1 /** @file exam_differentiation.cpp
3 * Tests for symbolic differentiation, including various functions. */
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25 static unsigned check_diff(const ex &e, const symbol &x,
26 const ex &d, unsigned nth=1)
28 ex ed = e.diff(x, nth);
29 if ((ed - d).compare(ex(0)) != 0) {
45 clog << "derivative of " << e << " by " << x << " returned "
46 << ed << " instead of " << d << endl;
47 clog << "returned:" << endl;
49 clog << endl << "instead of" << endl;
57 // Simple (expanded) polynomials
58 static unsigned exam_differentiation1(void)
61 symbol x("x"), y("y");
64 // construct bivariate polynomial e to be diff'ed:
65 e1 = pow(x, -2) * 3 + pow(x, -1) * 5 + 7 + x * 11 + pow(x, 2) * 13;
66 e2 = pow(y, -2) * 5 + pow(y, -1) * 7 + 11 + y * 13 + pow(y, 2) * 17;
67 e = (e1 * e2).expand();
70 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));
71 result += check_diff(e, x, d);
74 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));
75 result += check_diff(e, y, d);
78 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));
79 result += check_diff(e, x, d, 2);
82 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));
83 result += check_diff(e, y, d, 2);
88 // Trigonometric functions
89 static unsigned exam_differentiation2(void)
92 symbol x("x"), y("y"), a("a"), b("b");
95 // construct expression e to be diff'ed:
96 e1 = y*pow(x, 2) + a*x + b;
98 e = b*pow(e2, 2) + y*e2 + a;
100 d = 2*b*e2*cos(e1)*(2*x*y + a) + y*cos(e1)*(2*x*y + a);
101 result += check_diff(e, x, d);
103 d = 2*b*pow(cos(e1),2)*pow(2*x*y + a, 2) + 4*b*y*e2*cos(e1)
104 - 2*b*pow(e2,2)*pow(2*x*y + a, 2) - y*e2*pow(2*x*y + a, 2)
105 + 2*pow(y,2)*cos(e1);
106 result += check_diff(e, x, d, 2);
108 d = 2*b*e2*cos(e1)*pow(x, 2) + e2 + y*cos(e1)*pow(x, 2);
109 result += check_diff(e, y, d);
111 d = 2*b*pow(cos(e1),2)*pow(x,4) - 2*b*pow(e2,2)*pow(x,4)
112 + 2*cos(e1)*pow(x,2) - y*e2*pow(x,4);
113 result += check_diff(e, y, d, 2);
115 // construct expression e to be diff'ed:
117 e = b*pow(e2, 2) + y*e2 + a;
119 d = -2*b*e2*sin(e1)*(2*x*y + a) - y*sin(e1)*(2*x*y + a);
120 result += check_diff(e, x, d);
122 d = 2*b*pow(sin(e1),2)*pow(2*y*x + a,2) - 4*b*e2*sin(e1)*y
123 - 2*b*pow(e2,2)*pow(2*y*x + a,2) - y*e2*pow(2*y*x + a,2)
124 - 2*pow(y,2)*sin(e1);
125 result += check_diff(e, x, d, 2);
127 d = -2*b*e2*sin(e1)*pow(x,2) + e2 - y*sin(e1)*pow(x, 2);
128 result += check_diff(e, y, d);
130 d = -2*b*pow(e2,2)*pow(x,4) + 2*b*pow(sin(e1),2)*pow(x,4)
131 - 2*sin(e1)*pow(x,2) - y*e2*pow(x,4);
132 result += check_diff(e, y, d, 2);
138 static unsigned exam_differentiation3(void)
141 symbol x("x"), y("y"), a("a"), b("b");
144 // construct expression e to be diff'ed:
145 e1 = y*pow(x, 2) + a*x + b;
147 e = b*pow(e2, 2) + y*e2 + a;
149 d = 2*b*pow(e2, 2)*(2*x*y + a) + y*e2*(2*x*y + a);
150 result += check_diff(e, x, d);
152 d = 4*b*pow(e2,2)*pow(2*y*x + a,2) + 4*b*pow(e2,2)*y
153 + 2*pow(y,2)*e2 + y*e2*pow(2*y*x + a,2);
154 result += check_diff(e, x, d, 2);
156 d = 2*b*pow(e2,2)*pow(x,2) + e2 + y*e2*pow(x,2);
