| 1 | /*****************************************************************************\ |
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| 2 | * Computer Algebra System SINGULAR |
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| 3 | \*****************************************************************************/ |
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| 4 | /** @file facBivar.cc |
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| 5 | * |
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| 6 | * bivariate factorization over Q(a) |
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| 7 | * |
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| 8 | * @author Martin Lee |
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| 9 | * |
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| 10 | * @internal @version \$Id$ |
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| 11 | * |
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| 12 | **/ |
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| 13 | /*****************************************************************************/ |
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| 14 | |
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| 15 | #include "config.h" |
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| 16 | |
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| 17 | #include "assert.h" |
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| 18 | #include "debug.h" |
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| 19 | #include "timing.h" |
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| 20 | |
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| 21 | #include "cf_algorithm.h" |
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| 22 | #include "facFqBivar.h" |
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| 23 | #include "facBivar.h" |
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| 24 | #include "facHensel.h" |
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| 25 | #include "facMul.h" |
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| 26 | |
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| 27 | #ifdef HAVE_NTL |
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| 28 | TIMING_DEFINE_PRINT(uni_factorize) |
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| 29 | TIMING_DEFINE_PRINT(hensel_lift12) |
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| 30 | |
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| 31 | CFList conv (const CFFList& L) |
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| 32 | { |
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| 33 | CFList result; |
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| 34 | for (CFFListIterator i= L; i.hasItem(); i++) |
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| 35 | result.append (i.getItem().factor()); |
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| 36 | return result; |
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| 37 | } |
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| 38 | |
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| 39 | bool testPoint (const CanonicalForm& F, CanonicalForm& G, int i) |
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| 40 | { |
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| 41 | G= F (i, 2); |
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| 42 | if (G.inCoeffDomain() || degree (F, 1) > degree (G, 1)) |
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| 43 | return false; |
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| 44 | |
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| 45 | if (degree (gcd (deriv (G, G.mvar()), G)) > 0) |
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| 46 | return false; |
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| 47 | return true; |
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| 48 | } |
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| 49 | |
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| 50 | CanonicalForm evalPoint (const CanonicalForm& F, int& i) |
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| 51 | { |
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| 52 | Variable x= Variable (1); |
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| 53 | Variable y= Variable (2); |
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| 54 | CanonicalForm result; |
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| 55 | |
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| 56 | int k; |
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| 57 | |
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| 58 | if (i == 0) |
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| 59 | { |
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| 60 | if (testPoint (F, result, i)) |
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| 61 | return result; |
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| 62 | } |
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| 63 | do |
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| 64 | { |
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| 65 | if (i > 0) |
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| 66 | k= 1; |
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| 67 | else |
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| 68 | k= 2; |
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| 69 | while (k < 3) |
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| 70 | { |
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| 71 | if (k == 1) |
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| 72 | { |
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| 73 | if (testPoint (F, result, i)) |
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| 74 | return result; |
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| 75 | } |
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| 76 | else |
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| 77 | { |
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| 78 | if (testPoint (F, result, -i)) |
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| 79 | return result; |
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| 80 | } |
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| 81 | k++; |
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| 82 | } |
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| 83 | i++; |
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| 84 | } while (1); |
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| 85 | } |
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| 86 | |
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| 87 | CFList |
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| 88 | earlyFactorDetection0 (CanonicalForm& F, CFList& factors,int& adaptedLiftBound, |
