/** @File exam_pseries.cpp * * Series expansion test (Laurent and Taylor series). */ /* * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include "exams.h" static symbol x("x"); static unsigned check_series(const ex &e, const ex &point, const ex &d, int order = 8) { ex es = e.series(x==point, order); ex ep = ex_to(es).convert_to_poly(); if (!(ep - d).expand().is_zero()) { clog << "series expansion of " << e << " at " << point << " erroneously returned " << ep << " (instead of " << d << ")" << endl; clog << tree << (ep-d) << dflt; return 1; } return 0; } // Series expansion static unsigned exam_series1() { using GiNaC::log; symbol a("a"); symbol b("b"); unsigned result = 0; ex e, d; e = pow(a+b, x); d = 1 + Order(pow(x, 1)); result += check_series(e, 0, d, 1); e = sin(x); d = x - pow(x, 3) / 6 + pow(x, 5) / 120 - pow(x, 7) / 5040 + Order(pow(x, 8)); result += check_series(e, 0, d); e = cos(x); d = 1 - pow(x, 2) / 2 + pow(x, 4) / 24 - pow(x, 6) / 720 + Order(pow(x, 8)); result += check_series(e, 0, d); e = exp(x); d = 1 + x + pow(x, 2) / 2 + pow(x, 3) / 6 + pow(x, 4) / 24 + pow(x, 5) / 120 + pow(x, 6) / 720 + pow(x, 7) / 5040 + Order(pow(x, 8)); result += check_series(e, 0, d); e = pow(1 - x, -1); d = 1 + x + pow(x, 2) + pow(x, 3) + pow(x, 4) + pow(x, 5) + pow(x, 6) + pow(x, 7) + Order(pow(x, 8)); result += check_series(e, 0, d); e = x + pow(x, -1); d = x + pow(x, -1); result += check_series(e, 0, d); e = x + pow(x, -1); d = 2 + pow(x-1, 2) - pow(x-1, 3) + pow(x-1, 4) - pow(x-1, 5) + pow(x-1, 6) - pow(x-1, 7) + Order(pow(x-1, 8)); result += check_series(e, 1, d); e = pow(x + pow(x, 3), -1); d = pow(x, -1) - x + pow(x, 3) - pow(x, 5) + pow(x, 7) + Order(pow(x, 8)); result += check_series(e, 0, d); e = pow(pow(x, 2) + pow(x, 4), -1); d = pow(x, -2) - 1 + pow(x, 2) - pow(x, 4) + pow(x, 6) + Order(pow(x, 8)); result += check_series(e, 0, d); e = pow(sin(x), -2); d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + pow(x, 6) / 675 + Order(pow(x, 8)); result += check_series(e, 0, d); e = sin(x) / cos(x); d = x + pow(x, 3) / 3 + pow(x, 5) * 2/15 + pow(x, 7) * 17/315 + Order(pow(x, 8)); result += check_series(e, 0, d); e = cos(x) / sin(x); d = pow(x, -1) - x / 3 - pow(x, 3) / 45 - pow(x, 5) * 2/945 - pow(x, 7) / 4725 + Order(pow(x, 8)); result += check_series(e, 0, d); e = pow(numeric(2), x); ex t = log(2) * x; d = 1 + t + pow(t, 2) / 2 + pow(t, 3) / 6 + pow(t, 4) / 24 + pow(t, 5) / 120 + pow(t, 6) / 720 + pow(t, 7) / 5040 + Order(pow(x, 8)); result += check_series(e, 0, d.expand()); e = pow(Pi, x); t = log(Pi) * x; d = 1 + t + pow(t, 2) / 2 + pow(t, 3) / 6 + pow(t, 4) / 24 + pow(t, 5) / 120 + pow(t, 6) / 720 + pow(t, 7) / 5040 + Order(pow(x, 8)); result += check_series(e, 0, d.expand()); e = log(x); d = e; result += check_series(e, 0, d, 1); result += check_series(e, 0, d, 2); e = pow(x, 8) * pow(pow(x,3)+ pow(x + pow(x,3), 2), -2); d = pow(x, 4) - 2*pow(x, 5) + Order(pow(x, 6)); result += check_series(e, 0, d, 6); e = cos(x) * pow(sin(x)*(pow(x, 5) + 4 * pow(x, 2)), -3); d = pow(x, -9) / 64 - 3 * pow(x, -6) / 256 - pow(x, -5) / 960 + 535 * pow(x, -3) / 96768 + pow(x, -2) / 1280 - pow(x, -1) / 14400 - numeric(283, 129024) - 2143 * x / 5322240 + Order(pow(x, 2)); result += check_series(e, 