157 result += check_diff(e, y, d);
159 d = 4*b*pow(e2,2)*pow(x,4) + 2*e2*pow(x,2) + y*e2*pow(x,4);
160 result += check_diff(e, y, d, 2);
166 static unsigned exam_differentiation4(void)
169 symbol x("x"), y("y"), a("a"), b("b");
172 // construct expression e to be diff'ed:
173 e1 = y*pow(x, 2) + a*x + b;
175 e = b*pow(e2, 2) + y*e2 + a;
177 d = 2*b*e2*(2*x*y + a)/e1 + y*(2*x*y + a)/e1;
178 result += check_diff(e, x, d);
180 d = 2*b*pow((2*x*y + a),2)*pow(e1,-2) + 4*b*y*e2/e1
181 - 2*b*e2*pow(2*x*y + a,2)*pow(e1,-2) + 2*pow(y,2)/e1
182 - y*pow(2*x*y + a,2)*pow(e1,-2);
183 result += check_diff(e, x, d, 2);
185 d = 2*b*e2*pow(x,2)/e1 + e2 + y*pow(x,2)/e1;
186 result += check_diff(e, y, d);
188 d = 2*b*pow(x,4)*pow(e1,-2) - 2*b*e2*pow(e1,-2)*pow(x,4)
189 + 2*pow(x,2)/e1 - y*pow(x,4)*pow(e1,-2);
190 result += check_diff(e, y, d, 2);
195 // Functions with two variables
196 static unsigned exam_differentiation5(void)
199 symbol x("x"), y("y"), a("a"), b("b");
203 e1 = y*pow(x, 2) + a*x + b;
204 e2 = x*pow(y, 2) + b*y + a;
207 d = pow(y,2)*pow(pow(b+y*pow(x,2)+x*a,2)+pow(y*b+pow(y,2)*x+a,2),-1)*
209 +pow(pow(b+y*pow(x,2)+x*a,2)+pow(y*b+pow(y,2)*x+a,2),-1)*
210 (y*b+pow(y,2)*x+a)*(2*y*x+a);
211 result += check_diff(e, x, d);
217 static unsigned exam_differentiation6(void)
222 e = sin(x).series(x==0, 8);
223 d = cos(x).series(x==0, 7);
225 ed = series_to_poly(ed);
226 d = series_to_poly(d);
228 if ((ed - d).compare(ex(0)) != 0) {
229 clog << "derivative of " << e << " by " << x << " returned "
230 << ed << " instead of " << d << ")" << endl;
236 // Hashing can help a lot, if differentiation is done cleverly
237 static unsigned exam_differentiation7(void)
241 ex e = (P.diff(x) / P).diff(x, 2);
242 ex d = 6/P - 18*x/pow(P,2) - 54*pow(x,3)/pow(P,2) + 2/pow(P,3)
243 +18*pow(x,2)/pow(P,3) + 54*pow(x,4)/pow(P,3) + 54*pow(x,6)/pow(P,3);
245 if (!(e-d).expand().is_zero()) {
246 clog << "expanded second derivative of " << (P.diff(x) / P) << " by " << x
247 << " returned " << e.expand() << " instead of " << d << endl;
251 clog << "second derivative of " << (P.diff(x) / P) << " by " << x
252 << " has " << e.nops() << " operands. "
253 << "The result is still correct but not optimal: 3 are enough! "
254 << "(Hint: maybe the product rule for objects of class mul should be more careful about assembling the result?)" << endl;
260 unsigned exam_differentiation(void)
264 cout << "examining symbolic differentiation" << flush;
265 clog << "----------symbolic differentiation:" << endl;
267 result += exam_differentiation1(); cout << '.' << flush;
268 result += exam_differentiation2(); cout << '.' << flush;
269 result += exam_differentiation3(); cout << '.' << flush;
270 result += exam_differentiation4(); cout << '.' << flush;
271 result += exam_differentiation5(); cout << '.' << flush;
272 result += exam_differentiation6(); cout << '.' << flush;
273 result += exam_differentiation7(); cout << '.' << flush;
276 cout << " passed " << endl;
277 clog << "(no output)" << endl;
279 cout << " failed " << endl;