*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <vector>
return abs(arg);
}
+static ex abs_power(const ex & arg, const ex & exp)
+{
+ if (arg.is_equal(arg.conjugate()) && is_a<numeric>(exp) && ex_to<numeric>(exp).is_even())
+ return power(arg, exp);
+ else
+ return power(abs(arg), exp).hold();
+}
+
REGISTER_FUNCTION(abs, eval_func(abs_eval).
evalf_func(abs_evalf).
print_func<print_latex>(abs_print_latex).
print_func<print_csrc_float>(abs_print_csrc_float).
print_func<print_csrc_double>(abs_print_csrc_float).
- conjugate_func(abs_conjugate));
+ conjugate_func(abs_conjugate).
+ power_func(abs_power));
+
+//////////
+// Step function
+//////////
+
+static ex step_evalf(const ex & arg)
+{
+ if (is_exactly_a<numeric>(arg))
+ return step(ex_to<numeric>(arg));
+
+ return step(arg).hold();
+}
+
+static ex step_eval(const ex & arg)
+{
+ if (is_exactly_a<numeric>(arg))
+ return step(ex_to<numeric>(arg));
+
+ else if (is_exactly_a<mul>(arg) &&
+ is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
+ numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
+ if (oc.is_real()) {
+ if (oc > 0)
+ // step(42*x) -> step(x)
+ return step(arg/oc).hold();
+ else
+ // step(-42*x) -> step(-x)
+ return step(-arg/oc).hold();
+ }
+ if (oc.real().is_zero()) {
+ if (oc.imag() > 0)
+ // step(42*I*x) -> step(I*x)
+ return step(I*arg/oc).hold();
+ else
+ // step(-42*I*x) -> step(-I*x)
+ return step(-I*arg/oc).hold();
+ }
+ }
+
+ return step(arg).hold();
+}
+
+static ex step_series(const ex & arg,
+ const relational & rel,
+ int order,
+ unsigned options)
+{
+ const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
+ if (arg_pt.info(info_flags::numeric)
+ && ex_to<numeric>(arg_pt).real().is_zero()
+ && !(options & series_options::suppress_branchcut))
+ throw (std::domain_error("step_series(): on imaginary axis"));
+
+ epvector seq;
+ seq.push_back(expair(step(arg_pt), _ex0));
+ return pseries(rel,seq);
+}
+static ex step_power(const ex & arg, const ex & exp)
+{
+ if (exp.info(info_flags::positive))
+ return step(arg);
+
+ return power(step(arg), exp).hold();
+}
+
+static ex step_conjugate(const ex& arg)
+{
+ return step(arg);
+}
+
+REGISTER_FUNCTION(step, eval_func(step_eval).
+ evalf_func(step_evalf).
+ series_func(step_series).
+ conjugate_func(step_conjugate).
+ power_func(step_power));
//////////
// Complex sign
ex ser;
// manually construct the primitive expansion
for (int i=1; i<order; ++i)
- ser += pow(s,i) / pow(numeric(i), _num2);
+ ser += pow(s,i) / pow(numeric(i), *_num2_p);
// substitute the argument's series expansion
ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
// maybe that was terminating, so add a proper order term
if (y.is_integer()) {
if (y.is_nonneg_integer()) {
const unsigned N = y.to_int();
- if (N == 0) return _num0;
+ if (N == 0) return _ex0;
if (N == 1) return x;
ex t = x.expand();
for (unsigned i = 2; i <= N; ++i)
t = (t * (x + i - y - 1)).expand() / i;
return t;
} else
- return _num0;
+ return _ex0;
}
return binomial(x, y).hold();
return sollist;
}
+//////////
+// Find real root of f(x) numerically
+//////////
+
+const numeric
+fsolve(const ex& f_in, const symbol& x, const numeric& x1, const numeric& x2)
+{
+ if (!x1.is_real() || !x2.is_real()) {
+ throw std::runtime_error("fsolve(): interval not bounded by real numbers");
+ }
+ if (x1==x2) {
+ throw std::runtime_error("fsolve(): vanishing interval");
+ }
+ // xx[0] == left interval limit, xx[1] == right interval limit.
+ // fx[0] == f(xx[0]), fx[1] == f(xx[1]).
+ // We keep the root bracketed: xx[0]<xx[1] and fx[0]*fx[1]<0.
