GiNaC/ginac/numeric.h

/** @file numeric.h
 *
 *  Makes the interface to the underlying bignum package available. */

/*
 *  GiNaC Copyright (C) 1999 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 */

#ifndef __GINAC_NUMERIC_H__
#define __GINAC_NUMERIC_H__

#include <strstream>
#include <ginac/basic.h>
#include <ginac/ex.h>

class cl_N;     // We want to include cln.h only in numeric.cpp in order to 
                // avoid namespace pollution and keep compile-time low.

#ifndef NO_GINAC_NAMESPACE
namespace GiNaC {
#endif // ndef NO_GINAC_NAMESPACE

#define HASHVALUE_NUMERIC 0x80000001U

/** This class is used to instantiate a global object Digits which
 *  behaves just like Maple's Digits.  We need an object rather than a
 *  dumber basic type since as a side-effect we let it change
 *  cl_default_float_format when it gets changed.  The only other
 *  meaningful thing to do with it is converting it to an unsigned,
 *  for temprary storing its value e.g.  The user must not create an
 *  own working object of this class!  Since C++ forces us to make the
 *  class definition visible in order to use an object we put in a
 *  flag which prevents other objects of that class to be created. */
class _numeric_digits
{
// member functions
public:
    _numeric_digits();
    _numeric_digits& operator=(long prec);
    operator long();
    void print(ostream & os) const;
// member variables
private:
    long digits;
    static bool too_late;
};

/** This class is a wrapper around CLN-numbers within the GiNaC class
 *  hierarchy. Objects of this type may directly be created by the user.*/
class numeric : public basic
{
// friends
    friend numeric exp(numeric const & x);
    friend numeric log(numeric const & x);
    friend numeric sin(numeric const & x);
    friend numeric cos(numeric const & x);
    friend numeric tan(numeric const & x);
    friend numeric asin(numeric const & x);
    friend numeric acos(numeric const & x);
    friend numeric atan(numeric const & x);
    friend numeric atan(numeric const & y, numeric const & x);
    friend numeric sinh(numeric const & x);
    friend numeric cosh(numeric const & x);
    friend numeric tanh(numeric const & x);
    friend numeric asinh(numeric const & x);
    friend numeric acosh(numeric const & x);
    friend numeric atanh(numeric const & x);
    friend numeric zeta(numeric const & x);
    friend numeric bernoulli(numeric const & n);
    friend numeric abs(numeric const & x);
    friend numeric mod(numeric const & a, numeric const & b);
    friend numeric smod(numeric const & a, numeric const & b);
    friend numeric irem(numeric const & a, numeric const & b);
    friend numeric irem(numeric const & a, numeric const & b, numeric & q);
    friend numeric iquo(numeric const & a, numeric const & b);
    friend numeric iquo(numeric const & a, numeric const & b, numeric & r);
    friend numeric sqrt(numeric const & x);
    friend numeric isqrt(numeric const & x);
    friend numeric gcd(numeric const & a, numeric const & b);
    friend numeric lcm(numeric const & a, numeric const & b);
    friend numeric const & numZERO(void);
    friend numeric const & numONE(void);
    friend numeric const & numTWO(void);
    friend numeric const & numTHREE(void);
    friend numeric const & numMINUSONE(void);
    friend numeric const & numHALF(void);

// member functions

    // default constructor, destructor, copy constructor assignment
    // operator and helpers
public:
    numeric();
    ~numeric();
    numeric(numeric const & other);
    numeric const & operator=(numeric const & other);
protected:
    void copy(numeric const & other);
    void destroy(bool call_parent);

    // other constructors
public:
    explicit numeric(int i);
    explicit numeric(unsigned int i);
    explicit numeric(long i);
    explicit numeric(unsigned long i);
    explicit numeric(long numer, long denom);
    explicit numeric(double d);
    explicit numeric(char const *);
    numeric(cl_N const & z);

