* Tests for symbolic differentiation, including various functions. */
/*
- * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2001 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
#include "exams.h"
static unsigned check_diff(const ex &e, const symbol &x,
- const ex &d, unsigned nth=1)
+ const ex &d, unsigned nth=1)
{
- ex ed = e.diff(x, nth);
- if ((ed - d).compare(ex(0)) != 0) {
- switch (nth) {
- case 0:
- clog << "zeroth ";
- break;
- case 1:
- break;
- case 2:
- clog << "second ";
- break;
- case 3:
- clog << "third ";
- break;
- default:
- clog << nth << "th ";
- }
- clog << "derivative of " << e << " by " << x << " returned "
- << ed << " instead of " << d << endl;
- clog << "returned:" << endl;
- ed.printtree(clog);
- clog << endl << "instead of" << endl;
- d.printtree(clog);
+ ex ed = e.diff(x, nth);
+ if ((ed - d).compare(ex(0)) != 0) {
+ switch (nth) {
+ case 0:
+ clog << "zeroth ";
+ break;
+ case 1:
+ break;
+ case 2:
+ clog << "second ";
+ break;
+ case 3:
+ clog << "third ";
+ break;
+ default:
+ clog << nth << "th ";
+ }
+ clog << "derivative of " << e << " by " << x << " returned "
+ << ed << " instead of " << d << endl;
+ clog << "returned:" << endl;
+ ed.printtree(clog);
+ clog << endl << "instead of" << endl;
+ d.printtree(clog);
- return 1;
- }
- return 0;
+ return 1;
+ }
+ return 0;
}
// Simple (expanded) polynomials
static unsigned exam_differentiation1(void)
{
- unsigned result = 0;
- symbol x("x"), y("y");
- ex e1, e2, e, d;
-
- // construct bivariate polynomial e to be diff'ed:
- e1 = pow(x, -2) * 3 + pow(x, -1) * 5 + 7 + x * 11 + pow(x, 2) * 13;
- e2 = pow(y, -2) * 5 + pow(y, -1) * 7 + 11 + y * 13 + pow(y, 2) * 17;
- e = (e1 * e2).expand();
-
- // d e / dx:
- d = 121 - 55*pow(x,-2) - 66*pow(x,-3) - 30*pow(x,-3)*pow(y,-2)
- - 42*pow(x,-3)*pow(y,-1) - 78*pow(x,-3)*y
- - 102*pow(x,-3)*pow(y,2) - 25*pow(x,-2) * pow(y,-2)
- - 35*pow(x,-2)*pow(y,-1) - 65*pow(x,-2)*y
- - 85*pow(x,-2)*pow(y,2) + 77*pow(y,-1) + 143*y + 187*pow(y,2)
- + 130*x*pow(y,-2) + 182*pow(y,-1)*x + 338*x*y + 442*x*pow(y,2)
- + 55*pow(y,-2) + 286*x;
- result += check_diff(e, x, d);
-
- // d e / dy:
- d = 91 - 30*pow(x,-2)*pow(y,-3) - 21*pow(x,-2)*pow(y,-2)
- + 39*pow(x,-2) + 102*pow(x,-2)*y - 50*pow(x,-1)*pow(y,-3)
- - 35*pow(x,-1)*pow(y,-2) + 65*pow(x,-1) + 170*pow(x,-1)*y
- - 77*pow(y,-2)*x + 143*x + 374*x*y - 130*pow(y,-3)*pow(x,2)
- - 91*pow(y,-2)*pow(x,2) + 169*pow(x,2) + 442*pow(x,2)*y
