33 #define TRANSEXT_PRIVATES
38 #include "factory/factory.h"
62 #define ADD_COMPLEXITY 1
63 #define MULT_COMPLEXITY 2
64 #define DIFF_COMPLEXITY 2
65 #define BOUND_COMPLEXITY 10
68 #define NUMIS1(f) (p_IsOne(NUM(f), cf->extRing))
70 #define COM(f) (f)->complexity
77 #define ntTest(a) n_Test(a, cf)
81 #define ntRing cf->extRing
87 #define ntCoeffs cf->extRing->cf
95 BOOLEAN simpleTestsHaveAlreadyBeenPerformed);
145 if (IS0(a))
return TRUE;
147 const fraction t = (fraction)a;
150 const poly
num = NUM(t);
159 Print(
"ERROR in %s:%d: non-integer Q coeff in num. poly\n",
f,
l);
164 const poly
den = DEN(t);
174 Print(
"ERROR in %s:%d: non-integer Q coeff in den. poly\n",
f,
l);
183 Print(
"ERROR in %s:%d: constant den. poly / Zp\n",
f,
l);
191 Print(
"ERROR in %s:%d: non-monic den. poly / Zp\n",
f,
l);
205 Print(
"ERROR in %s:%d: 1 != GCD between num. & den. poly\n",
f,
l);
223 Print(
"negative sign of DEN. of a fraction in %s:%d\n",
f,
l);
256 Print(
"rational coeff in num: %s:%d\n",
f,
l);
267 Print(
"rational coeff in den.:%s:%d\n",
f,
l);
288 cf =
cf->extRing->cf;
306 fraction
f = (fraction)(*a);
322 if (a ==
b)
return TRUE;
323 if ((IS0(a)) && (!IS0(
b)))
return FALSE;
324 if ((IS0(
b)) && (!IS0(a)))
return FALSE;
327 fraction
fa = (fraction)a;
328 fraction
fb = (fraction)
b;
335 if (DENIS1(
fa) && DENIS1(
fb))
return TRUE;
336 if (DENIS1(
fa) && !DENIS1(
fb))
return FALSE;
337 if (!DENIS1(
fa) && DENIS1(
fb))
return FALSE;
364 if (IS0(a))
return NULL;
365 fraction
f = (fraction)a;
409 number c; number tmp;
418 lcmOfDenominators = tmp;
427 lcmOfDenominators = tmp;
448 gcdOfCoefficients = tmp;
457 gcdOfCoefficients = tmp;
462 number inverseOfGcdOfCoefficients =
n_Invers(gcdOfCoefficients,
476 if ((DEN(
f) !=
NULL) &&
498 if (IS0(a))
return NULL;
502 fraction
f = (fraction)a;
572 fraction
f = (fraction)a;
594 if( DEN (
f) !=
NULL )
662 fraction
f = (fraction)a;
671 fraction
f = (fraction)a;
685 fraction
f = (fraction)a;
765 if (IS0(a))
return 0;
767 fraction
f = (fraction)a;
768 if (!DENIS1(
f))
return 0;
770 const poly aAsPoly = NUM(
f);
788 if (IS0(a))
return FALSE;
789 fraction
f = (fraction)a;
803 fraction
fb = (fraction)
b;
808 fraction
fa = (fraction)a;
812 fraction
fa = (fraction)a;
815 number aDenCoeff =
NULL;
int aDenDeg = 0;
821 fraction
fb = (fraction)
b;
824 number bDenCoeff =
NULL;
int bDenDeg = 0;
830 if (aNumDeg-aDenDeg > bNumDeg-bDenDeg)
return TRUE;
831 if (aNumDeg-aDenDeg < bNumDeg-bDenDeg)
return FALSE;
848 const ring
A =
cf->extRing;
857 const int P =
rVar(
A);
862 for (
int nop=0; nop < P; nop ++)
865 if (nop!=P-1)
PrintS(
", ");
897 fraction t = (fraction) d;
900 WerrorS(
"expected differentiation by a variable");
906 WerrorS(
"expected differentiation by a variable");
912 fraction
fa = (fraction)a;
951 fraction
fa = (fraction)a;
952 fraction
fb = (fraction)
b;
992 fraction
fa = (fraction)a;
993 fraction
fb = (fraction)
b;
1029 if (IS0(a) || IS0(
b))
return NULL;
1031 fraction
fa = (fraction)a;
1032 fraction
fb = (fraction)
b;
1042 const poly da = DEN(
fa);
1043 const poly db = DEN(
fb);
1123 if (IS0(a))
return NULL;
1126 fraction
fa = (fraction)a;
1127 fraction