    // functions overriding virtual functions from bases classes
public:
    basic * duplicate() const;
    void print(ostream & os, unsigned precedence=0) const;
    void printraw(ostream & os) const;
    void printtree(ostream & os, unsigned indent) const;
    void printcsrc(ostream & os, unsigned type, unsigned precedence=0) const;
    bool info(unsigned inf) const;
    ex evalf(int level=0) const;
    ex diff(symbol const & s) const;
    ex normal(lst &sym_lst, lst &repl_lst, int level=0) const;
    numeric integer_content(void) const;
    ex smod(numeric const &xi) const;
    numeric max_coefficient(void) const;
protected:
    int compare_same_type(basic const & other) const;
    bool is_equal_same_type(basic const & other) const;
    unsigned calchash(void) const {
        hashvalue=HASHVALUE_NUMERIC;
        return HASHVALUE_NUMERIC;
    }

    // new virtual functions which can be overridden by derived classes
    // (none)

    // non-virtual functions in this class
public:
    numeric add(numeric const & other) const;
    numeric sub(numeric const & other) const;
    numeric mul(numeric const & other) const;
    numeric div(numeric const & other) const;
    numeric power(numeric const & other) const;
    numeric const & add_dyn(numeric const & other) const;
    numeric const & sub_dyn(numeric const & other) const;
    numeric const & mul_dyn(numeric const & other) const;
    numeric const & div_dyn(numeric const & other) const;
    numeric const & power_dyn(numeric const & other) const;
    numeric const & operator=(int i);
    numeric const & operator=(unsigned int i);
    numeric const & operator=(long i);
    numeric const & operator=(unsigned long i);
    numeric const & operator=(double d);
    numeric const & operator=(char const * s);
    /*
    numeric add_dyn(numeric const & other) const   { return add(other);   }
    numeric sub_dyn(numeric const & other) const   { return sub(other);   }
    numeric mul_dyn(numeric const & other) const   { return mul(other);   }
    numeric div_dyn(numeric const & other) const   { return div(other);   }
    numeric power_dyn(numeric const & other) const { return power(other); }
    */
    numeric inverse(void) const;
    int csgn(void) const;
    int compare(numeric const & other) const;
    bool is_equal(numeric const & other) const;
    bool is_zero(void) const;
    bool is_positive(void) const;
    bool is_negative(void) const;
    bool is_integer(void) const;
    bool is_pos_integer(void) const;
    bool is_nonneg_integer(void) const;
    bool is_even(void) const;
    bool is_odd(void) const;
    bool is_prime(void) const;
    bool is_rational(void) const;
    bool is_real(void) const;
    bool is_cinteger(void) const;
    bool is_crational(void) const;
    bool operator==(numeric const & other) const;
    bool operator!=(numeric const & other) const;
    bool operator<(numeric const & other) const;
    bool operator<=(numeric const & other) const;
    bool operator>(numeric const & other) const;
    bool operator>=(numeric const & other) const;
    int to_int(void) const;
    double to_double(void) const;
    numeric real(void) const;
    numeric imag(void) const;
    numeric numer(void) const;
    numeric denom(void) const;
    int int_length(void) const;

// member variables

protected:
    static unsigned precedence;
    cl_N *value;
};

// global constants

extern const numeric some_numeric;
extern const numeric I;
extern type_info const & typeid_numeric;
extern _numeric_digits Digits;

#define is_a_numeric_hash(x) ((x)==HASHVALUE_NUMERIC)
// may have to be changed to ((x)>=0x80000000U)

// global functions

numeric const & numZERO(void);
numeric const & numONE(void);
numeric const & numTWO(void);
numeric const & numMINUSONE(void);
numeric const & numHALF(void);

numeric exp(numeric const & x);
numeric log(numeric const & x);
numeric sin(numeric const & x);
numeric cos(numeric const & x);
numeric tan(numeric const & x);
numeric asin(numeric const & x);
numeric acos(numeric const & x);
numeric atan(numeric const & x);
numeric atan(numeric const & y, numeric const & x);
numeric sinh(numeric const & x);
numeric cosh(numeric const & x);
numeric tanh(numeric const & x);
numeric asinh(numeric const & x);
numeric acosh(numeric const & x);
numeric atanh(numeric const & x);
numeric zeta(numeric const & x);
numeric gamma(numeric const & x);
numeric psi(numeric const & x);
numeric psi(numeric const & n, numeric const & x);
numeric factorial(numeric const & n);
numeric doublefactorial(numeric const & n);
numeric binomial(numeric const & n, numeric const & k);
numeric bernoulli(numeric const & n);