+ numeric xx[2] = { x1<x2 ? x1 : x2,
+ x1<x2 ? x2 : x1 };
+ ex f;
+ if (is_a<relational>(f_in)) {
+ f = f_in.lhs()-f_in.rhs();
+ } else {
+ f = f_in;
+ }
+ const ex fx_[2] = { f.subs(x==xx[0]).evalf(),
+ f.subs(x==xx[1]).evalf() };
+ if (!is_a<numeric>(fx_[0]) || !is_a<numeric>(fx_[1])) {
+ throw std::runtime_error("fsolve(): function does not evaluate numerically");
+ }
+ numeric fx[2] = { ex_to<numeric>(fx_[0]),
+ ex_to<numeric>(fx_[1]) };
+ if (!fx[0].is_real() || !fx[1].is_real()) {
+ throw std::runtime_error("fsolve(): function evaluates to complex values at interval boundaries");
+ }
+ if (fx[0]*fx[1]>=0) {
+ throw std::runtime_error("fsolve(): function does not change sign at interval boundaries");
+ }
+
+ // The Newton-Raphson method has quadratic convergence! Simply put, it
+ // replaces x with x-f(x)/f'(x) at each step. -f/f' is the delta:
+ const ex ff = normal(-f/f.diff(x));
+ int side = 0; // Start at left interval limit.
+ numeric xxprev;
+ numeric fxprev;
+ do {
+ xxprev = xx[side];
+ fxprev = fx[side];
+ xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
+ fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
+ if ((side==0 && xx[0]<xxprev) || (side==1 && xx[1]>xxprev) || xx[0]>xx[1]) {
+ // Oops, Newton-Raphson method shot out of the interval.
+ // Restore, and try again with the other side instead!
+ xx[side] = xxprev;
+ fx[side] = fxprev;
+ side = !side;
+ xxprev = xx[side];
+ fxprev = fx[side];
+ xx[side] += ex_to<numeric>(ff.subs(x==xx[side]).evalf());
+ fx[side] = ex_to<numeric>(f.subs(x==xx[side]).evalf());
+ }
+ if ((fx[side]<0 && fx[!side]<0) || (fx[side]>0 && fx[!side]>0)) {
+ // Oops, the root isn't bracketed any more.
+ // Restore, and perform a bisection!
+ xx[side] = xxprev;
+ fx[side] = fxprev;
+
+ // Ah, the bisection! Bisections converge linearly. Unfortunately,
+ // they occur pretty often when Newton-Raphson arrives at an x too
+ // close to the result on one side of the interval and
+ // f(x-f(x)/f'(x)) turns out to have the same sign as f(x) due to
+ // precision errors! Recall that this function does not have a
+ // precision goal as one of its arguments but instead relies on
+ // x converging to a fixed point. We speed up the (safe but slow)
+ // bisection method by mixing in a dash of the (unsafer but faster)
+ // secant method: Instead of splitting the interval at the
+ // arithmetic mean (bisection), we split it nearer to the root as
+ // determined by the secant between the values xx[0] and xx[1].
+ // Don't set the secant_weight to one because that could disturb
+ // the convergence in some corner cases!
+ static const double secant_weight = 0.984375; // == 63/64 < 1
+ numeric xxmid = (1-secant_weight)*0.5*(xx[0]+xx[1])
+ + secant_weight*(xx[0]+fx[0]*(xx[0]-xx[1])/(fx[1]-fx[0]));
+ numeric fxmid = ex_to<numeric>(f.subs(x==xxmid).evalf());
+ if (fxmid.is_zero()) {
+ // Luck strikes...
+ return xxmid;
+ }
+ if ((fxmid<0 && fx[side]>0) || (fxmid>0 && fx[side]<0)) {
+ side = !side;
+ }
+ xxprev = xx[side];
+ fxprev = fx[side];
+ xx[side] = xxmid;
+ fx[side] = fxmid;
+ }
+ } while (xxprev!=xx[side]);
+ return xxprev;
+}
+
+
/* Force inclusion of functions from inifcns_gamma and inifcns_zeta
* for static lib (so ginsh will see them). */
unsigned force_include_tgamma = tgamma_SERIAL::serial;
git://www.ginac.de / ginac.git / blobdiff