0, d, 2); e = sqrt(1+x*x) * sqrt(1+2*x*x); d = 1 + Order(pow(x, 2)); result += check_series(e, 0, d, 2); e = pow(x, 4) * sin(a) + pow(x, 2); d = pow(x, 2) + Order(pow(x, 3)); result += check_series(e, 0, d, 3); e = log(a*x + b*x*x*log(x)); d = log(a*x) + b/a*log(x)*x - pow(b/a, 2)/2*pow(log(x)*x, 2) + Order(pow(x, 3)); result += check_series(e, 0, d, 3); e = pow((x+a), b); d = pow(a, b) + (pow(a, b)*b/a)*x + (pow(a, b)*b*b/a/a/2 - pow(a, b)*b/a/a/2)*pow(x, 2) + Order(pow(x, 3)); result += check_series(e, 0, d, 3); return result; } // Series addition static unsigned exam_series2() { unsigned result = 0; ex e, d; e = pow(sin(x), -1).series(x==0, 8) + pow(sin(-x), -1).series(x==0, 12); d = Order(pow(x, 8)); result += check_series(e, 0, d); return result; } // Series multiplication static unsigned exam_series3() { unsigned result = 0; ex e, d; e = sin(x).series(x==0, 8) * pow(sin(x), -1).series(x==0, 12); d = 1 + Order(pow(x, 7)); result += check_series(e, 0, d); return result; } // Series exponentiation static unsigned exam_series4() { unsigned result = 0; ex e, d; e = pow((2*cos(x)).series(x==0, 5), 2).series(x==0, 5); d = 4 - 4*pow(x, 2) + 4*pow(x, 4)/3 + Order(pow(x, 5)); result += check_series(e, 0, d); e = pow(tgamma(x), 2).series(x==0, 2); d = pow(x,-2) - 2*Euler/x + (pow(Pi,2)/6+2*pow(Euler,2)) + x*(-4*pow(Euler, 3)/3 -pow(Pi,2)*Euler/3 - 2*zeta(3)/3) + Order(pow(x, 2)); result += check_series(e, 0, d); return result; } // Order term handling static unsigned exam_series5() { unsigned result = 0; ex e, d; e = 1 + x + pow(x, 2) + pow(x, 3); d = Order(1); result += check_series(e, 0, d, 0); d = 1 + Order(x); result += check_series(e, 0, d, 1); d = 1 + x + Order(pow(x, 2)); result += check_series(e, 0, d, 2); d = 1 + x + pow(x, 2) + Order(pow(x, 3)); result += check_series(e, 0, d, 3); d = 1 + x + pow(x, 2) + pow(x, 3); result += check_series(e, 0, d, 4); return result; } // Series expansion of tgamma(-1) static unsigned exam_series6() { ex e = tgamma(2*x); ex d = pow(x+1,-1)*numeric(1,4) + pow(x+1,0)*(numeric(3,4) - numeric(1,2)*Euler) + pow(x+1,1)*(numeric(7,4) - numeric(3,2)*Euler + numeric(1,2)*pow(Euler,2) + numeric(1,12)*pow(Pi,2)) + pow(x+1,2)*(numeric(15,4) - numeric(7,2)*Euler - numeric(1,3)*pow(Euler,3) + numeric(1,4)*pow(Pi,2) + numeric(3,2)*pow(Euler,2) - numeric(1,6)*pow(Pi,2)*Euler - numeric(2,3)*zeta(3)) + pow(x+1,3)*(numeric(31,4) - pow(Euler,3) - numeric(15,2)*Euler + numeric(1,6)*pow(Euler,4) + numeric(7,2)*pow(Euler,2) + numeric(7,12)*pow(Pi,2) - numeric(1,2)*pow(Pi,2)*Euler - numeric(2)*zeta(3) + numeric(1,6)*pow(Euler,2)*pow(Pi,2) + numeric(1,40)*pow(Pi,4) + numeric(4,3)*zeta(3)*Euler) + Order(pow(x+1,4)); return check_series(e, -1, d, 4); } // Series expansion of tan(x==Pi/2) static unsigned exam_series7() { ex e = tan(x*Pi/2); ex d = pow(x-1,-1)/Pi*(-2) + pow(x-1,1)*Pi/6 + pow(x-1,3)*pow(Pi,3)/360 +pow(x-1,5)*pow(Pi,5)/15120 + pow(x-1,7)*pow(Pi,7)/604800 +Order(pow(x-1,9)); return check_series(e,1,d,9); } // Series expansion of log(sin(x==0)) static unsigned exam_series8() { ex e = log(sin(x)); ex d = log(x) - pow(x,2)/6 - pow(x,4)/180 - pow(x,6)/2835 - pow(x,8)/37800 + Order(pow(x,9)); return check_series(e,0,d,9); } // Series expansion of Li2(sin(x==0)) static