numeric abs(numeric const & x);
numeric mod(numeric const & a, numeric const & b);
numeric smod(numeric const & a, numeric const & b);
numeric irem(numeric const & a, numeric const & b);
numeric irem(numeric const & a, numeric const & b, numeric & q);
numeric iquo(numeric const & a, numeric const & b);
numeric iquo(numeric const & a, numeric const & b, numeric & r);
numeric sqrt(numeric const & x);
numeric isqrt(numeric const & x);

numeric gcd(numeric const & a, numeric const & b);
numeric lcm(numeric const & a, numeric const & b);

/** Exception thrown by numeric members to signal failure */
struct numeric_fail
{
    int failval;
    numeric_fail(int n) { failval = n; }
};

// wrapper functions around member functions
inline numeric inverse(numeric const & x)
{ return x.inverse(); }

inline bool csgn(numeric const & x)
{ return x.csgn(); }

inline bool is_zero(numeric const & x)
{ return x.is_zero(); }

inline bool is_positive(numeric const & x)
{ return x.is_positive(); }

inline bool is_integer(numeric const & x)
{ return x.is_integer(); }

inline bool is_pos_integer(numeric const & x)
{ return x.is_pos_integer(); }

inline bool is_nonneg_integer(numeric const & x)
{ return x.is_nonneg_integer(); }

inline bool is_even(numeric const & x)
{ return x.is_even(); }

inline bool is_odd(numeric const & x)
{ return x.is_odd(); }

inline bool is_prime(numeric const & x)
{ return x.is_prime(); }

inline bool is_rational(numeric const & x)
{ return x.is_rational(); }

inline bool is_real(numeric const & x)
{ return x.is_real(); }

inline bool is_cinteger(numeric const & x)
{ return x.is_cinteger(); }

inline bool is_crational(numeric const & x)
{ return x.is_crational(); }

inline numeric real(numeric const & x)
{ return x.real(); }

inline numeric imag(numeric const & x)
{ return x.imag(); }

inline numeric numer(numeric const & x)
{ return x.numer(); }

inline numeric denom(numeric const & x)
{ return x.denom(); }

// numeric evaluation functions for class constant objects:

ex PiEvalf(void);
ex EulerGammaEvalf(void);
ex CatalanEvalf(void);

// utility functions
inline const numeric &ex_to_numeric(const ex &e)
{
    return static_cast<const numeric &>(*e.bp);
}

#ifndef NO_GINAC_NAMESPACE
} // namespace GiNaC
#endif // ndef NO_GINAC_NAMESPACE

