[35aab3] | 1 | #ifndef MPR_NUMERIC_H |
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| 2 | #define MPR_NUMERIC_H |
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| 3 | /**************************************** |
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| 4 | * Computer Algebra System SINGULAR * |
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| 5 | ****************************************/ |
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| 6 | |
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| 7 | |
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| 8 | /* |
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| 9 | * ABSTRACT - multipolynomial resultants - numeric stuff |
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| 10 | * ( root finder, vandermonde system solver, simplex ) |
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| 11 | */ |
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| 12 | |
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| 13 | //-> include & define stuff |
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[6cc7f5] | 14 | #include <coeffs/numbers.h> |
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| 15 | #include <coeffs/mpr_global.h> |
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| 16 | #include <coeffs/mpr_complex.h> |
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[35aab3] | 17 | |
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| 18 | // define polish mode when finding roots |
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| 19 | #define PM_NONE 0 |
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| 20 | #define PM_POLISH 1 |
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| 21 | #define PM_CORRUPT 2 |
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| 22 | //<- |
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| 23 | |
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| 24 | //-> vandermonde system solver |
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| 25 | /** |
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| 26 | * vandermonde system solver for interpolating polynomials from their values |
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| 27 | */ |
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| 28 | class vandermonde |
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| 29 | { |
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| 30 | public: |
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| 31 | vandermonde( const long _cn, const long _n, |
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| 32 | const long _maxdeg, number *_p, const bool _homog = true ); |
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| 33 | ~vandermonde(); |
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| 34 | |
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| 35 | /** Solves the Vandermode linear system |
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| 36 | * \sum_{i=1}^{n} x_i^k-1 w_i = q_k, k=1,..,n. |
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| 37 | * Any computations are done using type number to get high pecision results. |
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| 38 | * @param q n-tuple of results (right hand of equations) |
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| 39 | * @return w n-tuple of coefficients of resulting polynomial, lowest deg first |
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| 40 | */ |
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| 41 | number * interpolateDense( const number * q ); |
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| 42 | |
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| 43 | poly numvec2poly(const number * q ); |
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| 44 | |
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| 45 | private: |
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| 46 | void init(); |
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| 47 | |
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| 48 | private: |
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| 49 | long n; // number of variables |
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| 50 | long cn; // real number of coefficients of poly to interpolate |
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| 51 | long maxdeg; // degree of the polynomial to interpolate |
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| 52 | long l; // max number of coefficients in poly of deg maxdeg = (maxdeg+1)^n |
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| 53 | |
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| 54 | number *p; // evaluation point |
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| 55 | number *x; // coefficients, determinend by init() from *p |
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| 56 | |
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| 57 | bool homog; |
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| 58 | }; |
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| 59 | //<- |
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| 60 | |
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| 61 | //-> rootContainer |
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| 62 | /** |
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| 63 | * complex root finder for univariate polynomials based on laguers algorithm |
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| 64 | */ |
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| 65 | class rootContainer |
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| 66 | { |
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| 67 | public: |
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| 68 | enum rootType { none, cspecial, cspecialmu, det, onepoly }; |
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| 69 | |
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| 70 | rootContainer(); |
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| 71 | ~rootContainer(); |
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| 72 | |
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| 73 | void fillContainer( number *_coeffs, number *_ievpoint, |
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| 74 | const int _var, const int _tdg, |
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| 75 | const rootType _rt, const int _anz ); |
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| 76 | |
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| 77 | bool solver( const int polishmode= PM_NONE ); |
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| 78 | |
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| 79 | poly getPoly(); |
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| 80 | |
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| 81 | //gmp_complex & operator[] ( const int i ); |
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[e2ad5d] | 82 | inline gmp_complex & operator[] ( const int i ) |
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| 83 | { |
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[35aab3] | 84 | return *theroots[i]; |
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| 85 | } |
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| 86 | gmp_complex & evPointCoord( const int i ); |
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| 87 | |
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[e2ad5d] | 88 | inline gmp_complex * getRoot( const int i ) |
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| 89 | { |
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[35aab3] | 90 | return theroots[i]; |
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| 91 | } |
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| 92 | |
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| 93 | bool swapRoots( const int from, const int to ); |
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| 94 | |
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| 95 | int getAnzElems() { return anz; } |
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| 96 | int getLDim() { return anz; } |
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| 97 | int getAnzRoots() { return tdg; } |
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| 98 | |
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| 99 | private: |
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| 100 | rootContainer( const rootContainer & v ); |
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| 101 | |
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| 102 | /** Given the degree tdg and the tdg+1 complex coefficients ad[0..tdg] |
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| 103 | * (generated from the number coefficients coeffs[0..tdg]) of the polynomial |
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| 104 | * this routine successively calls "laguer" and finds all m complex roots in |
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| 105 | * roots[0..tdg]. The bool var "polish" should be input as "true" if polishing |
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| 106 | * (also by "laguer") is desired, "false" if the roots will be subsequently |
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| 107 | * polished by other means. |
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| 108 | */ |
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| 109 | bool laguer_driver( gmp_complex ** a, gmp_complex ** roots, bool polish = true ); |
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| 110 | bool isfloat(gmp_complex **a); |
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| 111 | void divlin(gmp_complex **a, gmp_complex x, int j); |
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| 112 | void divquad(gmp_complex **a, gmp_complex x, int j); |
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| 113 | void solvequad(gmp_complex **a, gmp_complex **r, int &k, int &j); |
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| 114 | void sortroots(gmp_complex **roots, int r, int c, bool isf); |
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| 115 | void sortre(gmp_complex **r, int l, int u, int inc); |
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| 116 | |
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| 117 | /** Given the degree m and the m+1 complex coefficients a[0..m] of the |