fb = (fraction)
b;
1172 fraction
f = (fraction)a;
1178 const poly
den = DEN(
f);
1243 int expAbs =
exp;
if (expAbs < 0) expAbs = -expAbs;
1246 number
pow; number t;
1250 for (
int i = 2;
i <= expAbs;
i++)
1271 expAbs = expAbs / 2;
1301 fraction
f = (fraction)a;
1321 if( DEN(
f) !=
NULL )
1365 }
while(i<ntRing->
N);
1383 BOOLEAN simpleTestsHaveAlreadyBeenPerformed)
1387 fraction
f = (fraction)a;
1390 if (
COM(
f)==0)
return;
1392 if (!simpleTestsHaveAlreadyBeenPerformed)
1516 if( DEN(
f) !=
NULL )
1541 fraction
f = (fraction)a;
1566 fraction
f = (fraction)a;
1601 if ((DEN((fraction)a)!=
NULL)
1639 fraction
fb = (fraction)
b;
1641 fraction
fa = (fraction)a;
1655 number contentpa, contentpb, tmp;
1724 fraction
fa = (fraction)a;
1725 fraction
fb = (fraction)
b;
1740 number contentpa, contentpb, tmp;
1795 if (IS0(a))
return 0;
1796 fraction
f = (fraction)a;
1798 unsigned long noOfTerms = 0;
1799 unsigned long numDegree = 0;
1805 unsigned long denDegree = 0;
1813 unsigned long t= ((numDegree + denDegree)*(numDegree + denDegree) + 1) * noOfTerms;
1814 if (t>INT_MAX)
return INT_MAX;
1824 assume(src->rep == dst->extRing->cf->rep);
1834 fraction ff=(fraction)
res;
1836 else DEN(ff)=
p_NSet(nn,dst->extRing);
1848 poly
p=
p_NSet(nMap(a, src,dst->extRing->cf), dst->extRing);
1862 int n =
n_Int(a, src);
1863 number q =
n_Init(n, dst->extRing->cf);
1876 if (IS0(a))
return NULL;
1878 const ring rSrc =
cf->extRing;
1879 const ring rDst = dst->extRing;
1884 fraction
f = (fraction)a;
1905 if (IS0(a))
return NULL;
1907 const ring rSrc =
cf->extRing;
1908 const ring rDst = dst->extRing;
1911 fraction
f = (fraction)a;
1912 poly
g =
prMapR(NUM(
f), nMap, rSrc, rDst);
1943 h =
prMapR(DEN(
f), nMap, rSrc, rDst);
1994 return ntInit(
prMapR((poly)a, nMap,
cf->extRing, dst->extRing),dst);
2004 number q =
nlModP(a, src, dst->extRing->cf);
2012 poly
g =
p_NSet(q, dst->extRing);
2026 assume(src == dst->extRing->cf);
2027 poly
p =
p_One(dst->extRing);
2042 int n =
n_Int(a, src);
2043 number q =
n_Init(n, dst->extRing->cf);
2084 if (src->ch == dst->ch)
return ntMapPP;
2088 if (
h != 1)
return NULL;
2096 if (
rVar(src->extRing) >
rVar(dst->extRing))
2099 for (
int i = 0;
i <
rVar(src->extRing);
i++)
2105 if (src->extRing->cf==dst->extRing->cf)
2112 if (src->extRing->cf==dst->extRing->cf)
2124 if (n==
ntCopyAlg) printf(
"n=ntCopyAlg\n");
2125 else if (n==
ntCopyMap) printf(
"n=ntCopyMap\n");
2126 else if (n==
ntMapUP) printf(
"n=ntMapUP\n");
2127 else if (n==
ntMap0P) printf(
"n=ntMap0P\n");
2128 else if (n==
ntMapP0) printf(
"n=ntMapP0\n");
2129 else if (n==
ntMap00) printf(
"n=ntMap00\n");
2130 else if (n==
NULL) printf(
"n=NULL\n");
2131 else printf(
"n=?\n");
2138 if ((--
cf->extRing->ref) == 0)
2158 fraction
f = (fraction)n;
2165 if (IS0(a))
return -1;
2166 fraction
fa = (fraction)a;
2167 return cf->extRing->pFDeg(NUM(
fa),
cf->extRing);
2175 const ring
R =
cf->extRing;
2177 assume( 0 < iParameter && iParameter <=
rVar(
R) );
2198 const ring
R =
cf->extRing;
2201 fraction
f = (fraction)
m;
2203 if( DEN(
f) !=
NULL )
2214 return NUM((fraction)n);
2226 const ring
R =
cf->extRing;
2233 numberCollectionEnumerator.
