integral.cpp

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/*
 *  GiNaC Copyright (C) 1999-2011 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
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 */

#include "integral.h"
#include "numeric.h"
#include "symbol.h"
#include "add.h"
#include "mul.h"
#include "power.h"
#include "inifcns.h"
#include "wildcard.h"
#include "archive.h"
#include "registrar.h"
#include "utils.h"
#include "operators.h"
#include "relational.h"

using namespace std;

namespace GiNaC {

GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(integral, basic,
  print_func<print_dflt>(&integral::do_print).
  print_func<print_latex>(&integral::do_print_latex))



// default constructor

integral::integral()
        : 
        x((new symbol())->setflag(status_flags::dynallocated))
{}

// other constructors

// public

integral::integral(const ex & x_, const ex & a_, const ex & b_, const ex & f_)
        :  x(x_), a(a_), b(b_), f(f_)
{
    if (!is_a<symbol>(x)) {
        throw(std::invalid_argument("first argument of integral must be of type symbol"));
    }
}

// archiving

void integral::read_archive(const archive_node& n, lst& sym_lst)
{
    inherited::read_archive(n, sym_lst);
    n.find_ex("x", x, sym_lst);
    n.find_ex("a", a, sym_lst);
    n.find_ex("b", b, sym_lst);
    n.find_ex("f", f, sym_lst);
}

void integral::archive(archive_node & n) const
{
    inherited::archive(n);
    n.add_ex("x", x);
    n.add_ex("a", a);
    n.add_ex("b", b);
    n.add_ex("f", f);
}

// functions overriding virtual functions from base classes

void integral::do_print(const print_context & c, unsigned level) const
{
    c.s << "integral(";
    x.print(c);
    c.s << ",";
    a.print(c);
    c.s << ",";
    b.print(c);
    c.s << ",";
    f.print(c);
    c.s << ")";
}

void integral::do_print_latex(const print_latex & c, unsigned level) const
{
    string varname = ex_to<symbol>(x).get_name();
    if (level > precedence())
        c.s << "\\left(";
    c.s << "\\int_{";
    a.print(c);
    c.s << "}^{";
    b.print(c);
    c.s << "} d";
    if (varname.size() > 1)
        c.s << "\\," << varname << "\\:";
    else
        c.s << varname << "\\,";
    f.print(c,precedence());
    if (level > precedence())
        c.s << "\\right)";
}

int integral::compare_same_type(const basic & other) const
{
    GINAC_ASSERT(is_exactly_a<integral>(other));
    const integral &o = static_cast<const integral &>(other);

    int cmpval = x.compare(o.x);
    if (cmpval)
        return cmpval;
    cmpval = a.compare(o.a);
    if (cmpval)
        return cmpval;
    cmpval = b.compare(o.b);
    if (cmpval)
        return cmpval;
    return f.compare(o.f);
}

ex integral::eval(int level) const
{
    if ((level==1) && (flags & status_flags::evaluated))
        return *this;
    if (level == -max_recursion_level)
        throw(std::runtime_error("max recursion level reached"));

    ex eintvar = (level==1) ? x : x.eval(level-1);
    ex ea      = (level==1) ? a : a.eval(level-1);
    ex eb      = (level==1) ? b : b.eval(level-1);
    ex ef      = (level==1) ? f : f.eval(level-1);

    if (!ef.has(eintvar) && !haswild(ef))
        return eb*ef-ea*ef;

    if (ea==eb)
        return _ex0;

    if (are_ex_trivially_equal(eintvar,x) && are_ex_trivially_equal(ea,a)
            && are_ex_trivially_equal(eb,b) && are_ex_trivially_equal(ef,f))
        return this->hold();
    return (new integral(eintvar, ea, eb, ef))
        ->setflag(status_flags::dynallocated | status_flags::evaluated);
}

ex integral::evalf(int level) const
{
    ex ea;
    ex eb;
    ex ef;

    if (level==1) {
        ea = a;
        eb = b;
        ef = f;
    } else if (level == -max_recursion_level) {
        throw(runtime_error("max recursion level reached"));
    } else {
        ea = a.evalf(level-1);
        eb = b.evalf(level-1);
        ef = f.evalf(level-1);
    }

