# source:git/kernel/linear_algebra/minpoly.h@aa8a7e

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