/** @file inifcns.cpp
*
* Implementation of GiNaC's initially known functions. */
/*
* GiNaC Copyright (C) 1999-2007 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 <vector>
#include <stdexcept>
#include "inifcns.h"
#include "ex.h"
#include "constant.h"
#include "lst.h"
#include "matrix.h"
#include "mul.h"
#include "power.h"
#include "operators.h"
#include "relational.h"
#include "pseries.h"
#include "symbol.h"
#include "symmetry.h"
#include "utils.h"
namespace GiNaC {
//////////
// complex conjugate
//////////
static ex conjugate_evalf(const ex & arg)
{
if (is_exactly_a<numeric>(arg)) {
return ex_to<numeric>(arg).conjugate();
}
return conjugate_function(arg).hold();
}
static ex conjugate_eval(const ex & arg)
{
return arg.conjugate();
}
static void conjugate_print_latex(const ex & arg, const print_context & c)
{
c.s << "\\bar{"; arg.print(c); c.s << "}";
}
static ex conjugate_conjugate(const ex & arg)
{
return arg;
}
REGISTER_FUNCTION(conjugate_function, eval_func(conjugate_eval).
evalf_func(conjugate_evalf).
print_func<print_latex>(conjugate_print_latex).
conjugate_func(conjugate_conjugate).
set_name("conjugate","conjugate"));
//////////
// absolute value
//////////
static ex abs_evalf(const ex & arg)
{
if (is_exactly_a<numeric>(arg))
return abs(ex_to<numeric>(arg));
return abs(arg).hold();
}
static ex abs_eval(const ex & arg)
{
if (is_exactly_a<numeric>(arg))
return abs(ex_to<numeric>(arg));
else
return abs(arg).hold();
}
static void abs_print_latex(const ex & arg, const print_context & c)
{
c.s << "{|"; arg.print(c); c.s << "|}";
}
static void abs_print_csrc_float(const ex & arg, const print_context & c)
{
c.s << "fabs("; arg.print(c); c.s << ")";
}
static ex abs_conjugate(const ex & arg)
{
return abs(arg);
}
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));
//////////
// Complex sign
//////////
static ex csgn_evalf(const ex & arg)
{
if (is_exactly_a<numeric>(arg))
return csgn(ex_to<numeric>(arg));
return csgn(arg).hold();
}
static ex csgn_eval(const ex & arg)
{
if (is_exactly_a<numeric>(arg))
return csgn(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)
// csgn(42*x) -> csgn(x)
return csgn(arg/oc).hold();
else
// csgn(-42*x) -> -csgn(x)
return -csgn(arg/oc).hold();
}
if (oc.real().is_zero()) {
if (oc.imag() > 0)
// csgn(42*I*x) -> csgn(I*x)
return csgn(I*arg/oc).hold();
else
// csgn(-42*I*x) -> -csgn(I*x)
return -csgn(I*arg/oc).hold();
}
}
return csgn(arg).hold();
}
static ex csgn_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("csgn_series(): on imaginary axis"));
epvector seq;
seq.push_back(expair(csgn(arg_pt), _ex0));
return pseries(rel,seq);
}
static ex csgn_conjugate(const ex& arg)
{
return csgn(arg);
}
REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
evalf_func(csgn_evalf).
series_func(csgn_series).
conjugate_func(csgn_conjugate));
//////////
// Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
// This function is closely related to the unwinding number K, sometimes found
// in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
//////////
static ex eta_evalf(const ex &x, const ex &y)
{
// It seems like we basically have to replicate the eval function here,
// since the expression might not be fully evaluated yet.
