1 /** @File exam_pseries.cpp
3 * Series expansion test (Laurent and Taylor series). */
6 * GiNaC Copyright (C) 1999-2019 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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14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
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20 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
24 using namespace GiNaC;
31 static unsigned check_series(const ex &e, const ex &point, const ex &d, int order = 8)
33 ex es = e.series(x==point, order);
34 ex ep = ex_to<pseries>(es).convert_to_poly();
35 if (!(ep - d).expand().is_zero()) {
36 clog << "series expansion of " << e << " at " << point
37 << " erroneously returned " << ep << " (instead of " << d
39 clog << tree << (ep-d) << dflt;
46 static unsigned exam_series1()
56 d = 1 + Order(pow(x, 1));
57 result += check_series(e, 0, d, 1);
60 d = x - pow(x, 3) / 6 + pow(x, 5) / 120 - pow(x, 7) / 5040 + Order(pow(x, 8));
61 result += check_series(e, 0, d);
64 d = 1 - pow(x, 2) / 2 + pow(x, 4) / 24 - pow(x, 6) / 720 + Order(pow(x, 8));
65 result += check_series(e, 0, d);
68 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));
69 result += check_series(e, 0, d);
72 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));
73 result += check_series(e, 0, d);
77 result += check_series(e, 0, d);
80 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));
81 result += check_series(e, 1, d);
83 e = pow(x + pow(x, 3), -1);
84 d = pow(x, -1) - x + pow(x, 3) - pow(x, 5) + pow(x, 7) + Order(pow(x, 8));
85 result += check_series(e, 0, d);
87 e = pow(pow(x, 2) + pow(x, 4), -1);
88 d = pow(x, -2) - 1 + pow(x, 2) - pow(x, 4) + pow(x, 6) + Order(pow(x, 8));
89 result += check_series(e, 0, d);
92 d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + pow(x, 6) / 675 + Order(pow(x, 8));
93 result += check_series(e, 0, d);
96 d = x + pow(x, 3) / 3 + pow(x, 5) * 2/15 + pow(x, 7) * 17/315 + Order(pow(x, 8));
97 result += check_series(e, 0, d);
100 d = pow(x, -1) - x / 3 - pow(x, 3) / 45 - pow(x, 5) * 2/945 - pow(x, 7) / 4725 + Order(pow(x, 8));
101 result += check_series(e, 0, d);
103 e = pow(numeric(2), x);
105 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));
106 result += check_series(e, 0, d.expand());
110 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));
111 result += check_series(e, 0, d.expand());
115 result += check_series(e, 0, d, 1);
116 result += check_series(e, 0, d, 2);
118 e = pow(x, 8) * pow(pow(x,3)+ pow(x + pow(x,3), 2), -2);
119 d = pow(x, 4) - 2*pow(x, 5) + Order(pow(x, 6));
120 result += check_series(e, 0, d, 6);
122 e = cos(x) * pow(sin(x)*(pow(x, 5) + 4 * pow(x, 2)), -3);
123 d = pow(x, -9) / 64 - 3 * pow(x, -6) / 256 - pow(x, -5) / 960 + 535 * pow(x, -3) / 96768
124 + pow(x, -2) / 1280 - pow(x, -1) / 14400 - numeric(283, 129024) - 2143 * x / 5322240
126 result += check_series(e, 0, d, 2);
128 e = sqrt(1+x*x) * sqrt(1+2*x*x);
129 d = 1 + Order(pow(x, 2));
130 result += check_series(e, 0, d, 2);
132 e = pow(x, 4) * sin(a) + pow(x, 2);
133 d = pow(x, 2) + Order(pow(x, 3));
134 result += check_series(e, 0, d, 3);
136 e = log(a*x + b*x*x*log(x));
137 d = log(a*x) + b/a*log(x)*x - pow(b/a, 2)/2*pow(log(x)*x, 2) + Order(pow(x, 3));
