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optimization.cc
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optimization.cc
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/*
* optimization.cc
*
* Copyright 2021-2022 Luka Marohnić
*
* 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 3 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, see <http://www.gnu.org/licenses/>.
*
*/
#include "gen.h"
#include "giacPCH.h"
#include "giac.h"
#include "optimization.h"
#include "lpsolve.h"
#include "graphe.h"
#include "signalprocessing.h"
#include <bitset>
#include <string>
#ifdef HAVE_PARI_PARI_H
#include <pari/pari.h>
#endif
using namespace std;
#ifndef NO_NAMESPACE_GIAC
namespace giac {
#endif // ndef NO_NAMESPACE_GIAC
/* Create a temporary symbol "name<count>". */
gen temp_symb(const string &name,int count,GIAC_CONTEXT) {
gen ret=identificateur(" "+name+(count>=0?print_INT_(count):""));
_purge(ret,contextptr);
return ret;
}
/* Make a matrix of zeros. */
matrice zero_mat(int rows,int cols,GIAC_CONTEXT) {
return *_matrix(makesequence(rows,cols,0),contextptr)._VECTptr;
}
/* Copy matrix SRC to DEST, resizing the latter if necessary. */
void copy_matrice(const matrice &src,matrice &dest) {
int m=mrows(src),n=mcols(src);
if (int(dest.size())!=m) dest.resize(m);
const_iterateur it=src.begin(),itend=src.end();
iterateur jt=dest.begin();
for (;it!=itend;++it,++jt) {
if (jt->type!=_VECT)
*jt=vecteur(it->_VECTptr->begin(),it->_VECTptr->end());
else {
if (int(jt->_VECTptr->size())!=n) jt->_VECTptr->resize(n);
std::copy(it->_VECTptr->begin(),it->_VECTptr->end(),jt->_VECTptr->begin());
}
}
}
/* Return A+B (add two matrices). */
matrice madd(const matrice &a,const matrice &b) {
matrice ret;
ret.reserve(a.size());
const_iterateur at=a.begin(),bt=b.begin();
for (;at!=a.end()&&bt!=b.end();++at,++bt) {
ret.push_back(addvecteur(*at->_VECTptr,*bt->_VECTptr));
}
return ret;
}
void madd_inplace(matrice &a,const matrice &b,bool subt) {
iterateur at=a.begin(),ajt;
const_iterateur bt=b.begin(),bjt;
for (;at!=a.end()&&bt!=b.end();++at,++bt) {
ajt=at->_VECTptr->begin();
bjt=bt->_VECTptr->begin();
for (;ajt!=at->_VECTptr->end()&&bjt!=bt->_VECTptr->end();++ajt,++bjt) {
if (subt) *ajt-=*bjt; else *ajt+=*bjt;
}
}
}
/* Multiply each element of matrix B by A. */
matrice mscale(const gen &a,const matrice &b) {
matrice ret;
ret.reserve(b.size());
for (const_iterateur it=b.begin();it!=b.end();++it) {
ret.push_back(multvecteur(a,*it->_VECTptr));
}
return ret;
}
void mscale_inplace(const gen &a,matrice &b) {
for (iterateur it=b.begin();it!=b.end();++it) {
for (iterateur jt=it->_VECTptr->begin();jt!=it->_VECTptr->end();++jt) {
*jt=*jt*a;
}
}
}
/* Return A-B (subtract two matrices). */
matrice msub(const matrice &a,const matrice &b) {
return madd(a,mscale(-1,b));
}
void msub_inplace(matrice &a,const matrice &b) {
madd_inplace(a,b,true);
}
/* Return the index of the first minimal/maximal element in V */
int argminmax(const vecteur &v,bool min,GIAC_CONTEXT) {
gen val=min?plus_inf:minus_inf;
int ret=-1;
const_iterateur it,itstart=v.begin(),itend=v.end();
for (it=itstart;it!=itend;++it) {
if (is_strictly_greater(min?val:*it,min?*it:val,contextptr)) {
ret=it-itstart;
val=*it;
}
}
return ret;
}
int argmin(const vecteur &v,GIAC_CONTEXT) {
return argminmax(v,true,contextptr);
}
int argmax(const vecteur &v,GIAC_CONTEXT) {
return argminmax(v,false,contextptr);
}
/* Remove elements from V with indices in IND. */
void remove_elements_with_indices(vecteur &v,const set<int> &ind) {
set<int>::const_reverse_iterator it=ind.rbegin(),itend=ind.rend();
for (;it!=itend;++it)
v.erase(v.begin()+*it);
}
void remove_elements_with_indices(vecteur &v,const set<pair<int,gen> > &ind) {
set<pair<int,gen> >::const_reverse_iterator it=ind.rbegin(),itend=ind.rend();
for (;it!=itend;++it)
v.erase(v.begin()+it->first);
}
/* Return true iff g is an mapleconversion integer (corresponding to v) */
bool is_mcint(const gen &g,int v) {
return g.is_integer() && g.subtype==_INT_MAPLECONVERSION && (v<0 || g.val==v);
}
/* Return the index of g in v, or -1 if g is not contained in v. */
int indexof(const gen &g,const vecteur &v) {
const_iterateur it=find(v.begin(),v.end(),g);
if (it==v.end())
return -1;
return it-v.begin();
}
/* Return the list of indices of linearly dependent rows in matrix m.