if (x.info(info_flags::positive) || y.info(info_flags::positive))
return _ex0;
if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
const numeric nx = ex_to<numeric>(x);
const numeric ny = ex_to<numeric>(y);
const numeric nxy = ex_to<numeric>(x*y);
int cut = 0;
if (nx.is_real() && nx.is_negative())
cut -= 4;
if (ny.is_real() && ny.is_negative())
cut -= 4;
if (nxy.is_real() && nxy.is_negative())
cut += 4;
return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
(csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
}
return eta(x,y).hold();
}
static ex eta_eval(const ex &x, const ex &y)
{
// trivial: eta(x,c) -> 0 if c is real and positive
if (x.info(info_flags::positive) || y.info(info_flags::positive))
return _ex0;
if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
// don't call eta_evalf here because it would call Pi.evalf()!
const numeric nx = ex_to<numeric>(x);
const numeric ny = ex_to<numeric>(y);
const numeric nxy = ex_to<numeric>(x*y);
int cut = 0;
if (nx.is_real() && nx.is_negative())
cut -= 4;
if (ny.is_real() && ny.is_negative())
cut -= 4;
if (nxy.is_real() && nxy.is_negative())
cut += 4;
return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
(csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
}
return eta(x,y).hold();
}
static ex eta_series(const ex & x, const ex & y,
const relational & rel,
int order,
unsigned options)
{
const ex x_pt = x.subs(rel, subs_options::no_pattern);
const ex y_pt = y.subs(rel, subs_options::no_pattern);
if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
(y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
throw (std::domain_error("eta_series(): on discontinuity"));
epvector seq;
seq.push_back(expair(eta(x_pt,y_pt), _ex0));
return pseries(rel,seq);
}
static ex eta_conjugate(const ex & x, const ex & y)
{
return -eta(x,y);
}
REGISTER_FUNCTION(eta, eval_func(eta_eval).
evalf_func(eta_evalf).
series_func(eta_series).
latex_name("\\eta").
set_symmetry(sy_symm(0, 1)).
conjugate_func(eta_conjugate));
//////////
// dilogarithm
//////////
static ex Li2_evalf(const ex & x)
{
if (is_exactly_a<numeric>(x))
return Li2(ex_to<numeric>(x));
return Li2(x).hold();
}
static ex Li2_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// Li2(0) -> 0
if (x.is_zero())
return _ex0;
// Li2(1) -> Pi^2/6
if (x.is_equal(_ex1))
return power(Pi,_ex2)/_ex6;
// Li2(1/2) -> Pi^2/12 - log(2)^2/2
if (x.is_equal(_ex1_2))
return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
// Li2(-1) -> -Pi^2/12
if (x.is_equal(_ex_1))
return -power(Pi,_ex2)/_ex12;
// Li2(I) -> -Pi^2/48+Catalan*I
if (x.is_equal(I))
return power(Pi,_ex2)/_ex_48 + Catalan*I;
// Li2(-I) -> -Pi^2/48-Catalan*I
if (x.is_equal(-I))
return power(Pi,_ex2)/_ex_48 - Catalan*I;
// Li2(float)
if (!x.info(info_flags::crational))
return Li2(ex_to<numeric>(x));
}
return Li2(x).hold();
}
static ex Li2_deriv(const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param==0);
// d/dx Li2(x) -> -log(1-x)/x
return -log(_ex1-x)/x;
}
static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
{
const ex x_pt = x.subs(rel, subs_options::no_pattern);
if (x_pt.info(info_flags::numeric)) {
// First special case: x==0 (derivatives have poles)
if (x_pt.is_zero()) {
// method:
// The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
// simply substitute x==0. The limit, however, exists: it is 1.
// We also know all higher derivatives' limits:
// (d/dx)^n Li2(x) == n!/n^2.
// So the primitive series expansion is
// Li2(x==0) == x + x^2/4 + x^3/9 + ...
// and so on.
// We first construct such a primitive series expansion manually in
// a dummy symbol s and then insert the argument's series expansion
// for s. Reexpanding the resulting series returns the desired
// result.
const symbol s;
ex ser;
// manually construct the primitive expansion
for (int i=1; i<order; ++i)
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
epvector nseq;
nseq.push_back(expair(Order(_ex1), order));
ser += pseries(rel, nseq);
// reexpanding it will collapse the series again
return ser.series(rel, order);
// NB: Of course, this still does not allow us to compute anything
// like sin(Li2(x)).series(x==0,2), since then this code here is
// not reached and the derivative of sin(Li2(x)) doesn't allow the
// substitution x==0. Probably limits *are* needed for the general
// cases. In case L'Hospital's rule is implemented for limits and
// basic::series() takes care of this, this whole block is probably
// obsolete!