138 result += check_series(e, 0, d, 3);
141 d = pow(a, b) + (pow(a, b)*b/a)*x + (pow(a, b)*b*b/a/a/2 - pow(a, b)*b/a/a/2)*pow(x, 2) + Order(pow(x, 3));
142 result += check_series(e, 0, d, 3);
144 e = a * (1 / (x * sin(x)) - sin(x) / x);
145 d = a * pow(x, -2) + Order(pow(x, -1));
146 result += check_series(e, 0, d, -1);
152 static unsigned exam_series2()
157 e = pow(sin(x), -1).series(x==0, 8) + pow(sin(-x), -1).series(x==0, 12);
158 d = Order(pow(x, 8));
159 result += check_series(e, 0, d);
164 // Series multiplication
165 static unsigned exam_series3()
170 e = sin(x).series(x==0, 8) * pow(sin(x), -1).series(x==0, 12);
171 d = 1 + Order(pow(x, 7));
172 result += check_series(e, 0, d);
177 // Series exponentiation
178 static unsigned exam_series4()
184 e = pow((2*cos(x)).series(x==0, 5), 2).series(x==0, 5);
185 d = 4 - 4*pow(x, 2) + 4*pow(x, 4)/3 + Order(pow(x, 5));
186 result += check_series(e, 0, d);
188 e = pow(tgamma(x), 2).series(x==0, 2);
189 d = pow(x,-2) - 2*Euler/x + (pow(Pi,2)/6+2*pow(Euler,2))
190 + x*(-4*pow(Euler, 3)/3 -pow(Pi,2)*Euler/3 - 2*zeta(3)/3) + Order(pow(x, 2));
191 result += check_series(e, 0, d);
196 // Order term handling
197 static unsigned exam_series5()
202 e = 1 + x + pow(x, 2) + pow(x, 3);
204 result += check_series(e, 0, d, 0);
206 result += check_series(e, 0, d, 1);
207 d = 1 + x + Order(pow(x, 2));
208 result += check_series(e, 0, d, 2);
209 d = 1 + x + pow(x, 2) + Order(pow(x, 3));
210 result += check_series(e, 0, d, 3);
211 d = 1 + x + pow(x, 2) + pow(x, 3);
212 result += check_series(e, 0, d, 4);
216 // Series expansion of tgamma(-1)
217 static unsigned exam_series6()
221 ex d = pow(x+1,-1)*numeric(1,4) +
222 pow(x+1,0)*(numeric(3,4) -
223 numeric(1,2)*Euler) +
224 pow(x+1,1)*(numeric(7,4) -
226 numeric(1,2)*pow(Euler,2) +
227 numeric(1,12)*pow(Pi,2)) +
228 pow(x+1,2)*(numeric(15,4) -
230 numeric(1,3)*pow(Euler,3) +
231 numeric(1,4)*pow(Pi,2) +
232 numeric(3,2)*pow(Euler,2) -
233 numeric(1,6)*pow(Pi,2)*Euler -
234 numeric(2,3)*zeta(3)) +
235 pow(x+1,3)*(numeric(31,4) - pow(Euler,3) -
236 numeric(15,2)*Euler +
237 numeric(1,6)*pow(Euler,4) +
238 numeric(7,2)*pow(Euler,2) +
239 numeric(7,12)*pow(Pi,2) -
240 numeric(1,2)*pow(Pi,2)*Euler -
242 numeric(1,6)*pow(Euler,2)*pow(Pi,2) +
243 numeric(1,40)*pow(Pi,4) +
244 numeric(4,3)*zeta(3)*Euler) +
246 return check_series(e, -1, d, 4);
249 // Series expansion of tan(x==Pi/2)
250 static unsigned exam_series7()
253 ex d = pow(x-1,-1)/Pi*(-2) + pow(x-1,1)*Pi/6 + pow(x-1,3)*pow(Pi,3)/360
254 +pow(x-1,5)*pow(Pi,5)/15120 + pow(x-1,7)*pow(Pi,7)/604800
256 return check_series(e,1,d,9);
259 // Series expansion of log(sin(x==0))
260 static unsigned exam_series8()
263 ex d = log(x) - pow(x,2)/6 - pow(x,4)/180 - pow(x,6)/2835 - pow(x,8)/37800 + Order(pow(x,9));
264 return check_series(e,0,d,9);
267 // Series expansion of Li2(sin(x==0))
268 static unsigned exam_series9()
271 ex d = x + pow(x,2)/4 - pow(x,3)/18 - pow(x,4)/48
272 - 13*pow(x,5)/1800 - pow(x,6)/360 - 23*pow(x,7)/21168
274 return check_series(e,0,d,8);
277 // Series expansion of Li2((x==2)^2), caring about branch-cut
278 static unsigned exam_series10()
282 ex e = Li2(pow(x,2));