* The returned list is sorted in ascending order. */
vector<int> linearly_dependent_rows(const matrice &m,GIAC_CONTEXT) {
int nc=mcols(m);
std::set<int> found;
vector<int> res;
res.reserve(m.size());
matrice mf=mtran(*_rref(mtran(*_evalf(m,contextptr)._VECTptr),contextptr)._VECTptr);
for (const_iterateur it=mf.begin();it!=mf.end();++it) {
int i=0;
for (;i<nc && is_zero(it->_VECTptr->at(i));++i);
if (i<nc && is_one(it->_VECTptr->at(i)) && found.find(i)==found.end())
found.insert(i);
else res.push_back(it-mf.begin());
}
std::sort(res.begin(),res.end());
return res;
}
/* Return true if variables in V are all identifiers. */
bool ckvars(const vecteur &v,GIAC_CONTEXT) {
vecteur vp;
for (const_iterateur it=v.begin();it!=v.end();++it) {
if (it->type!=_IDNT || _eval(*it,contextptr)!=*it)
return false;
}
return true;
}
/* Set assumptions on VARS w.r.t. BNDS. If OPEN=true, exclude bounds from range. */
void bound_variables(const vecteur &vars,const vecteur &bnds,bool open,GIAC_CONTEXT) {
assert(vars.size()==bnds.size() && (vars.empty() || (ckmatrix(bnds) && bnds.front()._VECTptr->size()==2)));
for (int i=vars.size();i-->0;) {
const gen &v=vars[i],&lb=bnds[i]._VECTptr->front(),&ub=bnds[i]._VECTptr->back();
if (!is_inf(lb) && !is_inf(ub))
giac_assume(symb_and(open?symb_superieur_strict(v,lb):symb_superieur_egal(v,lb),
open?symb_inferieur_strict(v,ub):symb_inferieur_egal(v,ub)),contextptr);
else if (!is_inf(lb))
giac_assume(open?symb_superieur_strict(v,lb):symb_superieur_egal(v,lb),contextptr);
else if (!is_inf(ub))
giac_assume(open?symb_inferieur_strict(v,ub):symb_inferieur_egal(v,ub),contextptr);
}
}
void purge_variables(const vecteur &vars,GIAC_CONTEXT) {
/* Purge identifiers in VARS. */
for (const_iterateur it=vars.begin();it!=vars.end();++it) {
assert(it->type==_IDNT);
_purge(*it,contextptr);
}
}
/* Sort a list of elements such that letters and numbers are treated separately.
* Useful for sorting lists of numbered identifiers, such as [x1,x2,...,x10,x11,...]. */
vecteur sort_identifiers(const vecteur &v,GIAC_CONTEXT) {
if (v.empty())
return v;
vecteur strv=*_apply(makesequence(at_string,v),contextptr)._VECTptr,snv;
snv.reserve(strv.size());
for (const_iterateur it=strv.begin();it!=strv.end();++it) {
if (it->type!=_STRNG)
return v;
const string &s=*it->_STRNGptr;
size_t i=s.find_last_not_of("0123456789");
string str_id=i==string::npos?"":s.substr(0,i+1),str_num=i==string::npos?s:s.substr(i+1);
gen id=str_id.size()==0?undef:string2gen(str_id,false);
gen num=_expr(string2gen(str_num,false),contextptr);
vecteur sn;
if (!is_undef(id))
sn.push_back(id);
if (!is_undef(num))
sn.push_back(num);
else if (!is_undef(id))
sn.push_back(0);
snv.push_back(sn);
}
return *mtran(*_sort(mtran(makevecteur(snv,v)),contextptr)._VECTptr)[1]._VECTptr;
}
/* Convert bounds given in matrix form [[L1,U1],[L2,U2],..] to the sequence
* x1=L1..U1,x2=L2..U2,.. */
gen _box_constraints(const gen &g,GIAC_CONTEXT) {
if (g.type==_STRNG && g.subtype==-1) return g;
if (g.type!=_VECT || g.subtype!=_SEQ__VECT || g._VECTptr->size()!=2 ||
g._VECTptr->front().type!=_VECT || g._VECTptr->back().type!=_VECT ||
g._VECTptr->front()._VECTptr->empty() || g._VECTptr->back()._VECTptr->empty())
return generrtype(gettext("Expected a sequence of two nonempty lists"));
const vecteur &x=*g._VECTptr->front()._VECTptr;
const vecteur &b=*g._VECTptr->back()._VECTptr;
if (!ckmatrix(b,false) || b.size()!=x.size() || b.front()._VECTptr->size()!=2)
return generrdim(gettext("Invalid list of bounds"));
matrice B=mtran(b);
gen intrv=_zip(makesequence(at_interval,B.front(),B.back()),contextptr);
return change_subtype(_zip(makesequence(at_equal,x,intrv),contextptr),_SEQ__VECT);
}
static const char _box_constraints_s []="box_constraints";
static define_unary_function_eval (__box_constraints,&_box_constraints,_box_constraints_s);
define_unary_function_ptr5(at_box_constraints,alias_at_box_constraints,&__box_constraints,0,true)
/* Return the sum of elements in V (exclude the last one if DROP_LAST=true). */
int ipdiff::sum_ivector(const ivector &v,bool drop_last) {
int res=0;
for (ivector_iter it=v.begin();it!=v.end()-drop_last?1:0;++it) {