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| 89 | DegreePattern& degs, bool& success, int deg) |
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| 90 | { |
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| 91 | DegreePattern bufDegs1= degs; |
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| 92 | DegreePattern bufDegs2; |
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| 93 | CFList result; |
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| 94 | CFList T= factors; |
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| 95 | CanonicalForm buf= F; |
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| 96 | CanonicalForm LCBuf= LC (buf, Variable (1)); |
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| 97 | CanonicalForm g, quot; |
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| 98 | CanonicalForm M= power (F.mvar(), deg); |
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| 99 | adaptedLiftBound= 0; |
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| 100 | int d= degree (F) + degree (LCBuf, F.mvar()); |
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| 101 | for (CFListIterator i= factors; i.hasItem(); i++) |
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| 102 | { |
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| 103 | if (!bufDegs1.find (degree (i.getItem(), 1))) |
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| 104 | continue; |
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| 105 | else |
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| 106 | { |
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| 107 | g= i.getItem() (0, 1); |
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| 108 | g *= LCBuf; |
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| 109 | g= mod (g, M); |
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| 110 | if (fdivides (LC (g), LCBuf)) |
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| 111 | { |
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| 112 | g= mulMod2 (i.getItem(), LCBuf, M); |
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| 113 | g /= content (g, Variable (1)); |
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| 114 | if (fdivides (g, buf, quot)) |
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| 115 | { |
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| 116 | result.append (g); |
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| 117 | buf= quot; |
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| 118 | d -= degree (g) + degree (LC (g, Variable (1)), F.mvar()); |
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| 119 | LCBuf= LC (buf, Variable (1)); |
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| 120 | T= Difference (T, CFList (i.getItem())); |
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| 121 | |
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| 122 | // compute new possible degree pattern |
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| 123 | bufDegs2= DegreePattern (T); |
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| 124 | bufDegs1.intersect (bufDegs2); |
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| 125 | bufDegs1.refine (); |
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| 126 | if (bufDegs1.getLength() <= 1) |
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| 127 | { |
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| 128 | result.append (buf); |
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| 129 | break; |
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| 130 | } |
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| 131 | } |
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| 132 | } |
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| 133 | } |
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| 134 | } |
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| 135 | adaptedLiftBound= d + 1; |
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| 136 | if (d < deg) |
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| 137 | { |
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| 138 | factors= T; |
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| 139 | degs= bufDegs1; |
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| 140 | F= buf; |
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| 141 | success= true; |
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| 142 | } |
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| 143 | if (bufDegs1.getLength() <= 1) |
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| 144 | degs= bufDegs1; |
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| 145 | return result; |
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| 146 | } |
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| 147 | |
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| 148 | |
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| 149 | CFList |
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| 150 | henselLiftAndEarly0 (CanonicalForm& A, bool& earlySuccess, CFList& |
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| 151 | earlyFactors, DegreePattern& degs, int& liftBound, |
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| 152 | const CFList& uniFactors, const CanonicalForm& eval) |
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| 153 | { |
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| 154 | int sizeOfLiftPre; |
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| 155 | int * liftPre= getLiftPrecisions (A, sizeOfLiftPre, degree (LC (A, 1), 2)); |
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| 156 | |
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| 157 | Variable x= Variable (1); |
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| 158 | Variable y= Variable (2); |
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| 159 | CFArray Pi; |
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| 160 | CFList diophant; |
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| 161 | CFList bufUniFactors= uniFactors; |
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| 162 | bufUniFactors.insert (LC (A, x)); |
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| 163 | CFMatrix M= CFMatrix (liftBound, bufUniFactors.length() - 1); |
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| 164 | earlySuccess= false; |
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| 165 | int newLiftBound= 0; |
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| 166 | int smallFactorDeg= tmin (11, liftPre [sizeOfLiftPre- 1] + 1);//this is a tunable parameter |
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| 167 | if (smallFactorDeg >= liftBound || degree (A,y) <= 4) |
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| 168 | henselLift12 (A, bufUniFactors, liftBound, Pi, diophant, M); |