Reset();
2235 if( !numberCollectionEnumerator.
MoveNext() )
2248 number &n = numberCollectionEnumerator.
Current();
2252 fraction
f = (fraction)n;
2256 const poly
den = DEN(
f);
2260 const poly
num = NUM(
f);
2270 while( numberCollectionEnumerator.
MoveNext() ) ;
2279 numberCollectionEnumerator.
Reset();
2280 while (numberCollectionEnumerator.
MoveNext() )
2282 number &n = numberCollectionEnumerator.
Current();
2283 const number t =
ntDiv(n, c,
cf);
2314 numberCollectionEnumerator.
Reset();
2316 if( !numberCollectionEnumerator.
MoveNext() )
2327 const ring
R =
cf->extRing;
2336 number &n = numberCollectionEnumerator.
Current();
2344 const poly
den = NUM(
f);
2371 while( numberCollectionEnumerator.
MoveNext() );
2381 numberCollectionEnumerator.
Reset();
2385 while (numberCollectionEnumerator.
MoveNext() )
2387 number &n = numberCollectionEnumerator.
Current();
2394 fraction
f = (fraction)t;
2397 const poly
den = DEN(
f);
2416 numberCollectionEnumerator.
Reset();
2417 while (numberCollectionEnumerator.
MoveNext() )
2419 number &n = numberCollectionEnumerator.
Current();
2420 fraction
f = (fraction)n;
2424 const poly
den = DEN(
f);
2444 NUM((fraction)c) =
__p_Mult_nn(NUM((fraction)c), d,
R);
2456 poly *P=(poly*)
omAlloc(rl*
sizeof(poly*));
2457 number *X=(number *)
omAlloc(rl*
sizeof(number));
2461 for(
i=0;
i<rl;
i++) P[
i]=
p_Copy(NUM((fraction)(
x[
i])),
cf->extRing);
2520 cf->factoryVarOffset =
R->cf->factoryVarOffset +
rVar(
R);
2573 cf->iNumberOfParameters =
rVar(
R);
2574 cf->pParameterNames = (
const char**)
R->names;
2576 cf->has_simple_Inverse=
FALSE;
Rational pow(const Rational &a, int e)
Concrete implementation of enumerators over polynomials.
char * naCoeffString(const coeffs r)
char * naCoeffName(const coeffs r)
const CanonicalForm CFMap CFMap & N
const CanonicalForm const CanonicalForm const CanonicalForm const CanonicalForm & cand
CanonicalForm convSingPFactoryP(poly p, const ring r)
poly convFactoryPSingP(const CanonicalForm &f, const ring r)
poly singclap_pdivide(poly f, poly g, const ring r)
poly singclap_gcd_and_divide(poly &f, poly &g, const ring r)
clears denominators of f and g, divides by gcd(f,g)
poly singclap_gcd_r(poly f, poly g, const ring r)
poly singclap_gcd(poly f, poly g, const ring r)
destroys f and g
This is a polynomial enumerator for simple iteration over coefficients of polynomials.
go into polynomials over an alg. extension recursively
virtual reference Current()=0
Gets the current element in the collection (read and write).
virtual void Reset()=0
Sets the enumerator to its initial position: -1, which is before the first element in the collection.
virtual bool MoveNext()=0
Advances the enumerator to the next element of the collection. returns true if the enumerator was suc...
Templated enumerator interface for simple iteration over a generic collection of T's.