- - 110*pow(y,-3)*x - 70*pow(y,-3) + 238*y - 49*pow(y,-2);
- result += check_diff(e, y, d);
-
- // d^2 e / dx^2:
- d = 286 + 90*pow(x,-4)*pow(y,-2) + 126*pow(x,-4)*pow(y,-1)
- + 234*pow(x,-4)*y + 306*pow(x,-4)*pow(y,2)
- + 50*pow(x,-3)*pow(y,-2) + 70*pow(x,-3)*pow(y,-1)
- + 130*pow(x,-3)*y + 170*pow(x,-3)*pow(y,2)
- + 130*pow(y,-2) + 182*pow(y,-1) + 338*y + 442*pow(y,2)
- + 198*pow(x,-4) + 110*pow(x,-3);
- result += check_diff(e, x, d, 2);
-
- // d^2 e / dy^2:
- d = 238 + 90*pow(x,-2)*pow(y,-4) + 42*pow(x,-2)*pow(y,-3)
- + 102*pow(x,-2) + 150*pow(x,-1)*pow(y,-4)
- + 70*pow(x,-1)*pow(y,-3) + 170*pow(x,-1) + 330*x*pow(y,-4)
- + 154*x*pow(y,-3) + 374*x + 390*pow(x,2)*pow(y,-4)
- + 182*pow(x,2)*pow(y,-3) + 442*pow(x,2) + 210*pow(y,-4)
- + 98*pow(y,-3);
- result += check_diff(e, y, d, 2);
-
- return result;
+ unsigned result = 0;
+ symbol x("x"), y("y");
+ ex e1, e2, e, d;
+
+ // construct bivariate polynomial e to be diff'ed:
+ e1 = pow(x, -2) * 3 + pow(x, -1) * 5 + 7 + x * 11 + pow(x, 2) * 13;
+ e2 = pow(y, -2) * 5 + pow(y, -1) * 7 + 11 + y * 13 + pow(y, 2) * 17;
+ e = (e1 * e2).expand();
+
+ // d e / dx:
+ d = ex("121-55/x^2-66/x^3-30/x^3/y^2-42/x^3/y-78/x^3*y-102/x^3*y^2-25/x^2/y^2-35/x^2/y-65/x^2*y-85/x^2*y^2+77/y+143*y+187*y^2+130*x/y^2+182/y*x+338*x*y+442*x*y^2+55/y^2+286*x",lst(x,y));
+ result += check_diff(e, x, d);
+
+ // d e / dy:
+ d = ex("91-30/x^2/y^3-21/x^2/y^2+39/x^2+102/x^2*y-50/x/y^3-35/x/y^2+65/x+170/x*y-77*x/y^2+143*x+374*x*y-130/y^3*x^2-91/y^2*x^2+169*x^2+442*x^2*y-110/y^3*x-70/y^3+238*y-49/y^2",lst(x,y));
+ result += check_diff(e, y, d);
+
+ // d^2 e / dx^2:
+ d = ex("286+90/x^4/y^2+126/x^4/y+234/x^4*y+306/x^4*y^2+50/x^3/y^2+70/x^3/y+130/x^3*y+170/x^3*y^2+130/y^2+182/y+338*y+442*y^2+198/x^4+110/x^3",lst(x,y));
+ result += check_diff(e, x, d, 2);
+
+ // d^2 e / dy^2:
+ d = ex("238+90/x^2/y^4+42/x^2/y^3+102/x^2+150/x/y^4+70/x/y^3+170/x+330*x/y^4+154*x/y^3+374*x+390*x^2/y^4+182*x^2/y^3+442*x^2+210/y^4+98/y^3",lst(x,y));
+ result += check_diff(e, y, d, 2);
+
+ return result;
}
// Trigonometric functions
static unsigned exam_differentiation2(void)
{
- unsigned result = 0;
- symbol x("x"), y("y"), a("a"), b("b");
- ex e1, e2, e, d;
-
- // construct expression e to be diff'ed:
- e1 = y*pow(x, 2) + a*x + b;
- e2 = sin(e1);
- e = b*pow(e2, 2) + y*e2 + a;
-
- d = 2*b*e2*cos(e1)*(2*x*y + a) + y*cos(e1)*(2*x*y + a);
- result += check_diff(e, x, d);