unsigned exam_series9() { ex e = Li2(sin(x)); ex d = x + pow(x,2)/4 - pow(x,3)/18 - pow(x,4)/48 - 13*pow(x,5)/1800 - pow(x,6)/360 - 23*pow(x,7)/21168 + Order(pow(x,8)); return check_series(e,0,d,8); } // Series expansion of Li2((x==2)^2), caring about branch-cut static unsigned exam_series10() { using GiNaC::log; ex e = Li2(pow(x,2)); ex d = Li2(4) + (-log(3) + I*Pi*csgn(I-I*pow(x,2))) * (x-2) + (numeric(-2,3) + log(3)/4 - I*Pi/4*csgn(I-I*pow(x,2))) * pow(x-2,2) + (numeric(11,27) - log(3)/12 + I*Pi/12*csgn(I-I*pow(x,2))) * pow(x-2,3) + (numeric(-155,648) + log(3)/32 - I*Pi/32*csgn(I-I*pow(x,2))) * pow(x-2,4) + Order(pow(x-2,5)); return check_series(e,2,d,5); } // Series expansion of logarithms around branch points static unsigned exam_series11() { using GiNaC::log; unsigned result = 0; ex e, d; symbol a("a"); e = log(x); d = log(x); result += check_series(e,0,d,5); e = log(3/x); d = log(3)-log(x); result += check_series(e,0,d,5); e = log(3*pow(x,2)); d = log(3)+2*log(x); result += check_series(e,0,d,5); // These ones must not be expanded because it would result in a branch cut // running in the wrong direction. (Other systems tend to get this wrong.) e = log(-x); d = e; result += check_series(e,0,d,5); e = log(I*(x-123)); d = e; result += check_series(e,123,d,5); e = log(a*x); d = e; // we don't know anything about a! result += check_series(e,0,d,5); e = log((1-x)/x); d = log(1-x) - (x-1) + pow(x-1,2)/2 - pow(x-1,3)/3 + pow(x-1,4)/4 + Order(pow(x-1,5)); result += check_series(e,1,d,5); return result; } // Series expansion of other functions around branch points static unsigned exam_series12() { using GiNaC::log; unsigned result = 0; ex e, d; // NB: Mma and Maple give different results, but they agree if one // takes into account that by assumption |x|<1. e = atan(x); d = (I*log(2)/2-I*log(1+I*x)/2) + (x-I)/4 + I*pow(x-I,2)/16 + Order(pow(x-I,3)); result += check_series(e,I,d,3); // NB: here, at -I, Mathematica disagrees, but it is wrong -- they // pick up a complex phase by incorrectly expanding logarithms. e = atan(x); d = (-I*log(2)/2+I*log(1-I*x)/2) + (x+I)/4 - I*pow(x+I,2)/16 + Order(pow(x+I,3)); result += check_series(e,-I,d,3); // This is basically the same as above, the branch point is at +/-1: e = atanh(x); d = (-log(2)/2+log(x+1)/2) + (x+1)/4 + pow(x+1,2)/16 + Order(pow(x+1,3)); result += check_series(e,-1,d,3); return result; } unsigned exam_pseries() { unsigned result = 0; cout << "examining series expansion" << flush; clog << "----------series expansion:" << endl; result += exam_series1(); cout << '.' << flush; result += exam_series2(); cout << '.' << flush; result += exam_series3(); cout << '.' << flush; result += exam_series4(); cout << '.' << flush; result += exam_series5(); cout << '.' << flush; result += exam_series6(); cout << '.' << flush; result += exam_series7(); cout << '.' << flush; result += exam_series8(); cout << '.' << flush; result += exam_series9(); cout << '.' << flush; result += exam_series10(); cout << '.' << flush; result += exam_series11(); cout << '.' << flush; result += exam_series12(); cout << '.' << flush; if (!result) { cout << " passed " << endl; clog << "(no output)" << endl; } else { cout << " failed " << endl; } return result; }