/** @file exam_differentiation.cpp
*
* Tests for symbolic differentiation, including various functions. */
/*
* GiNaC Copyright (C) 1999-2007 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 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;
clog << "----------symbolic differentiation:" << endl;
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;
if (!result) {
cout << " passed " << endl;
clog << "(no output)" << endl;
} else {
cout << " failed " << endl;
}
return result;
}
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