-/** @file exam_pseries.cpp
+/** @File exam_pseries.cpp
*
* Series expansion test (Laurent and Taylor series). */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
static unsigned check_series(const ex &e, const ex &point, const ex &d, int order = 8)
{
- ex es = e.series(x, point, order);
- ex ep = ex_to_pseries(es).convert_to_poly();
- if (!(ep - d).is_zero()) {
- clog << "series expansion of " << e << " at " << point
- << " erroneously returned " << ep << " (instead of " << d
- << ")" << endl;
- (ep-d).printtree(clog);
- return 1;
- }
- return 0;
+ ex es = e.series(x==point, order);
+ ex ep = ex_to<pseries>(es).convert_to_poly();
+ if (!(ep - d).is_zero()) {
+ clog << "series expansion of " << e << " at " << point
+ << " erroneously returned " << ep << " (instead of " << d
+ << ")" << endl;
+ clog << tree << (ep-d) << dflt;
+ return 1;
+ }
+ return 0;
}
// Series expansion
-static unsigned exam_series1(void)
+static unsigned exam_series1()
{
- unsigned result = 0;
- ex e, d;
-
- e = sin(x);
- d = x - pow(x, 3) / 6 + pow(x, 5) / 120 - pow(x, 7) / 5040 + Order(pow(x, 8));
- result += check_series(e, 0, d);
-
- e = cos(x);
- d = 1 - pow(x, 2) / 2 + pow(x, 4) / 24 - pow(x, 6) / 720 + Order(pow(x, 8));
- result += check_series(e, 0, d);
-
- e = exp(x);
- 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));
- result += check_series(e, 0, d);
-
- e = pow(1 - x, -1);
- 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));
- result += check_series(e, 0, d);
-
- e = x + pow(x, -1);
- d = x + pow(x, -1);
- result += check_series(e, 0, d);
-
- e = x + pow(x, -1);
- 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));
- result += check_series(e, 1, d);
-
- e = pow(x + pow(x, 3), -1);
- d = pow(x, -1) - x + pow(x, 3) - pow(x, 5) + Order(pow(x, 7));
- result += check_series(e, 0, d);
-
- e = pow(pow(x, 2) + pow(x, 4), -1);
- d = pow(x, -2) - 1 + pow(x, 2) - pow(x, 4) + Order(pow(x, 6));
- result += check_series(e, 0, d);
-
- e = pow(sin(x), -2);
- d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + Order(pow(x, 5));
- result += check_series(e, 0, d);
-
- e = sin(x) / cos(x);
- d = x + pow(x, 3) / 3 + pow(x, 5) * 2/15 + pow(x, 7) * 17/315 + Order(pow(x, 8));
- result += check_series(e, 0, d);
-
- e = cos(x) / sin(x);
- d = pow(x, -1) - x / 3 - pow(x, 3) / 45 - pow(x, 5) * 2/945 + Order(pow(x, 6));
- result += check_series(e, 0, d);
-
- e = pow(numeric(2), x);
- ex t = log(ex(2)) * x;
- 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));
- result += check_series(e, 0, d.expand());
-
- e = pow(Pi, x);
- t = log(Pi) * x;
- 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));
- result += check_series(e, 0, d.expand());
-
- return result;
+ using GiNaC::log;
+
+ unsigned result = 0;
+ ex e, d;
+
+ e = sin(x);
+ d = x - pow(x, 3) / 6 + pow(x, 5) / 120 - pow(x, 7) / 5040 + Order(pow(x, 8));
+ result += check_series(e, 0, d);
+
+ e = cos(x);
+ d = 1 - pow(x, 2) / 2 + pow(x, 4) / 24 - pow(x, 6) / 720 + Order(pow(x, 8));
+ result += check_series(e, 0, d);
+
+ e = exp(x);
+ 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));
+ result += check_series(e, 0, d);
+
+ e = pow(1 - x, -1);
+ 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));
+ result += check_series(e, 0, d);
+
+ e = x + pow(x, -1);
+ d = x + pow(x, -1);
+ result += check_series(e, 0, d);
+
+ e = x + pow(x, -1);