#endif // ndef __GINAC_NUMERIC_H__
1	cbauer	1.1	/** @file numeric.h
2			*
3	cbauer	1.4	* Makes the interface to the underlying bignum package available. */
4
5			/*
6	cbauer	1.2	* GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
7			*
8			* This program is free software; you can redistribute it and/or modify
9			* it under the terms of the GNU General Public License as published by
10			* the Free Software Foundation; either version 2 of the License, or
11			* (at your option) any later version.
12			*
13			* This program is distributed in the hope that it will be useful,
14			* but WITHOUT ANY WARRANTY; without even the implied warranty of
15			* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16			* GNU General Public License for more details.
17			*
18			* You should have received a copy of the GNU General Public License
19			* along with this program; if not, write to the Free Software
20			* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
21			*/
22	cbauer	1.1
23	cbauer	1.2	#ifndef __GINAC_NUMERIC_H__
24			#define __GINAC_NUMERIC_H__
25	cbauer	1.1
26			#include <strstream>
27	cbauer	1.3	#include <ginac/basic.h>
28	cbauer	1.5	#include <ginac/ex.h>
29	cbauer	1.1
30			class cl_N; // We want to include cln.h only in numeric.cpp in order to
31			// avoid namespace pollution and keep compile-time low.
32
33	frink	1.12	#ifndef NO_GINAC_NAMESPACE
34	cbauer	1.5	namespace GiNaC {
35	frink	1.12	#endif // ndef NO_GINAC_NAMESPACE
36	cbauer	1.5
37			#define HASHVALUE_NUMERIC 0x80000001U
38
39	cbauer	1.1	/** This class is used to instantiate a global object Digits which
40			* behaves just like Maple's Digits. We need an object rather than a
41			* dumber basic type since as a side-effect we let it change
42			* cl_default_float_format when it gets changed. The only other
43			* meaningful thing to do with it is converting it to an unsigned,
44			* for temprary storing its value e.g. The user must not create an
45			* own working object of this class! Since C++ forces us to make the
46			* class definition visible in order to use an object we put in a
47			* flag which prevents other objects of that class to be created. */
48			class _numeric_digits
49			{
50			// member functions
51			public:
52			_numeric_digits();
53			_numeric_digits& operator=(long prec);
54			operator long();
55			void print(ostream & os) const;
56			// member variables
57			private:
58			long digits;
59			static bool too_late;
60			};
61
62			/** This class is a wrapper around CLN-numbers within the GiNaC class
63			* hierarchy. Objects of this type may directly be created by the user.*/
64			class numeric : public basic
65			{
66			// friends
67			friend numeric exp(numeric const & x);
68			friend numeric log(numeric const & x);
69			friend numeric sin(numeric const & x);
70			friend numeric cos(numeric const & x);
71			friend numeric tan(numeric const & x);
72			friend numeric asin(numeric const & x);
73			friend numeric acos(numeric const & x);
74			friend numeric atan(numeric const & x);
75			friend numeric atan(numeric const & y, numeric const & x);
76			friend numeric sinh(numeric const & x);
77			friend numeric cosh(numeric const & x);
78			friend numeric tanh(numeric const & x);
79			friend numeric asinh(numeric const & x);
80			friend numeric acosh(numeric const & x);
81			friend numeric atanh(numeric const & x);
82	kreckel	1.14	friend numeric zeta(numeric const & x);
83	kreckel	1.8	friend numeric bernoulli(numeric const & n);
84	cbauer	1.1	friend numeric abs(numeric const & x);
85			friend numeric mod(numeric const & a, numeric const & b);
86			friend numeric smod(numeric const & a, numeric const & b);
87			friend numeric irem(numeric const & a, numeric const & b);
88			friend numeric irem(numeric const & a, numeric const & b, numeric & q);
89			friend numeric iquo(numeric const & a, numeric const & b);
90			friend numeric iquo(numeric const & a, numeric const & b, numeric & r);
91			friend numeric sqrt(numeric const & x);
92			friend numeric isqrt(numeric const & x);
93			friend numeric gcd(numeric const & a, numeric const & b);
94			friend numeric lcm(numeric const & a, numeric const & b);
95			friend numeric const & numZERO(void);
96			friend numeric const & numONE(void);
97			friend numeric const & numTWO(void);
98			friend numeric const & numTHREE(void);
99			friend numeric const & numMINUSONE(void);
100			friend numeric const & numHALF(void);
101
