[a5f15a] | 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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[9a6c4a] | 6 | |
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[c7f3b7] | 7 | /* $Id: mpr_numeric.h,v 1.5 1999-12-02 23:03:52 wenk Exp $ */ |
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[a5f15a] | 8 | |
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[e858e7] | 9 | /* |
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[a5f15a] | 10 | * ABSTRACT - multipolynomial resultants - numeric stuff |
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[e858e7] | 11 | * ( root finder, vandermonde system solver, simplex ) |
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[a5f15a] | 12 | */ |
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| 13 | |
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| 14 | //-> include & define stuff |
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| 15 | #include "numbers.h" |
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| 16 | #include "mpr_global.h" |
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| 17 | #include "mpr_complex.h" |
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| 18 | |
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| 19 | // define polish mode when finding roots |
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[9a6c4a] | 20 | #define PM_NONE 0 |
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| 21 | #define PM_POLISH 1 |
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| 22 | #define PM_CORRUPT 2 |
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[a5f15a] | 23 | //<- |
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| 24 | |
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| 25 | //-> vandermonde system solver |
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| 26 | /** |
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| 27 | * vandermonde system solver for interpolating polynomials from their values |
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| 28 | */ |
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| 29 | class vandermonde |
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| 30 | { |
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| 31 | public: |
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[e858e7] | 32 | vandermonde( const long _cn, const long _n, |
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[a5f15a] | 33 | const long _maxdeg, number *_p, const bool _homog = true ); |
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| 34 | ~vandermonde(); |
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[e858e7] | 35 | |
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| 36 | /** Solves the Vandermode linear system |
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| 37 | * \sum_{i=1}^{n} x_i^k-1 w_i = q_k, k=1,..,n. |
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[a5f15a] | 38 | * Any computations are done using type number to get high pecision results. |
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| 39 | * @param q n-tuple of results (right hand of equations) |
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| 40 | * @return w n-tuple of coefficients of resulting polynomial, lowest deg first |
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| 41 | */ |
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| 42 | number * interpolateDense( const number * q ); |
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| 43 | |
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| 44 | poly numvec2poly(const number * q ); |
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| 45 | |
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| 46 | private: |
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| 47 | void init(); |
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| 48 | |
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| 49 | private: |
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| 50 | long n; // number of variables |
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| 51 | long cn; // real number of coefficients of poly to interpolate |
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| 52 | long maxdeg; // degree of the polynomial to interpolate |
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| 53 | long l; // max number of coefficients in poly of deg maxdeg = (maxdeg+1)^n |
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| 54 | |
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| 55 | number *p; // evaluation point |
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| 56 | number *x; // coefficients, determinend by init() from *p |
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| 57 | |
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| 58 | bool homog; |
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| 59 | }; |
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| 60 | //<- |
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| 61 | |
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| 62 | //-> rootContainer |
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| 63 | /** |
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| 64 | * complex root finder for univariate polynomials based on laguers algorithm |
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| 65 | */ |
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| 66 | class rootContainer |
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| 67 | { |
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| 68 | public: |
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| 69 | enum rootType { none, cspecial, cspecialmu, det, onepoly }; |
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| 70 | |
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| 71 | rootContainer(); |
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| 72 | ~rootContainer(); |
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| 73 | |
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[e858e7] | 74 | void fillContainer( number *_coeffs, number *_ievpoint, |
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| 75 | const int _var, const int _tdg, |
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[a5f15a] | 76 | const rootType _rt, const int _anz ); |
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| 77 | |
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| 78 | bool solver( const int polishmode= PM_NONE ); |
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| 79 | |
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| 80 | poly getPoly(); |
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| 81 | |
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| 82 | //gmp_complex & operator[] ( const int i ); |
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| 83 | inline gmp_complex & operator[] ( const int i ) { |
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| 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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| 88 | inline gmp_complex * getRoot( const int i ) { |
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| 89 | return theroots[i]; |
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| 90 | } |
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| 91 | |
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| 92 | bool swapRoots( const int from, const int to ); |
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| 93 | |
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| 94 | int getAnzElems() { return anz; } |
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| 95 | int getLDim() { return anz; } |
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| 96 | int getAnzRoots() { return tdg; } |
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| 97 | |
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| 98 | private: |
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| 99 | rootContainer( const rootContainer & v ); |
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| 100 | |
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[e858e7] | 101 | /** Given the degree tdg and the tdg+1 complex coefficients ad[0..tdg] |
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[a5f15a] | 102 | * (generated from the number coefficients coeffs[0..tdg]) of the polynomial |
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[e858e7] | 103 | * this routine successively calls "laguer" and finds all m complex roots in |
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[a5f15a] | 104 | * roots[0..tdg]. The bool var "polish" should be input as "true" if polishing |
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| 105 | * (also by "laguer") is desired, "false" if the roots will be subsequently |
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| 106 | * polished by other means. |
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| 107 | */ |
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| 108 | bool laguer_driver( gmp_complex ** a, gmp_complex ** roots, bool polish = true ); |
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| 109 | |
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| 110 | /** Given the degree m and the m+1 complex coefficients a[0..m] of the |