+ 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));
+ result += check_series(e, 1, d);
+
+ e = pow(x + pow(x, 3), -1);
+ d = pow(x, -1) - x + pow(x, 3) - pow(x, 5) + Order(pow(x, 7));
+ result += check_series(e, 0, d);
+
+ e = pow(pow(x, 2) + pow(x, 4), -1);
+ d = pow(x, -2) - 1 + pow(x, 2) - pow(x, 4) + Order(pow(x, 6));
+ result += check_series(e, 0, d);
+
+ e = pow(sin(x), -2);
+ d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + Order(pow(x, 5));
+ result += check_series(e, 0, d);
+
+ e = sin(x) / cos(x);
+ d = x + pow(x, 3) / 3 + pow(x, 5) * 2/15 + pow(x, 7) * 17/315 + Order(pow(x, 8));
+ result += check_series(e, 0, d);
+
+ e = cos(x) / sin(x);
+ d = pow(x, -1) - x / 3 - pow(x, 3) / 45 - pow(x, 5) * 2/945 + Order(pow(x, 6));
+ result += check_series(e, 0, d);
+
+ e = pow(numeric(2), x);
+ ex t = log(2) * x;
+ 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));
+ result += check_series(e, 0, d.expand());
+
+ e = pow(Pi, x);
+ t = log(Pi) * x;
+ 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));
+ result += check_series(e, 0, d.expand());
+
+ e = log(x);
+ d = e;
+ result += check_series(e, 0, d, 1);
+ result += check_series(e, 0, d, 2);
+
+ return result;
}
// Series addition
-static unsigned exam_series2(void)
+static unsigned exam_series2()
{
- unsigned result = 0;
- ex e, d;
-
- e = pow(sin(x), -1).series(x, 0, 8) + pow(sin(-x), -1).series(x, 0, 12);
- d = Order(pow(x, 6));
- result += check_series(e, 0, d);
-
- return result;
+ unsigned result = 0;
+ ex e, d;
+
+ e = pow(sin(x), -1).series(x==0, 8) + pow(sin(-x), -1).series(x==0, 12);
+ d = Order(pow(x, 6));
+ result += check_series(e, 0, d);
+
+ return result;
}
// Series multiplication
-static unsigned exam_series3(void)
+static unsigned exam_series3()
+{
+ unsigned result = 0;
+ ex e, d;
+
+ e = sin(x).series(x==0, 8) * pow(sin(x), -1).series(x==0, 12);
+ d = 1 + Order(pow(x, 7));
+ result += check_series(e, 0, d);
+
+ return result;
+}
+
+// Series exponentiation
+static unsigned exam_series4()
{
- unsigned result = 0;
- ex e, d;
-
- e = sin(x).series(x, 0, 8) * pow(sin(x), -1).series(x, 0, 12);
- d = 1 + Order(pow(x, 7));
- result += check_series(e, 0, d);
-
- return result;
+ unsigned result = 0;
+ ex e, d;
+
+ e = pow((2*cos(x)).series(x==0, 5), 2).series(x==0, 5);
+ d = 4 - 4*pow(x, 2) + 4*pow(x, 4)/3 + Order(pow(x, 5));
+ result += check_series(e, 0, d);
+
+ e = pow(tgamma(x), 2).series(x==0, 3);
+ d = pow(x,-2) - 2*Euler/x + (pow(Pi,2)/6+2*pow(Euler,2)) + Order(x);
+ result += check_series(e, 0, d);
+
+ return result;
}
// Order term handling
-static unsigned exam_series4(void)
+static unsigned exam_series5()
+{
+ unsigned result = 0;
+ ex e, d;
+
+ e = 1 + x + pow(x, 2) + pow(x, 3);
+ d = Order(1);
+ result += check_series(e, 0, d, 0);
+ d = 1 + Order(x);
+ result += check_series(e, 0, d, 1);
+ d = 1 + x + Order(pow(x, 2));
+ result += check_series(e, 0, d, 2);
+ d = 1 + x + pow(x, 2) + Order(pow(x, 3));
+ result += check_series(e, 0, d, 3);
+ d = 1 + x + pow(x, 2) + pow(x, 3);
+ result += check_series(e, 0, d, 4);
+ return result;
+}
+
+// Series expansion of tgamma(-1)
+static unsigned exam_series6()
+{
+ ex e = tgamma(2*x);
+ ex d = pow(x+1,-1)*numeric(1,4) +
+ pow(x+1,0)*(numeric(3,4) -
+ numeric(1,2)*Euler) +
+ pow(x+1,1)*(numeric(7,4) -
+ numeric(3,2)*Euler +
+ numeric(1,2)*pow(Euler,2) +
+ numeric(1,12)*pow(Pi,2)) +