    // 12.34 is just an arbitrary number used to check whether a number
    // results after subsituting a number for the integration variable.
    if (is_exactly_a<numeric>(ea) && is_exactly_a<numeric>(eb) 
            && is_exactly_a<numeric>(ef.subs(x==12.34).evalf())) {
            return adaptivesimpson(x, ea, eb, ef);
    }

    if (are_ex_trivially_equal(a, ea) && are_ex_trivially_equal(b, eb)
                && are_ex_trivially_equal(f, ef))
            return *this;
        else
            return (new integral(x, ea, eb, ef))
                ->setflag(status_flags::dynallocated);
}

int integral::max_integration_level = 15;
ex integral::relative_integration_error = 1e-8;

ex subsvalue(const ex & var, const ex & value, const ex & fun)
{
    ex result = fun.subs(var==value).evalf();
    if (is_a<numeric>(result))
        return result;
    throw logic_error("integrand does not evaluate to numeric");
}

struct error_and_integral
{
    error_and_integral(const ex &err, const ex &integ)
        :error(err), integral(integ){}
    ex error;
    ex integral;
};

struct error_and_integral_is_less
{
    bool operator()(const error_and_integral &e1,const error_and_integral &e2) const
    {
        int c = e1.integral.compare(e2.integral);
        if(c < 0)
            return true;
        if(c > 0)
            return false;
        return ex_is_less()(e1.error, e2.error);
    }
};

typedef map<error_and_integral, ex, error_and_integral_is_less> lookup_map;

ex adaptivesimpson(const ex & x, const ex & a_in, const ex & b_in, const ex & f, const ex & error)
{
    // Check whether boundaries and error are numbers.
    ex a = is_exactly_a<numeric>(a_in) ? a_in : a_in.evalf();
    ex b = is_exactly_a<numeric>(b_in) ? b_in : b_in.evalf();
    if(!is_exactly_a<numeric>(a) || !is_exactly_a<numeric>(b))
        throw std::runtime_error("For numerical integration the boundaries of the integral should evalf into numbers.");
    if(!is_exactly_a<numeric>(error))
        throw std::runtime_error("For numerical integration the error should be a number.");

    // Use lookup table to be potentially much faster.
    static lookup_map lookup;
    static symbol ivar("ivar");
    ex lookupex = integral(ivar,a,b,f.subs(x==ivar));
    lookup_map::iterator emi = lookup.find(error_and_integral(error, lookupex));
    if (emi!=lookup.end())
        return emi->second;

    ex app = 0;
    int i = 1;
    exvector avec(integral::max_integration_level+1);
    exvector hvec(integral::max_integration_level+1);
    exvector favec(integral::max_integration_level+1);
    exvector fbvec(integral::max_integration_level+1);
    exvector fcvec(integral::max_integration_level+1);
    exvector svec(integral::max_integration_level+1);
    exvector errorvec(integral::max_integration_level+1);
    vector<int> lvec(integral::max_integration_level+1);

    avec[i] = a;
    hvec[i] = (b-a)/2;
    favec[i] = subsvalue(x, a, f);
    fcvec[i] = subsvalue(x, a+hvec[i], f);
    fbvec[i] = subsvalue(x, b, f);
    svec[i] = hvec[i]*(favec[i]+4*fcvec[i]+fbvec[i])/3;
    lvec[i] = 1;
    errorvec[i] = error*abs(svec[i]);

    while (i>0) {
        ex fd = subsvalue(x, avec[i]+hvec[i]/2, f);
        ex fe = subsvalue(x, avec[i]+3*hvec[i]/2, f);
        ex s1 = hvec[i]*(favec[i]+4*fd+fcvec[i])/6;
        ex s2 = hvec[i]*(fcvec[i]+4*fe+fbvec[i])/6;
        ex nu1 = avec[i];
        ex nu2 = favec[i];
        ex nu3 = fcvec[i];
        ex nu4 = fbvec[i];
        ex nu5 = hvec[i];
        // hopefully prevents a crash if the function is zero sometimes.
        ex nu6 = max(errorvec[i], abs(s1+s2)*error);
        ex nu7 = svec[i];
        int nu8 = lvec[i];
        --i;
        if (abs(ex_to<numeric>(s1+s2-nu7)) <= nu6)
            app+=(s1+s2);
        else {
            if (nu8>=integral::max_integration_level)
                throw runtime_error("max integration level reached");
            ++i;
            avec[i] = nu1+nu5;
            favec[i] = nu3;
            fcvec[i] = fe;
            fbvec[i] = nu4;
            hvec[i] = nu5/2;
            errorvec[i]=nu6/2;
            svec[i] = s2;
            lvec[i] = nu8+1;
            ++i;
            avec[i] = nu1;
            favec[i] = nu2;
            fcvec[i] = fd;
            fbvec[i] = nu3;
            hvec[i] = hvec[i-1];
            errorvec[i]=errorvec[i-1];
            svec[i] = s1;
            lvec[i] = lvec[i-1];
        }
    }