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| 169 | else if (sizeOfLiftPre > 1 && sizeOfLiftPre < 30) |
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| 170 | { |
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| 171 | henselLift12 (A, bufUniFactors, smallFactorDeg, Pi, diophant, M); |
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| 172 | earlyFactors= earlyFactorDetection0 (A, bufUniFactors, newLiftBound, |
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| 173 | degs, earlySuccess, |
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| 174 | smallFactorDeg); |
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| 175 | if (degs.getLength() > 1 && !earlySuccess && |
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| 176 | smallFactorDeg != liftPre [sizeOfLiftPre-1] + 1) |
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| 177 | { |
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| 178 | if (newLiftBound > liftPre[sizeOfLiftPre-1]+1) |
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| 179 | { |
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| 180 | bufUniFactors.insert (LC (A, x)); |
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| 181 | henselLiftResume12 (A, bufUniFactors, smallFactorDeg, |
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| 182 | liftPre[sizeOfLiftPre-1] + 1, Pi, diophant, M); |
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| 183 | earlyFactors= earlyFactorDetection0 (A, bufUniFactors, newLiftBound, |
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| 184 | degs, earlySuccess, liftPre[sizeOfLiftPre-1] + 1); |
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| 185 | } |
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| 186 | } |
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| 187 | else if (earlySuccess) |
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| 188 | liftBound= newLiftBound; |
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| 189 | int i= sizeOfLiftPre - 1; |
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| 190 | while (degs.getLength() > 1 && !earlySuccess && i - 1 >= 0) |
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| 191 | { |
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| 192 | if (newLiftBound > liftPre[i] + 1) |
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| 193 | { |
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| 194 | bufUniFactors.insert (LC (A, x)); |
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| 195 | henselLiftResume12 (A, bufUniFactors, liftPre[i] + 1, |
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| 196 | liftPre[i-1] + 1, Pi, diophant, M); |
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| 197 | earlyFactors= earlyFactorDetection0 (A, bufUniFactors, newLiftBound, |
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| 198 | degs, earlySuccess, liftPre[i-1] + 1); |
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| 199 | } |
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| 200 | else |
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| 201 | { |
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| 202 | liftBound= newLiftBound; |
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| 203 | break; |
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| 204 | } |
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| 205 | i--; |
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| 206 | } |
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| 207 | if (earlySuccess) |
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| 208 | liftBound= newLiftBound; |
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| 209 | //after here all factors are lifted to liftPre[sizeOfLiftPre-1] |
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| 210 | } |
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| 211 | else |
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| 212 | { |
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| 213 | henselLift12 (A, bufUniFactors, smallFactorDeg, Pi, diophant, M); |
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| 214 | earlyFactors= earlyFactorDetection0 (A, bufUniFactors, newLiftBound, |
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| 215 | degs, earlySuccess, |
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| 216 | smallFactorDeg); |
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| 217 | int i= 1; |
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| 218 | while ((degree (A,y)/4)*i + 4 <= smallFactorDeg) |
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| 219 | i++; |
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| 220 | if (degs.getLength() > 1 && !earlySuccess) |
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| 221 | { |
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| 222 | bufUniFactors.insert (LC (A, x)); |
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| 223 | henselLiftResume12 (A, bufUniFactors, smallFactorDeg, |
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| 224 | (degree (A, y)/4)*i + 4, Pi, diophant, M); |
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| 225 | earlyFactors= earlyFactorDetection0 (A, bufUniFactors, newLiftBound, |
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| 226 | degs, earlySuccess, (degree (A, y)/4)*i + 4); |
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| 227 | } |
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| 228 | while (degs.getLength() > 1 && !earlySuccess && i < 4) |
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| 229 | { |
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| 230 | if (newLiftBound > (degree (A, y)/4)*i + 4) |
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| 231 | { |
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| 232 | bufUniFactors.insert (LC (A, x)); |
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| 233 | henselLiftResume12 (A, bufUniFactors, (degree (A,y)/4)*i + 4, |
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| 234 | (degree (A, y)/4)*(i+1) + 4, Pi, diophant, M); |
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| 235 | earlyFactors= earlyFactorDetection0 (A, bufUniFactors, newLiftBound, |
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| 236 | degs, earlySuccess, (degree (A, y)/4)*(i+1) + 4); |
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| 237 | } |
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| 238 | else |
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| 239 | { |
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| 240 | liftBound= newLiftBound; |
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| 241 | break; |
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| 242 | } |
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| 243 | i++; |
