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#define | TRANSEXT_PRIVATES 1 |
| ABSTRACT: numbers in an algebraic extension field K[a] / < f(a) > Assuming that we have a coeffs object cf, then these numbers are polynomials in the polynomial ring K[a] represented by cf->extRing. More...
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#define | naTest(a) naDBTest(a,__FILE__,__LINE__,cf) |
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#define | naRing cf->extRing |
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#define | naCoeffs cf->extRing->cf |
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#define | naMinpoly naRing->qideal->m[0] |
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#define | n2pTest(a) n2pDBTest(a,__FILE__,__LINE__,cf) |
| ABSTRACT: numbers as polys in the ring K[a] Assuming that we have a coeffs object cf, then these numbers are polynomials in the polynomial ring K[a] represented by cf->extRing. More...
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#define | n2pRing cf->extRing |
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#define | n2pCoeffs cf->extRing->cf |
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BOOLEAN | naDBTest (number a, const char *f, const int l, const coeffs r) |
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BOOLEAN | naGreaterZero (number a, const coeffs cf) |
| forward declarations More...
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BOOLEAN | naGreater (number a, number b, const coeffs cf) |
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BOOLEAN | naEqual (number a, number b, const coeffs cf) |
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BOOLEAN | naIsOne (number a, const coeffs cf) |
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BOOLEAN | naIsMOne (number a, const coeffs cf) |
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number | naInit (long i, const coeffs cf) |
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number | naNeg (number a, const coeffs cf) |
| this is in-place, modifies a More...
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number | naInvers (number a, const coeffs cf) |
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number | naAdd (number a, number b, const coeffs cf) |
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number | naSub (number a, number b, const coeffs cf) |
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number | naMult (number a, number b, const coeffs cf) |
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number | naDiv (number a, number b, const coeffs cf) |
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void | naPower (number a, int exp, number *b, const coeffs cf) |
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number | naCopy (number a, const coeffs cf) |
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void | naWriteLong (number a, const coeffs cf) |
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void | naWriteShort (number a, const coeffs cf) |
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number | naGetDenom (number &a, const coeffs cf) |
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number | naGetNumerator (number &a, const coeffs cf) |
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number | naGcd (number a, number b, const coeffs cf) |
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void | naDelete (number *a, const coeffs cf) |
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void | naCoeffWrite (const coeffs cf, BOOLEAN details) |
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const char * | naRead (const char *s, number *a, const coeffs cf) |
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static BOOLEAN | naCoeffIsEqual (const coeffs cf, n_coeffType n, void *param) |
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static void | p_Monic (poly p, const ring r) |
| returns NULL if p == NULL, otherwise makes p monic by dividing by its leading coefficient (only done if this is not already 1); this assumes that we are over a ground field so that division is well-defined; modifies p More...
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static poly | p_GcdHelper (poly &p, poly &q, const ring r) |
| see p_Gcd; additional assumption: deg(p) >= deg(q); must destroy p and q (unless one of them is returned) More...
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static poly | p_Gcd (const poly p, const poly q, const ring r) |
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static poly | p_ExtGcdHelper (poly &p, poly &pFactor, poly &q, poly &qFactor, ring r) |
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poly | p_ExtGcd (poly p, poly &pFactor, poly q, poly &qFactor, ring r) |
| assumes that p and q are univariate polynomials in r, mentioning the same variable; assumes a global monomial ordering in r; assumes that not both p and q are NULL; returns the gcd of p and q; moreover, afterwards pFactor and qFactor contain appropriate factors such that gcd(p, q) = p * pFactor + q * qFactor; leaves p and q unmodified More...