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| 111 | * polynomial, and given the complex value x, this routine improves x by |
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| 112 | * Laguerre's method until it converges, within the achievable roundoff limit, |
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| 113 | * to a root of the given polynomial. The number of iterations taken is |
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| 114 | * returned at its. |
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| 115 | */ |
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| 116 | void laguer(gmp_complex ** a, int m, gmp_complex * x, int * its); |
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[e858e7] | 117 | |
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[a5f15a] | 118 | int var; |
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| 119 | int tdg; |
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| 120 | |
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| 121 | number * coeffs; |
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| 122 | number * ievpoint; |
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| 123 | rootType rt; |
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| 124 | |
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| 125 | gmp_complex ** theroots; |
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[e858e7] | 126 | |
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[a5f15a] | 127 | int anz; |
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| 128 | bool found_roots; |
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| 129 | }; |
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| 130 | //<- |
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| 131 | |
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| 132 | //-> class rootArranger |
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[e858e7] | 133 | class rootArranger |
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[a5f15a] | 134 | { |
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[e858e7] | 135 | public: |
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| 136 | rootArranger( rootContainer ** _roots, |
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| 137 | rootContainer ** _mu, |
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[a5f15a] | 138 | const int _howclean = PM_CORRUPT ); |
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| 139 | ~rootArranger() {} |
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| 140 | |
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| 141 | void solve_all(); |
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| 142 | void arrange(); |
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| 143 | |
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| 144 | lists listOfRoots( const unsigned int oprec ); |
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| 145 | |
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| 146 | const bool success() { return found_roots; } |
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| 147 | |
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| 148 | private: |
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| 149 | rootArranger( const rootArranger & ); |
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| 150 | |
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| 151 | rootContainer ** roots; |
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| 152 | rootContainer ** mu; |
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| 153 | |
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| 154 | int howclean; |
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| 155 | int rc,mc; |
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| 156 | bool found_roots; |
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| 157 | }; |
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| 158 | //<- |
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| 159 | |
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| 160 | //-> simplex computation |
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| 161 | // (used by sparse matrix construction) |
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| 162 | #define SIMPLEX_EPS 1.0e-12 |
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| 163 | |
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[c7f3b7] | 164 | /** Linear Programming / Linear Optimization using Simplex - Algorithm |
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| 165 | * |
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| 166 | * On output, the tableau LiPM is indexed by two arrays of integers. |
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| 167 | * ipsov[j] contains, for j=1..m, the number i whose original variable |
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| 168 | * is now represented by row j+1 of LiPM. (left-handed vars in solution) |
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| 169 | * (first row is the one with the objective function) |
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| 170 | * izrov[j] contains, for j=1..n, the number i whose original variable |
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| 171 | * x_i is now a right-handed variable, rep. by column j+1 of LiPM. |
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| 172 | * These vars are all zero in the solution. The meaning of n<i<n+m1+m2 |
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| 173 | * is the same as above. |
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| 174 | */ |
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| 175 | class simplex |
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| 176 | { |
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| 177 | public: |
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| 178 | |
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| 179 | int m; // number of constraints, make sure m == m1 + m2 + m3 !! |
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| 180 | int n; // # of independent variables |
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| 181 | int m1,m2,m3; // constraints <=, >= and == |
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| 182 | int icase; // == 0: finite solution found; |
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| 183 | // == +1 objective funtion unbound; == -1: no solution |
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| 184 | int *izrov,*iposv; |
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| 185 | |
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| 186 | mprfloat **LiPM; // the matrix (of size [m+2, n+1]) |
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| 187 | |
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| 188 | /** #rows should be >= m+2, #cols >= n+1 |
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| 189 | */ |
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| 190 | simplex( int rows, int cols ); |
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| 191 | ~simplex(); |
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| 192 | |
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| 193 | BOOLEAN mapFromMatrix( matrix m ); |
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| 194 | matrix mapToMatrix( matrix m ); |
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| 195 | intvec * posvToIV(); |
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| 196 | intvec * zrovToIV(); |
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| 197 | |
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| 198 | void compute(); |
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| 199 | |
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| 200 | private: |
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| 201 | simplex( const simplex & ); |
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| 202 | void simp1( mprfloat **a, int mm, int ll[], int nll, int iabf, int *kp, mprfloat *bmax ); |
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| 203 | void simp2( mprfloat **a, int n, int l2[], int nl2, int *ip, int kp, mprfloat *q1 ); |
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| 204 | void simp3( mprfloat **a, int i1, int k1, int ip, int kp ); |
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| 205 | |
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| 206 | int LiPM_cols,LiPM_rows; |
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| 207 | }; |
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| 208 | |
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[a5f15a] | 209 | //<- |
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| 210 | |
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| 211 | #endif MPR_NUMERIC_H |
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| 212 | |
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| 213 | // local Variables: *** |
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| 214 | // folded-file: t *** |
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| 215 | // compile-command-1: "make installg" *** |
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| 216 | // compile-command-2: "make install" *** |
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[e858e7] | 217 | // End: *** |
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