 Pi PARI 1.38 Wijngaarden F. Villegas D. Zagier H. Cohen 16 \p 28 sumpos(k=1, 1 / k^2) Pi^2/6 sumalt zet(s) = sumalt(k=1, (-1)^(k-1) / k^s) / (1-2^(1-s)) s 1 zet(s) zeta(s) sumalt s zet(s) zet(-1) zet(-2) zet(-1.5) zeta zet(-14.5) zet(i) zeta(i) sumalt(n=1, (-1)^n/(n+I)) intnum(t=1,2, 1/t) log(2) PARI h(z) = e z z r 1 h (z) 2iπ h(z) dz C r C r r fun(z) = { } local(u); u = exp(z); (u-1) / (u-z) zero(r) = r/(2*pi) * intnum(t=0, 2*Pi, fun(r*exp(i*t)) * exp(i*t)) u fun GP 0 16 Cohen, Villegas, Zagier. Convergence Acceleration of Alternating Series, Experimental Mathematics, vol. 9, No. 1, (2000) 17 Chapter Sums, product, integrals and similar functions 17

18 u 18 zero(r) z = r exp(i t) \p 9 zero(1) zero(1.5) z x y z = x + iy e z z = 0 e 2x = x 2 + y 2 e x cos(y) = x y = ± e 2x x 2 e x cos( e 2x x 2 ) = x fun(x) = { local(u); u = exp(x); u*cos(sqrt(u^2 - x^2)) - x } fun(0) fun(1) 28 x0 = solve(x=0,1, fun(x)) x z = x0 + I*sqrt(exp(2*x0) - x0^2) exp(z) - z abs(z) solve solve solve(t=14,15, real(zeta(1/2 + I*t))) 19 t=14 t=15 solve(t=14,15, imag(zeta(1/2 + I*t))) solve(t=14,14.2, real(zeta(1/2 + I*t))) 8 - Functions Related to Polynomials and Power Series exact inexact 18 u 19 roots must be bracketed in solve. 18

19 gcd(x^2-1, x^2-3*x + 2) x - 1 gcd(x^2-1., x^2-3.*x + 2.) 3 PARI gcd(x - Pi, x^2-6*zeta(2)) x - Pi PARI π GCD polresultant(x - Pi, x^2-6*zeta(2)) 0 pol = polcyclo(15) 15 ϕ(15) = 8 r = polroots(pol) 8 28 \b polroots pol 15 rc = r^15. rc = r^ rr = round(real(rc)) sqrt(norml2(rc-rr)) rr 1 rc - rr L norml2 L2 pol = x^5 + x^4 + 2*x^3-2*x^2-4*x -3 factor(pol) pol Q Z fun(p) = factorpadic(pol,p,10) 10 Q p pol fun(p) factor(poldisc(pol)) fun(5) fun(11) fun(23) fun(37) lf(p) = lift(factormod(pol,p)) lf(2) lf(11) lf(23) lf(37) F p pol2 = x^3 + x^2 + x - 1 fq = factorff(pol2, 3, t^3 + t^2 + t - 1) centerlift(lift(fq)) θ t^3 + t^2 + t - 1 F 3 (θ) pol2 F 27 Gal(F 27 /F 3 ) u u 3 3 t

20 pol3 = x^4-4*x^ fn = factornf(pol3,t^2+1) pol3 t^2+1 Q(i) lift(fn) t i = 1 fn2 = factornf(pol3, x^2 + 1) 21 PARI lift(fn2) x*x PARI x x lift C R polroots F p factormod F p k factorff Q p factorpadic Q Z factor factornf nffactor factor pol = x^2+1 factor pol pol * 1. pol * (1+0.*I) pol * Mod(1,2) pol * Mod(1,Mod(1,3)*(t^2+1)) padicappr rootsmod polrootspadic polsturm polsym(pol3,20) pol3 k = 20 pol3 k polroots 2 k 8*x + prod(n=1,39, if(n%4, 1 - x^n, 1), 1 + O(x^40))^8 x O(x^40) n=39 x^40 \ps 122 default(seriesprecision, 122) d = x * eta(x)^24 Ramanujan 122 τ 2 d%2 PARI 21 incorrect polynomial in rnf function. 20