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void | heuristicReduce (poly &p, poly reducer, const coeffs cf) |
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void | definiteReduce (poly &p, poly reducer, const coeffs cf) |
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static coeffs | nCoeff_bottom (const coeffs r, int &height) |
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BOOLEAN | naIsZero (number a, const coeffs cf) |
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long | naInt (number &a, const coeffs cf) |
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number | napNormalizeHelper (number b, const coeffs cf) |
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number | naLcmContent (number a, number b, const coeffs cf) |
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int | naSize (number a, const coeffs cf) |
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void | naNormalize (number &a, const coeffs cf) |
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number | naConvFactoryNSingN (const CanonicalForm n, const coeffs cf) |
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CanonicalForm | naConvSingNFactoryN (number n, BOOLEAN, const coeffs cf) |
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number | naMap00 (number a, const coeffs src, const coeffs dst) |
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number | naMapZ0 (number a, const coeffs src, const coeffs dst) |
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number | naMapP0 (number a, const coeffs src, const coeffs dst) |
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number | naCopyTrans2AlgExt (number a, const coeffs src, const coeffs dst) |
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number | naMap0P (number a, const coeffs src, const coeffs dst) |
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number | naMapPP (number a, const coeffs src, const coeffs dst) |
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number | naMapUP (number a, const coeffs src, const coeffs dst) |
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number | naGenMap (number a, const coeffs cf, const coeffs dst) |
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number | naGenTrans2AlgExt (number a, const coeffs cf, const coeffs dst) |
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nMapFunc | naSetMap (const coeffs src, const coeffs dst) |
| Get a mapping function from src into the domain of this type (n_algExt) More...
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int | naParDeg (number a, const coeffs cf) |
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number | naParameter (const int iParameter, const coeffs cf) |
| return the specified parameter as a number in the given alg. field More...
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int | naIsParam (number m, const coeffs cf) |
| if m == var(i)/1 => return i, More...
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static void | naClearContent (ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs cf) |
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void | naClearDenominators (ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs cf) |
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void | naKillChar (coeffs cf) |
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char * | naCoeffString (const coeffs r) |
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char * | naCoeffName (const coeffs r) |
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number | naChineseRemainder (number *x, number *q, int rl, BOOLEAN, CFArray &inv_cache, const coeffs cf) |
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number | naFarey (number p, number n, const coeffs cf) |
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BOOLEAN | naInitChar (coeffs cf, void *infoStruct) |
| Initialize the coeffs object. More...
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BOOLEAN | n2pDBTest (number a, const char *f, const int l, const coeffs r) |
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void | n2pNormalize (number &a, const coeffs cf) |
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number | n2pMult (number a, number b, const coeffs cf) |
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number | n2pDiv (number a, number b, const coeffs cf) |
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void | n2pPower (number a, int exp, number *b, const coeffs cf) |
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const char * | n2pRead (const char *s, number *a, const coeffs cf) |
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static BOOLEAN | n2pCoeffIsEqual (const coeffs cf, n_coeffType n, void *param) |
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char * | n2pCoeffString (const coeffs cf) |
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char * | n2pCoeffName (const coeffs cf) |
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void | n2pCoeffWrite (const coeffs cf, BOOLEAN details) |
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number | n2pInvers (number a, const coeffs cf) |
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BOOLEAN | n2pInitChar (coeffs cf, void *infoStruct) |
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◆ n2pCoeffs
#define n2pCoeffs cf->extRing->cf |
◆ n2pRing
#define n2pRing cf->extRing |
◆ n2pTest
ABSTRACT: numbers as polys in the ring K[a] Assuming that we have a coeffs object cf, then these numbers are polynomials in the polynomial ring K[a] represented by cf->extRing.
IMPORTANT ASSUMPTIONS: 1.) So far we assume that cf->extRing is a valid polynomial ring
Definition at line 1513 of file algext.cc.
◆ naCoeffs
#define naCoeffs cf->extRing->cf |
◆ naMinpoly
#define naMinpoly naRing->qideal->m[0] |
◆ naRing
#define naRing cf->extRing |
◆ naTest
#define naTest |
( |
|
a | ) |
naDBTest(a,__FILE__,__LINE__,cf) |
◆ TRANSEXT_PRIVATES
#define TRANSEXT_PRIVATES 1 |
ABSTRACT: numbers in an algebraic extension field K[a] / < f(a) > Assuming that we have a coeffs object cf, then these numbers are polynomials in the polynomial ring K[a] represented by cf->extRing.