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| 244 | } |
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| 245 | if (earlySuccess) |
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| 246 | liftBound= newLiftBound; |
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| 247 | } |
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| 248 | |
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| 249 | return bufUniFactors; |
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| 250 | } |
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| 251 | |
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| 252 | CFList biFactorize (const CanonicalForm& F, const Variable& v) |
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| 253 | { |
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| 254 | if (F.inCoeffDomain()) |
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| 255 | return CFList(F); |
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| 256 | |
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| 257 | bool extension= (v.level() != 1); |
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| 258 | CanonicalForm A; |
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| 259 | CanonicalForm multiplier= 1; |
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| 260 | if (isOn (SW_RATIONAL)) |
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| 261 | A= F*bCommonDen (F); |
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| 262 | else |
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| 263 | A= F; |
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| 264 | |
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| 265 | if (A.isUnivariate()) |
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| 266 | { |
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| 267 | CFFList buf; |
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| 268 | if (extension) |
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| 269 | buf= factorize (A, v); |
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| 270 | else |
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| 271 | buf= factorize (A, true); |
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| 272 | CFList result= conv (buf); |
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| 273 | if (result.getFirst().inCoeffDomain()) |
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| 274 | result.removeFirst(); |
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| 275 | return result; |
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| 276 | } |
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| 277 | |
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| 278 | CFMap N; |
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| 279 | A= compress (A, N); |
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| 280 | Variable y= A.mvar(); |
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| 281 | |
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| 282 | if (y.level() > 2) return CFList (F); |
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| 283 | Variable x= Variable (1); |
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| 284 | |
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| 285 | CanonicalForm contentAx= content (A, x); |
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| 286 | CanonicalForm contentAy= content (A); |
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| 287 | |
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| 288 | A= A/(contentAx*contentAy); |
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| 289 | CFFList contentAxFactors, contentAyFactors; |
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| 290 | |
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| 291 | if (extension) |
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| 292 | { |
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| 293 | if (!contentAx.inCoeffDomain()) |
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| 294 | { |
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| 295 | contentAxFactors= factorize (contentAx, v); |
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| 296 | if (contentAxFactors.getFirst().factor().inCoeffDomain()) |
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| 297 | contentAxFactors.removeFirst(); |
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| 298 | } |
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| 299 | if (!contentAy.inCoeffDomain()) |
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| 300 | { |
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| 301 | contentAyFactors= factorize (contentAy, v); |
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| 302 | if (contentAyFactors.getFirst().factor().inCoeffDomain()) |
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| 303 | contentAyFactors.removeFirst(); |
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| 304 | } |
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| 305 | } |
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| 306 | else |
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| 307 | { |
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| 308 | if (!contentAx.inCoeffDomain()) |
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| 309 | { |
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| 310 | contentAxFactors= factorize (contentAx, true); |
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| 311 | if (contentAxFactors.getFirst().factor().inCoeffDomain()) |
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| 312 | contentAxFactors.removeFirst(); |
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| 313 | } |
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| 314 | if (!contentAy.inCoeffDomain()) |
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| 315 | { |
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| 316 | contentAyFactors= factorize (contentAy, true); |
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| 317 | if (contentAyFactors.getFirst().factor().inCoeffDomain()) |
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| 318 | contentAyFactors.removeFirst(); |
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| 319 | } |
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| 320 | } |
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| 321 | |
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| 322 | //check trivial case |
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| 323 | if (degree (A) == 1 || degree (A, 1) == 1) |
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| 324 | { |
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| 325 | CFList factors; |
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| 326 | factors.append (A); |
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| 327 | |