21 22 lift(mod(1,2) * d) Antwarp III 23 centerlift(mod(1,3) * d) Antwerp III 9 - Working with Elliptic Curves ellinit e = ellinit([6,-3,9,-16,-14]) y 2 + 6xy + 9y = x 3 3x 2 16x 14?. ell ellinit init clgpinit 24 e.disc r = ellglobalred(e) r[1] 37 4 e = ellchangecurve(e, r[2]) e c p e y 2 + y = x 3 x e q = [0, 0] ellisoncurve(e, q) q torsion point ellheight(e, q) q - canonical Neron-Tate height q for(k = 1, 20, print(ellpow(e, q,k))) ellheight(e, ellpow(e, q,20)) / ellheight(e, q) 400 = forbidden division t SER % t INT. 23 Volume 3 of the Proceedings of the International Summer School on Modular functions of one variable and arithmetial applications 24 clgp class group 21

22 ellorder(e, q) 0 q ell ob ob init e = ellinit([0,-1,1,0,0]) y 2 +y = x 3 x 2 11 ellglobalred(e) e q = [0, 0] e ellheight(e, q) q for(k=1,5, print(ellpow(e, q,k))) q 5 [0] ellorder(e, q) e = ellinit([0,0,1,-7,6]) y 2 +y = x 3 7x + 6 ellglobalred(e) 5077 elltors(e) e.roots C Y 2 = X 3 7X + 25/4 (X, Y ) = (x, y + 1/2) 3 (x, y) x x 3 (x, y) (x, y 1) ellordinate point.gp \r points { } v=[]; for (x = -3, 1000, s = ellordinate(e,x); if (#s, if cardinality of \kbd{s} is non-zero v = concat(v, [[x,s[1]]]) ) ); v \\ /*.. */ points.gp 25 /* e * 25 EUC OK 22

23 * e * */ order hv = ellheight(e, v) canonical height iv = vecsort(hv,, 1); hv = vecextract(hv, iv); v = vecextract( v, iv); \\ - 4 m = ellheightmatrix(e, vecextract(v,[1,2,3,4])); matdet(m) 4 m kernel matker(m) elladd(e, v[1],v[3]) v[4] 4 vp = [v[1],v[2],v[3]]~; m = ellheightmatrix(e,vp); matdet(m) e 3 ellbil Q v[1] v[2] v[3] matsolve(m, ellbil(e, vp,q)) matker w = vector(18,k, matsolve(m, ellbil(e, vp,v[k]))) wr = round(w); sqrt(norml2(w - wr)) wr v e v1 = [1,0]; v2 = [2,0]; z1 = ellpointtoz(e, v1) z2 = ellpointtoz(e, v2) 23

24 v1 v2 elladd(e, v1,v2) ellztopoint(e, z1 + z2) C e group law one f = x * Ser(ellan(e, 30)) Γ 0 (5077) 2 e Wiles modul = elltaniyama(e) u=modul[1]; v=modul[2]; (v^2 + v) - (u^3-7*u + 6) x * u / (2*v + 1) f GP u v x = e 2iπτ Γ 0 (5077) q = [0,0] 5 e = ellinit([0,-1,1,0,0]) 11 ellglobalred(e) 26 elllseries(e, 1,-11,1.1) elllseries(e, 1,-11,1) 6 ls = elllseries(e, 1,11,1) elllseries(e, 1,11,1.1) 27 1 e L Birch - (Swinnerton-Dyer ls L(E, 1) = Ω c III E tors 2 Ω E c p c p III - Tate-Shafarevich E tors E torsion group Ω e.omega[1] e.roots 3 Ω 2 * e.omega[1] c ellglobalred(e) 1 E 5 torsell(e) 28 5 ls * 25/e[15] III 1 26 expected character: ) instead of: elllseries(e,1,-11,1.1) elltors(e) 24