IMPORTANT ASSUMPTIONS: 1.) So far we assume that cf->extRing is a valid polynomial ring in exactly one variable, i.e., K[a], where K is allowed to be any field (representable in SINGULAR and which may itself be some extension field, thus allowing for extension towers). 2.) Moreover, this implementation assumes that cf->extRing->qideal is not NULL but an ideal with at least one non-zero generator which may be accessed by cf->extRing->qideal->m[0] and which represents the minimal polynomial f(a) of the extension variable 'a' in K[a]. 3.) As soon as an std method for polynomial rings becomes availabe, all reduction steps modulo f(a) should be replaced by a call to std. Moreover, in this situation one can finally move from K[a] / < f(a) > to K[a_1, ..., a_s] / I, with I some zero-dimensional ideal in K[a_1, ..., a_s] given by a lex Gröbner basis. The code in algext.h and algext.cc is then capable of computing in K[a_1, ..., a_s] / I.
Definition at line 50 of file algext.cc.
◆ definiteReduce()
void definiteReduce |
( |
poly & |
p, |
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poly |
reducer, |
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const coeffs |
cf |
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) |
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◆ heuristicReduce()
void heuristicReduce |
( |
poly & |
p, |
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poly |
reducer, |
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const coeffs |
cf |
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) |
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◆ n2pCoeffIsEqual()
◆ n2pCoeffName()
Definition at line 1624 of file algext.cc.
1632 l+=(strlen(
p[
i])+1);
1637 snprintf(
s,strlen(cf_s)+2,
"%s",cf_s);
1648 else { tt[0]=
']'; strcat(
s,tt); }
◆ n2pCoeffString()
Definition at line 1596 of file algext.cc.
1604 l+=(strlen(
p[
i])+1);
1607 char *
s=(
char *)
omAlloc(
l+5+strlen(cf_s));
1609 snprintf(
s,strlen(cf_s)+2,
"%s",cf_s);
1620 else { tt[0]=
']'; strcat(
s,tt); }
◆ n2pCoeffWrite()
Definition at line 1652 of file algext.cc.
1657 const ring
A =
cf->extRing;
1660 PrintS(
"// polynomial ring as coefficient ring :\n");
◆ n2pDBTest()
◆ n2pDiv()
◆ n2pInitChar()
first check whether cf->extRing != NULL and delete old ring???
Definition at line 1680 of file algext.cc.
1694 const ring
R = e->
r;
1752 cf->iNumberOfParameters =
rVar(
R);
1753 cf->pParameterNames = (
const char**)
R->names;
◆ n2pInvers()
◆ n2pMult()
Definition at line 1543 of file algext.cc.
1549 return (number)aTimesB;
◆ n2pNormalize()
◆ n2pPower()
void n2pPower |
( |
number |
a, |
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int |
exp, |
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number * |
b, |
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const coeffs |
cf |
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) |
| |
◆ n2pRead()
Definition at line 1567 of file algext.cc.
1572 *a = (number)aAsPoly;
◆ naAdd()
Definition at line 437 of file algext.cc.
445 return (number)aPlusB;
◆ naChineseRemainder()
Definition at line 1377 of file algext.cc.
1379 poly *P=(poly*)
omAlloc(rl*
sizeof(poly*));
1380 number *X=(number *)
omAlloc(rl*
sizeof(number));
◆ naClearContent()
Definition at line 1104 of file algext.cc.
1110 const ring
R =
cf->extRing;
1116 numberCollectionEnumerator.
Reset();
1118 if( !numberCollectionEnumerator.
MoveNext() )
1127 int s1;
int s=2147483647;
1131 int normalcount = 0;
1137 number& n = numberCollectionEnumerator.
Current();
1150 }
while (numberCollectionEnumerator.
MoveNext() );
1157 numberCollectionEnumerator.
Reset();
1160 while (numberCollectionEnumerator.
MoveNext() )
1162 number& n = numberCollectionEnumerator.
Current();
1165 if( (--normalcount) <= 0)
1215 numberCollectionEnumerator.
Reset();
1218 while (numberCollectionEnumerator.
MoveNext() )
1220 number& n = numberCollectionEnumerator.
Current();
1231 n = (number)
p_Mult_q(cInverse, (poly)n,
R);
◆ naClearDenominators()
◆ naCoeffIsEqual()
Definition at line 678 of file algext.cc.
695 const ideal mi =
naRing->qideal;
697 const ideal ii = e->
r->qideal;
◆ naCoeffName()
Definition at line 1354 of file algext.cc.