res+=*it;
}
return res;
}
/* Return 1 if G is a <(=) contraint, 2 if G is a >(=) constraint, and 0 otherwise. */
int which_ineq(const gen &g) {
if (g.is_symb_of_sommet(at_inferieur_egal) || g.is_symb_of_sommet(at_inferieur_strict))
return 1;
if (g.is_symb_of_sommet(at_superieur_egal) || g.is_symb_of_sommet(at_superieur_strict))
return 2;
return 0;
}
/* Compute the intersection RES of two sorted lists A and B, return the size of RES. */
int intersect(const vector<int> &a,const vector<int> &b,vector<int> &res) {
res.resize(a.size()>b.size()?a.size():b.size());
vector<int>::iterator it=set_intersection(a.begin(),a.end(),b.begin(),b.end(),res.begin());
res.resize(it-res.begin());
return res.size();
}
/* Compute the union RES of two sorted lists A and B, return the size of RES. */
int unite(const vector<int> &a,const vector<int> &b,vector<int> &res) {
res.resize(a.size()+b.size());
vector<int>::iterator it=set_union(a.begin(),a.end(),b.begin(),b.end(),res.begin());
res.resize(it-res.begin());
return res.size();
}
/* Simplify expression G. */
gen simp(const gen &g,GIAC_CONTEXT) {
if (g.type==_VECT) {
vecteur res;
for (const_iterateur it=g._VECTptr->begin();it!=g._VECTptr->end();++it) {
res.push_back(simp(*it,contextptr));
}
return change_subtype(res,g.subtype);
}
log_output_redirect lor(contextptr);
gen ret=recursive_normal(g,contextptr);
#if 0
if (_evalf(g,contextptr).is_approx()) {
gen ret_simp=_simplify(g,contextptr);
if (!has_rootof(ret_simp))
return ret_simp;
if (has_rootof(ret)) {
ret=ratnormal(g,contextptr);
return has_rootof(ret)?ret_simp:ret;
}
}
#endif
return ret;
}
/* Return true if the expression E contains a non-elementary derivative w.r.t. X,
* which must be an identifier or a list of identifiers. */
bool has_diff(const gen &e,const gen &x) {
if (x.type==_VECT) {
for (const_iterateur it=x._VECTptr->begin();it!=x._VECTptr->end();++it) {
if (has_diff(e,*it))
return true;
}
return false;
}
assert(x.type==_IDNT);
if (e.type==_VECT) {
for (const_iterateur it=e._VECTptr->begin();it!=e._VECTptr->end();++it) {
if (has_diff(*it,x))
return true;
}
return false;
}
if (e.type!=_SYMB)
return false;
if ((e.is_symb_of_sommet(at_derive) && e._SYMBptr->feuille.type==_VECT &&
e._SYMBptr->feuille._VECTptr->size()>=2 && e._SYMBptr->feuille._VECTptr->at(1)==x) ||
(e.is_symb_of_sommet(at_of) && e._SYMBptr->feuille._VECTptr->front().is_symb_of_sommet(at_derive) &&
e._SYMBptr->feuille._VECTptr->back().type==_VECT && contains(*e._SYMBptr->feuille._VECTptr->back()._VECTptr,x)))
return true;
return has_diff(e._SYMBptr->feuille,x);
}
/* Return true iff G contains any of the following:
* - absolute values
* - piecewise, when and ifte functions
* - Heaviside and Dirac functions
* - max, min, floor, ceil, round functions
* - sign function
*/
bool has_breaks(const gen &g) {
if (g.type==_VECT) {
for (const_iterateur it=g._VECTptr->begin();it!=g._VECTptr->end();++it) {
if (has_breaks(*it))
return true;
}
return false;
}
if (g.type!=_SYMB)
return false;
const gen &s=g._SYMBptr->sommet;
const gen &f=g._SYMBptr->feuille;
if (s==at_abs || s==at_piecewise || s==at_when || s==at_ifte || s==at_Heaviside || s==at_Dirac || s==at_min ||
s==at_max || s==at_floor || s==at_ceil || s==at_round || s==at_sign)
return true;
return has_breaks(f);
}
/* Brent's algorithm (implemented by John Burkardt), code below */
double local_min_rc(double &a,double &b,int &status,double value);
/* Return true if G = A*X^3+B*X^2+C*X+D. */
bool is_cubic_wrt(const gen &g,const gen &x,gen &a,gen &b,gen &c,gen &d,GIAC_CONTEXT) {
if (is_quadratic_wrt(derive(g,x,contextptr),x,a,b,c,contextptr)) {
a=a/3;
b=b/2;
d=g-a*x*x*x-b*x*x-c*x;
return true;
}
return false;
}
/* Return true if G = A*X^4+B*X^3+C*X^2+D*x+E. */
bool is_quartic_wrt(const gen &g,const gen &x,gen &a,gen &b,gen &c,gen &d,gen &e,GIAC_CONTEXT) {
if (is_cubic_wrt(derive(g,x,contextptr),x,a,b,c,d,contextptr)) {
a=a/4;
b=b/3;
c=c/2;
e=g-a*x*x*x*x-b*x*x*x-c*x*x-d*x;
return true;
}
return false;
}
/* Return true if G is a quadratic with positive leading coefficient and
* negative discriminant. */
bool is_positive_quadratic(const gen &g,bool strict,GIAC_CONTEXT) {