-
- d = 2*b*pow(cos(e1),2)*pow(2*x*y + a, 2) + 4*b*y*e2*cos(e1)
- - 2*b*pow(e2,2)*pow(2*x*y + a, 2) - y*e2*pow(2*x*y + a, 2)
- + 2*pow(y,2)*cos(e1);
- result += check_diff(e, x, d, 2);
-
- d = 2*b*e2*cos(e1)*pow(x, 2) + e2 + y*cos(e1)*pow(x, 2);
- result += check_diff(e, y, d);
+ unsigned result = 0;
+ symbol x("x"), y("y"), a("a"), b("b");
+ ex e1, e2, e, d;
+
+ // construct expression e to be diff'ed:
+ e1 = y*pow(x, 2) + a*x + b;
+ e2 = sin(e1);
+ e = b*pow(e2, 2) + y*e2 + a;
+
+ d = 2*b*e2*cos(e1)*(2*x*y + a) + y*cos(e1)*(2*x*y + a);
+ result += check_diff(e, x, d);
+
+ d = 2*b*pow(cos(e1),2)*pow(2*x*y + a, 2) + 4*b*y*e2*cos(e1)
+ - 2*b*pow(e2,2)*pow(2*x*y + a, 2) - y*e2*pow(2*x*y + a, 2)
+ + 2*pow(y,2)*cos(e1);
+ result += check_diff(e, x, d, 2);
+
+ d = 2*b*e2*cos(e1)*pow(x, 2) + e2 + y*cos(e1)*pow(x, 2);
+ result += check_diff(e, y, d);
- d = 2*b*pow(cos(e1),2)*pow(x,4) - 2*b*pow(e2,2)*pow(x,4)
- + 2*cos(e1)*pow(x,2) - y*e2*pow(x,4);
- result += check_diff(e, y, d, 2);
-
- // construct expression e to be diff'ed:
- e2 = cos(e1);
- e = b*pow(e2, 2) + y*e2 + a;
-
- d = -2*b*e2*sin(e1)*(2*x*y + a) - y*sin(e1)*(2*x*y + a);
- result += check_diff(e, x, d);
-
- d = 2*b*pow(sin(e1),2)*pow(2*y*x + a,2) - 4*b*e2*sin(e1)*y
- - 2*b*pow(e2,2)*pow(2*y*x + a,2) - y*e2*pow(2*y*x + a,2)
- - 2*pow(y,2)*sin(e1);
- result += check_diff(e, x, d, 2);
-
- d = -2*b*e2*sin(e1)*pow(x,2) + e2 - y*sin(e1)*pow(x, 2);
- result += check_diff(e, y, d);
-
- d = -2*b*pow(e2,2)*pow(x,4) + 2*b*pow(sin(e1),2)*pow(x,4)
- - 2*sin(e1)*pow(x,2) - y*e2*pow(x,4);
- result += check_diff(e, y, d, 2);
+ d = 2*b*pow(cos(e1),2)*pow(x,4) - 2*b*pow(e2,2)*pow(x,4)
+ + 2*cos(e1)*pow(x,2) - y*e2*pow(x,4);
+ result += check_diff(e, y, d, 2);
+
+ // construct expression e to be diff'ed:
+ e2 = cos(e1);
+ e = b*pow(e2, 2) + y*e2 + a;
+
+ d = -2*b*e2*sin(e1)*(2*x*y + a) - y*sin(e1)*(2*x*y + a);
+ result += check_diff(e, x, d);
+
+ d = 2*b*pow(sin(e1),2)*pow(2*y*x + a,2) - 4*b*e2*sin(e1)*y
+ - 2*b*pow(e2,2)*pow(2*y*x + a,2) - y*e2*pow(2*y*x + a,2)
+ - 2*pow(y,2)*sin(e1);
+ result += check_diff(e, x, d, 2);
+
+ d = -2*b*e2*sin(e1)*pow(x,2) + e2 - y*sin(e1)*pow(x, 2);
+ result += check_diff(e, y, d);
+
+ d = -2*b*pow(e2,2)*pow(x,4) + 2*b*pow(sin(e1),2)*pow(x,4)
+ - 2*sin(e1)*pow(x,2) - y*e2*pow(x,4);
+ result += check_diff(e, y, d, 2);
return result;
}
-
+
// exp function
static unsigned exam_differentiation3(void)
