/** @file exam_differentiation.cpp * * Tests for symbolic differentiation, including various functions. */ /* * GiNaC Copyright (C) 1999-2008 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 #include "ginac.h" using namespace std; using namespace GiNaC; static unsigned check_diff(const ex &e, const symbol &x, const ex &d, unsigned nth=1) { ex ed = e.diff(x, nth); if (!(ed - d).is_zero()) { switch (nth) { case 0: clog << "zeroth "; break; case 1: break; case 2: clog << "second "; break; case 3: clog << "third "; break; default: clog << nth << "th "; } clog << "derivative of " << e << " by " << x << " returned " << ed << " instead of " << d << endl; clog << "returned:" << endl; clog << tree << ed << "instead of\n" << d << dflt; return 1; } return 0; } // Simple (expanded) polynomials static unsigned exam_differentiation1() { unsigned result = 0; symbol x("x"), y("y"); ex e1, e2, e, d; // construct bivariate polynomial e to be diff'ed: e1 = pow(x, -2) * 3 + pow(x, -1) * 5 + 7 + x * 11 + pow(x, 2) * 13; e2 = pow(y, -2) * 5 + pow(y, -1) * 7 + 11 + y * 13 + pow(y, 2) * 17; e = (e1 * e2).expand(); // d e / dx: d = ex("121-55/x^2-66/x^3-30/x^3/y^2-42/x^3/y-78/x^3*y-102/x^3*y^2-25/x^2/y^2-35/x^2/y-65/x^2*y-85/x^2*y^2+77/y+143*y+187*y^2+130*x/y^2+182/y*x+338*x*y+442*x*y^2+55/y^2+286*x",lst(x,y)); result += check_diff(e, x, d); // d e / dy: d = ex("91-30/x^2/y^3-21/x^2/y^2+39/x^2+102/x^2*y-50/x/y^3-35/x/y^2+65/x+170/x*y-77*x/y^2+143*x+374*x*y-130/y^3*x^2-91/y^2*x^2+169*x^2+442*x^2*y-110/y^3*x-70/y^3+238*y-49/y^2",lst(x,y)); result += check_diff(e, y, d); // d^2 e / dx^2: d = ex("286+90/x^4/y^2+126/x^4/y+234/x^4*y+306/x^4*y^2+50/x^3/y^2+70/x^3/y+130/x^3*y+170/x^3*y^2+130/y^2+182/y+338*y+442*y^2+198/x^4+110/x^3",lst(x,y)); result += check_diff(e, x, d, 2); // d^2 e / dy^2: d = ex("238+90/x^2/y^4+42/x^2/y^3+102/x^2+150/x/y^4+70/x/y^3+170/x+330*x/y^4+154*x/y^3+374*x+390*x^2/y^4+182*x^2/y^3+442*x^2+210/y^4+98/y^3",lst(x,y)); result += check_diff(e, y, d, 2); return result; } // Trigonometric functions static unsigned exam_differentiation2() { unsigned result = 0; symbol x("x"), y("y"), a("a"), b("b"); ex e1, e2, e, d; // construct expression e to be diff'ed: e1 = y*pow(x, 2) + a*x + b; e2 = sin(e1); e = b*pow(e2, 2) + y*e2 + a; d = 2*b*e2*cos(e1)*(2*x*y + a) + y*cos(e1)*(2*x*y + a); result += check_diff(e, x, d); d = 2*b*pow(cos(e1),2)*pow(2*x*y + a, 2) + 4*b*y*e2*cos(e1) - 2*b*pow(e2,2)*pow(2*x*y + a, 2) - y*e2*pow(2*x*y + a, 2) + 2*pow(y,2)*cos(e1); result += check_diff(e, x, d, 2); d = 2*b*e2*cos(e1)*pow(x, 2) + e2 + y*cos(e1)*pow(x, 2); result += check_diff(e, y, d); d = 2*b*pow(cos(e1),2)*pow(x,4) - 2*b*pow(e2,2)*pow(x,4) + 2*cos(e1)*pow(x,2) - y*e2*pow(x,4); result += check_diff(e, y, d, 2); // construct expression e to be diff'ed: e2 = cos(e1); e = b*pow(e2, 2) + y*e2 + a; d = -2*b*e2*sin(e1)*(2*x*y + a) - y*sin(e1)*(2*x*y + a); result += check_diff(e, x, d); d = 2*b*pow(sin(e1),2)*pow(2*y*x + a,2) - 4*b*e2*sin(e1)*y - 2*b*pow(e2,2)*pow(2*y*x + a,2) - y*e2*pow(2*y*x + a,2) - 2*pow(y,2)*sin(e1); result += check_diff(e, x, d, 2); d = -2*b*e2*sin(e1)*pow(x,2) + e2 - y*sin(e1)*pow(x, 2); result += check_diff(e, y, d); d = -2*b*pow(e2,2)*pow(x,4) + 2*b*pow(sin(e1),2)*pow(x,4) - 2*sin(e1)*pow(x,2) - y*e2*pow(x,4); result += check_diff(e, y, d, 2); return result; } // exp function static unsigned exam_differentiation3() { unsigned result = 0; symbol x("x"), y("y"), a("a"), b("b"); ex e1, e2, e, d; // construct