1 /** @File exam_pseries.cpp
3 * Series expansion test (Laurent and Taylor series). */
6 * GiNaC Copyright (C) 1999-2003 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
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
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.
18 * You should have received a copy of the GNU General Public License
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 clog << tree << (ep-d) << dflt;
42 static unsigned exam_series1(void)
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) + Order(pow(x, 7));
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) + Order(pow(x, 6));
79 result += check_series(e, 0, d);
82 d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + Order(pow(x, 5));
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 + Order(pow(x, 6));
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);
112 static unsigned exam_series2(void)
117 e = pow(sin(x), -1).series(x==0, 8) + pow(sin(-x), -1).series(x==0, 12);
118 d = Order(pow(x, 6));
119 result += check_series(e, 0, d);
124 // Series multiplication
125 static unsigned exam_series3(void)
130 e = sin(x).series(x==0, 8) * pow(sin(x), -1).series(x==0, 12);
131 d = 1 + Order(pow(x, 7));
132 result += check_series(e, 0, d);
137 // Series exponentiation
138 static unsigned exam_series4(void)
143 e = pow((2*cos(x)).series(x==0, 5), 2).series(x==0, 5);
144 d = 4 - 4*pow(x, 2) + 4*pow(x, 4)/3 + Order(pow(x, 5));
145 result += check_series(e, 0, d);
147 e = pow(tgamma(x), 2).series(x==0, 3);
148 d = pow(x,-2) - 2*Euler/x + (pow(Pi,2)/6+2*pow(Euler,2)) + Order(x);
149 result += check_series(e, 0, d);
154 // Order term handling
155 static unsigned exam_series5(void)
160 e = 1 + x + pow(x, 2) + pow(x, 3);
162 result += check_series(e, 0, d, 0);
164 result += check_series(e, 0, d, 1);
165 d = 1 + x + Order(pow(x, 2));
166 result += check_series(e, 0, d, 2);
167 d = 1 + x + pow(x, 2) + Order(pow(x, 3));
168 result += check_series(e, 0, d, 3);
169 d = 1 + x + pow(x, 2) + pow(x, 3);
170 result += check_series(e, 0, d, 4);
174 // Series expansion of tgamma(-1)
175 static unsigned exam_series6(void)
178 ex d = pow(x+1,-1)*numeric(1,4) +
179 pow(x+1,0)*(numeric(3,4) -
180 numeric(1,2)*Euler) +
181 pow(x+1,1)*(numeric(7,4) -
183 numeric(1,2)*pow(Euler,2) +
184 numeric(1,12)*pow(Pi,2)) +
185 pow(x+1,2)*(numeric(15,4) -
187 numeric(1,3)*pow(Euler,3) +
188 numeric(1,4)*pow(Pi,2) +
189 numeric(3,2)*pow(Euler,2) -
190 numeric(1,6)*pow(Pi,2)*Euler -
191 numeric(2,3)*zeta(3)) +
192 pow(x+1,3)*(numeric(31,4) - pow(Euler,3) -
193 numeric(15,2)*Euler +
194 numeric(1,6)*pow(Euler,4) +
195 numeric(7,2)*pow(Euler,2) +
196 numeric(7,12)*pow(Pi,2) -
197 numeric(1,2)*pow(Pi,2)*Euler -
199 numeric(1,6)*pow(Euler,2)*pow(Pi,2) +
200 numeric(1,40)*pow(Pi,4) +
201 numeric(4,3)*zeta(3)*Euler) +
203 return check_series(e, -1, d, 4);
206 // Series expansion of tan(x==Pi/2)
207 static unsigned exam_series7(void)
210 ex d = pow(x-1,-1)/Pi*(-2) + pow(x-1,1)*Pi/6 + pow(x-1,3)*pow(Pi,3)/360
211 +pow(x-1,5)*pow(Pi,5)/15120 + pow(x-1,7)*pow(Pi,7)/604800
213 return check_series(e,1,d,8);
216 // Series expansion of log(sin(x==0))
217 static unsigned exam_series8(void)
220 ex d = log(x) - pow(x,2)/6 - pow(x,4)/180 - pow(x,6)/2835
222 return check_series(e,0,d,8);
225 // Series expansion of Li2(sin(x==0))
226 static unsigned exam_series9(void)
229 ex d = x + pow(x,2)/4 - pow(x,3)/18 - pow(x,4)/48
230 - 13*pow(x,5)/1800 - pow(x,6)/360 - 23*pow(x,7)/21168
232 return check_series(e,0,d,8);
235 // Series expansion of Li2((x==2)^2), caring about branch-cut
236 static unsigned exam_series10(void)
240 ex e = Li2(pow(x,2));
241 ex d = Li2(4) + (-log(3) + I*Pi*csgn(I-I*pow(x,2))) * (x-2)
242 + (numeric(-2,3) + log(3)/4 - I*Pi/4*csgn(I-I*pow(x,2))) * pow(x-2,2)
243 + (numeric(11,27) - log(3)/12 + I*Pi/12*csgn(I-I*pow(x,2))) * pow(x-2,3)
244 + (numeric(-155,648) + log(3)/32 - I*Pi/32*csgn(I-I*pow(x,2))) * pow(x-2,4)
246 return check_series(e,2,d,5);
249 // Series expansion of logarithms around branch points
250 static unsigned exam_series11(void)
260 result += check_series(e,0,d,5);
264 result += check_series(e,0,d,5);
268 result += check_series(e,0,d,5);
270 // These ones must not be expanded because it would result in a branch cut
271 // running in the wrong direction. (Other systems tend to get this wrong.)
274 result += check_series(e,0,d,5);
278 result += check_series(e,123,d,5);
281 d = e; // we don't know anything about a!
282 result += check_series(e,0,d,5);
285 d = log(1-x) - (x-1) + pow(x-1,2)/2 - pow(x-1,3)/3 + Order(pow(x-1,4));
286 result += check_series(e,1,d,4);
291 // Series expansion of other functions around branch points
292 static unsigned exam_series12(void)
299 // NB: Mma and Maple give different results, but they agree if one
300 // takes into account that by assumption |x|<1.
302 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));
303 result += check_series(e,I,d,3);
305 // NB: here, at -I, Mathematica disagrees, but it is wrong -- they
306 // pick up a complex phase by incorrectly expanding logarithms.
308 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));
309 result += check_series(e,-I,d,3);
311 // This is basically the same as above, the branch point is at +/-1:
313 d = (-log(2)/2+log(x+1)/2) + (x+1)/4 + pow(x+1,2)/16 + Order(pow(x+1,3));
314 result += check_series(e,-1,d,3);
320 unsigned exam_pseries(void)
324 cout << "examining series expansion" << flush;
325 clog << "----------series expansion:" << endl;
327 result += exam_series1(); cout << '.' << flush;
328 result += exam_series2(); cout << '.' << flush;
329 result += exam_series3(); cout << '.' << flush;
330 result += exam_series4(); cout << '.' << flush;
331 result += exam_series5(); cout << '.' << flush;
332 result += exam_series6(); cout << '.' << flush;
333 result += exam_series7(); cout << '.' << flush;
334 result += exam_series8(); cout << '.' << flush;
335 result += exam_series9(); cout << '.' << flush;
336 result += exam_series10(); cout << '.' << flush;
337 result += exam_series11(); cout << '.' << flush;
338 result += exam_series12(); cout << '.' << flush;
341 cout << " passed " << endl;
342 clog << "(no output)" << endl;
344 cout << " failed " << endl;