[2024f6a] | 1 | /*********************************************************************************** |
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[f613a24] | 2 | * Author: Sebastian Jambor, 2011 * |
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| 3 | * (C) GPL (e-mail from June 6, 2012, 17:00:31 MESZ) * |
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[2024f6a] | 4 | * sebastian@momo.math.rwth-aachen.de * |
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| 5 | * * |
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| 6 | * Implementation of an algorithm to compute the minimal polynomial of a * |
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| 7 | * square matrix A \in \F_p^{n \times n}. * |
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| 8 | * * |
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| 9 | * Let V_1, \dotsc, V_k \in \F_p^{1 \times n} be vectors such that * |
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| 10 | * V_1, V_1*A, V_1*A^2, \dotsc, V_2, V_2*A, V_2*A^2, \dotsc * |
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| 11 | * generate \F_p^{1 \times n}. * |
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| 12 | * Let mpV_i be the monic polynomial of smallest degree such that * |
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| 13 | * V_i*mpV_i(A) = 0. * |
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| 14 | * Then the minimal polynomial of A is the least common multiple of the mpV_i. * |
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| 15 | * * |
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| 16 | * * |
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| 17 | * The algorithm uses two classes: * |
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| 18 | * * |
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| 19 | * 1. LinearDependencyMatrix * |
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| 20 | * This is used to find a linear dependency between the vectors V, V*A, \ldotsc. * |
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| 21 | * To to this, it has an internal n \times (2n + 1) matrix. * |
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| 22 | * Every time a new row VA^i is inserted, it is reduced via Gauss' Algorithm, * |
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| 23 | * using right hand sides. If VA^i is reduced to zero, then the vectors are * |
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| 24 | * linearly dependend, and the dependency can be read of at the right hand sides. * |
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| 25 | * * |
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| 26 | * Example: Compute the minimal polynomial of A = [[0,1],[1,1]] with V = [1,0] * |
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| 27 | * over F_5. * |
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| 28 | * Then LinearDependencyMatrix will be: * |
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| 29 | * After the first step (i.e., after inserting V = [1,0]): * |
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| 30 | * ( 1 0 | 1 0 0 ) * |
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| 31 | * After the second step (i.e., after inserting VA = [0,1]): * |
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| 32 | * ( 1 0 | 1 0 0 ) * |
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| 33 | * ( 0 1 | 0 1 0 ) * |
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| 34 | * In the third step, where VA^2 = [1,1] is inserted, the row * |
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| 35 | * ( 1 1 | 0 0 1 ) * |
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| 36 | * is reduced to * |
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| 37 | * ( 0 0 | 4 4 1) * |
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| 38 | * Thus VA^2 + 4*VA + 4*V = 0, so mpV = t^2 + 4*t + 4. * |
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| 39 | * * |
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| 40 | * * |
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| 41 | * * |
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| 42 | * 2. NewVectorMatrix * |
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| 43 | * If one vector V_1 is not enough to compute the minimal polynomial, i.e. the * |
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| 44 | * vectors V_1, V_1*A, V_1*A^2, \dotsc don't generate \F_p^{1 \times n}, then * |
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| 45 | * we have to find a vector V_2 which is not in the span of the V_1*A^i. * |
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| 46 | * This is done with NewVectorMatrix, which simply holds a reduced n \times n * |
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| 47 | * matrix, where the rows generate the span of the V_jA^i. * |
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| 48 | * To find a vector which is not in the span, simply take the k-th standard * |
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| 49 | * vector, where k is not a pivot element of A. * |
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| 50 | * * |
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| 51 | * * |
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| 52 | * For efficiency reasons, the matrix entries in LinearDependencyMatrix * |
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| 53 | * and NewVectorMatrix are not initialized to zero. Instead, a variable rows * |
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| 54 | * is used to indicate the number of rows which are nonzero (all further * |
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| 55 | * rows are regarded as zero rows). Furthermore, the array pivots stores the * |
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| 56 | * pivot entries of the rows, i.e., pivots[i] indicates the position of the * |
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| 57 | * first non-zero entry in the i-th row, which is normalized to 1. * |
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| 58 | ***********************************************************************************/ |
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| 59 | |
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| 60 | |
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| 61 | |
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| 62 | |
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| 63 | #ifndef MINPOLY_H |
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| 64 | #define MINPOLY_H |
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| 65 | |
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[ec9db9] | 66 | //#include<iostream> |
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[2024f6a] | 67 | |
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| 68 | class NewVectorMatrix; |
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| 69 | |