+ pow(x+1,2)*(numeric(15,4) -
+ numeric(7,2)*Euler -
+ numeric(1,3)*pow(Euler,3) +
+ numeric(1,4)*pow(Pi,2) +
+ numeric(3,2)*pow(Euler,2) -
+ numeric(1,6)*pow(Pi,2)*Euler -
+ numeric(2,3)*zeta(3)) +
+ pow(x+1,3)*(numeric(31,4) - pow(Euler,3) -
+ numeric(15,2)*Euler +
+ numeric(1,6)*pow(Euler,4) +
+ numeric(7,2)*pow(Euler,2) +
+ numeric(7,12)*pow(Pi,2) -
+ numeric(1,2)*pow(Pi,2)*Euler -
+ numeric(2)*zeta(3) +
+ numeric(1,6)*pow(Euler,2)*pow(Pi,2) +
+ numeric(1,40)*pow(Pi,4) +
+ numeric(4,3)*zeta(3)*Euler) +
+ Order(pow(x+1,4));
+ return check_series(e, -1, d, 4);
+}
+
+// Series expansion of tan(x==Pi/2)
+static unsigned exam_series7()
+{
+ ex e = tan(x*Pi/2);
+ ex d = pow(x-1,-1)/Pi*(-2) + pow(x-1,1)*Pi/6 + pow(x-1,3)*pow(Pi,3)/360
+ +pow(x-1,5)*pow(Pi,5)/15120 + pow(x-1,7)*pow(Pi,7)/604800
+ +Order(pow(x-1,8));
+ return check_series(e,1,d,8);
+}
+
+// Series expansion of log(sin(x==0))
+static unsigned exam_series8()
{
- unsigned result = 0;
- ex e, d;
-
- e = 1 + x + pow(x, 2) + pow(x, 3);
- d = Order(1);
- result += check_series(e, 0, d, 0);
- d = 1 + Order(x);
- result += check_series(e, 0, d, 1);
- d = 1 + x + Order(pow(x, 2));
- result += check_series(e, 0, d, 2);
- d = 1 + x + pow(x, 2) + Order(pow(x, 3));
- result += check_series(e, 0, d, 3);
- d = 1 + x + pow(x, 2) + pow(x, 3);
- result += check_series(e, 0, d, 4);
- return result;
+ ex e = log(sin(x));
+ ex d = log(x) - pow(x,2)/6 - pow(x,4)/180 - pow(x,6)/2835
+ +Order(pow(x,8));
+ return check_series(e,0,d,8);
}
-// Series of special functions
-static unsigned exam_series5(void)
+// Series expansion of Li2(sin(x==0))
+static unsigned exam_series9()
{
- unsigned result = 0;
- ex e, d;
-
- // gamma(-1):
- e = gamma(2*x);
- d = pow(x+1,-1)*numeric(1,4) +
- pow(x+1,0)*(numeric(3,4) -
- numeric(1,2)*EulerGamma) +
- pow(x+1,1)*(numeric(7,4) -
- numeric(3,2)*EulerGamma +
- numeric(1,2)*pow(EulerGamma,2) +
- numeric(1,12)*pow(Pi,2)) +
- pow(x+1,2)*(numeric(15,4) -
- numeric(7,2)*EulerGamma -
- numeric(1,3)*pow(EulerGamma,3) +
- numeric(1,4)*pow(Pi,2) +
- numeric(3,2)*pow(EulerGamma,2) -
- numeric(1,6)*pow(Pi,2)*EulerGamma -
- numeric(2,3)*zeta(3)) +
- pow(x+1,3)*(numeric(31,4) - pow(EulerGamma,3) -
- numeric(15,2)*EulerGamma +
- numeric(1,6)*pow(EulerGamma,4) +
- numeric(7,2)*pow(EulerGamma,2) +
- numeric(7,12)*pow(Pi,2) -
- numeric(1,2)*pow(Pi,2)*EulerGamma -
- numeric(2)*zeta(3) +
- numeric(1,6)*pow(EulerGamma,2)*pow(Pi,2) +
- numeric(1,40)*pow(Pi,4) +
- numeric(4,3)*zeta(3)*EulerGamma) +
- Order(pow(x+1,4));
- result += check_series(e, -1, d, 4);
-
- // tan(Pi/2)
- e = tan(x*Pi/2);
- d = pow(x-1,-1)/Pi*(-2) +
- pow(x-1,1)*Pi/6 +
- pow(x-1,3)*pow(Pi,3)/360 +
- pow(x-1,5)*pow(Pi,5)/15120 +
- pow(x-1,7)*pow(Pi,7)/604800 +
- Order(pow(x-1,8));
- result += check_series(e,1,d,8);
-
- return result;
+ ex e = Li2(sin(x));
+ ex d = x + pow(x,2)/4 - pow(x,3)/18 - pow(x,4)/48
+ - 13*pow(x,5)/1800 - pow(x,6)/360 - 23*pow(x,7)/21168
+ + Order(pow(x,8));
+ return check_series(e,0,d,8);
}
-unsigned exam_pseries(void)
+// Series expansion of Li2((x==2)^2), caring about branch-cut
+static unsigned exam_series10()
+{
+ using GiNaC::log;
+
+ ex e = Li2(pow(x,2));
+ ex d = Li2(4) + (-log(3) + I*Pi*csgn(I-I*pow(x,2))) * (x-2)
+ + (numeric(-2,3) + log(3)/4 - I*Pi/4*csgn(I-I*pow(x,2))) * pow(x-2,2)
+ + (numeric(11,27) - log(3)/12 + I*Pi/12*csgn(I-I*pow(x,2))) * pow(x-2,3)