102			// member functions
103
104			// default constructor, destructor, copy constructor assignment
105			// operator and helpers
106			public:
107			numeric();
108			~numeric();
109			numeric(numeric const & other);
110			numeric const & operator=(numeric const & other);
111			protected:
112			void copy(numeric const & other);
113			void destroy(bool call_parent);
114
115			// other constructors
116			public:
117			explicit numeric(int i);
118			explicit numeric(unsigned int i);
119			explicit numeric(long i);
120			explicit numeric(unsigned long i);
121			explicit numeric(long numer, long denom);
122			explicit numeric(double d);
123			explicit numeric(char const *);
124			numeric(cl_N const & z);
125
126			// functions overriding virtual functions from bases classes
127			public:
128			basic * duplicate() const;
129	kreckel	1.14	void print(ostream & os, unsigned precedence=0) const;
130	cbauer	1.1	void printraw(ostream & os) const;
131			void printtree(ostream & os, unsigned indent) const;
132			void printcsrc(ostream & os, unsigned type, unsigned precedence=0) const;
133			bool info(unsigned inf) const;
134			ex evalf(int level=0) const;
135			ex diff(symbol const & s) const;
136			ex normal(lst &sym_lst, lst &repl_lst, int level=0) const;
137			numeric integer_content(void) const;
138			ex smod(numeric const &xi) const;
139			numeric max_coefficient(void) const;
140			protected:
141			int compare_same_type(basic const & other) const;
142			bool is_equal_same_type(basic const & other) const;
143			unsigned calchash(void) const {
144			hashvalue=HASHVALUE_NUMERIC;
145			return HASHVALUE_NUMERIC;
146			}
147
148			// new virtual functions which can be overridden by derived classes
149			// (none)
150
151			// non-virtual functions in this class
152			public:
153			numeric add(numeric const & other) const;
154			numeric sub(numeric const & other) const;
155			numeric mul(numeric const & other) const;
156			numeric div(numeric const & other) const;
157			numeric power(numeric const & other) const;
158			numeric const & add_dyn(numeric const & other) const;
159			numeric const & sub_dyn(numeric const & other) const;
160			numeric const & mul_dyn(numeric const & other) const;
161			numeric const & div_dyn(numeric const & other) const;
162			numeric const & power_dyn(numeric const & other) const;
163			numeric const & operator=(int i);
164			numeric const & operator=(unsigned int i);
165			numeric const & operator=(long i);
166			numeric const & operator=(unsigned long i);
167			numeric const & operator=(double d);
168			numeric const & operator=(char const * s);
169			/*
170			numeric add_dyn(numeric const & other) const { return add(other); }
171			numeric sub_dyn(numeric const & other) const { return sub(other); }
172			numeric mul_dyn(numeric const & other) const { return mul(other); }
173			numeric div_dyn(numeric const & other) const { return div(other); }
174			numeric power_dyn(numeric const & other) const { return power(other); }
175			*/
176			numeric inverse(void) const;
177	kreckel	1.6	int csgn(void) const;
178	cbauer	1.1	int compare(numeric const & other) const;
179			bool is_equal(numeric const & other) const;
180			bool is_zero(void) const;
181			bool is_positive(void) const;
182			bool is_negative(void) const;
183			bool is_integer(void) const;
184			bool is_pos_integer(void) const;
185			bool is_nonneg_integer(void) const;
186			bool is_even(void) const;
187			bool is_odd(void) const;
188			bool is_prime(void) const;
189			bool is_rational(void) const;
190			bool is_real(void) const;
191	kreckel	1.13	bool is_cinteger(void) const;
192			bool is_crational(void) const;
193	cbauer	1.1	bool operator==(numeric const & other) const;
194			bool operator!=(numeric const & other) const;
195			bool operator<(numeric const & other) const;
196			bool operator<=(numeric const & other) const;
197			bool operator>(numeric const & other) const;
198			bool operator>=(numeric const & other) const;
199			int to_int(void) const;
200			double to_double(void) const;
201			numeric real(void) const;
202			numeric imag(void) const;
203			numeric numer(void) const;
204			numeric denom(void) const;
205			int int_length(void) const;
206
207			// member variables
208
209			protected:
210			static unsigned precedence;
211			cl_N *value;
212			};
213
214			// global constants
215
216			extern const numeric some_numeric;
217			extern const numeric I;
218			extern type_info const & typeid_numeric;
