1 /** @File exam_pseries.cpp
3 * Series expansion test (Laurent and Taylor series). */
6 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
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13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
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19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
27 static unsigned check_series(const ex &e, const ex &point, const ex &d, int order = 8)
29 ex es = e.series(x==point, order);
30 ex ep = ex_to<pseries>(es).convert_to_poly();
31 if (!(ep - d).is_zero()) {
32 clog << "series expansion of " << e << " at " << point
33 << " erroneously returned " << ep << " (instead of " << d
35 (ep-d).printtree(clog);
42 static unsigned exam_series1(void)
48 d = x - pow(x, 3) / 6 + pow(x, 5) / 120 - pow(x, 7) / 5040 + Order(pow(x, 8));
49 result += check_series(e, 0, d);
52 d = 1 - pow(x, 2) / 2 + pow(x, 4) / 24 - pow(x, 6) / 720 + Order(pow(x, 8));
53 result += check_series(e, 0, d);
56 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));
57 result += check_series(e, 0, d);
60 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));
61 result += check_series(e, 0, d);
65 result += check_series(e, 0, d);
68 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));
69 result += check_series(e, 1, d);
71 e = pow(x + pow(x, 3), -1);
72 d = pow(x, -1) - x + pow(x, 3) - pow(x, 5) + Order(pow(x, 7));
73 result += check_series(e, 0, d);
75 e = pow(pow(x, 2) + pow(x, 4), -1);
76 d = pow(x, -2) - 1 + pow(x, 2) - pow(x, 4) + Order(pow(x, 6));
77 result += check_series(e, 0, d);
80 d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + Order(pow(x, 5));
81 result += check_series(e, 0, d);
84 d = x + pow(x, 3) / 3 + pow(x, 5) * 2/15 + pow(x, 7) * 17/315 + Order(pow(x, 8));
85 result += check_series(e, 0, d);
88 d = pow(x, -1) - x / 3 - pow(x, 3) / 45 - pow(x, 5) * 2/945 + Order(pow(x, 6));
89 result += check_series(e, 0, d);
91 e = pow(numeric(2), x);
93 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));
94 result += check_series(e, 0, d.expand());
98 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));
99 result += check_series(e, 0, d.expand());
103 result += check_series(e, 0, d, 1);
104 result += check_series(e, 0, d, 2);
110 static unsigned exam_series2(void)
115 e = pow(sin(x), -1).series(x==0, 8) + pow(sin(-x), -1).series(x==0, 12);
116 d = Order(pow(x, 6));
117 result += check_series(e, 0, d);
122 // Series multiplication
123 static unsigned exam_series3(void)
128 e = sin(x).series(x==0, 8) * pow(sin(x), -1).series(x==0, 12);
129 d = 1 + Order(pow(x, 7));
130 result += check_series(e, 0, d);
135 // Series exponentiation
136 static unsigned exam_series4(void)
141 e = pow((2*cos(x)).series(x==0, 5), 2).series(x==0, 5);
142 d = 4 - 4*pow(x, 2) + 4*pow(x, 4)/3 + Order(pow(x, 5));
143 result += check_series(e, 0, d);
145 e = pow(tgamma(x), 2).series(x==0, 3);
146 d = pow(x,-2) - 2*Euler/x + (pow(Pi,2)/6+2*pow(Euler,2)) + Order(x);
147 result += check_series(e, 0, d);
152 // Order term handling
153 static unsigned exam_series5(void)
158 e = 1 + x + pow(x, 2) + pow(x, 3);
160 result += check_series(e, 0, d, 0);
162 result += check_series(e, 0, d, 1);
163 d = 1 + x + Order(pow(x, 2));
164 result += check_series(e, 0, d, 2);
165 d = 1 + x + pow(x, 2) + Order(pow(x, 3));
166 result += check_series(e, 0, d, 3);
167 d = 1 + x + pow(x, 2) + pow(x, 3);
168 result += check_series(e, 0, d, 4);
172 // Series expansion of tgamma(-1)
173 static unsigned exam_series6(void)
176 ex d = pow(x+1,-1)*numeric(1,4) +
177 pow(x+1,0)*(numeric(3,4) -
178 numeric(1,2)*Euler) +
179 pow(x+1,1)*(numeric(7,4) -
181 numeric(1,2)*pow(Euler,2) +
182 numeric(1,12)*pow(Pi,2)) +
183 pow(x+1,2)*(numeric(15,4) -
185 numeric(1,3)*pow(Euler,3) +
186 numeric(1,4)*pow(Pi,2) +