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| 328 | appendSwapDecompress (factors, conv (contentAxFactors), |
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| 329 | conv (contentAyFactors), false, false, N); |
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| 330 | |
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| 331 | normalize (factors); |
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| 332 | return factors; |
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| 333 | } |
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| 334 | |
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| 335 | //trivial case |
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| 336 | CFList factors; |
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| 337 | if (A.inCoeffDomain()) |
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| 338 | { |
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| 339 | append (factors, conv (contentAxFactors)); |
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| 340 | append (factors, conv (contentAyFactors)); |
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| 341 | decompress (factors, N); |
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| 342 | return factors; |
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| 343 | } |
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| 344 | else if (A.isUnivariate()) |
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| 345 | { |
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| 346 | if (extension) |
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| 347 | factors= conv (factorize (A, v)); |
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| 348 | else |
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| 349 | factors= conv (factorize (A, true)); |
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| 350 | append (factors, conv (contentAxFactors)); |
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| 351 | append (factors, conv (contentAyFactors)); |
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| 352 | decompress (factors, N); |
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| 353 | return factors; |
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| 354 | } |
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| 355 | |
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| 356 | bool swap= false; |
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| 357 | if (degree (A) > degree (A, x)) |
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| 358 | { |
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| 359 | A= swapvar (A, y, x); |
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| 360 | swap= true; |
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| 361 | } |
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| 362 | |
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| 363 | CanonicalForm Aeval, bufAeval, buf; |
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| 364 | CFList uniFactors, list, bufUniFactors; |
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| 365 | DegreePattern degs; |
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| 366 | DegreePattern bufDegs; |
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| 367 | |
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| 368 | CanonicalForm Aeval2, bufAeval2; |
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| 369 | CFList bufUniFactors2, list2, uniFactors2; |
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| 370 | DegreePattern degs2; |
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| 371 | DegreePattern bufDegs2; |
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| 372 | bool swap2= false; |
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| 373 | |
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| 374 | // several univariate factorizations to obtain more information about the |
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| 375 | // degree pattern therefore usually less combinations have to be tried during |
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| 376 | // the recombination process |
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| 377 | int factorNums= 2; |
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| 378 | int subCheck1= substituteCheck (A, x); |
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| 379 | int subCheck2= substituteCheck (A, y); |
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| 380 | buf= swapvar (A,x,y); |
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| 381 | int evaluation, evaluation2, bufEvaluation= 0, bufEvaluation2= 0; |
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| 382 | for (int i= 0; i < factorNums; i++) |
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| 383 | { |
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| 384 | bufAeval= A; |
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| 385 | bufAeval= evalPoint (A, bufEvaluation); |
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| 386 | |
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| 387 | bufAeval2= buf; |
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| 388 | bufAeval2= evalPoint (buf, bufEvaluation2); |
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| 389 | |
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| 390 | |
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| 391 | // univariate factorization |
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| 392 | TIMING_START (uni_factorize); |
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| 393 | |
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| 394 | if (extension) |
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| 395 | bufUniFactors= conv (factorize (bufAeval, v)); |
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| 396 | else |
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| 397 | bufUniFactors= conv (factorize (bufAeval, true)); |
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| 398 | TIMING_END_AND_PRINT (uni_factorize, |
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| 399 | "time for univariate factorization: "); |
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| 400 | DEBOUTLN (cerr, "Lc (bufAeval)*prod (bufUniFactors)== bufAeval " << |
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| 401 | (prod (bufUniFactors)*Lc (bufAeval) == bufAeval)); |
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| 402 | |
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| 403 | TIMING_START (uni_factorize); |
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| 404 | if (extension) |
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| 405 | bufUniFactors2= conv (factorize (bufAeval2, v)); |
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| 406 | else |