{
- unsigned result = 0;
- symbol x("x"), y("y"), a("a"), b("b");
- ex e1, e2, e, d;
+ unsigned result = 0;
+ symbol x("x"), y("y"), a("a"), b("b");
+ ex e1, e2, e, d;
- // construct expression e to be diff'ed:
- e1 = y*pow(x, 2) + a*x + b;
- e2 = exp(e1);
- e = b*pow(e2, 2) + y*e2 + a;
-
- d = 2*b*pow(e2, 2)*(2*x*y + a) + y*e2*(2*x*y + a);
- result += check_diff(e, x, d);
-
- d = 4*b*pow(e2,2)*pow(2*y*x + a,2) + 4*b*pow(e2,2)*y
- + 2*pow(y,2)*e2 + y*e2*pow(2*y*x + a,2);
- result += check_diff(e, x, d, 2);
-
- d = 2*b*pow(e2,2)*pow(x,2) + e2 + y*e2*pow(x,2);
- result += check_diff(e, y, d);
-
- d = 4*b*pow(e2,2)*pow(x,4) + 2*e2*pow(x,2) + y*e2*pow(x,4);
- result += check_diff(e, y, d, 2);
+ // construct expression e to be diff'ed:
+ e1 = y*pow(x, 2) + a*x + b;
+ e2 = exp(e1);
+ e = b*pow(e2, 2) + y*e2 + a;
+
+ d = 2*b*pow(e2, 2)*(2*x*y + a) + y*e2*(2*x*y + a);
+ result += check_diff(e, x, d);
+
+ d = 4*b*pow(e2,2)*pow(2*y*x + a,2) + 4*b*pow(e2,2)*y
+ + 2*pow(y,2)*e2 + y*e2*pow(2*y*x + a,2);
+ result += check_diff(e, x, d, 2);
+
+ d = 2*b*pow(e2,2)*pow(x,2) + e2 + y*e2*pow(x,2);
+ result += check_diff(e, y, d);
+
+ d = 4*b*pow(e2,2)*pow(x,4) + 2*e2*pow(x,2) + y*e2*pow(x,4);
+ result += check_diff(e, y, d, 2);
return result;
}
// log functions
static unsigned exam_differentiation4(void)
{
- unsigned result = 0;
- symbol x("x"), y("y"), a("a"), b("b");
- ex e1, e2, e, d;
-
- // construct expression e to be diff'ed:
- e1 = y*pow(x, 2) + a*x + b;
- e2 = log(e1);
- e = b*pow(e2, 2) + y*e2 + a;
-
- d = 2*b*e2*(2*x*y + a)/e1 + y*(2*x*y + a)/e1;
- result += check_diff(e, x, d);
-
- d = 2*b*pow((2*x*y + a),2)*pow(e1,-2) + 4*b*y*e2/e1
- - 2*b*e2*pow(2*x*y + a,2)*pow(e1,-2) + 2*pow(y,2)/e1
- - y*pow(2*x*y + a,2)*pow(e1,-2);
- result += check_diff(e, x, d, 2);
-
- d = 2*b*e2*pow(x,2)/e1 + e2 + y*pow(x,2)/e1;
- result += check_diff(e, y, d);
-
- d = 2*b*pow(x,4)*pow(e1,-2) - 2*b*e2*pow(e1,-2)*pow(x,4)
- + 2*pow(x,2)/e1 - y*pow(x,4)*pow(e1,-2);
- result += check_diff(e, y, d, 2);
+ unsigned result = 0;
+ symbol x("x"), y("y"), a("a"), b("b");
+ ex e1, e2, e, d;
+
+ // construct expression e to be diff'ed:
+ e1 = y*pow(x, 2) + a*x + b;
+ e2 = log(e1);
+ e = b*pow(e2, 2) + y*e2 + a;
+
+ d = 2*b*e2*(2*x*y + a)/e1 + y*(2*x*y + a)/e1;
+ result += check_diff(e, x, d);
+
+ d = 2*b*pow((2*x*y + a),2)*pow(e1,-2) + 4*b*y*e2/e1
+ - 2*b*e2*pow(2*x*y + a,2)*pow(e1,-2) + 2*pow(y,2)/e1