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| 118 | * polynomial, and given the complex value x, this routine improves x by |
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| 119 | * Laguerre's method until it converges, within the achievable roundoff limit, |
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| 120 | * to a root of the given polynomial. The number of iterations taken is |
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| 121 | * returned at its. |
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| 122 | */ |
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| 123 | void laguer(gmp_complex ** a, int m, gmp_complex * x, int * its, bool type); |
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| 124 | void computefx(gmp_complex **a, gmp_complex x, int m, |
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| 125 | gmp_complex &f0, gmp_complex &f1, gmp_complex &f2, |
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| 126 | gmp_float &ex, gmp_float &ef); |
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| 127 | void computegx(gmp_complex **a, gmp_complex x, int m, |
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| 128 | gmp_complex &f0, gmp_complex &f1, gmp_complex &f2, |
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| 129 | gmp_float &ex, gmp_float &ef); |
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| 130 | void checkimag(gmp_complex *x, gmp_float &e); |
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| 131 | |
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| 132 | int var; |
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| 133 | int tdg; |
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| 134 | |
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| 135 | number * coeffs; |
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| 136 | number * ievpoint; |
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| 137 | rootType rt; |
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| 138 | |
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| 139 | gmp_complex ** theroots; |
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| 140 | |
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| 141 | int anz; |
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| 142 | bool found_roots; |
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| 143 | }; |
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| 144 | //<- |
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| 145 | |
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[213d64] | 146 | class slists; typedef slists * lists; |
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| 147 | |
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[35aab3] | 148 | //-> class rootArranger |
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| 149 | class rootArranger |
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| 150 | { |
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| 151 | public: |
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[213d64] | 152 | friend lists listOfRoots( rootArranger*, const unsigned int oprec ); |
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| 153 | |
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[35aab3] | 154 | rootArranger( rootContainer ** _roots, |
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| 155 | rootContainer ** _mu, |
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| 156 | const int _howclean = PM_CORRUPT ); |
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| 157 | ~rootArranger() {} |
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| 158 | |
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| 159 | void solve_all(); |
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| 160 | void arrange(); |
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| 161 | |
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| 162 | const bool success() { return found_roots; } |
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| 163 | |
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| 164 | private: |
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| 165 | rootArranger( const rootArranger & ); |
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| 166 | |
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| 167 | rootContainer ** roots; |
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| 168 | rootContainer ** mu; |
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| 169 | |
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| 170 | int howclean; |
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| 171 | int rc,mc; |
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| 172 | bool found_roots; |
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| 173 | }; |
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[08e15e7] | 174 | |
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| 175 | |
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| 176 | |
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[35aab3] | 177 | //<- |
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| 178 | |
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| 179 | //-> simplex computation |
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| 180 | // (used by sparse matrix construction) |
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| 181 | #define SIMPLEX_EPS 1.0e-12 |
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| 182 | |
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| 183 | /** Linear Programming / Linear Optimization using Simplex - Algorithm |
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| 184 | * |
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| 185 | * On output, the tableau LiPM is indexed by two arrays of integers. |
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| 186 | * ipsov[j] contains, for j=1..m, the number i whose original variable |
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| 187 | * is now represented by row j+1 of LiPM. (left-handed vars in solution) |
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| 188 | * (first row is the one with the objective function) |
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| 189 | * izrov[j] contains, for j=1..n, the number i whose original variable |
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| 190 | * x_i is now a right-handed variable, rep. by column j+1 of LiPM. |
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| 191 | * These vars are all zero in the solution. The meaning of n<i<n+m1+m2 |
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| 192 | * is the same as above. |
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| 193 | */ |
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| 194 | class simplex |
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| 195 | { |
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| 196 | public: |
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| 197 | |
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| 198 | int m; // number of constraints, make sure m == m1 + m2 + m3 !! |
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| 199 | int n; // # of independent variables |
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| 200 | int m1,m2,m3; // constraints <=, >= and == |
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| 201 | int icase; // == 0: finite solution found; |
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| 202 | // == +1 objective funtion unbound; == -1: no solution |
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| 203 | int *izrov,*iposv; |
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| 204 | |
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| 205 | mprfloat **LiPM; // the matrix (of size [m+2, n+1]) |
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| 206 | |
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| 207 | /** #rows should be >= m+2, #cols >= n+1 |
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| 208 | */ |
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| 209 | simplex( int rows, int cols ); |
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| 210 | ~simplex(); |
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| 211 | |
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| 212 | BOOLEAN mapFromMatrix( matrix m ); |
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| 213 | matrix mapToMatrix( matrix m ); |
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| 214 | intvec * posvToIV(); |
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| 215 | intvec * zrovToIV(); |
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| 216 | |
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| 217 | void compute(); |
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| 218 | |
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| 219 | private: |
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| 220 | simplex( const simplex & ); |
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| 221 | void simp1( mprfloat **a, int mm, int ll[], int nll, int iabf, int *kp, mprfloat *bmax ); |
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| 222 | void simp2( mprfloat **a, int n, int l2[], int nl2, int *ip, int kp, mprfloat *q1 ); |
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| 223 | void simp3( mprfloat **a, int i1, int k1, int ip, int kp ); |
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| 224 | |
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| 225 | int LiPM_cols,LiPM_rows; |
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| 226 | }; |
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| 227 | |
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| 228 | //<- |
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| 229 | |
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[91f1a3] | 230 | #endif /*MPR_NUMERIC_H*/ |
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[35aab3] | 231 | |
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| 232 | // local Variables: *** |
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| 233 | // folded-file: t *** |
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| 234 | // compile-command-1: "make installg" *** |
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| 235 | // compile-command-2: "make install" *** |
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| 236 | // End: *** |
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