219			extern _numeric_digits Digits;
220
221			#define is_a_numeric_hash(x) ((x)==HASHVALUE_NUMERIC)
222			// may have to be changed to ((x)>=0x80000000U)
223
224			// global functions
225
226			numeric const & numZERO(void);
227			numeric const & numONE(void);
228			numeric const & numTWO(void);
229			numeric const & numMINUSONE(void);
230			numeric const & numHALF(void);
231
232			numeric exp(numeric const & x);
233			numeric log(numeric const & x);
234			numeric sin(numeric const & x);
235			numeric cos(numeric const & x);
236			numeric tan(numeric const & x);
237			numeric asin(numeric const & x);
238			numeric acos(numeric const & x);
239			numeric atan(numeric const & x);
240			numeric atan(numeric const & y, numeric const & x);
241			numeric sinh(numeric const & x);
242			numeric cosh(numeric const & x);
243			numeric tanh(numeric const & x);
244			numeric asinh(numeric const & x);
245			numeric acosh(numeric const & x);
246			numeric atanh(numeric const & x);
247	kreckel	1.7	numeric zeta(numeric const & x);
248	cbauer	1.1	numeric gamma(numeric const & x);
249	kreckel	1.10	numeric psi(numeric const & x);
250	kreckel	1.7	numeric psi(numeric const & n, numeric const & x);
251	cbauer	1.1	numeric factorial(numeric const & n);
252			numeric doublefactorial(numeric const & n);
253			numeric binomial(numeric const & n, numeric const & k);
254	kreckel	1.8	numeric bernoulli(numeric const & n);
255	cbauer	1.1
256			numeric abs(numeric const & x);
257			numeric mod(numeric const & a, numeric const & b);
258			numeric smod(numeric const & a, numeric const & b);
259			numeric irem(numeric const & a, numeric const & b);
260			numeric irem(numeric const & a, numeric const & b, numeric & q);
261			numeric iquo(numeric const & a, numeric const & b);
262			numeric iquo(numeric const & a, numeric const & b, numeric & r);
263			numeric sqrt(numeric const & x);
264			numeric isqrt(numeric const & x);
265
266			numeric gcd(numeric const & a, numeric const & b);
267			numeric lcm(numeric const & a, numeric const & b);
268
269			/** Exception thrown by numeric members to signal failure */
270			struct numeric_fail
271			{
272			int failval;
273			numeric_fail(int n) { failval = n; }
274			};
275
276			// wrapper functions around member functions
277			inline numeric inverse(numeric const & x)
278			{ return x.inverse(); }
279	kreckel	1.6
280			inline bool csgn(numeric const & x)
281			{ return x.csgn(); }
282	cbauer	1.1
283			inline bool is_zero(numeric const & x)
284			{ return x.is_zero(); }
285
286			inline bool is_positive(numeric const & x)
287			{ return x.is_positive(); }
288
289			inline bool is_integer(numeric const & x)
290			{ return x.is_integer(); }
291
292			inline bool is_pos_integer(numeric const & x)
293			{ return x.is_pos_integer(); }
294
295			inline bool is_nonneg_integer(numeric const & x)
296			{ return x.is_nonneg_integer(); }
297
298			inline bool is_even(numeric const & x)
299			{ return x.is_even(); }
300
301			inline bool is_odd(numeric const & x)
302			{ return x.is_odd(); }
303
304			inline bool is_prime(numeric const & x)
305			{ return x.is_prime(); }
306
307			inline bool is_rational(numeric const & x)
308			{ return x.is_rational(); }
309
310			inline bool is_real(numeric const & x)
311			{ return x.is_real(); }
312
313	kreckel	1.14	inline bool is_cinteger(numeric const & x)
314			{ return x.is_cinteger(); }
315
316			inline bool is_crational(numeric const & x)
317			{ return x.is_crational(); }
318
319	cbauer	1.1	inline numeric real(numeric const & x)
320			{ return x.real(); }
321
322			inline numeric imag(numeric const & x)
323			{ return x.imag(); }
324
325			inline numeric numer(numeric const & x)
326			{ return x.numer(); }
327
328			inline numeric denom(numeric const & x)
329			{ return x.denom(); }
330
331	kreckel	1.11	// numeric evaluation functions for class constant objects:
332
333	cbauer	1.1	ex PiEvalf(void);
334			ex EulerGammaEvalf(void);
335			ex CatalanEvalf(void);
336
337	cbauer	1.5	// utility functions
338			inline const numeric &ex_to_numeric(const ex &e)
339			{
340	kreckel	1.14	return static_cast<const numeric &>(*e.bp);
341	cbauer	1.5	}
342	cbauer	1.1
343	frink	1.12	#ifndef NO_GINAC_NAMESPACE
344	cbauer	1.5	} // namespace GiNaC
345	frink	1.12	#endif // ndef NO_GINAC_NAMESPACE
346	cbauer	1.1
347	cbauer	1.2	#endif // ndef __GINAC_NUMERIC_H__
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