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| 407 | bufUniFactors2= conv (factorize (bufAeval2, true)); |
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| 408 | TIMING_END_AND_PRINT (uni_factorize, |
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| 409 | "time for univariate factorization in y: "); |
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| 410 | DEBOUTLN (cerr, "Lc (Aeval2)*prod (uniFactors2)== Aeval2 " << |
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| 411 | (prod (bufUniFactors2)*Lc (bufAeval2) == bufAeval2)); |
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| 412 | |
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| 413 | if (bufUniFactors.getFirst().inCoeffDomain()) |
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| 414 | bufUniFactors.removeFirst(); |
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| 415 | if (bufUniFactors2.getFirst().inCoeffDomain()) |
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| 416 | bufUniFactors2.removeFirst(); |
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| 417 | if (bufUniFactors.length() == 1 || bufUniFactors2.length() == 1) |
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| 418 | { |
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| 419 | factors.append (A); |
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| 420 | |
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| 421 | appendSwapDecompress (factors, conv (contentAxFactors), |
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| 422 | conv (contentAyFactors), swap, swap2, N); |
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| 423 | |
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| 424 | if (isOn (SW_RATIONAL)) |
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| 425 | normalize (factors); |
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| 426 | return factors; |
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| 427 | } |
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| 428 | |
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| 429 | if (i == 0) |
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| 430 | { |
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| 431 | if (subCheck1 > 0) |
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| 432 | { |
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| 433 | int subCheck= substituteCheck (bufUniFactors); |
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| 434 | |
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| 435 | if (subCheck > 1 && (subCheck1%subCheck == 0)) |
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| 436 | { |
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| 437 | CanonicalForm bufA= A; |
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| 438 | subst (bufA, bufA, subCheck, x); |
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| 439 | factors= biFactorize (bufA, v); |
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| 440 | reverseSubst (factors, subCheck, x); |
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| 441 | appendSwapDecompress (factors, conv (contentAxFactors), |
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| 442 | conv (contentAyFactors), swap, swap2, N); |
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| 443 | if (isOn (SW_RATIONAL)) |
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| 444 | normalize (factors); |
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| 445 | return factors; |
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| 446 | } |
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| 447 | } |
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| 448 | |
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| 449 | if (subCheck2 > 0) |
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| 450 | { |
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| 451 | int subCheck= substituteCheck (bufUniFactors2); |
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| 452 | |
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| 453 | if (subCheck > 1 && (subCheck2%subCheck == 0)) |
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| 454 | { |
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| 455 | CanonicalForm bufA= A; |
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| 456 | subst (bufA, bufA, subCheck, y); |
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| 457 | factors= biFactorize (bufA, v); |
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| 458 | reverseSubst (factors, subCheck, y); |
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| 459 | appendSwapDecompress (factors, conv (contentAxFactors), |
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| 460 | conv (contentAyFactors), swap, swap2, N); |
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| 461 | if (isOn (SW_RATIONAL)) |
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| 462 | normalize (factors); |
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| 463 | return factors; |
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| 464 | } |
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| 465 | } |
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| 466 | } |
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| 467 | |
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| 468 | // degree analysis |
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| 469 | bufDegs = DegreePattern (bufUniFactors); |
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| 470 | bufDegs2= DegreePattern (bufUniFactors2); |
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| 471 | |
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| 472 | if (i == 0) |
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| 473 | { |
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| 474 | Aeval= bufAeval; |
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| 475 | evaluation= bufEvaluation; |
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| 476 | uniFactors= bufUniFactors; |
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| 477 | degs= bufDegs; |
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| 478 | Aeval2= bufAeval2; |
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| 479 | evaluation2= bufEvaluation2; |
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| 480 | uniFactors2= bufUniFactors2; |
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| 481 | degs2= bufDegs2; |
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| 482 | } |
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| 483 | else |
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| 484 | { |
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| 485 | degs.intersect (bufDegs); |
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| 486 | degs2.intersect (bufDegs2); |
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| 487 | if (bufUniFactors2.length() < uniFactors2.length()) |