1361 l+=(strlen(
p[
i])+1);
1365 snprintf(
s,10+1,
"%d",r->ch);
◆ naCoeffString()
Definition at line 1331 of file algext.cc.
1338 l+=(strlen(
p[
i])+1);
1342 snprintf(
s,10+1,
"%d",r->ch);
◆ naCoeffWrite()
Definition at line 387 of file algext.cc.
391 const ring
A =
cf->extRing;
400 const int P =
rVar(
A);
405 for (
int nop=0; nop < P; nop ++)
408 if (nop!=P-1)
PrintS(
", ");
413 const ideal I =
A->qideal;
◆ naConvFactoryNSingN()
◆ naConvSingNFactoryN()
◆ naCopy()
◆ naCopyTrans2AlgExt()
Definition at line 890 of file algext.cc.
894 fraction
fa=(fraction)a;
927 number t=
naDiv ((number)
p,(number)q, dst);
932 WerrorS (
"mapping denominator to zero");
◆ naDBTest()
◆ naDelete()
Definition at line 278 of file algext.cc.
280 if (*a ==
NULL)
return;
282 poly aAsPoly = (poly)(*a);
◆ naDiv()
Definition at line 469 of file algext.cc.
480 return (number)aDivB;
◆ naEqual()
◆ naFarey()
◆ naGcd()
◆ naGenMap()
Definition at line 972 of file algext.cc.
976 const ring rSrc =
cf->extRing;
977 const ring rDst = dst->extRing;
981 poly
g =
prMapR(
f, nMap, rSrc, rDst);
◆ naGenTrans2AlgExt()
Definition at line 987 of file algext.cc.
991 const ring rSrc =
cf->extRing;
992 const ring rDst = dst->extRing;
995 fraction
f = (fraction)a;
996 poly
g =
prMapR(NUM(
f), nMap, rSrc, rDst);
1002 h =
prMapR(DEN(
f), nMap, rSrc, rDst);
◆ naGetDenom()
◆ naGetNumerator()
◆ naGreater()
Definition at line 358 of file algext.cc.
372 if (aDeg>bDeg)
return TRUE;
373 if (aDeg<bDeg)
return FALSE;
◆ naGreaterZero()
forward declarations
Definition at line 378 of file algext.cc.
◆ naInit()
◆ naInitChar()
Initialize the coeffs object.
first check whether cf->extRing != NULL and delete old ring???
Definition at line 1397 of file algext.cc.
1409 (e->
r->qideal->m[0] !=
NULL) );
1415 const ring
R = e->
r;
1480 cf->iNumberOfParameters =
rVar(
R);
1481 cf->pParameterNames = (
const char**)
R->names;
1483 cf->has_simple_Inverse=
R->cf->has_simple_Inverse;
◆ naInt()
Definition at line 345 of file algext.cc.
348 poly aAsPoly = (poly)a;
◆ naInvers()
Definition at line 818 of file algext.cc.
823 poly aFactor =
NULL; poly mFactor =
NULL; poly theGcd =
NULL;
839 WerrorS(
"zero divisor found - your minpoly is not irreducible");
844 return (number)(aFactor);
◆ naIsMOne()
Definition at line 323 of file algext.cc.
326 poly aAsPoly = (poly)a;
◆ naIsOne()
Definition at line 315 of file algext.cc.
318 poly aAsPoly = (poly)a;
◆ naIsParam()
if m == var(i)/1 => return i,
Definition at line 1093 of file algext.cc.
1097 const ring
R =
cf->extRing;
◆ naIsZero()
◆ naKillChar()
Definition at line 1325 of file algext.cc.
1327 if ((--
cf->extRing->ref) == 0)
◆ naLcmContent()
number naLcmContent |
( |
number |
a, |
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number |
b, |
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const coeffs |
cf |
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) |
| |
◆ naMap00()
◆ naMap0P()
Definition at line 938 of file algext.cc.
943 number q =
nlModP(a, src, dst->extRing->cf);
◆ naMapP0()
◆ naMapPP()
◆ naMapUP()
◆ naMapZ0()
◆ naMult()
Definition at line 459 of file algext.cc.