vecteur p=*_lname(g,contextptr)._VECTptr;
gen a,b,c,d;
unsigned mt=100;
for (const_iterateur it=p.begin();it!=p.end();++it) {
if (!is_quadratic_wrt(g,*it,a,b,c,contextptr))
continue;
d=4*a*c-b*b;
if (!is_exactly_zero(a) && is_positive_safe(a,false,mt,contextptr) && is_positive_safe(d,strict,mt,contextptr))
return true;
}
return false;
}
/* Return true if G is a (strictly) positive quartic. */
bool is_positive_quartic(const gen &g,bool strict,GIAC_CONTEXT) {
vecteur p=*_lname(g,contextptr)._VECTptr;
gen a,b,c,d,e,dsc,dsc0,D,P,R;
unsigned mt=100;
for (const_iterateur it=p.begin();it!=p.end();++it) {
if (!is_quartic_wrt(g,*it,a,b,c,d,e,contextptr) || is_positive_safe(-a,false,mt,contextptr))
continue;
dsc=((256*a*e-192*b*d-128*c*c)*e*e+(144*c*e-27*d*d)*d*d)*a*a+(144*c*e*e-6*d*d*e)*a*b*b
+(-80*c*d*e+18*d*d*d)*a*b*c+(16*c*e-4*d*d)*a*c*c*c+(-27*b*e*e+18*c*d*e-4*d*d*d)*b*b*b+b*b*c*c*d*d;
dsc0=c*c-3*b*d+12*a*e;
D=64*a*a*a*e-16*a*a*c*c+16*a*b*b*c-16*a*a*b*d-3*b*b*b*b;
P=8*a*c-3*b*b;
R=b*b*b+8*d*a*a-4*a*b*c;
bool pos_dsc=is_positive_safe(dsc,true,mt,contextptr),zero_dsc=is_exactly_zero(dsc);
bool pos_P=is_positive_safe(P,true,mt,contextptr),neg_P=is_positive_safe(-P,true,mt,contextptr),pos_D=is_positive_safe(D,true,mt,contextptr);
bool zero_R=is_exactly_zero(R),zero_D=is_exactly_zero(D),zero_dsc0=is_exactly_zero(dsc0);
if ((pos_dsc && pos_D && pos_P) || (zero_dsc && zero_D && zero_R && pos_P) ||
(!strict && (zero_dsc && ((pos_D || (pos_P && (!zero_D || !zero_R))) || (zero_D && (neg_P || zero_dsc0))))))
return true;
}
return false;
}
/* Return true if G does not change sign (but may be zero). */
bool is_const_sign(const gen &g,bool pos,GIAC_CONTEXT) {
return is_positive_safe(pos?g:-g,false,100,contextptr) ||
is_positive_quadratic(pos?g:-g,false,contextptr) ||
is_positive_quartic(pos?g:-g,false,contextptr);
}
/* Return true if g is *strictly* positive/negative given pos=true/false.
* LIN should be set to true. */
bool is_const_sign_strict(const gen &g,bool pos,bool lin,GIAC_CONTEXT) {
if (is_real_number(g,contextptr) && ((pos && is_strictly_positive(to_real_number(g,contextptr),contextptr)) ||
(!pos && is_strictly_positive(-to_real_number(g,contextptr),contextptr))))
return true;
if (g.is_symb_of_sommet(at_neg))
return is_const_sign_strict(g._SYMBptr->feuille,!pos,lin,contextptr);
if (g.is_symb_of_sommet(at_abs))
return is_const_sign_strict(g._SYMBptr->feuille,true,lin,contextptr) || is_const_sign_strict(g._SYMBptr->feuille,false,lin,contextptr);
gen eps=exact(epsilon(contextptr),contextptr);
if ((pos?is_greater(g,eps,contextptr):is_greater(-eps,g,contextptr)) ||
is_positive_quadratic(pos?g:-g,true,contextptr) || is_positive_quartic(pos?g:-g,true,contextptr))
return true;
if (!lin && g.is_symb_of_sommet(at_prod) && g._SYMBptr->feuille.type==_VECT) {
const vecteur &f=*g._SYMBptr->feuille._VECTptr;
vecteur s(f.size(),0);
for (const_iterateur it=f.begin();it!=f.end();++it) {
gen fac=*it,exponent;
bool power=false;
if (it->is_symb_of_sommet(at_pow) && (exponent=_abs(it->_SYMBptr->feuille._VECTptr->back(),contextptr)).is_integer()) {
power=true;
fac=it->_SYMBptr->feuille._VECTptr->front();
}
if (is_const_sign_strict(fac,true,false,contextptr))
s[it-f.begin()]=1;
else if (is_const_sign_strict(fac,false,false,contextptr))
s[it-f.begin()]=power && exponent.val%2==0?1:-1;
}
gen ps=_product(s,contextptr);
if (is_zero(ps))
return false;
return pos?is_one(ps):is_minus_one(ps);
}
vecteur terms(1,g);
if (g.is_symb_of_sommet(at_plus) && g._SYMBptr->feuille.type==_VECT)
terms=*g._SYMBptr->feuille._VECTptr;
bool surplus=false;
gen rest(0);
for (const_iterateur it=terms.begin();it!=terms.end();++it) {
gen l=lin?_lin(*it,contextptr):*it;
if ((pos && (l.is_symb_of_sommet(at_exp) || l.is_symb_of_sommet(at_cosh))) ||
((l.is_symb_of_sommet(at_atan) || l.is_symb_of_sommet(at_asin) || l.is_symb_of_sommet(at_atanh) || l.is_symb_of_sommet(at_asinh))
&& is_const_sign_strict(l._SYMBptr->feuille,pos,true,contextptr)) ||
(pos && l.is_symb_of_sommet(at_acosh) && is_const_sign_strict(1-l._SYMBptr->feuille,true,true,contextptr)) ||
(pos && (l.is_symb_of_sommet(at_ln) || l.is_symb_of_sommet(at_acosh))