+ - y*pow(2*x*y + a,2)*pow(e1,-2);
+ result += check_diff(e, x, d, 2);
+
+ d = 2*b*e2*pow(x,2)/e1 + e2 + y*pow(x,2)/e1;
+ result += check_diff(e, y, d);
+
+ d = 2*b*pow(x,4)*pow(e1,-2) - 2*b*e2*pow(e1,-2)*pow(x,4)
+ + 2*pow(x,2)/e1 - y*pow(x,4)*pow(e1,-2);
+ result += check_diff(e, y, d, 2);
return result;
}
// Functions with two variables
static unsigned exam_differentiation5(void)
{
- unsigned result = 0;
- symbol x("x"), y("y"), a("a"), b("b");
- ex e1, e2, e, d;
-
- // test atan2
- e1 = y*pow(x, 2) + a*x + b;
- e2 = x*pow(y, 2) + b*y + a;
- e = atan2(e1,e2);
- /*
- d = pow(y,2)*(-b-y*pow(x,2)-a*x)/(pow(b+y*pow(x,2)+a*x,2)+pow(x*pow(y,2)+b*y+a,2))
- +(2*y*x+a)/((x*pow(y,2)+b*y+a)*(1+pow(b*y*pow(x,2)+a*x,2)/pow(x*pow(y,2)+b*y+a,2)));
- */
- /*
- d = ((a+2*y*x)*pow(y*b+pow(y,2)*x+a,-1)-(a*x+b+y*pow(x,2))*
- pow(y*b+pow(y,2)*x+a,-2)*pow(y,2))*
- pow(1+pow(a*x+b+y*pow(x,2),2)*pow(y*b+pow(y,2)*x+a,-2),-1);
- */
- /*
- d = pow(1+pow(a*x+b+y*pow(x,2),2)*pow(y*b+pow(y,2)*x+a,-2),-1)
- *pow(y*b+pow(y,2)*x+a,-1)*(a+2*y*x)
- +pow(y,2)*(-a*x-b-y*pow(x,2))*
- pow(pow(y*b+pow(y,2)*x+a,2)+pow(a*x+b+y*pow(x,2),2),-1);
- */
- d = pow(y,2)*pow(pow(b+y*pow(x,2)+x*a,2)+pow(y*b+pow(y,2)*x+a,2),-1)*
- (-b-y*pow(x,2)-x*a)+
- pow(pow(b+y*pow(x,2)+x*a,2)+pow(y*b+pow(y,2)*x+a,2),-1)*
- (y*b+pow(y,2)*x+a)*(2*y*x+a);
- result += check_diff(e, x, d);
-
- return result;
+ unsigned result = 0;
+ symbol x("x"), y("y"), a("a"), b("b");
+ ex e1, e2, e, d;
+
+ // test atan2
+ e1 = y*pow(x, 2) + a*x + b;
+ e2 = x*pow(y, 2) + b*y + a;
+ e = atan2(e1,e2);
+
+ d = pow(y,2)*pow(pow(b+y*pow(x,2)+x*a,2)+pow(y*b+pow(y,2)*x+a,2),-1)*
+ (-b-y*pow(x,2)-x*a)
+ +pow(pow(b+y*pow(x,2)+x*a,2)+pow(y*b+pow(y,2)*x+a,2),-1)*
+ (y*b+pow(y,2)*x+a)*(2*y*x+a);
+ result += check_diff(e, x, d);
+
+ return result;
}
// Series
static unsigned exam_differentiation6(void)
{
- symbol x("x");
- ex e, d, ed;
-
- e = sin(x).series(x, 0, 8);
- d = cos(x).series(x, 0, 7);
- ed = e.diff(x);
- ed = series_to_poly(ed);
- d = series_to_poly(d);
-
- if ((ed - d).compare(ex(0)) != 0) {
- clog << "derivative of " << e << " by " << x << " returned "
- << ed << " instead of " << d << ")" << endl;
- return 1;
- }
- return 0;
+ symbol x("x");
+ ex e, d, ed;
+
+ e = sin(x).series(x==0, 8);
+ d = cos(x).series(x==0, 7);
+ ed = e.diff(x);
+ ed = series_to_poly(ed);
+ d = series_to_poly(d);
+
+ if ((ed - d).compare(ex(0)) != 0) {