466 return (number)aTimesB;
◆ naNeg()
this is in-place, modifies a
Definition at line 332 of file algext.cc.
◆ naNormalize()
◆ naParameter()
return the specified parameter as a number in the given alg. field
Definition at line 1078 of file algext.cc.
1082 const ring
R =
cf->extRing;
1084 assume( 0 < iParameter && iParameter <=
rVar(
R) );
◆ naParDeg()
Definition at line 1070 of file algext.cc.
1072 if (a ==
NULL)
return -1;
1074 return cf->extRing->pFDeg(aa,
cf->extRing);
◆ napNormalizeHelper()
number napNormalizeHelper |
( |
number |
b, |
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const coeffs |
cf |
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) |
| |
◆ naPower()
void naPower |
( |
number |
a, |
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int |
exp, |
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number * |
b, |
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const coeffs |
cf |
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) |
| |
Definition at line 493 of file algext.cc.
508 int expAbs =
exp;
if (expAbs < 0) expAbs = -expAbs;
511 poly
pow; poly aAsPoly = (poly)a;
515 for (
int i = 2;
i <= expAbs;
i++)
545 number n = (number)
pow;
◆ naRead()
Definition at line 606 of file algext.cc.
611 *a = (number)aAsPoly;
◆ naSetMap()
Get a mapping function from src into the domain of this type (n_algExt)
Q or Z --> Q(a)
Z --> Q(a)
Z/p --> Q(a)
Q --> Z/p(a)
Z --> Z/p(a)
Z/p --> Z/p(a)
Z/u --> Z/p(a)
default
Definition at line 1017 of file algext.cc.
1043 if (src->ch == dst->ch)
return naMapPP;
1047 if (
h != 1)
return NULL;
1059 else if ((nMap!=
NULL) && (strcmp(
rRingVar(0,src->extRing),
rRingVar(0,dst->extRing))==0) && (
rVar (src->extRing) ==
rVar (dst->extRing)))
◆ naSize()
Definition at line 712 of file algext.cc.
714 if (a ==
NULL)
return 0;
715 poly aAsPoly = (poly)a;
717 while (aAsPoly !=
NULL)
◆ naSub()
Definition at line 448 of file algext.cc.
453 if (a ==
NULL)
return (number)minusB;
456 return (number)aMinusB;
◆ naWriteLong()
Definition at line 570 of file algext.cc.
577 poly aAsPoly = (poly)a;
◆ naWriteShort()
Definition at line 588 of file algext.cc.
595 poly aAsPoly = (poly)a;
◆ nCoeff_bottom()
◆ p_ExtGcd()
poly p_ExtGcd |
( |
poly |
p, |
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poly & |
pFactor, |
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poly |
q, |
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poly & |
qFactor, |
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ring |
r |
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) |
| |
assumes that p and q are univariate polynomials in r, mentioning the same variable; assumes a global monomial ordering in r; assumes that not both p and q are NULL; returns the gcd of p and q; moreover, afterwards pFactor and qFactor contain appropriate factors such that gcd(p, q) = p * pFactor + q * qFactor; leaves p and q unmodified
Definition at line 216 of file algext.cc.
221 { a = q;
b =
p; aCorrespondsToP =
FALSE; }
223 poly aFactor =
NULL; poly bFactor =
NULL;
225 if (aCorrespondsToP) { pFactor = aFactor; qFactor = bFactor; }
226 else { pFactor = bFactor; qFactor = aFactor; }
◆ p_ExtGcdHelper()
static poly p_ExtGcdHelper |
( |
poly & |
p, |
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poly & |
pFactor, |
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poly & |
q, |
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poly & |
qFactor, |
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ring |
r |
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) |
| |
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inlinestatic |
◆ p_Gcd()
◆ p_GcdHelper()
static poly p_GcdHelper |
( |
poly & |
p, |
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poly & |
q, |
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const ring |
r |
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) |
| |
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inlinestatic |
see p_Gcd; additional assumption: deg(p) >= deg(q); must destroy p and q (unless one of them is returned)
Definition at line 145 of file algext.cc.