&& is_const_sign_strict(l._SYMBptr->feuille-1,true,true,contextptr)) ||
(is_real_number(*it,contextptr) && is_strictly_positive((pos?1:-1)*to_real_number(*it,contextptr),contextptr)) ||
(lin && is_const_sign_strict(l,pos,false,contextptr)))
surplus=true;
else if (surplus && is_const_sign(*it,pos,contextptr))
;
else rest+=*it;
}
if (!surplus)
return false;
return is_const_sign(rest,pos,contextptr);
}
bool is_definitely_positive(const gen &g,GIAC_CONTEXT) {
return is_const_sign_strict(g,true,true,contextptr);
}
/* Return true if im(g)=0. */
bool has_imag(const gen &g,GIAC_CONTEXT) {
if (g.type==_VECT) {
for (const_iterateur it=g._VECTptr->begin();it!=g._VECTptr->end();++it) {
if (has_imag(*it,contextptr))
return true;
}
return false;
}
if (is_inf(g) || is_real_number(g,contextptr))
return false;
gen gr,gi;
reim(g,gr,gi,contextptr);
return !is_zero(simp(gi,contextptr),contextptr);
}
/* Make piecewise sub-expressions nested, e.g. replace
* piecewise(cond1,g1,cond2,g2,...,otherwise) by
* piecewise(cond1,g1,piecewise(cond2,g2,piecewise(...,piecewise(condn,gn,otherwise))...)). */
gen make_piecewise_nested(const gen &g) {
if (g.type==_VECT) {
vecteur res;
for (const_iterateur it=g._VECTptr->begin();it!=g._VECTptr->end();++it) {
res.push_back(make_piecewise_nested(*it));
}
return change_subtype(res,g.subtype);
} else if (g.type==_SYMB) {
const gen &f=g._SYMBptr->feuille;
if (g.is_symb_of_sommet(at_piecewise) && f.type==_VECT && f._VECTptr->size()>3) {
const gen &cond=f._VECTptr->front(),&h=f._VECTptr->at(1);
vecteur rest(f._VECTptr->begin()+2,f._VECTptr->end());
return symbolic(at_piecewise,makesequence(make_piecewise_nested(cond),make_piecewise_nested(h),
make_piecewise_nested(symbolic(at_piecewise,change_subtype(rest,_SEQ__VECT)))));
}
return symbolic(g._SYMBptr->sommet,make_piecewise_nested(f));
} else return g;
}
/* Replace the first found piecewise expression in G with PCW and store its
* arguments in FEU. */
bool find_piecewise(gen &g,const gen &pcw,vecteur &feu) {
if (g.type==_VECT) {
for (iterateur it=g._VECTptr->begin();it!=g._VECTptr->end();++it) {
if (find_piecewise(*it,pcw,feu))
return true;
}
} else if (g.type==_SYMB) {
gen &f=g._SYMBptr->feuille;
if (g._SYMBptr->sommet==at_piecewise) {
feu=*f._VECTptr;
g=pcw;
return true;
}
return find_piecewise(f,pcw,feu);
}
return false;
}
/* Replace the first found absolute value in G with ASYMB and store its
* argument in VAL. */
bool find_abs(gen &g,const gen &asymb,gen &val) {
if (g.type==_VECT) {
for (iterateur it=g._VECTptr->begin();it!=g._VECTptr->end();++it) {
if (find_abs(*it,asymb,val))
return true;
}
} else if (g.type==_SYMB) {
gen &f=g._SYMBptr->feuille;
if (g._SYMBptr->sommet==at_abs) {
val=f;
g=asymb;
return true;
}
return find_abs(f,asymb,val);
}
return false;
}
/* Return true if E is rational with respect to variables in VARS. */
bool is_rational_wrt_vars(const gen &e,const vecteur &vars) {
for (const_iterateur it=vars.begin();it!=vars.end();++it) {
vecteur l(rlvarx(e,*it));
if (l.size()>1)
return false;
}
return true;
}
/* Return true if E is linear with respect to variables in VARS. */
bool is_linear_wrt_vars(const gen &e,const vecteur &vars,GIAC_CONTEXT) {
gen a,b;
for (const_iterateur it=vars.begin();it!=vars.end();++it) {
if (!is_linear_wrt(e,*it,a,b,contextptr) || !is_constant_wrt_vars(a,vars,contextptr))
return false;
}
return true;
}
/* Return true if E is quadratic with respect to variables in VARS. */
bool is_quadratic_wrt_vars(const gen &e,const vecteur &vars,GIAC_CONTEXT) {
gen a,b,c;
for (const_iterateur it=vars.begin();it!=vars.end();++it) {
if (!is_quadratic_wrt(e,*it,a,b,c,contextptr) || !is_constant_wrt_vars(a,vars,contextptr) || !is_linear_wrt_vars(b,vars,contextptr))
return false;
}
return true;
}
/* Return an uniform random real between A and B. */
gen rand_uniform(const gen &a,const gen &b,GIAC_CONTEXT) {
gen fa=to_real_number(a,contextptr),fb=to_real_number(b,contextptr);
if (is_zero(b-a,contextptr))
return fa;
return _rand(is_greater(b,a,contextptr)?makesequence(fa,fb):makesequence(fb,fa),contextptr);
}
/* Generate a normal variable with parameters MU=0 and SIGMA.
