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
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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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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());
105 static unsigned exam_series2(void)
110 e = pow(sin(x), -1).series(x==0, 8) + pow(sin(-x), -1).series(x==0, 12);
111 d = Order(pow(x, 6));
112 result += check_series(e, 0, d);
117 // Series multiplication
118 static unsigned exam_series3(void)
123 e = sin(x).series(x==0, 8) * pow(sin(x), -1).series(x==0, 12);
124 d = 1 + Order(pow(x, 7));
125 result += check_series(e, 0, d);
130 // Order term handling
131 static unsigned exam_series4(void)
136 e = 1 + x + pow(x, 2) + pow(x, 3);
138 result += check_series(e, 0, d, 0);
140 result += check_series(e, 0, d, 1);
141 d = 1 + x + Order(pow(x, 2));
142 result += check_series(e, 0, d, 2);
143 d = 1 + x + pow(x, 2) + Order(pow(x, 3));
144 result += check_series(e, 0, d, 3);
145 d = 1 + x + pow(x, 2) + pow(x, 3);
146 result += check_series(e, 0, d, 4);
150 // Series expansion of tgamma(-1)
151 static unsigned exam_series5(void)
154 ex d = pow(x+1,-1)*numeric(1,4) +
155 pow(x+1,0)*(numeric(3,4) -
156 numeric(1,2)*Euler) +
157 pow(x+1,1)*(numeric(7,4) -
159 numeric(1,2)*pow(Euler,2) +
160 numeric(1,12)*pow(Pi,2)) +
161 pow(x+1,2)*(numeric(15,4) -
163 numeric(1,3)*pow(Euler,3) +
164 numeric(1,4)*pow(Pi,2) +
165 numeric(3,2)*pow(Euler,2) -
166 numeric(1,6)*pow(Pi,2)*Euler -
167 numeric(2,3)*zeta(3)) +
168 pow(x+1,3)*(numeric(31,4) - pow(Euler,3) -
169 numeric(15,2)*Euler +
170 numeric(1,6)*pow(Euler,4) +
171 numeric(7,2)*pow(Euler,2) +
172 numeric(7,12)*pow(Pi,2) -
173 numeric(1,2)*pow(Pi,2)*Euler -
175 numeric(1,6)*pow(Euler,2)*pow(Pi,2) +
176 numeric(1,40)*pow(Pi,4) +
177 numeric(4,3)*zeta(3)*Euler) +
179 return check_series(e, -1, d, 4);
182 // Series expansion of tan(x==Pi/2)
183 static unsigned exam_series6(void)
186 ex d = pow(x-1,-1)/Pi*(-2) + pow(x-1,1)*Pi/6 + pow(x-1,3)*pow(Pi,3)/360
187 +pow(x-1,5)*pow(Pi,5)/15120 + pow(x-1,7)*pow(Pi,7)/604800
189 return check_series(e,1,d,8);
192 // Series expansion of log(sin(x==0))
193 static unsigned exam_series7(void)
196 ex d = log(x) - pow(x,2)/6 - pow(x,4)/180 - pow(x,6)/2835
198 return check_series(e,0,d,8);
201 // Series expansion of Li2(sin(x==0))
202 static unsigned exam_series8(void)
205 ex d = x + pow(x,2)/4 - pow(x,3)/18 - pow(x,4)/48
206 - 13*pow(x,5)/1800 - pow(x,6)/360 - 23*pow(x,7)/21168
208 return check_series(e,0,d,8);
211 // Series expansion of Li2((x==2)^2), caring about branch-cut
212 static unsigned exam_series9(void)
214 ex e = Li2(pow(x,2));
215 ex d = Li2(4) + (-log(3) + I*Pi*csgn(I-I*pow(x,2))) * (x-2)
216 + (numeric(-2,3) + log(3)/4 - I*Pi/4*csgn(I-I*pow(x,2))) * pow(x-2,2)
217 + (numeric(11,27) - log(3)/12 + I*Pi/12*csgn(I-I*pow(x,2))) * pow(x-2,3)
218 + (numeric(-155,648) + log(3)/32 - I*Pi/32*csgn(I-I*pow(x,2))) * pow(x-2,4)
220 return check_series(e,2,d,5);
223 // Series expansion of logarithms around branch points
224 static unsigned exam_series10(void)
232 result += check_series(e,0,d,5);
236 result += check_series(e,0,d,5);
240 result += check_series(e,0,d,5);
242 // These ones must not be expanded because it would result in a branch cut
243 // running in the wrong direction. (Other systems tend to get this wrong.)
246 result += check_series(e,0,d,5);
250 result += check_series(e,123,d,5);
253 d = e; // we don't know anything about a!
254 result += check_series(e,0,d,5);
257 d = log(1-x) - (x-1) + pow(x-1,2)/2 - pow(x-1,3)/3 + Order(pow(x-1,4));
258 result += check_series(e,1,d,4);
263 // Series expansion of other functions around branch points
264 static unsigned exam_series11(void)
269 // NB: Mma and Maple give different results, but they agree if one
270 // takes into account that by assumption |x|<1.
272 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));
273 result += check_series(e,I,d,3);
275 // NB: here, at -I, Mathematica disagrees, but it is wrong -- they
276 // pick up a complex phase by incorrectly expanding logarithms.
278 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));
279 result += check_series(e,-I,d,3);
281 // This is basically the same as above, the branch point is at +/-1:
283 d = (-log(2)/2+log(x+1)/2) + (x+1)/4 + pow(x+1,2)/16 + Order(pow(x+1,3));
284 result += check_series(e,-1,d,3);
290 unsigned exam_pseries(void)
294 cout << "examining series expansion" << flush;
295 clog << "----------series expansion:" << endl;
297 result += exam_series1(); cout << '.' << flush;
298 result += exam_series2(); cout << '.' << flush;
299 result += exam_series3(); cout << '.' << flush;
300 result += exam_series4(); cout << '.' << flush;
301 result += exam_series5(); cout << '.' << flush;
302 result += exam_series6(); cout << '.' << flush;
303 result += exam_series7(); cout << '.' << flush;
304 result += exam_series8(); cout << '.' << flush;
305 result += exam_series9(); cout << '.' << flush;
306 result += exam_series10(); cout << '.' << flush;
307 result += exam_series11(); cout << '.' << flush;
310 cout << " passed " << endl;
311 clog << "(no output)" << endl;
313 cout << " failed " << endl;