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| 70 | class LinearDependencyMatrix { |
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| 71 | friend class NewVectorMatrix; |
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| 72 | private: |
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| 73 | unsigned p; |
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| 74 | unsigned long n; |
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| 75 | unsigned long **matrix; |
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| 76 | unsigned long *tmprow; |
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| 77 | unsigned *pivots; |
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| 78 | unsigned rows; |
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| 79 | |
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| 80 | public: |
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| 81 | LinearDependencyMatrix(unsigned n, unsigned long p); |
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| 82 | ~LinearDependencyMatrix(); |
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| 83 | |
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| 84 | // reset the matrix, so that we can use it to find another linear dependence |
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| 85 | // Note: there is no need to reinitalize the matrix and vectors! |
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| 86 | void resetMatrix(); |
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[ec9db9] | 87 | |
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[2024f6a] | 88 | |
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| 89 | // return the first nonzero entry in row (only the first n entries are checked, |
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| 90 | // regardless of the size, since we will also apply this for rows with |
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| 91 | // right hand sides). |
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| 92 | // If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector) |
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| 93 | int firstNonzeroEntry(unsigned long *row); |
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| 94 | |
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| 95 | void reduceTmpRow(); |
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| 96 | |
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| 97 | void normalizeTmp(unsigned i); |
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[ec9db9] | 98 | |
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[2024f6a] | 99 | bool findLinearDependency(unsigned long* newRow, unsigned long* dep); |
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| 100 | |
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[abcc692] | 101 | //friend std::ostream& operator<<(std::ostream& out, const LinearDependencyMatrix& mat); |
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[2024f6a] | 102 | }; |
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| 103 | |
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| 104 | |
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| 105 | // This class is used to find a new vector for the next step in the |
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| 106 | // minimal polynomial algorithm. |
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| 107 | class NewVectorMatrix { |
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| 108 | private: |
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| 109 | unsigned p; |
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| 110 | unsigned long n; |
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| 111 | unsigned long **matrix; |
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| 112 | unsigned *pivots; |
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[f613a24] | 113 | unsigned *nonPivots; |
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[2024f6a] | 114 | unsigned rows; |
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| 115 | |
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| 116 | public: |
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| 117 | NewVectorMatrix(unsigned n, unsigned long p); |
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| 118 | ~NewVectorMatrix(); |
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[ec9db9] | 119 | |
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[2024f6a] | 120 | // return the first nonzero entry in row (only the first n entries are checked, |
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| 121 | // regardless of the size, since we will also apply this for rows with |
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| 122 | // right hand sides). |
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| 123 | // If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector) |
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| 124 | int firstNonzeroEntry(unsigned long *row); |
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| 125 | |
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[abcc692] | 126 | // // let piv be the pivot position of row i. then this method eliminates entry piv of row |
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| 127 | // void subtractIthRow(unsigned long *row, unsigned i); |
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| 128 | |
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[2024f6a] | 129 | void normalizeRow(unsigned long *row, unsigned i); |
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| 130 | |
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| 131 | void insertRow(unsigned long* row); |
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| 132 | |
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| 133 | // insert each row of the matrix |
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| 134 | void insertMatrix(LinearDependencyMatrix& mat); |
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| 135 | |
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| 136 | // Finds the smallest integer between 0 and n-1, which is not a pivot position. |
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| 137 | // If no such number exists, return -1. |
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| 138 | int findSmallestNonpivot(); |
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[f613a24] | 139 | |
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| 140 | int findLargestNonpivot(); |
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[2024f6a] | 141 | }; |
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[ec9db9] | 142 | |
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[2024f6a] | 143 | |
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| 144 | // compute the minimal polynomial of matrix \in \F_p^{n \times n}. |
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| 145 | // The result is an array of length n + 1, where the i-th entry represents the i-th coefficient |