}
// second special case: x==1 (branch point)
if (x_pt.is_equal(_ex1)) {
// method:
// construct series manually in a dummy symbol s
const symbol s;
ex ser = zeta(_ex2);
// manually construct the primitive expansion
for (int i=1; i<order; ++i)
ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
// 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
epvector nseq;
nseq.push_back(expair(Order(_ex1), order));
ser += pseries(rel, nseq);
// reexpanding it will collapse the series again
return ser.series(rel, order);
}
// third special case: x real, >=1 (branch cut)
if (!(options & series_options::suppress_branchcut) &&
ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
// method:
// This is the branch cut: assemble the primitive series manually
// and then add the corresponding complex step function.
const symbol &s = ex_to<symbol>(rel.lhs());
const ex point = rel.rhs();
const symbol foo;
epvector seq;
// zeroth order term:
seq.push_back(expair(Li2(x_pt), _ex0));
// compute the intermediate terms:
ex replarg = series(Li2(x), s==foo, order);
for (size_t i=1; i<replarg.nops()-1; ++i)
seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
// append an order term:
seq.push_back(expair(Order(_ex1), replarg.nops()-1));
return pseries(rel, seq);
}
}
// all other cases should be safe, by now:
throw do_taylor(); // caught by function::series()
}
REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
evalf_func(Li2_evalf).
derivative_func(Li2_deriv).
series_func(Li2_series).
latex_name("\\mbox{Li}_2"));
//////////
// trilogarithm
//////////
static ex Li3_eval(const ex & x)
{
if (x.is_zero())
return x;
return Li3(x).hold();
}
REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
latex_name("\\mbox{Li}_3"));
//////////
// Derivatives of Riemann's Zeta-function zetaderiv(0,x)==zeta(x)
//////////
static ex zetaderiv_eval(const ex & n, const ex & x)
{
if (n.info(info_flags::numeric)) {
// zetaderiv(0,x) -> zeta(x)
if (n.is_zero())
return zeta(x);
}
return zetaderiv(n, x).hold();
}
static ex zetaderiv_deriv(const ex & n, const ex & x, unsigned deriv_param)
{
GINAC_ASSERT(deriv_param<2);
if (deriv_param==0) {
// d/dn zeta(n,x)
throw(std::logic_error("cannot diff zetaderiv(n,x) with respect to n"));
}
// d/dx psi(n,x)
return zetaderiv(n+1,x);
}
REGISTER_FUNCTION(zetaderiv, eval_func(zetaderiv_eval).
derivative_func(zetaderiv_deriv).
latex_name("\\zeta^\\prime"));
//////////
// factorial
//////////
static ex factorial_evalf(const ex & x)
{
return factorial(x).hold();
}
static ex factorial_eval(const ex & x)
{
if (is_exactly_a<numeric>(x))
return factorial(ex_to<numeric>(x));
else
return factorial(x).hold();
}
static ex factorial_conjugate(const ex & x)
{
return factorial(x);
}
REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
evalf_func(factorial_evalf).
conjugate_func(factorial_conjugate));
//////////
// binomial
//////////
static ex binomial_evalf(const ex & x, const ex & y)
{
return binomial(x, y).hold();
}
static ex binomial_sym(const ex & x, const numeric & y)
{
if (y.is_integer()) {
if (y.is_nonneg_integer()) {
const unsigned N = y.to_int();
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 _ex0;
}
return binomial(x, y).hold();
}
static ex binomial_eval(const ex & x, const ex &y)
{
if (is_exactly_a<numeric>(y)) {
if (is_exactly_a<numeric>(x) && ex_to<numeric>(x).is_integer())
return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
else
return binomial_sym(x, ex_to<numeric>(y));
} else
return binomial(x, y).hold();
}
// At the moment the numeric evaluation of a binomail function always
// gives a real number, but if this would be implemented using the gamma
// function, also complex conjugation should be changed (or rather, deleted).