Coefficient rings, fields and other domains suitable for Singular polynomials.
static FORCE_INLINE number n_Mult(number a, number b, const coeffs r)
return the product of 'a' and 'b', i.e., a*b
static FORCE_INLINE long n_Int(number &n, const coeffs r)
conversion of n to an int; 0 if not possible in Z/pZ: the representing int lying in (-p/2 ....
static FORCE_INLINE number n_Copy(number n, const coeffs r)
return a copy of 'n'
static FORCE_INLINE number n_NormalizeHelper(number a, number b, const coeffs r)
assume that r is a quotient field (otherwise, return 1) for arguments (a1/a2,b1/b2) return (lcm(a1,...
static FORCE_INLINE number n_GetDenom(number &n, const coeffs r)
return the denominator of n (if elements of r are by nature not fractional, result is 1)
static FORCE_INLINE void n_CoeffWrite(const coeffs r, BOOLEAN details=TRUE)
output the coeff description
static FORCE_INLINE BOOLEAN nCoeff_is_Extension(const coeffs r)
#define n_Test(a, r)
BOOLEAN n_Test(number a, const coeffs r)
@ n_Q
rational (GMP) numbers
@ n_transExt
used for all transcendental extensions, i.e., the top-most extension in an extension tower is transce...
static FORCE_INLINE number n_Gcd(number a, number b, const coeffs r)
in Z: return the gcd of 'a' and 'b' in Z/nZ, Z/2^kZ: computed as in the case Z in Z/pZ,...
static FORCE_INLINE number n_Invers(number a, const coeffs r)
return the multiplicative inverse of 'a'; raise an error if 'a' is not invertible
static FORCE_INLINE BOOLEAN n_GreaterZero(number n, const coeffs r)
ordered fields: TRUE iff 'n' is positive; in Z/pZ: TRUE iff 0 < m <= roundedBelow(p/2),...
static FORCE_INLINE BOOLEAN n_IsMOne(number n, const coeffs r)
TRUE iff 'n' represents the additive inverse of the one element, i.e. -1.
static FORCE_INLINE nMapFunc n_SetMap(const coeffs src, const coeffs dst)
set the mapping function pointers for translating numbers from src to dst
static FORCE_INLINE number n_InpNeg(number n, const coeffs r)
in-place negation of n MUST BE USED: n = n_InpNeg(n) (no copy is returned)
static FORCE_INLINE BOOLEAN nCoeff_has_simple_inverse(const coeffs r)
TRUE, if the computation of the inverse is fast, i.e. prefer leading coeff. 1 over content.
static FORCE_INLINE number n_Div(number a, number b, const coeffs r)
return the quotient of 'a' and 'b', i.e., a/b; raises an error if 'b' is not invertible in r exceptio...
static FORCE_INLINE BOOLEAN nCoeff_is_Q(const coeffs r)
static FORCE_INLINE BOOLEAN n_Greater(number a, number b, const coeffs r)
ordered fields: TRUE iff 'a' is larger than 'b'; in Z/pZ: TRUE iff la > lb, where la and lb are the l...
static FORCE_INLINE BOOLEAN n_IsZero(number n, const coeffs r)
TRUE iff 'n' represents the zero element.
static FORCE_INLINE void n_ClearDenominators(ICoeffsEnumerator &numberCollectionEnumerator, number &d, const coeffs r)
(inplace) Clears denominators on a collection of numbers number d is the LCM of all the coefficient d...
static FORCE_INLINE BOOLEAN nCoeff_is_Q_or_BI(const coeffs r)
static FORCE_INLINE n_coeffType getCoeffType(const coeffs r)
Returns the type of coeffs domain.
static FORCE_INLINE void n_Delete(number *p, const coeffs r)
delete 'p'
static FORCE_INLINE BOOLEAN nCoeff_is_Zp(const coeffs r)
static FORCE_INLINE number n_Init(long i, const coeffs r)
a number representing i in the given coeff field/ring r
static FORCE_INLINE void n_ClearContent(ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs r)
Computes the content and (inplace) divides it out on a collection of numbers number c is the content ...
static FORCE_INLINE number n_GetNumerator(number &n, const coeffs r)
return the numerator of n (if elements of r are by nature not fractional, result is n)
@ n_rep_gap_rat
(number), see longrat.h
@ n_rep_gap_gmp
(), see rinteger.h, new impl.
@ n_rep_rat_fct
(fraction), see transext.h
static FORCE_INLINE number n_SubringGcd(number a, number b, const coeffs r)
number(* nMapFunc)(number a, const coeffs src, const coeffs dst)
maps "a", which lives in src, into dst
static FORCE_INLINE void n_Normalize(number &n, const coeffs r)
inplace-normalization of n; produces some canonical representation of n;
static FORCE_INLINE BOOLEAN n_IsOne(number n, const coeffs r)
TRUE iff 'n' represents the one element.