◆ p_Monic()
static void p_Monic |
( |
poly |
p, |
|
|
const ring |
r |
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) |
| |
|
inlinestatic |
returns NULL if p == NULL, otherwise makes p monic by dividing by its leading coefficient (only done if this is not already 1); this assumes that we are over a ground field so that division is well-defined; modifies p
assumes that p and q are univariate polynomials in r, mentioning the same variable; assumes a global monomial ordering in r; assumes that not both p and q are NULL; returns the gcd of p and q; leaves p and q unmodified
Definition at line 120 of file algext.cc.
122 if (
p ==
NULL)
return;
123 number n =
n_Init(1, r->cf);
static FORCE_INLINE n_coeffType getCoeffType(const coeffs r)
Returns the type of coeffs domain.
int dReportError(const char *fmt,...)
char * n2pCoeffString(const coeffs cf)
static FORCE_INLINE BOOLEAN nCoeff_is_Zp(const coeffs r)
void StringAppendS(const char *st)
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 void p_Monic(poly p, const ring r)
returns NULL if p == NULL, otherwise makes p monic by dividing by its leading coefficient (only done ...
#define p_SetCoeff0(p, n, r)
void p_Normalize(poly p, const ring r)
virtual reference Current()=0
Gets the current element in the collection (read and write).
void p_Write0(poly p, ring lmRing, ring tailRing)
nMapFunc naSetMap(const coeffs src, const coeffs dst)
Get a mapping function from src into the domain of this type (n_algExt)
void naCoeffWrite(const coeffs cf, BOOLEAN details)
number napNormalizeHelper(number b, const coeffs cf)
static BOOLEAN rCanShortOut(const ring r)
const CanonicalForm int const CFList const Variable & y
BOOLEAN naIsOne(number a, const coeffs cf)
char * naCoeffName(const coeffs r)
void n2pCoeffWrite(const coeffs cf, BOOLEAN details)
const char * n2pRead(const char *s, number *a, const coeffs cf)
void n2pPower(number a, int exp, number *b, const coeffs cf)
number naDiv(number a, number b, const coeffs cf)
number naGenMap(number a, const coeffs cf, const coeffs dst)
number naGcd(number a, number b, const coeffs cf)
int p_Var(poly m, const ring r)
static FORCE_INLINE char * nCoeffString(const coeffs cf)
TODO: make it a virtual method of coeffs, together with: Decompose & Compose, rParameter & rPar.
number n2pInvers(number a, const coeffs cf)
number naChineseRemainder(number *x, number *q, int rl, BOOLEAN, CFArray &inv_cache, const coeffs cf)
static poly p_Neg(poly p, const ring r)
int naParDeg(number a, const coeffs cf)
number naInvers(number a, const coeffs cf)
number naFarey(number p, number n, const coeffs cf)
static BOOLEAN length(leftv result, leftv arg)
number naMap00(number a, const coeffs src, const coeffs dst)
poly p_ChineseRemainder(poly *xx, number *x, number *q, int rl, CFArray &inv_cache, const ring R)
poly singclap_pdivide(poly f, poly g, const ring r)
void naWriteLong(number a, const coeffs cf)
static FORCE_INLINE BOOLEAN nCoeff_is_Q_or_BI(const coeffs r)
static coeffs nCoeff_bottom(const coeffs r, int &height)
number ndCopyMap(number a, const coeffs aRing, const coeffs r)
poly prMapR(poly src, nMapFunc nMap, ring src_r, ring dest_r)
static FORCE_INLINE void n_Delete(number *p, const coeffs r)
delete 'p'
BOOLEAN naDBTest(number a, const char *f, const int l, const coeffs r)
number naMapPP(number a, const coeffs src, const coeffs dst)
number naAdd(number a, number b, const coeffs cf)
static FORCE_INLINE BOOLEAN n_IsZero(number n, const coeffs r)
TRUE iff 'n' represents the zero element.
static FORCE_INLINE int n_NumberOfParameters(const coeffs r)
Returns the number of parameters.
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_IsOne(number n, const coeffs r)
TRUE iff 'n' represents the one element.