25 10 - Working in Quadratic Number Fields Q D isfundamental(d) ω D D/2 (1 + D)/2 a + bω w = quadgen(d) w ω w w w1 = quadgen(-23); w2 = quadgen(-15); w1 w2 w PARI ax 2 +bxy +cy 2 3 (a, b, c) Qfb(a,b,c) PARI qfbred Shanks NUCOMP qfbnumcomp qfbnupow qfbclassno qfbclassno Hurwitz class group quadclassunit qfbclassno(-10007) GP 77 PARI qfbclassno quadclassunit Generalized Riemann Hypothesis GRH (a, b, c) a = 2 f = qfbprimeform(-10007, 2) 77 f^77 1 f 11 7 f sqrt(10007)/pi * prodeuler(p=2,500, 1./(1 - kronecker(-10007,p)/p)) kronecker 1 25

26 1./(1 - \dots) PARI PARI prodeuler D = 3299 qfbclassno 27 f = qfbprimeform(-3299, 3) 2 3 f 9 9 f f = qfbprimeform(-3299, 3) quadclassunit quadclassunit(-3299) GRH GRH quadclassunit(-3299,,[1,6]),, GP GRH bnfinit GPH bnf = bnfinit(x^ ); bnfcertify(bnf) 1 bnf init ellinit \bnfcertify quadclassunit?. bnf.clgp 27 GP %.no bnf.clgp.no bnf.no Q 3299 Hilbert quadhilbert(-3299) 27 D > 0 26

27 quadgen (a, b, c) 4 (a, b, c, d) ax 2 + bxy + cy 2 a b c d Archimedean component d Lenstra Qfb(a,b,c,d) (a, b, c) d 0. qfbred d qfbcompraw qfbpowraw qfbprimeform qfbred qfbclassno quadregulator quadclassunit qfbclassno quadregulator O( D) 10 quadclassunit d = 3 * 3299; qfbclassno(d) 3 d = Sholz 3 f = qfbred(qfbprimeform(d,2), 2) 2 qfbref 2 f g = f; for(i=1,20, g = qfbred(g, 3); print(g)) 3 ±1 f f 3 f^3 αz K α f3 = qfbpowraw(f, 3) f 3 8 = 2 3 α ±8 1 27

28 reach the unit form qfbred(%,1) 6 c = component(%, 4) α σ(α) = ±e2c σ a = sqrt(8 * exp(2*c)) sa = 8 / a α σ(α) α a sa α 8 σ(α) sa p = x^2 - round(a - sa)*x - 8 α d α α = ( d)/2 polred modreverse 2 2 α f3 D = 10^7 + 1 quadclassunit(d,,[1,6]) 5 GRH 1 regulator qfbclass quadregulartor bnfinit bnfcertify 5 bnfcertify fundamental unit bnfinit R/ log(10) 1147 quadunit(d) 500 unit form D 0 u = qfbred(qfb(1,1,(1 - D)/4), 2) 0 qfbred f = qfbred(u, 1) principal cycle f f /4.253 = 621 f^ f / f^517 principal form 28

29 517 PARI 500 u f = qfbred(u, 1) f compact representaion p quadhilbert Working on General Number Fields 12 GP - GP Programming 13 - Plotting PARI X windows system 29 PostScript gnuplot 2 /* */ parametric = 1; no_x_axis = 8; points = 64; recursive = 2; no_y_axis = 16; points_lines = 128; norescale = 4; no_frame = 32; splines = 256; /* */ nw = 0; se = 4; relative = 1; sw = 2; ne = 6; /* */ /* */ bottom = 0; /* */ left = 0; /* */ hgap = 16; vcenter = 4; center = 1; vgap = 32; 29 X window system window s 29

30 top = 8; right = 2; \p ploth(x = -2, 2, sin(x^7)) sin 1 1 PARI sin(x 7 ) ploth(x = -2, 2, sin(x^7),recursive) PARI ploth(x = -2, 2, sin(x*7), recursive) 3 [ 2, 2] PARI ploth(x = -2, 2, sin(x*7), recursive, 16) PARI ploth ζ( it) ploth(t = 100, 110, real(zeta(0.5+i*t)), /*empty*/, 1000)