BOOLEAN fa(leftv res, leftv args)
BOOLEAN fb(leftv res, leftv args)
const CanonicalForm int s
void WerrorS(const char *s)
number nlModP(number q, const coeffs, const coeffs Zp)
static FORCE_INLINE BOOLEAN nlIsInteger(number q, const coeffs r)
static number & pGetCoeff(poly p)
return an alias to the leading coefficient of p assumes that p != NULL NOTE: not copy
gmp_float exp(const gmp_float &a)
The main handler for Singular numbers which are suitable for Singular polynomials.
number ndCopyMap(number a, const coeffs aRing, const coeffs r)
const char *const nDivBy0
#define omFreeSize(addr, size)
#define omFreeBin(addr, bin)
#define omGetSpecBin(size)
poly p_Diff(poly a, int k, const ring r)
poly p_Farey(poly p, number N, const ring r)
poly p_ISet(long i, const ring r)
returns the poly representing the integer i
poly p_Div_nn(poly p, const number n, const ring r)
void p_Normalize(poly p, const ring r)
void p_Norm(poly p1, const ring r)
const char * p_Read(const char *st, poly &rc, const ring r)
int p_Var(poly m, const ring r)
poly p_Sub(poly p1, poly p2, const ring r)
poly p_NSet(number n, const ring r)
returns the poly representing the number n, destroys n
poly p_ChineseRemainder(poly *xx, number *x, number *q, int rl, CFArray &inv_cache, const ring R)
BOOLEAN p_EqualPolys(poly p1, poly p2, const ring r)
static poly p_Neg(poly p, const ring r)
static poly p_Add_q(poly p, poly q, const ring r)
static poly p_Mult_q(poly p, poly q, const ring r)
void p_Write(poly p, ring lmRing, ring tailRing)
static unsigned long p_SetExp(poly p, const unsigned long e, const unsigned long iBitmask, const int VarOffset)
set a single variable exponent @Note: VarOffset encodes the position in p->exp
static void p_ExpVectorDiff(poly pr, poly p1, poly p2, const ring r)
void p_String0Long(const poly p, ring lmRing, ring tailRing)
print p in a long way
void p_String0Short(const poly p, ring lmRing, ring tailRing)
print p in a short way, if possible
static void p_Setm(poly p, const ring r)
static number p_SetCoeff(poly p, number n, ring r)
static BOOLEAN p_LmIsConstantComp(const poly p, const ring r)
static long p_GetExp(const poly p, const unsigned long iBitmask, const int VarOffset)
get a single variable exponent @Note: the integer VarOffset encodes:
static BOOLEAN p_LmIsConstant(const poly p, const ring r)
static BOOLEAN p_IsOne(const poly p, const ring R)
either poly(1) or gen(k)?!
static BOOLEAN p_IsConstant(const poly p, const ring r)
static void p_Delete(poly *p, const ring r)
static unsigned pLength(poly a)
static poly pp_Mult_qq(poly p, poly q, const ring r)
static poly p_LmFreeAndNext(poly p, ring)
static poly p_Copy(poly p, const ring r)
returns a copy of p
static long p_Totaldegree(poly p, const ring r)
#define __p_Mult_nn(p, n, r)
void p_wrp(poly p, ring lmRing, ring tailRing)
poly prMapR(poly src, nMapFunc nMap, ring src_r, ring dest_r)
poly prCopyR(poly p, ring src_r, ring dest_r)
void StringAppendS(const char *st)
void PrintS(const char *s)
void rDelete(ring r)
unconditionally deletes fields in r
BOOLEAN rEqual(ring r1, ring r2, BOOLEAN qr)
returns TRUE, if r1 equals r2 FALSE, otherwise Equality is determined componentwise,...
static char * rRingVar(short i, const ring r)
static BOOLEAN rCanShortOut(const ring r)
static short rVar(const ring r)
#define rVar(r) (r->N)
static poly convert(const number &n)
static int ntSize(number a, const coeffs cf)
static BOOLEAN ntDBTest(number a, const char *f, const int l, const coeffs r)
static number ntParameter(const int iParameter, const coeffs cf)
return the specified parameter as a number in the given trans.ext.