    lookup[error_and_integral(error, lookupex)]=app;
    return app;
}

int integral::degree(const ex & s) const
{
    return ((b-a)*f).degree(s);
}

int integral::ldegree(const ex & s) const
{
    return ((b-a)*f).ldegree(s);
}

ex integral::eval_ncmul(const exvector & v) const
{
    return f.eval_ncmul(v);
}

size_t integral::nops() const
{
    return 4;
}

ex integral::op(size_t i) const
{
    GINAC_ASSERT(i<4);

    switch (i) {
        case 0:
            return x;
        case 1:
            return a;
        case 2:
            return b;
        case 3:
            return f;
        default:
            throw (std::out_of_range("integral::op() out of range"));
    }
}

ex & integral::let_op(size_t i)
{
    ensure_if_modifiable();
    switch (i) {
        case 0:
            return x;
        case 1:
            return a;
        case 2:
            return b;
        case 3:
            return f;
        default:
            throw (std::out_of_range("integral::let_op() out of range"));
    }
}

ex integral::expand(unsigned options) const
{
    if (options==0 && (flags & status_flags::expanded))
        return *this;

    ex newa = a.expand(options);
    ex newb = b.expand(options);
    ex newf = f.expand(options);

    if (is_a<add>(newf)) {
        exvector v;
        v.reserve(newf.nops());
        for (size_t i=0; i<newf.nops(); ++i)
            v.push_back(integral(x, newa, newb, newf.op(i)).expand(options));
        return ex(add(v)).expand(options);
    }

    if (is_a<mul>(newf)) {
        ex prefactor = 1;
        ex rest = 1;
        for (size_t i=0; i<newf.nops(); ++i)
            if (newf.op(i).has(x))
                rest *= newf.op(i);
            else
                prefactor *= newf.op(i);
        if (prefactor != 1)
            return (prefactor*integral(x, newa, newb, rest)).expand(options);
    }

    if (are_ex_trivially_equal(a, newa) && are_ex_trivially_equal(b, newb)
            && are_ex_trivially_equal(f, newf)) {
        if (options==0)
            this->setflag(status_flags::expanded);
        return *this;
    }

    const basic & newint = (new integral(x, newa, newb, newf))
        ->setflag(status_flags::dynallocated);
    if (options == 0)
        newint.setflag(status_flags::expanded);
    return newint;
}

ex integral::derivative(const symbol & s) const
{
    if (s==x)
        throw(logic_error("differentiation with respect to dummy variable"));
    return b.diff(s)*f.subs(x==b)-a.diff(s)*f.subs(x==a)+integral(x, a, b, f.diff(s));
}

unsigned integral::return_type() const
{
    return f.return_type();
}

return_type_t integral::return_type_tinfo() const
{
    return f.return_type_tinfo();
}

ex integral::conjugate() const
{
    ex conja = a.conjugate();
    ex conjb = b.conjugate();
    ex conjf = f.conjugate().subs(x.conjugate()==x);

    if (are_ex_trivially_equal(a, conja) && are_ex_trivially_equal(b, conjb)
            && are_ex_trivially_equal(f, conjf))
        return *this;

    return (new integral(x, conja, conjb, conjf))
        ->setflag(status_flags::dynallocated);
}

ex integral::eval_integ() const
{
    if (!(flags & status_flags::expanded))
        return this->expand().eval_integ();
    
    if (f==x)
        return b*b/2-a*a/2;
    if (is_a<power>(f) && f.op(0)==x) {
        if (f.op(1)==-1)
            return log(b/a);
        if (!f.op(1).has(x)) {
            ex primit = power(x,f.op(1)+1)/(f.op(1)+1);
            return primit.subs(x==b)-primit.subs(x==a);
        }
    }

    return *this;
}

GINAC_BIND_UNARCHIVER(integral);
} // namespace GiNaC