1 /** @File exam_pseries.cpp
3 * Series expansion test (Laurent and Taylor series). */
6 * GiNaC Copyright (C) 1999-2004 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).expand().is_zero()) {
32 clog << "series expansion of " << e << " at " << point
33 << " erroneously returned " << ep << " (instead of " << d
35 clog << tree << (ep-d) << dflt;
42 static unsigned exam_series1()
50 d = x - pow(x, 3) / 6 + pow(x, 5) / 120 - pow(x, 7) / 5040 + Order(pow(x, 8));
51 result += check_series(e, 0, d);
54 d = 1 - pow(x, 2) / 2 + pow(x, 4) / 24 - pow(x, 6) / 720 + Order(pow(x, 8));
55 result += check_series(e, 0, d);
58 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));
59 result += check_series(e, 0, d);
62 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));
63 result += check_series(e, 0, d);
67 result += check_series(e, 0, d);
70 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));
71 result += check_series(e, 1, d);
73 e = pow(x + pow(x, 3), -1);
74 d = pow(x, -1) - x + pow(x, 3) - pow(x, 5) + pow(x, 7) + Order(pow(x, 8));
75 result += check_series(e, 0, d);
77 e = pow(pow(x, 2) + pow(x, 4), -1);
78 d = pow(x, -2) - 1 + pow(x, 2) - pow(x, 4) + pow(x, 6) + Order(pow(x, 8));
79 result += check_series(e, 0, d);
82 d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + pow(x, 6) / 675 + Order(pow(x, 8));
83 result += check_series(e, 0, d);
86 d = x + pow(x, 3) / 3 + pow(x, 5) * 2/15 + pow(x, 7) * 17/315 + Order(pow(x, 8));
87 result += check_series(e, 0, d);
90 d = pow(x, -1) - x / 3 - pow(x, 3) / 45 - pow(x, 5) * 2/945 - pow(x, 7) / 4725 + Order(pow(x, 8));
91 result += check_series(e, 0, d);
93 e = pow(numeric(2), x);
95 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));
96 result += check_series(e, 0, d.expand());
100 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));
101 result += check_series(e, 0, d.expand());
105 result += check_series(e, 0, d, 1);
106 result += check_series(e, 0, d, 2);
108 e = pow(x, 8) * pow(pow(x,3)+ pow(x + pow(x,3), 2), -2);
109 d = pow(x, 4) - 2*pow(x, 5) + Order(pow(x, 6));
110 result += check_series(e, 0, d, 6);
112 e = cos(x) * pow(sin(x)*(pow(x, 5) + 4 * pow(x, 2)), -3);
113 d = pow(x, -9) / 64 - 3 * pow(x, -6) / 256 - pow(x, -5) / 960 + 535 * pow(x, -3) / 96768
114 + pow(x, -2) / 1280 - pow(x, -1) / 14400 - numeric(283, 129024) - 2143 * x / 5322240
116 result += check_series(e, 0, d, 2);
118 e = sqrt(1+x*x) * sqrt(1+2*x*x);
119 d = 1 + Order(pow(x, 2));
120 result += check_series(e, 0, d, 2);
123 e = pow(x, 4) * sin(a) + pow(x, 2);
124 d = pow(x, 2) + Order(pow(x, 3));
125 result += check_series(e, 0, d, 3);
128 e = log(a*x + b*x*x*log(x));
129 d = log(a*x) + b/a*log(x)*x - pow(b/a, 2)/2*pow(log(x)*x, 2) + Order(pow(x, 3));
130 result += check_series(e, 0, d, 3);
136 static unsigned exam_series2()
141 e = pow(sin(x), -1).series(x==0, 8) + pow(sin(-x), -1).series(x==0, 12);
142 d = Order(pow(x, 8));
143 result += check_series(e, 0, d);
148 // Series multiplication
149 static unsigned exam_series3()
154 e = sin(x).series(x==0, 8) * pow(sin(x), -1).series(x==0, 12);
155 d = 1 + Order(pow(x, 7));
156 result += check_series(e, 0, d);
161 // Series exponentiation
162 static unsigned exam_series4()
167 e = pow((2*cos(x)).series(x==0, 5), 2).series(x==0, 5);
168 d = 4 - 4*pow(x, 2) + 4*pow(x, 4)/3 + Order(pow(x, 5));
169 result += check_series(e, 0, d);
171 e = pow(tgamma(x), 2).series(x==0, 2);
172 d = pow(x,-2) - 2*Euler/x + (pow(Pi,2)/6+2*pow(Euler,2))
173 + x*(-4*pow(Euler, 3)/3 -pow(Pi,2)*Euler/3 - 2*zeta(3)/3) + Order(pow(x, 2));
174 result += check_series(e, 0, d);
179 // Order term handling
180 static unsigned exam_series5()
185 e = 1 + x + pow(x, 2) + pow(x, 3);
187 result += check_series(e, 0, d, 0);
189 result += check_series(e, 0, d, 1);
190 d = 1 + x + Order(pow(x, 2));
191 result += check_series(e, 0, d, 2);
192 d = 1 + x + pow(x, 2) + Order(pow(x, 3));
193 result += check_series(e, 0, d, 3);
194 d = 1 + x + pow(x, 2) + pow(x, 3);
195 result += check_series(e, 0, d, 4);
199 // Series expansion of tgamma(-1)
200 static unsigned exam_series6()