* If ABSOLUT=true, return the absolute value. */
gen rand_normal(const gen &sigma,bool absolut,GIAC_CONTEXT) {
assert(is_positive(sigma,contextptr));
gen s=to_real_number(sigma,contextptr);
if (is_zero(sigma,contextptr))
return 0;
gen r=_randNorm(makesequence(0,s),contextptr);
return absolut?_abs(r,contextptr):r;
}
/* Generate a multi-dimensional normal random variable.
* SIGMA is either a single number or a vector of the same length as MU. */
void rand_multinormal(const vecteur &mu,const gen &sigma,vecteur &res,GIAC_CONTEXT) {
int n=mu.size();
res.resize(n);
iterateur it=res.begin(),itend=res.end();
const_iterateur mt=mu.begin();
if (sigma.type==_VECT) {
const_iterateur st=sigma._VECTptr->begin();
for (;it!=itend;++it,++st,++mt) {
*it=_randNorm(makesequence(*mt,*st),contextptr);
}
} else for (;it!=itend;++it,++mt) {
*it=_randNorm(makesequence(*mt,sigma),contextptr);
}
}
/* Generate categorical random integer variable. */
int rand_categorical(const vecteur &weights,GIAC_CONTEXT) {
gen c=rand_uniform(0,_sum(weights,contextptr),contextptr),s=0;
int ret=0;
for (const_iterateur it=weights.begin();it!=weights.end();++it,++ret) {
s+=*it;
if (is_greater(s,c,contextptr))
return ret;
}
return ret-1;
}
/* TODO */
gen eval_continuous(const gen &g,const vecteur &x,const vecteur &bnds,const vecteur &a,GIAC_CONTEXT) {
if (g.type==_VECT) {
if (ckmatrix(g)) {
const matrice &mat=*g._VECTptr;
int nr=mat.size(),nc=mat.front()._VECTptr->size();
matrice ret=zero_mat(nr,nc,contextptr);
for (int i=0;i<nr;++i) {
for (int j=0;j<nc;++j) {
ret[i]._VECTptr->at(j)=eval_continuous(mat[i][j],x,bnds,a,contextptr);
}
}
return ret;
} else {
vecteur ret;
for (const_iterateur it=g._VECTptr->begin();it!=g._VECTptr->end();++it) {
ret.push_back(eval_continuous(*it,x,bnds,a,contextptr));
}
return change_subtype(ret,g.subtype);
}
}
assert(x.size()==a.size());
if (is_constant_wrt_vars(g,x,contextptr))
return g;
return simp(subst(g,x,a,false,contextptr),contextptr);
/* TODO: implement multivariable limits */
}
/* Filter critical points in CV. Discard those being complex, those not satisfying
* the inequality constraints and those for which f (and its extension) is not defined. */
void filter_cpts(vecteur &cv,const vecteur &vars,const vecteur &bnds,const vecteur &ineq,bool open,const gen &f,GIAC_CONTEXT) {
bound_variables(vars,bnds,open,contextptr);
for (int j=cv.size();j-->0;) {
gen val,&cpt=cv[j].type==_VECT && !cv[j]._VECTptr->empty() && cv[j]._VECTptr->front().type==_VECT?cv[j]._VECTptr->front():cv[j];
vecteur cp=gen2vecteur(cpt);
cp.resize(vars.size());
if (has_inf_or_undef(cp) || has_imag(cp,contextptr)) {
cv.erase(cv.begin()+j);
continue;
}
if (is_constant_wrt_vars(cp,vars,contextptr)) {
int i=ineq.size();
for (;i-->0;) {
gen g=simp(subst(ineq[i],vars,cp,false,contextptr),contextptr);
if (!(open?is_strictly_positive(-g,contextptr)
:((is_approx(g) && is_zero(_pow(makesequence(_abs(g,contextptr),1.2),contextptr),contextptr))
|| is_positive(-g,contextptr))))
break;
}
if (i>=0 || is_undef(val=simp(eval_continuous(f,vars,bnds,cp,contextptr),contextptr)) || has_imag(val,contextptr)) {
cv.erase(cv.begin()+j);
continue;
}
} else if (is_undef(val=simp(subst(f,vars,cp,false,contextptr),contextptr)) ||
!is_constant_wrt_vars(val,vars,contextptr) || has_imag(val,contextptr)) {
cv.erase(cv.begin()+j);
continue;
}
cpt=simp(cpt,contextptr);
}
purge_variables(vars,contextptr);
}
/* Call _solve without printing any messages.
* If an error occurs, an empty list is returned. */
vecteur solve_quiet(const gen &e,const gen &x,GIAC_CONTEXT) {
gen sol;
try {
log_output_redirect lor(contextptr);
sol=_solve(makesequence(e,x),contextptr);
if (sol.type!=_VECT)
return vecteur(0);
} catch (const std::runtime_error &e) {
return vecteur(0);
}
return *sol._VECTptr;
}
int var_index=0;
/* Remove strictly positive or strictly negative factors from G and return the result. */
gen remove_nonzero_factors(const gen &g,GIAC_CONTEXT) {
gen f(g);
bool is_neg=false;
if (f.is_symb_of_sommet(at_neg)) {
f=f._SYMBptr->feuille;
is_neg=true;
}
if (f.is_symb_of_sommet(at_prod) && f._SYMBptr->feuille.type==_VECT) {
const vecteur &fv=*f._SYMBptr->feuille._VECTptr;
gen p(1);
for (const_iterateur jt=fv.begin();jt!=fv.end();++jt) {
if (is_const_sign_strict(*jt,true,true,contextptr) || is_const_sign_strict(*jt,false,true,contextptr))
continue;
else p=*jt*p;
}
f=p;
} else if (is_const_sign_strict(f,true,true,contextptr) || is_const_sign_strict(f,false,true,contextptr))
f=1;
return (is_neg?-1:1)*f;
}
/* Solve a system of equations.