+ + (numeric(-155,648) + log(3)/32 - I*Pi/32*csgn(I-I*pow(x,2))) * pow(x-2,4)
+ + Order(pow(x-2,5));
+ return check_series(e,2,d,5);
+}
+
+// Series expansion of logarithms around branch points
+static unsigned exam_series11()
+{
+ using GiNaC::log;
+
+ unsigned result = 0;
+ ex e, d;
+ symbol a("a");
+
+ e = log(x);
+ d = log(x);
+ result += check_series(e,0,d,5);
+
+ e = log(3/x);
+ d = log(3)-log(x);
+ result += check_series(e,0,d,5);
+
+ e = log(3*pow(x,2));
+ d = log(3)+2*log(x);
+ result += check_series(e,0,d,5);
+
+ // These ones must not be expanded because it would result in a branch cut
+ // running in the wrong direction. (Other systems tend to get this wrong.)
+ e = log(-x);
+ d = e;
+ result += check_series(e,0,d,5);
+
+ e = log(I*(x-123));
+ d = e;
+ result += check_series(e,123,d,5);
+
+ e = log(a*x);
+ d = e; // we don't know anything about a!
+ result += check_series(e,0,d,5);
+
+ e = log((1-x)/x);
+ d = log(1-x) - (x-1) + pow(x-1,2)/2 - pow(x-1,3)/3 + Order(pow(x-1,4));
+ result += check_series(e,1,d,4);
+
+ return result;
+}
+
+// Series expansion of other functions around branch points
+static unsigned exam_series12()
+{
+ using GiNaC::log;
+
+ unsigned result = 0;
+ ex e, d;
+
+ // NB: Mma and Maple give different results, but they agree if one
+ // takes into account that by assumption |x|<1.
+ e = atan(x);
+ 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));
+ result += check_series(e,I,d,3);
+
+ // NB: here, at -I, Mathematica disagrees, but it is wrong -- they
+ // pick up a complex phase by incorrectly expanding logarithms.
+ e = atan(x);
+ 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));
+ result += check_series(e,-I,d,3);
+
+ // This is basically the same as above, the branch point is at +/-1:
+ e = atanh(x);
+ d = (-log(2)/2+log(x+1)/2) + (x+1)/4 + pow(x+1,2)/16 + Order(pow(x+1,3));
+ result += check_series(e,-1,d,3);
+
+ return result;
+}
+
+
+unsigned exam_pseries()
{
- unsigned result = 0;
-
- cout << "examining series expansion" << flush;
- clog << "----------series expansion:" << endl;
-
- result += exam_series1(); cout << '.' << flush;
- result += exam_series2(); cout << '.' << flush;
- result += exam_series3(); cout << '.' << flush;
- result += exam_series4(); cout << '.' << flush;
- result += exam_series5(); cout << '.' << flush;
-
- if (!result) {
- cout << " passed " << endl;
- clog << "(no output)" << endl;
- } else {
- cout << " failed " << endl;
- }
- return result;
+ unsigned result = 0;
+
+ cout << "examining series expansion" << flush;
+ clog << "----------series expansion:" << endl;
+
+ result += exam_series1(); cout << '.' << flush;
+ result += exam_series2(); cout << '.' << flush;
+ result += exam_series3(); cout << '.' << flush;
+ result += exam_series4(); cout << '.' << flush;
+ result += exam_series5(); cout << '.' << flush;
+ result += exam_series6(); cout << '.' << flush;
+ result += exam_series7(); cout << '.' << flush;
+ result += exam_series8(); cout << '.' << flush;
+ result += exam_series9(); cout << '.' << flush;
+ result += exam_series10(); cout << '.' << flush;
+ result += exam_series11(); cout << '.' << flush;
+ result += exam_series12(); cout << '.' << flush;
+
+ if (!result) {
+ cout << " passed " << endl;
+ clog << "(no output)" << endl;
+ } else {
+ cout << " failed " << endl;
+ }
+ return result;
}
git://www.ginac.de / ginac.git / blobdiff