203 ex d = pow(x+1,-1)*numeric(1,4) +
204 pow(x+1,0)*(numeric(3,4) -
205 numeric(1,2)*Euler) +
206 pow(x+1,1)*(numeric(7,4) -
208 numeric(1,2)*pow(Euler,2) +
209 numeric(1,12)*pow(Pi,2)) +
210 pow(x+1,2)*(numeric(15,4) -
212 numeric(1,3)*pow(Euler,3) +
213 numeric(1,4)*pow(Pi,2) +
214 numeric(3,2)*pow(Euler,2) -
215 numeric(1,6)*pow(Pi,2)*Euler -
216 numeric(2,3)*zeta(3)) +
217 pow(x+1,3)*(numeric(31,4) - pow(Euler,3) -
218 numeric(15,2)*Euler +
219 numeric(1,6)*pow(Euler,4) +
220 numeric(7,2)*pow(Euler,2) +
221 numeric(7,12)*pow(Pi,2) -
222 numeric(1,2)*pow(Pi,2)*Euler -
224 numeric(1,6)*pow(Euler,2)*pow(Pi,2) +
225 numeric(1,40)*pow(Pi,4) +
226 numeric(4,3)*zeta(3)*Euler) +
228 return check_series(e, -1, d, 4);
231 // Series expansion of tan(x==Pi/2)
232 static unsigned exam_series7()
235 ex d = pow(x-1,-1)/Pi*(-2) + pow(x-1,1)*Pi/6 + pow(x-1,3)*pow(Pi,3)/360
236 +pow(x-1,5)*pow(Pi,5)/15120 + pow(x-1,7)*pow(Pi,7)/604800
238 return check_series(e,1,d,9);
241 // Series expansion of log(sin(x==0))
242 static unsigned exam_series8()
245 ex d = log(x) - pow(x,2)/6 - pow(x,4)/180 - pow(x,6)/2835 - pow(x,8)/37800 + Order(pow(x,9));
246 return check_series(e,0,d,9);
249 // Series expansion of Li2(sin(x==0))
250 static unsigned exam_series9()
253 ex d = x + pow(x,2)/4 - pow(x,3)/18 - pow(x,4)/48
254 - 13*pow(x,5)/1800 - pow(x,6)/360 - 23*pow(x,7)/21168
256 return check_series(e,0,d,8);
259 // Series expansion of Li2((x==2)^2), caring about branch-cut
260 static unsigned exam_series10()
264 ex e = Li2(pow(x,2));
265 ex d = Li2(4) + (-log(3) + I*Pi*csgn(I-I*pow(x,2))) * (x-2)
266 + (numeric(-2,3) + log(3)/4 - I*Pi/4*csgn(I-I*pow(x,2))) * pow(x-2,2)
267 + (numeric(11,27) - log(3)/12 + I*Pi/12*csgn(I-I*pow(x,2))) * pow(x-2,3)
268 + (numeric(-155,648) + log(3)/32 - I*Pi/32*csgn(I-I*pow(x,2))) * pow(x-2,4)
270 return check_series(e,2,d,5);
273 // Series expansion of logarithms around branch points
274 static unsigned exam_series11()
284 result += check_series(e,0,d,5);
288 result += check_series(e,0,d,5);
292 result += check_series(e,0,d,5);
294 // These ones must not be expanded because it would result in a branch cut
295 // running in the wrong direction. (Other systems tend to get this wrong.)
298 result += check_series(e,0,d,5);
302 result += check_series(e,123,d,5);
305 d = e; // we don't know anything about a!
306 result += check_series(e,0,d,5);
309 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));
310 result += check_series(e,1,d,5);
315 // Series expansion of other functions around branch points
316 static unsigned exam_series12()
323 // NB: Mma and Maple give different results, but they agree if one
324 // takes into account that by assumption |x|<1.
326 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));
327 result += check_series(e,I,d,3);
329 // NB: here, at -I, Mathematica disagrees, but it is wrong -- they
330 // pick up a complex phase by incorrectly expanding logarithms.
332 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));
333 result += check_series(e,-I,d,3);
335 // This is basically the same as above, the branch point is at +/-1:
337 d = (-log(2)/2+log(x+1)/2) + (x+1)/4 + pow(x+1,2)/16 + Order(pow(x+1,3));
338 result += check_series(e,-1,d,3);
344 unsigned exam_pseries()
348 cout << "examining series expansion" << flush;
349 clog << "----------series expansion:" << endl;
351 result += exam_series1(); cout << '.' << flush;
352 result += exam_series2(); cout << '.' << flush;
353 result += exam_series3(); cout << '.' << flush;
354 result += exam_series4(); cout << '.' << flush;
355 result += exam_series5(); cout << '.' << flush;
356 result += exam_series6(); cout << '.' << flush;
357 result += exam_series7(); cout << '.' << flush;
358 result += exam_series8(); cout << '.' << flush;
359 result += exam_series9(); cout << '.' << flush;
360 result += exam_series10(); cout << '.' << flush;
361 result += exam_series11(); cout << '.' << flush;
362 result += exam_series12(); cout << '.' << flush;
365 cout << " passed " << endl;
366 clog << "(no output)" << endl;
368 cout << " failed " << endl;