* This function is based on _solve but handles cases where a variable
* is found inside trigonometric, hyperbolic or exponential functions. */
vecteur solve2(const vecteur &e_orig,const vecteur &vars_orig,GIAC_CONTEXT) {
int m=e_orig.size(),n=vars_orig.size(),i=0;
vecteur e_orig_simp=*expexpand(expand(_pow2exp(e_orig,contextptr),contextptr),contextptr)._VECTptr;
for (iterateur it=e_orig_simp.begin();it!=e_orig_simp.end();++it) {
if (is_equal(*it))
*it=equal2diff(*it);
gen f=(it->type==_SYMB?_factor(*it,contextptr):*it);
gen num=remove_nonzero_factors(_numer(f,contextptr),contextptr);
gen den=remove_nonzero_factors(_denom(f,contextptr),contextptr);
*it=num/den;
}
for (;i<m;++i) {
if (!is_rational_wrt_vars(e_orig_simp[i],vars_orig))
break;
}
if (n==1 || i==m)
return solve_quiet(e_orig_simp,vars_orig,contextptr);
vecteur e(*halftan(_texpand(hyp2exp(e_orig_simp,contextptr),contextptr),contextptr)._VECTptr);
vecteur lv(*exact(lvar(_evalf(lvar(e),contextptr)),contextptr)._VECTptr);
vecteur deps(n),depvars(n,gen(0));
vecteur vars(vars_orig);
const_iterateur it=lv.begin();
for (;it!=lv.end();++it) {
i=0;
for (;i<n;++i) {
if (is_undef(vars[i]))
continue;
if (*it==(deps[i]=vars[i]) ||
*it==(deps[i]=exp(vars[i],contextptr)) ||
is_exactly_zero(simp(*it-(deps[i]=tan(vars[i]/2,contextptr)),contextptr))) {
vars[i]=undef;
depvars[i]=temp_symb("depvar",i,contextptr);
break;
}
}
if (i==n)
break;
}
if (it!=lv.end() || contains(depvars,gen(0)))
return solve_quiet(e_orig_simp,vars_orig,contextptr);
vecteur e_subs=subst(e,deps,depvars,false,contextptr);
vecteur sol=solve_quiet(e_subs,depvars,contextptr);
vecteur ret;
for (const_iterateur it=sol.begin();it!=sol.end();++it) {
vecteur r(n);
i=0;
for (;i<n;++i) {
gen c(it->_VECTptr->at(i));
if (deps[i].type==_IDNT)
r[i]=c;
else if (deps[i].is_symb_of_sommet(at_exp) && is_strictly_positive(c,contextptr))
r[i]=simp(ln(c,contextptr),contextptr);
else if (deps[i].is_symb_of_sommet(at_tan))
r[i]=simp(2*atan(c,contextptr),contextptr);
else
break;
}
if (i==n)
ret.push_back(r);
}
return ret;
}
/* Solve the system of equations e=0 w.r.t. variables in v, including the solutions of a
* continuous extension too. It is assumed that e is exact. */
vecteur zeros_ext(const vecteur &e,const vecteur &v,const vecteur &bnds,GIAC_CONTEXT) {
vecteur e_numer=*_apply(makesequence(at_numer,e),contextptr)._VECTptr;
vecteur e_denom=*_apply(makesequence(at_denom,e),contextptr)._VECTptr;
vecteur sol=solve2(e_numer,v,contextptr),res;
for (const_iterateur it=sol.begin();it!=sol.end();++it) {
bool ok=true;
if (!is_approx(*it)) {
for (const_iterateur jt=e.begin();ok && jt!=e.end();++jt) {
if (!is_constant_wrt_vars(e_denom[jt-e.begin()],v,contextptr) &&
is_zero(simp(subst(e_denom[jt-e.begin()],v,*(it->_VECTptr),false,contextptr),contextptr))) {
ok=is_zero(simp(eval_continuous(*jt,v,bnds,*(it->_VECTptr),contextptr),contextptr));
}
}
}
if (ok)
res.push_back(*it);
}
return res;
}
/* Return true if G = (A and B) for some A, B. */
bool is_conjunction(const gen &g) {
return g.is_symb_of_sommet(at_and) || g.is_symb_of_sommet(at_et);
}
/* Get bounds on variables in VARS using the inequalities INEQ. */
bool get_variable_bounds(const vecteur &vars,vecteur &ineq,vecteur &bnds,GIAC_CONTEXT) {
gen a,b,c,vmin,vmax;
bnds=zero_mat(vars.size(),2,contextptr);
for (const_iterateur it=vars.begin();it!=vars.end();++it) {
vecteur &bnd=*bnds[it-vars.begin()]._VECTptr;
vmin=bnd.front()=minus_inf;
vmax=bnd.back()=plus_inf;
vecteur as;
vector<int> ind;
for (const_iterateur jt=ineq.begin();jt!=ineq.end();++jt) {
if ((is_linear_wrt(*jt,*it,a,b,contextptr) || (is_quadratic_wrt(*jt,*it,a,b,c,contextptr) && is_constant_wrt_vars(c,vars,contextptr))) &&
is_constant_wrt_vars(a,vars,contextptr) && is_constant_wrt_vars(b,vars,contextptr) && _lname(*jt,contextptr)._VECTptr->size()==1) {