expression e to be diff'ed: e1 = y*pow(x, 2) + a*x + b; e2 = exp(e1); e = b*pow(e2, 2) + y*e2 + a; d = 2*b*pow(e2, 2)*(2*x*y + a) + y*e2*(2*x*y + a); result += check_diff(e, x, d); d = 4*b*pow(e2,2)*pow(2*y*x + a,2) + 4*b*pow(e2,2)*y + 2*pow(y,2)*e2 + y*e2*pow(2*y*x + a,2); result += check_diff(e, x, d, 2); d = 2*b*pow(e2,2)*pow(x,2) + e2 + y*e2*pow(x,2); result += check_diff(e, y, d); d = 4*b*pow(e2,2)*pow(x,4) + 2*e2*pow(x,2) + y*e2*pow(x,4); result += check_diff(e, y, d, 2); return result; } // log functions static unsigned exam_differentiation4() { unsigned result = 0; symbol x("x"), y("y"), a("a"), b("b"); ex e1, e2, e, d; // construct expression e to be diff'ed: e1 = y*pow(x, 2) + a*x + b; e2 = log(e1); e = b*pow(e2, 2) + y*e2 + a; d = 2*b*e2*(2*x*y + a)/e1 + y*(2*x*y + a)/e1; result += check_diff(e, x, d); d = 2*b*pow((2*x*y + a),2)*pow(e1,-2) + 4*b*y*e2/e1 - 2*b*e2*pow(2*x*y + a,2)*pow(e1,-2) + 2*pow(y,2)/e1 - y*pow(2*x*y + a,2)*pow(e1,-2); result += check_diff(e, x, d, 2); d = 2*b*e2*pow(x,2)/e1 + e2 + y*pow(x,2)/e1; result += check_diff(e, y, d); d = 2*b*pow(x,4)*pow(e1,-2) - 2*b*e2*pow(e1,-2)*pow(x,4) + 2*pow(x,2)/e1 - y*pow(x,4)*pow(e1,-2); result += check_diff(e, y, d, 2); return result; } // Functions with two variables static unsigned exam_differentiation5() { unsigned result = 0; symbol x("x"), y("y"), a("a"), b("b"); ex e1, e2, e, d; // test atan2 e1 = y*pow(x, 2) + a*x + b; e2 = x*pow(y, 2) + b*y + a; e = atan2(e1,e2); d = pow(y,2)*pow(pow(b+y*pow(x,2)+x*a,2)+pow(y*b+pow(y,2)*x+a,2),-1)* (-b-y*pow(x,2)-x*a) +pow(pow(b+y*pow(x,2)+x*a,2)+pow(y*b+pow(y,2)*x+a,2),-1)* (y*b+pow(y,2)*x+a)*(2*y*x+a); result += check_diff(e, x, d); return result; } // Series static unsigned exam_differentiation6() { symbol x("x"); ex e, d, ed; e = sin(x).series(x==0, 8); d = cos(x).series(x==0, 7); ed = e.diff(x); ed = series_to_poly(ed); d = series_to_poly(d); if (!(ed - d).is_zero()) { clog << "derivative of " << e << " by " << x << " returned " << ed << " instead of " << d << ")" << endl; return 1; } return 0; } // Hashing can help a lot, if differentiation is done cleverly static unsigned exam_differentiation7() { symbol x("x"); ex P = x + pow(x,3); ex e = (P.diff(x) / P).diff(x, 2); ex d = 6/P - 18*x/pow(P,2) - 54*pow(x,3)/pow(P,2) + 2/pow(P,3) +18*pow(x,2)/pow(P,3) + 54*pow(x,4)/pow(P,3) + 54*pow(x,6)/pow(P,3); if (!(e-d).expand().is_zero()) { clog << "expanded second derivative of " << (P.diff(x) / P) << " by " << x << " returned " << e.expand() << " instead of " << d << endl; return 1; } if (e.nops() > 3) { clog << "second derivative of " << (P.diff(x) / P) << " by " << x << " has " << e.nops() << " operands. " << "The result is still correct but not optimal: 3 are enough! " << "(Hint: maybe the product rule for objects of class mul should be more careful about assembling the result?)" << endl; return 1; } return 0; } unsigned exam_differentiation() { unsigned result = 0; cout << "examining symbolic differentiation" << flush; result += exam_differentiation1(); cout << '.' << flush; result += exam_differentiation2(); cout << '.' << flush; result += exam_differentiation3(); cout << '.' << flush; result += exam_differentiation4(); cout << '.' << flush; result += exam_differentiation5(); cout << '.' << flush; result += exam_differentiation6(); cout << '.' << flush; result += exam_differentiation7(); cout << '.' << flush; return result; } int main(int argc, char** argv) { return exam_differentiation(); }