CanonicalForm naConvSingNFactoryN(number n, BOOLEAN, const coeffs cf)
#define __p_Mult_nn(p, n, r)
static FORCE_INLINE void n_Normalize(number &n, const coeffs r)
inplace-normalization of n; produces some canonical representation of n;
void p_String0Short(const poly p, ring lmRing, ring tailRing)
print p in a short way, if possible
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,...
number n2pMult(number a, number b, const coeffs cf)
poly gcd_over_Q(poly f, poly g, const ring r)
helper routine for calling singclap_gcd_r
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
virtual void Reset()=0
Sets the enumerator to its initial position: -1, which is before the first element in the collection.
number naLcmContent(number a, number b, const coeffs cf)
static FORCE_INLINE BOOLEAN nCoeff_is_Q(const coeffs r)
BOOLEAN rEqual(ring r1, ring r2, BOOLEAN qr)
returns TRUE, if r1 equals r2 FALSE, otherwise Equality is determined componentwise,...
static FORCE_INLINE void n_CoeffWrite(const coeffs r, BOOLEAN details=TRUE)
output the coeff description
static poly p_Copy(poly p, const ring r)
returns a copy of p
poly p_Power(poly p, int i, const ring r)
number naCopy(number a, const coeffs cf)
void definiteReduce(poly &p, poly reducer, const coeffs cf)
static short rVar(const ring r)
#define rVar(r) (r->N)
used to represent polys as coeffcients
#define n2pTest(a)
ABSTRACT: numbers as polys in the ring K[a] Assuming that we have a coeffs object cf,...
struct for passing initialization parameters to naInitChar
poly p_PermPoly(poly p, const int *perm, const ring oldRing, const ring dst, nMapFunc nMap, const int *par_perm, int OldPar, BOOLEAN use_mult)
number(* nMapFunc)(number a, const coeffs src, const coeffs dst)
maps "a", which lives in src, into dst
const char *const nDivBy0
void PrintS(const char *s)
#define omFreeSize(addr, size)
static FORCE_INLINE BOOLEAN nCoeff_is_algExt(const coeffs r)
TRUE iff r represents an algebraic extension field.
static char * rRingVar(short i, const ring r)
void naClearDenominators(ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs cf)
BOOLEAN naGreaterZero(number a, const coeffs cf)
forward declarations
BOOLEAN naGreater(number a, number b, const coeffs cf)
int naSize(number a, const coeffs cf)
char * n2pCoeffName(const coeffs cf)
static void naClearContent(ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs cf)
number naParameter(const int iParameter, const coeffs cf)
return the specified parameter as a number in the given alg. field
number naGetDenom(number &a, const coeffs cf)
number ndGcd(number, number, const coeffs r)
void p_String0Long(const poly p, ring lmRing, ring tailRing)
print p in a long way
poly p_Farey(poly p, number N, const ring r)
number naGetNumerator(number &a, const coeffs cf)
long naInt(number &a, const coeffs cf)
static poly pp_Mult_qq(poly p, poly q, const ring r)
number naMult(number a, number b, const coeffs cf)
BOOLEAN fa(leftv res, leftv args)
number naMapUP(number a, const coeffs src, const coeffs dst)
char * naCoeffString(const coeffs r)
static FORCE_INLINE number n_Mult(number a, number b, const coeffs r)
return the product of 'a' and 'b', i.e., a*b
number naConvFactoryNSingN(const CanonicalForm n, const coeffs cf)
static FORCE_INLINE number n_Init(long i, const coeffs r)
a number representing i in the given coeff field/ring r
void naPower(number a, int exp, number *b, const coeffs cf)
const char * p_Read(const char *st, poly &rc, const ring r)
number nlModP(number q, const coeffs, const coeffs Zp)
static poly p_Init(const ring r, omBin bin)
void naWriteShort(number a, const coeffs cf)
number n2pDiv(number a, number b, const coeffs cf)
poly p_PolyDiv(poly &p, const poly divisor, const BOOLEAN needResult, const ring r)
assumes that p and divisor are univariate polynomials in r, mentioning the same variable; assumes div...
void naKillChar(coeffs cf)
static FORCE_INLINE BOOLEAN nCoeff_is_Extension(const coeffs r)
static FORCE_INLINE BOOLEAN nCoeff_is_transExt(const coeffs r)
TRUE iff r represents a transcendental extension field.
static poly p_ExtGcdHelper(poly &p, poly &pFactor, poly &q, poly &qFactor, ring r)
static FORCE_INLINE const char ** n_ParameterNames(const coeffs r)
Returns a (const!) pointer to (const char*) names of parameters.