31 ploth(t = 100, 110, real(zeta(0.5+i*t)), recursive) ploth(t = 100, 110, real(zeta(0.5+i*t)), splines, 100) ζ f(t) = z=zeta(0.5+i*t); [real(z),imag(z)] ploth(t = 100, 110, f(t),, 1000) f(t) t ζ( it) ploth(t = 100, 110, f(t), parametric, 1000) ploth(t = 100, 110, f(t), parametric+splines, 100) PARI ploth(t = 100, 110, f(t), parametric+splines+points_lines, 100) t ploth(t = 10000, 10004, f(t), parametric+splines+points_lines, 50) ploth PARI sin(x 7 ) rectangles rectwindows 0 15 rectangles rectangle plotinit(2) plotscale(2, 0,1, 0,1) x y

32 plotmove(2, 0,0) plotbox(2, 1,1) % plotinit(3, 0.88, 0.88, relative) plotrecth(3, X = -2, 2, sin(x^7), recursive) PARI rectangle 2 3 plotdraw([2, 0,0, 3, 0.1,0.02], relative) ploth gnuplot x y rectangle 2 (0.1, 0.1) (0.1, 0.98) (0.98, 0.1) (0.98, 0.98) x plotmove(2, 0.1,0.1) plotstring(2, "-2.000", left+top+vgap) sin X 7 y plotrecth x y plotinit(3, 0.88, 0.88, relative) lims = plotrecth(3, X = -2, 2, sin(x^7), recursive) \p 3 \\ $3$ limits = vector(4,i, Str(lims[i])) plotinit(2); plotscale(2, 0,1, 0,1) plotmove(2, 0,0); plotbox(2, 1,1) plotmove(2, 0.1,0.1); plotstring(2, limits[1], left+top+vgap) plotstring(2, limits[3], bottom+vgap+right+hgap) plotmove(2, 0.98,0.1); plotstring(2, limits[2], right+top+vgap) plotmove(2, 0.1,0.98); plotstring(2, limits[4], right+hgap+top) plotdraw([2, 0,0, 3, 0.1,0.02], relative) PARI ζ(1/2 + it) 32

33 rectangle 0 rectangle 1 rectangle 0 rectangle 3 rectangle 2 rectangle 3 plotrecth plotinit(0, 0.9,0.9, relative) plotrecth(0, t=100,110, f(t), parametric, 300) plotrecth(0, t=100,110, f(t), parametric+splines+points_lines+norescale, 30); plotrecth(0, t=100,110, f(t), parametric+splines+points_lines+norescale, 20); plotdraw([0, 0.05,0.05], relative) plotrect plotinit(0, 0.9,0.9, relative) plotpointtype(-1, 0) \\ plotpointsize(0, 0.4) \\ plotrecth(0, t=100,110, f(t), parametric+points, 300) plotpointsize(0, 1) \\ plotlinetype(-1, 1) \\ plotpointtype(-1, 1) \\ curve_only = norescale + no_frame + no_x_axis + no_y_axis plotrecth(0, t=100,110,f(t), parametric+splines+points_lines+curve_only, 30); plotlinetype(-1, 2) \\ plotpointtype(-1, 2) \\ plotrecth(0, t=100,110,f(t), parametric+splines+points_lines+curve_only, 20); plotdraw([0, 0.05,0.05], relative) plotoinetype plotpointtype plotpointsize PM OS/2 gnuplot PARI 33

34 gnuplot plotfile("plot.tex") plotterm("texdraw") plot.tex gnuplot texdraw plotfile("-") PARI plotfile("manypl1.gif") \\ plotterm("gif=300,200") wpoints = plothsizes()[1] \\$300 \times 200$ { for( k=1,6, plotfile("manypl" k ".gif"); ploth(x = -1, 3, sin(x^k),, wpoints) ) } sin x k, k = GIF manypl1.gif...manypl6.gif 30 PARI PS plotdraw psdraw ploth psploth default psfile PARI PS ζ(1/2 + it) 16 PARI 16 PARI plotcopy 4 A T 2 A = 2; \\ accumulator T = 3; \\ temporary target plotinit(a); plotscale(a, 0, 1, 0, 1) plotinit(t, 0.42, 0.42, relative); plotrecth(t, x= -5, 5, sin(x), recursive) plotcopy(t, 2, 0.05, 0.05, relative + nw) plotmove(a, /2, /2) plotstring(a,"graph", center + vcenter) 30 gnuplot 34