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| 146 | // of the minimal polynomial. |
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| 147 | // |
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| 148 | // result should be deleted with delete[] |
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| 149 | unsigned long* computeMinimalPolynomial(unsigned long** matrix, unsigned n, unsigned long p); |
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| 150 | |
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| 151 | |
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| 152 | |
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| 153 | ///////////////////////////////// |
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| 154 | // auxiliary methods |
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| 155 | ///////////////////////////////// |
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| 156 | |
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| 157 | |
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| 158 | // compute x^(-1) mod p |
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| 159 | // |
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[abcc692] | 160 | // NOTE: this uses long long instead of unsigned long, for the XEA to work. |
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[2024f6a] | 161 | // This shouldn't be a problem, since p has to be < 2^31 for the multiplication to work anyway. |
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[abcc692] | 162 | // |
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| 163 | // There is no need to distinguish between 32bit and 64bit architectures: On 64bit, long long |
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| 164 | // is the same as long, and on 32bit, we need long long so that the variables can hold negative values. |
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| 165 | unsigned long modularInverse(long long x, long long p); |
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[2024f6a] | 166 | |
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[f613a24] | 167 | void vectorMatrixMult(unsigned long* vec, unsigned long **mat, unsigned **nonzeroIndices, unsigned *nonzeroCounts, unsigned long* result, unsigned n, unsigned long p); |
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[2024f6a] | 168 | |
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| 169 | // a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1, |
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| 170 | // representing polynomials \sum_i a[i]t^i \in \F_p[t]. |
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| 171 | // After this method, a will be a mod q. |
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| 172 | // Method will change dega accordingly. |
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| 173 | void rem(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq); |
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| 174 | |
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| 175 | // a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1, |
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| 176 | // representing polynomials \sum_i a[i]t^i \in \F_p[t]. |
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| 177 | // After this method, a will be a / q. |
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| 178 | // Method will change dega accordingly. |
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| 179 | void quo(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq); |
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| 180 | |
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| 181 | |
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[ec9db9] | 182 | // NOTE: since we don't know the size of result (the list can be longer than the degree of the polynomial), |
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[2024f6a] | 183 | // every entry has to be preinitialized to zero! |
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| 184 | void mult(unsigned long* result, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb); |
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| 185 | |
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| 186 | |
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| 187 | // g = gcd(a,b). |
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| 188 | // returns deg(g) |
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| 189 | // |
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| 190 | // NOTE: since we don't know the size of g, every entry has to be preinitialized to zero! |
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| 191 | int gcd(unsigned long* g, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb); |
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| 192 | |
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| 193 | // l = lcm(a,b). |
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| 194 | // returns deg(l) |
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| 195 | // |
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| 196 | // has side effects for a |
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| 197 | // |
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| 198 | // NOTE: since we don't know the size of l, every entry has to be preinitialized to zero! |
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| 199 | int lcm(unsigned long* l, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb); |
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| 200 | |
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[abcc692] | 201 | |
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| 202 | // method suggested by Hans Schoenemann to multiply elements in finite fields |
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| 203 | // on 32bit and 64bit machines |
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| 204 | static inline unsigned long multMod(unsigned long a, unsigned long b, unsigned long p) |
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| 205 | { |
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| 206 | #if SIZEOF_LONG == 4 |
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| 207 | #define ULONG64 (unsigned long long) |
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| 208 | #else |
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| 209 | #define ULONG64 (unsigned long) |
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| 210 | #endif |
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| 211 | return (unsigned long)((ULONG64 a)*(ULONG64 b) % (ULONG64 p)); |
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| 212 | } |
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| 213 | |
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[2024f6a] | 214 | #endif // MINPOLY_H |
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