187 numeric(3,2)*pow(Euler,2) -
188 numeric(1,6)*pow(Pi,2)*Euler -
189 numeric(2,3)*zeta(3)) +
190 pow(x+1,3)*(numeric(31,4) - pow(Euler,3) -
191 numeric(15,2)*Euler +
192 numeric(1,6)*pow(Euler,4) +
193 numeric(7,2)*pow(Euler,2) +
194 numeric(7,12)*pow(Pi,2) -
195 numeric(1,2)*pow(Pi,2)*Euler -
197 numeric(1,6)*pow(Euler,2)*pow(Pi,2) +
198 numeric(1,40)*pow(Pi,4) +
199 numeric(4,3)*zeta(3)*Euler) +
201 return check_series(e, -1, d, 4);
204 // Series expansion of tan(x==Pi/2)
205 static unsigned exam_series7(void)
208 ex d = pow(x-1,-1)/Pi*(-2) + pow(x-1,1)*Pi/6 + pow(x-1,3)*pow(Pi,3)/360
209 +pow(x-1,5)*pow(Pi,5)/15120 + pow(x-1,7)*pow(Pi,7)/604800
211 return check_series(e,1,d,8);
214 // Series expansion of log(sin(x==0))
215 static unsigned exam_series8(void)
218 ex d = log(x) - pow(x,2)/6 - pow(x,4)/180 - pow(x,6)/2835
220 return check_series(e,0,d,8);
223 // Series expansion of Li2(sin(x==0))
224 static unsigned exam_series9(void)
227 ex d = x + pow(x,2)/4 - pow(x,3)/18 - pow(x,4)/48
228 - 13*pow(x,5)/1800 - pow(x,6)/360 - 23*pow(x,7)/21168
230 return check_series(e,0,d,8);
233 // Series expansion of Li2((x==2)^2), caring about branch-cut
234 static unsigned exam_series10(void)
236 ex e = Li2(pow(x,2));
237 ex d = Li2(4) + (-log(3) + I*Pi*csgn(I-I*pow(x,2))) * (x-2)
238 + (numeric(-2,3) + log(3)/4 - I*Pi/4*csgn(I-I*pow(x,2))) * pow(x-2,2)
239 + (numeric(11,27) - log(3)/12 + I*Pi/12*csgn(I-I*pow(x,2))) * pow(x-2,3)
240 + (numeric(-155,648) + log(3)/32 - I*Pi/32*csgn(I-I*pow(x,2))) * pow(x-2,4)
242 return check_series(e,2,d,5);
245 // Series expansion of logarithms around branch points
246 static unsigned exam_series11(void)
254 result += check_series(e,0,d,5);
258 result += check_series(e,0,d,5);
262 result += check_series(e,0,d,5);
264 // These ones must not be expanded because it would result in a branch cut
265 // running in the wrong direction. (Other systems tend to get this wrong.)
268 result += check_series(e,0,d,5);
272 result += check_series(e,123,d,5);
275 d = e; // we don't know anything about a!
276 result += check_series(e,0,d,5);
279 d = log(1-x) - (x-1) + pow(x-1,2)/2 - pow(x-1,3)/3 + Order(pow(x-1,4));
280 result += check_series(e,1,d,4);
285 // Series expansion of other functions around branch points
286 static unsigned exam_series12(void)
291 // NB: Mma and Maple give different results, but they agree if one
292 // takes into account that by assumption |x|<1.
294 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));
295 result += check_series(e,I,d,3);
297 // NB: here, at -I, Mathematica disagrees, but it is wrong -- they
298 // pick up a complex phase by incorrectly expanding logarithms.
300 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));
301 result += check_series(e,-I,d,3);
303 // This is basically the same as above, the branch point is at +/-1:
305 d = (-log(2)/2+log(x+1)/2) + (x+1)/4 + pow(x+1,2)/16 + Order(pow(x+1,3));
306 result += check_series(e,-1,d,3);
312 unsigned exam_pseries(void)
316 cout << "examining series expansion" << flush;
317 clog << "----------series expansion:" << endl;
319 result += exam_series1(); cout << '.' << flush;
320 result += exam_series2(); cout << '.' << flush;
321 result += exam_series3(); cout << '.' << flush;
322 result += exam_series4(); cout << '.' << flush;
323 result += exam_series5(); cout << '.' << flush;
324 result += exam_series6(); cout << '.' << flush;
325 result += exam_series7(); cout << '.' << flush;
326 result += exam_series8(); cout << '.' << flush;
327 result += exam_series9(); cout << '.' << flush;
328 result += exam_series10(); cout << '.' << flush;
329 result += exam_series11(); cout << '.' << flush;
330 result += exam_series12(); cout << '.' << flush;
333 cout << " passed " << endl;
334 clog << "(no output)" << endl;
336 cout << " failed " << endl;