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| 488 | { |
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| 489 | uniFactors2= bufUniFactors2; |
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| 490 | Aeval2= bufAeval2; |
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| 491 | evaluation2= bufEvaluation2; |
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| 492 | } |
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| 493 | if (bufUniFactors.length() < uniFactors.length()) |
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| 494 | { |
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| 495 | if (evaluation != 0) |
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| 496 | { |
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| 497 | uniFactors= bufUniFactors; |
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| 498 | Aeval= bufAeval; |
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| 499 | evaluation= bufEvaluation; |
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| 500 | } |
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| 501 | } |
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| 502 | } |
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| 503 | bufEvaluation++; |
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| 504 | bufEvaluation2++; |
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| 505 | } |
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| 506 | |
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| 507 | if (evaluation != 0 && (uniFactors.length() > uniFactors2.length() || |
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| 508 | (uniFactors.length() == uniFactors2.length() |
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| 509 | && degs.getLength() > degs2.getLength()))) |
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| 510 | { |
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| 511 | degs= degs2; |
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| 512 | uniFactors= uniFactors2; |
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| 513 | evaluation= evaluation2; |
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| 514 | Aeval= Aeval2; |
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| 515 | A= buf; |
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| 516 | swap2= true; |
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| 517 | } |
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| 518 | |
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| 519 | |
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| 520 | if (degs.getLength() == 1) // A is irreducible |
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| 521 | { |
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| 522 | factors.append (A); |
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| 523 | appendSwapDecompress (factors, conv (contentAxFactors), |
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| 524 | conv (contentAyFactors), swap, swap2, N); |
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| 525 | if (isOn (SW_RATIONAL)) |
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| 526 | normalize (factors); |
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| 527 | return factors; |
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| 528 | } |
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| 529 | |
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| 530 | if (!extension) |
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| 531 | { |
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| 532 | for (CFListIterator i= uniFactors; i.hasItem(); i++) |
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| 533 | i.getItem() /= lc (i.getItem()); |
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| 534 | } |
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| 535 | |
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| 536 | A= A (y + evaluation, y); |
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| 537 | |
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| 538 | int liftBound= degree (A, y) + 1; |
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| 539 | |
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| 540 | ExtensionInfo dummy= ExtensionInfo (false); |
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| 541 | bool earlySuccess= false; |
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| 542 | CFList earlyFactors; |
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| 543 | TIMING_START (fac_hensel_lift); |
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| 544 | //maybe one should use a multiple of LC (A,1) and try a nonmonic lifting here? |
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| 545 | uniFactors= henselLiftAndEarly |
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| 546 | (A, earlySuccess, earlyFactors, degs, liftBound, |
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| 547 | uniFactors, dummy, evaluation); |
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| 548 | TIMING_END_AND_PRINT (fac_hensel_lift, "time for hensel lifting: "); |
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| 549 | DEBOUTLN (cerr, "lifted factors= " << uniFactors); |
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| 550 | |
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| 551 | CanonicalForm MODl= power (y, liftBound); |
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| 552 | |
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| 553 | factors= factorRecombination (uniFactors, A, MODl, degs, 1, |
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| 554 | uniFactors.length()/2); |
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| 555 | |
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| 556 | if (earlySuccess) |
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| 557 | factors= Union (earlyFactors, factors); |
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| 558 | else if (!earlySuccess && degs.getLength() == 1) |
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| 559 | factors= earlyFactors; |
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| 560 | |
|---|
| 561 | for (CFListIterator i= factors; i.hasItem(); i++) |
|---|
| 562 | i.getItem()= i.getItem() (y - evaluation, y); |
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| 563 | |
|---|
| 564 | appendSwapDecompress (factors, conv (contentAxFactors), |
|---|
| 565 | conv (contentAyFactors), swap, swap2, N); |
|---|
| 566 | if (isOn (SW_RATIONAL)) |
|---|
| 567 | normalize (factors); |
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| 568 | |
|---|
| 569 | return factors; |
|---|
| 570 | } |
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| 571 | |
|---|
| 572 | #endif |
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