CanonicalForm convSingPFactoryP(poly p, const ring r)
gmp_float exp(const gmp_float &a)
number naInit(long i, const coeffs cf)
const char * naRead(const char *s, number *a, const coeffs cf)
void rDelete(ring r)
unconditionally deletes fields in r
const CanonicalForm const CanonicalForm const CanonicalForm const CanonicalForm & cand
long p_Deg(poly a, const ring r)
static void p_Delete(poly *p, const ring r)
static poly p_Add_q(poly p, poly q, const ring r)
BOOLEAN singclap_extgcd(poly f, poly g, poly &res, poly &pa, poly &pb, const ring r)
virtual bool MoveNext()=0
Advances the enumerator to the next element of the collection. returns true if the enumerator was suc...
number naSub(number a, number b, const coeffs cf)
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_Invers(number a, const coeffs r)
return the multiplicative inverse of 'a'; raise an error if 'a' is not invertible
BOOLEAN p_EqualPolys(poly p1, poly p2, const ring r)
void rWrite(ring r, BOOLEAN details)
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...
void naNormalize(number &a, const coeffs cf)
number naGenTrans2AlgExt(number a, const coeffs cf, const coeffs dst)
void heuristicReduce(poly &p, poly reducer, const coeffs cf)
poly p_NSet(number n, const ring r)
returns the poly representing the number n, destroys n
static BOOLEAN naCoeffIsEqual(const coeffs cf, n_coeffType n, void *param)
static BOOLEAN n2pCoeffIsEqual(const coeffs cf, n_coeffType n, void *param)
poly singclap_gcd_r(poly f, poly g, const ring r)
BOOLEAN naIsZero(number a, const coeffs cf)
static number p_SetCoeff(poly p, number n, ring r)
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 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 number n_Copy(number n, const coeffs r)
return a copy of 'n'
Rational pow(const Rational &a, int e)
void WerrorS(const char *s)
number naMapP0(number a, const coeffs src, const coeffs dst)
static FORCE_INLINE number n_SubringGcd(number a, number b, const coeffs r)
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 BOOLEAN nCoeff_is_Q_algext(const coeffs r)
is it an alg. ext. of Q?
number naNeg(number a, const coeffs cf)
this is in-place, modifies a
static void p_Setm(poly p, const ring r)
BOOLEAN rSamePolyRep(ring r1, ring r2)
returns TRUE, if r1 and r2 represents the monomials in the same way FALSE, otherwise this is an analo...
static long p_Totaldegree(poly p, const ring r)
static BOOLEAN p_IsConstant(const poly p, const ring r)
void naDelete(number *a, const coeffs cf)
number naCopyTrans2AlgExt(number a, const coeffs src, const coeffs dst)
const CanonicalForm int s
static poly p_GcdHelper(poly &p, poly &q, const ring r)
see p_Gcd; additional assumption: deg(p) >= deg(q); must destroy p and q (unless one of them is retur...
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 nMapFunc n_SetMap(const coeffs src, const coeffs dst)
set the mapping function pointers for translating numbers from src to dst
poly p_ISet(long i, const ring r)
returns the poly representing the integer i
static poly p_Mult_q(poly p, poly q, const ring r)
BOOLEAN naIsMOne(number a, const coeffs cf)
static number & pGetCoeff(poly p)
return an alias to the leading coefficient of p assumes that p != NULL NOTE: not copy
BOOLEAN naEqual(number a, number b, const coeffs cf)
#define n_Test(a, r)
BOOLEAN n_Test(number a, const coeffs r)
poly convFactoryPSingP(const CanonicalForm &f, const ring r)
go into polynomials over an alg. extension recursively
number naMapZ0(number a, const coeffs src, const coeffs dst)
void n2pNormalize(number &a, const coeffs cf)
used for all algebraic extensions, i.e., the top-most extension in an extension tower is algebraic
number naMap0P(number a, const coeffs src, const coeffs dst)
(), see rinteger.h, new impl.