static void ntWriteLong(number a, const coeffs cf)
static BOOLEAN ntCoeffIsEqual(const coeffs cf, n_coeffType n, void *param)
static void heuristicGcdCancellation(number a, const coeffs cf)
forward declarations
number ntDiff(number a, number d, const coeffs cf)
nMapFunc ntSetMap(const coeffs src, const coeffs dst)
Get a mapping function from src into the domain of this type (n_transExt)
static number ntSub(number a, number b, const coeffs cf)
static number ntCopyAlg(number a, const coeffs cf, const coeffs dst)
static long ntInt(number &a, const coeffs cf)
static number ntMapPP(number a, const coeffs src, const coeffs dst)
static void ntDelete(number *a, const coeffs cf)
static BOOLEAN ntIsOne(number a, const coeffs cf)
static void ntNormalizeDen(fraction result, const ring R)
BOOLEAN ntInitChar(coeffs cf, void *infoStruct)
Initialize the coeffs object.
static BOOLEAN ntIsZero(number a, const coeffs cf)
static number ntGetNumerator(number &a, const coeffs cf)
TODO: normalization of a!?
static CanonicalForm ntConvSingNFactoryN(number n, BOOLEAN, const coeffs cf)
static void ntClearDenominators(ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs cf)
static BOOLEAN ntIsMOne(number a, const coeffs cf)
static number ntMapZ0(number a, const coeffs src, const coeffs dst)
static void ntCoeffWrite(const coeffs cf, BOOLEAN details)
static number ntMap0P(number a, const coeffs src, const coeffs dst)
#define NUMIS1(f)
TRUE iff num. represents 1.
static number ntGcd(number a, number b, const coeffs cf)
static number ntConvFactoryNSingN(const CanonicalForm n, const coeffs cf)
static void ntClearContent(ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs cf)
static number ntInvers(number a, const coeffs cf)
static number ntNormalizeHelper(number a, number b, const coeffs cf)
static void ntPower(number a, int exp, number *b, const coeffs cf)
number ntInit(long i, const coeffs cf)
static number ntMapP0(number a, const coeffs src, const coeffs dst)
static number ntGetDenom(number &a, const coeffs cf)
TODO: normalization of a!?
static const char * ntRead(const char *s, number *a, const coeffs cf)
static number ntMapUP(number a, const coeffs src, const coeffs dst)
#define MULT_COMPLEXITY
complexity increase due to * and /
static void ntKillChar(coeffs cf)
static void ntWriteShort(number a, const coeffs cf)
static number ntDiv(number a, number b, const coeffs cf)
static number ntCopy(number a, const coeffs cf)
static int ntParDeg(number a, const coeffs cf)
static BOOLEAN ntGreaterZero(number a, const coeffs cf)
static number ntChineseRemainder(number *x, number *q, int rl, BOOLEAN, CFArray &inv_cache, const coeffs cf)
static number ntGenMap(number a, const coeffs cf, const coeffs dst)
int ntIsParam(number m, const coeffs cf)
if m == var(i)/1 => return i,
static number ntMap00(number a, const coeffs src, const coeffs dst)
static number ntMult(number a, number b, const coeffs cf)
#define ADD_COMPLEXITY
complexity increase due to + and -
static number ntGenAlg(number a, const coeffs cf, const coeffs dst)
static number ntNeg(number a, const coeffs cf)
this is in-place, modifies a
static void definiteGcdCancellation(number a, const coeffs cf, BOOLEAN simpleTestsHaveAlreadyBeenPerformed)
modifies a
static number ntAdd(number a, number b, const coeffs cf)
static number ntFarey(number p, number n, const coeffs cf)
static number ntCopyMap(number a, const coeffs cf, const coeffs dst)
static void ntNormalize(number &a, const coeffs cf)
static BOOLEAN ntEqual(number a, number b, const coeffs cf)
static coeffs nCoeff_bottom(const coeffs r, int &height)
#define DIFF_COMPLEXITY
complexity increase due to * and /
static BOOLEAN ntGreater(number a, number b, const coeffs cf)
static void handleNestedFractionsOverQ(fraction f, const coeffs cf)
#define BOUND_COMPLEXITY
maximum complexity of a number
struct for passing initialization parameters to naInitChar