static ex binomial_conjugate(const ex & x, const ex & y)
{
return binomial(x,y);
}
REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
evalf_func(binomial_evalf).
conjugate_func(binomial_conjugate));
//////////
// Order term function (for truncated power series)
//////////
static ex Order_eval(const ex & x)
{
if (is_exactly_a<numeric>(x)) {
// O(c) -> O(1) or 0
if (!x.is_zero())
return Order(_ex1).hold();
else
return _ex0;
} else if (is_exactly_a<mul>(x)) {
const mul &m = ex_to<mul>(x);
// O(c*expr) -> O(expr)
if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
return Order(x / m.op(m.nops() - 1)).hold();
}
return Order(x).hold();
}
static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
{
// Just wrap the function into a pseries object
epvector new_seq;
GINAC_ASSERT(is_a<symbol>(r.lhs()));
const symbol &s = ex_to<symbol>(r.lhs());
new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
return pseries(r, new_seq);
}
static ex Order_conjugate(const ex & x)
{
return Order(x);
}
// Differentiation is handled in function::derivative because of its special requirements
REGISTER_FUNCTION(Order, eval_func(Order_eval).
series_func(Order_series).
latex_name("\\mathcal{O}").
conjugate_func(Order_conjugate));
//////////
// Solve linear system
//////////
ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
{
// solve a system of linear equations
if (eqns.info(info_flags::relation_equal)) {
if (!symbols.info(info_flags::symbol))
throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
const ex sol = lsolve(lst(eqns),lst(symbols));
GINAC_ASSERT(sol.nops()==1);
GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
return sol.op(0).op(1); // return rhs of first solution
}
// syntax checks
if (!eqns.info(info_flags::list)) {
throw(std::invalid_argument("lsolve(): 1st argument must be a list or an equation"));
}
for (size_t i=0; i<eqns.nops(); i++) {
if (!eqns.op(i).info(info_flags::relation_equal)) {
throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
}
}
if (!symbols.info(info_flags::list)) {
throw(std::invalid_argument("lsolve(): 2nd argument must be a list or a symbol"));
}
for (size_t i=0; i<symbols.nops(); i++) {
if (!symbols.op(i).info(info_flags::symbol)) {
throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
}
}
// build matrix from equation system
matrix sys(eqns.nops(),symbols.nops());
matrix rhs(eqns.nops(),1);
matrix vars(symbols.nops(),1);
for (size_t r=0; r<eqns.nops(); r++) {
const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
ex linpart = eq;
for (size_t c=0; c<symbols.nops(); c++) {
const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
linpart -= co*symbols.op(c);
sys(r,c) = co;
}
linpart = linpart.expand();
rhs(r,0) = -linpart;
}
// test if system is linear and fill vars matrix
for (size_t i=0; i<symbols.nops(); i++) {
vars(i,0) = symbols.op(i);
if (sys.has(symbols.op(i)))
throw(std::logic_error("lsolve: system is not linear"));
if (rhs.has(symbols.op(i)))
throw(std::logic_error("lsolve: system is not linear"));
}
matrix solution;
try {
solution = sys.solve(vars,rhs,options);
} catch (const std::runtime_error & e) {
// Probably singular matrix or otherwise overdetermined system:
// It is consistent to return an empty list
return lst();
}
GINAC_ASSERT(solution.cols()==1);
GINAC_ASSERT(solution.rows()==symbols.nops());
// return list of equations of the form lst(var1==sol1,var2==sol2,...)
lst sollist;
for (size_t i=0; i<symbols.nops(); i++)
sollist.append(symbols.op(i)==solution(i,0));
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;
unsigned force_include_zeta1 = zeta1_SERIAL::serial;
} // namespace GiNaC
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