283 ex d = Li2(4) + (-log(3) + I*Pi*csgn(I-I*pow(x,2))) * (x-2)
284 + (numeric(-2,3) + log(3)/4 - I*Pi/4*csgn(I-I*pow(x,2))) * pow(x-2,2)
285 + (numeric(11,27) - log(3)/12 + I*Pi/12*csgn(I-I*pow(x,2))) * pow(x-2,3)
286 + (numeric(-155,648) + log(3)/32 - I*Pi/32*csgn(I-I*pow(x,2))) * pow(x-2,4)
288 return check_series(e,2,d,5);
291 // Series expansion of logarithms around branch points
292 static unsigned exam_series11()
302 result += check_series(e,0,d,5);
306 result += check_series(e,0,d,5);
310 result += check_series(e,0,d,5);
312 // These ones must not be expanded because it would result in a branch cut
313 // running in the wrong direction. (Other systems tend to get this wrong.)
316 result += check_series(e,0,d,5);
320 result += check_series(e,123,d,5);
323 d = e; // we don't know anything about a!
324 result += check_series(e,0,d,5);
327 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));
328 result += check_series(e,1,d,5);
333 // Series expansion of other functions around branch points
334 static unsigned exam_series12()
342 // NB: Mma and Maple give different results, but they agree if one
343 // takes into account that by assumption |x|<1.
345 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));
346 result += check_series(e,I,d,3);
348 // NB: here, at -I, Mathematica disagrees, but it is wrong -- they
349 // pick up a complex phase by incorrectly expanding logarithms.
351 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));
352 result += check_series(e,-I,d,3);
354 // This is basically the same as above, the branch point is at +/-1:
356 d = (-log(2)/2+log(x+1)/2) + (x+1)/4 + pow(x+1,2)/16 + Order(pow(x+1,3));
357 result += check_series(e,-1,d,3);
362 // Test of the patch of Stefan Weinzierl that prevents an infinite loop if
363 // a factor in a product is a complicated way of writing zero.
364 static unsigned exam_series13()
368 ex e = (new mul(pow(2,x), (1/x*(-(1+x)/(1-x)) + (1+x)/x/(1-x)))
369 )->setflag(status_flags::evaluated);
371 result += check_series(e,0,d,1);
376 // Test if (1+x)^(1/x) can be expanded.
377 static unsigned exam_series14()
381 ex e = pow(1+x, sin(x)/x);
382 ex d = 1 + x - pow(x,3)/6 + Order(pow(x,4));
384 result += check_series(e,0,d,4);
385 } catch (const pole_error& err) {
386 clog << "series expansion of " << e << " at 0 raised an exception." << endl;
393 unsigned exam_pseries()
397 cout << "examining series expansion" << flush;
399 result += exam_series1(); cout << '.' << flush;
400 result += exam_series2(); cout << '.' << flush;
401 result += exam_series3(); cout << '.' << flush;
402 result += exam_series4(); cout << '.' << flush;
403 result += exam_series5(); cout << '.' << flush;
404 result += exam_series6(); cout << '.' << flush;
405 result += exam_series7(); cout << '.' << flush;
406 result += exam_series8(); cout << '.' << flush;
407 result += exam_series9(); cout << '.' << flush;
408 result += exam_series10(); cout << '.' << flush;
409 result += exam_series11(); cout << '.' << flush;
410 result += exam_series12(); cout << '.' << flush;
411 result += exam_series13(); cout << '.' << flush;
412 result += exam_series14(); cout << '.' << flush;
417 int main(int argc, char** argv)
419 return exam_pseries();