+ clog << "derivative of " << e << " by " << x << " returned "
+ << ed << " instead of " << d << ")" << endl;
+ return 1;
+ }
+ return 0;
+}
+
+// Hashing can help a lot, if differentiation is done cleverly
+static unsigned exam_differentiation7(void)
+{
+ symbol x("x");
+ ex P = x + pow(x,3);
+ ex e = (P.diff(x) / P).diff(x, 2);
+ ex d = 6/P - 18*x/pow(P,2) - 54*pow(x,3)/pow(P,2) + 2/pow(P,3)
+ +18*pow(x,2)/pow(P,3) + 54*pow(x,4)/pow(P,3) + 54*pow(x,6)/pow(P,3);
+
+ if (!(e-d).expand().is_zero()) {
+ clog << "expanded second derivative of " << (P.diff(x) / P) << " by " << x
+ << " returned " << e.expand() << " instead of " << d << endl;
+ return 1;
+ }
+ if (e.nops() > 3) {
+ clog << "second derivative of " << (P.diff(x) / P) << " by " << x
+ << " has " << e.nops() << " operands. "
+ << "The result is still correct but not optimal: 3 are enough! "
+ << "(Hint: maybe the product rule for objects of class mul should be more careful about assembling the result?)" << endl;
+ return 1;
+ }
+ return 0;
}
unsigned exam_differentiation(void)
{
- unsigned result = 0;
-
- cout << "examining symbolic differentiation" << flush;
- clog << "----------symbolic differentiation:" << endl;
-
- result += exam_differentiation1(); cout << '.' << flush;
- result += exam_differentiation2(); cout << '.' << flush;
- result += exam_differentiation3(); cout << '.' << flush;
- result += exam_differentiation4(); cout << '.' << flush;
- result += exam_differentiation5(); cout << '.' << flush;
- result += exam_differentiation6(); cout << '.' << flush;
-
- if (!result) {
- cout << " passed " << endl;
- clog << "(no output)" << endl;
- } else {
- cout << " failed " << endl;
- }
- return result;
+ unsigned result = 0;
+
+ cout << "examining symbolic differentiation" << flush;
+ clog << "----------symbolic differentiation:" << endl;
+
+ result += exam_differentiation1(); cout << '.' << flush;
+ result += exam_differentiation2(); cout << '.' << flush;
+ result += exam_differentiation3(); cout << '.' << flush;
+ result += exam_differentiation4(); cout << '.' << flush;
+ result += exam_differentiation5(); cout << '.' << flush;
+ result += exam_differentiation6(); cout << '.' << flush;
+ result += exam_differentiation7(); cout << '.' << flush;
+
+ if (!result) {
+ cout << " passed " << endl;
+ clog << "(no output)" << endl;
+ } else {
+ cout << " failed " << endl;
+ }
+ return result;
}
git://www.ginac.de / ginac.git / blobdiff