35 plotinit(t, 0.42, 0.42, relative); plotrecth(t, x= -5, 5, [sin(x),cos(2*x)], 0) plotcopy(t, 2, 0.05, 0.05, relative + ne) plotmove(a, /2, /2) plotstring(a,"multigraph", center + vcenter) plotinit(t, 0.42, 0.42, relative); plotrecth(t, x= -5, 5, [sin(3*x), cos(2*x)], parametric) plotcopy(t, 2, 0.05, 0.05, relative + sw) plotmove(a, /2, 0.5) plotstring(a,"parametric", center + vcenter) plotinit(t, 0.42, 0.42, relative); plotrecth(t, x= -5, 5, [sin(x), cos(x), sin(3*x),cos(2*x)], parametric) plotcopy(t, 2, 0.05, 0.05, relative + se) plotmove(a, /2, 0.5) plotstring(a,"multiparametric", center + vcenter) plotlinetype(a, 3) plotmove(a, 0, 0) plotbox(a, 1, 1) plotdraw([a, 0, 0]) \\ psdraw([a, 0, 0], relative) \\ Rectangle A accumulator rectangle T plotrecth Rectangle T geographic plotcopy 0 sin x 0 xlim = 13; ordlim = 25; f(x) = sin(x); 35

36 default(seriesprecision,ordlim) farray(t) = vector((ordlim+1)/2, k, truncate( f(1.*t + O(t^(2*k+1)) ))) FARRAY = farray( t); \\ t t farray(x) f(x) FARRAY f(x) plotinit(3, 0.9, 0.9/1.2, 1); curve_only = no_x_axis+no_y_axis+no_frame; lims = plotrecth(3, x= -xlim, xlim, f(x), recursive+curve_only,16); h = lims[4] - lims[3]; 31 plotinit(4, 0.9,0.9, relative); plotscale(4, lims[1], lims[2], lims[3] - h/10, lims[4] + h/10) plotrecth(4, x = -xlim, xlim, subst(farray,t,x), norescale); plotclip(4) plotrecth(...norescale) xlim 13 -xlim...xlim plotclip plotinit(2) plotscale(2, 0.0, 1.0, 0.0, 1.0) plotmove(2,0.5,0.975) plotstring(2,"multiple Taylor Approximations",center+vcenter) plotdraw([2, 0, 0, 3, 0.05, /12, 4, 0.05, 0.05], relative) 3 PARI plotscale x y x y x y [0, 1] (0,0) (0,1) 31 3 plotrecth 40 36

37 PARI PARI 6 plotsizes ploth plotstring relative plothsizes gnuplot plotterm("dumb") 14 ponpoko web [1] 14.1 Mac OS X v Apple / PowerBook G4(867MHz, 640MB ) GP/PARI CALCULATOR ver (development CHANGES-1.907) (readline v4.3 enabled, extended help available) gnuplot ptex Version p3.1.2 (sjis) (Web2C 7.4.5) platex2e 2001/09/04 +0 (based on LaTeX2e 2001/06/01 patch level 0) 37

38 dvipdfmx TEX ptex package for Mac OS X PARI/GP PARI/GP CVS Unix Mac OS X xterm kterm rxvt % % cvs -d :pserver:cvs@pari.math.u-bordeaux.fr:/home/cvs login % cvs -z3 -d :pserver:cvs@pari.math.u-bordeaux.fr:/home/cvs checkout pari pari GNUPLOT gnuplot CVS PARI/GP configure make install % % cvs -d:pserver:anonymous@cvs.sourceforge.net:/cvsroot/gnuplot login % cvs -z3 -d:pserver:anonymous@cvs.sourceforge.net:/cvsroot/gnuplot co gnuplot % cd gnuplot %./configure % make % cd src % ar cr libgnuplot.a version.o util.o term.o bitmap.o stdfn.o % sudo cp libgnuplot.a /usr/local/lib PARI/GP Configure & make Configure libgnuplot.a CVS PARI/GP gnuplot ver j

39 [1] [2] 39

2.4.2 if Par

2.4.2 if Par Pari-GP 1 Pari-gp 2 1.1 Pari-gp............... 3 2 gp 4 2.1 gp............................. 5 2.1.1 gp................... 5 2.1.2...................... 6 2.1.3............... 7 2.1.4......................