as.push_back(symb_inferieur_egal(*jt,0));
ind.push_back(jt-ineq.begin());
}
}
if (!as.empty()) {
as=solve_quiet(as,*it,contextptr); // solve a system of inequalities in one variable
if (as.size()==1 && as.front().type!=_IDNT) {
std::sort(ind.begin(),ind.end());
for (vector<int>::const_reverse_iterator jt=ind.rbegin();jt!=ind.rend();++jt) {
ineq.erase(ineq.begin()+*jt);
}
const gen &s=as.front();
if (s.is_symb_of_sommet(at_inferieur_egal) && s._SYMBptr->feuille._VECTptr->front()==*it)
vmax=s._SYMBptr->feuille._VECTptr->back();
else if (s.is_symb_of_sommet(at_superieur_egal) && s._SYMBptr->feuille._VECTptr->front()==*it)
vmin=s._SYMBptr->feuille._VECTptr->back();
else if (is_conjunction(s) && s._SYMBptr->feuille._VECTptr->size()==2 &&
s._SYMBptr->feuille._VECTptr->front().is_symb_of_sommet(at_superieur_egal) &&
s._SYMBptr->feuille._VECTptr->front()._SYMBptr->feuille._VECTptr->front()==*it &&
s._SYMBptr->feuille._VECTptr->back().is_symb_of_sommet(at_inferieur_egal) &&
s._SYMBptr->feuille._VECTptr->back()._SYMBptr->feuille._VECTptr->front()==*it) {
vmin=s._SYMBptr->feuille._VECTptr->front()._SYMBptr->feuille._VECTptr->back();
vmax=s._SYMBptr->feuille._VECTptr->back()._SYMBptr->feuille._VECTptr->back();
} else if (is_conjunction(s) && s._SYMBptr->feuille._VECTptr->size()==2 &&
s._SYMBptr->feuille._VECTptr->front().is_symb_of_sommet(at_inferieur_egal) &&
s._SYMBptr->feuille._VECTptr->front()._SYMBptr->feuille._VECTptr->front()==*it &&
s._SYMBptr->feuille._VECTptr->back().is_symb_of_sommet(at_superieur_egal) &&
s._SYMBptr->feuille._VECTptr->back()._SYMBptr->feuille._VECTptr->front()==*it) {
vmax=s._SYMBptr->feuille._VECTptr->front()._SYMBptr->feuille._VECTptr->back();
vmin=s._SYMBptr->feuille._VECTptr->back()._SYMBptr->feuille._VECTptr->back();
}
if (!is_inf(vmin))
ineq.push_back(vmin-*it);
if (!is_inf(vmax))
ineq.push_back(*it-vmax);
} else if (as.empty())
return false; // infeasible
} else continue;
bnd.front()=vmin;
bnd.back()=vmax;
}
return true;
}
/*
*
******* IPDIFF CLASS IMPLEMENTATION *******
*
*/
ipdiff::ipdiff(const gen &f_orig,const vecteur &g_orig,const vecteur &vars_orig,GIAC_CONTEXT) {
ctx=contextptr;
f=f_orig;
g=g_orig;
vars=vars_orig;
ord=0;
nconstr=g.size();
nvars=vars.size()-nconstr;
assert(nvars>0);
pdv[ivector(nvars,0)]=f; // make the zeroth order derivative initially available
}
void ipdiff::ipartition(int m,int n,ivectors &c,const ivector &p) {
for (int i=0;i<n;++i) {
if (!p.empty() && p[i]!=0)
continue;
ivector r;
if (p.empty())
r.resize(n,0);
else r=p;
for (int j=0;j<m;++j) {
++r[i];
int s=sum_ivector(r);
if (s==m && find(c.begin(),c.end(),r)==c.end())
c.push_back(r);
else if (s<m)
ipartition(m,n,c,r);
else break;
}
}
}
ipdiff::diffterms ipdiff::derive_diffterms(const diffterms &terms,ivector &sig) {
while (!sig.empty() && sig.back()==0) {
sig.pop_back();
}
if (sig.empty())
return terms;
int k=sig.size()-1,p;
diffterms tv;
ivector u(nvars+1,0);
for (diffterms::const_iterator it=terms.begin();it!=terms.end();++it) {
int c=it->second;
diffterm t(it->first);
const ivector_map &h_orig=it->first.second;
++t.first.at(k);
tv[t]+=c;
--t.first.at(k);
ivector_map h(h_orig);
for (ivector_map::const_iterator jt=h_orig.begin();jt!=h_orig.end();++jt) {
ivector v=jt->first;
if ((p=jt->second)==0)
continue;
if (p==1)
h.erase(h.find(v));
else
--h[v];
++v[k];
++h[v];
t.second=h;
tv[t]+=c*p;
--h[v];
--v[k];
++h[v];
}
t.second=h_orig;
for (int i=0;i<nconstr;++i) {
++t.first.at(nvars+i);
u[k]=1;
u.back()=i;
++t.second[u];
tv[t]+=c;
--t.first.at(nvars+i);
--t.second[u];
u[k]=0;
}
}
--sig.back();
return derive_diffterms(tv,sig);
}
const gen &ipdiff::get_pd(const pd_map &pds,const ivector &sig) const {
try {
return pds.at(sig);
}