# source:git/Singular/minpoly.h@2024f6a

spielwiese
Last change on this file since 2024f6a was 2024f6a, checked in by Frank Seelisch <seelisch@…>, 12 years ago
added minpoly code by Sebastian Jambor, accessable through extra.cc git-svn-id: file:///usr/local/Singular/svn/trunk@14238 2c84dea3-7e68-4137-9b89-c4e89433aadc
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1/***********************************************************************************
2 * Copyright (C) 2011 Sebastian Jambor                                             *
3 * sebastian@momo.math.rwth-aachen.de                                              *
4 *                                                                                 *
5 * Implementation of an algorithm to compute the minimal polynomial of a           *
6 * square matrix A \in \F_p^{n \times n}.                                          *
7 *                                                                                 *
8 * Let V_1, \dotsc, V_k \in \F_p^{1 \times n} be vectors such that                 *
9 * V_1, V_1*A, V_1*A^2, \dotsc, V_2, V_2*A, V_2*A^2, \dotsc                        *
10 * generate \F_p^{1 \times n}.                                                     *
11 * Let mpV_i be the monic polynomial of smallest degree such that                  *
12 * V_i*mpV_i(A) = 0.                                                               *
13 * Then the minimal polynomial of A is the least common multiple of the mpV_i.     *
14 *                                                                                 *
15 *                                                                                 *
16 * The algorithm uses two classes:                                                 *
17 *                                                                                 *
18 * 1. LinearDependencyMatrix                                                       *
19 * This is used to find a linear dependency between the vectors V, V*A, \ldotsc.   *
20 * To to this, it has an internal n \times (2n + 1) matrix.                        *
21 * Every time a new row VA^i is inserted, it is reduced via Gauss' Algorithm,      *
22 * using right hand sides. If VA^i is reduced to zero, then the vectors are        *
23 * linearly dependend, and the dependency can be read of at the right hand sides.  *
24 *                                                                                 *
25 * Example: Compute the minimal polynomial of A = [[0,1],[1,1]] with V = [1,0]     *
26 * over F_5.                                                                       *
27 * Then LinearDependencyMatrix will be:                                            *
28 * After the first step (i.e., after inserting V = [1,0]):                         *
29 *       ( 1 0 | 1 0 0 )                                                           *
30 * After the second step (i.e., after inserting VA = [0,1]):                       *
31 *       ( 1 0 | 1 0 0 )                                                           *
32 *       ( 0 1 | 0 1 0 )                                                           *
33 * In the third step, where VA^2 = [1,1] is inserted, the row                      *
34 *       ( 1 1 | 0 0 1 )                                                           *
35 * is reduced to                                                                   *
36 *       ( 0 0 | 4 4 1)                                                            *
37 * Thus VA^2 + 4*VA + 4*V = 0, so mpV = t^2 + 4*t + 4.                             *
38 *                                                                                 *
39 *                                                                                 *
40 *                                                                                 *
41 * 2. NewVectorMatrix                                                              *
42 * If one vector V_1 is not enough to compute the minimal polynomial, i.e. the     *
43 * vectors V_1, V_1*A, V_1*A^2, \dotsc don't generate \F_p^{1 \times n}, then      *
44 * we have to find a vector V_2 which is not in the span of the V_1*A^i.           *
45 * This is done with NewVectorMatrix, which simply holds a reduced n \times n      *
46 * matrix, where the rows generate the span of the V_jA^i.                         *
47 * To find a vector which is not in the span, simply take the k-th standard        *
48 * vector, where k is not a pivot element of A.                                    *
49 *                                                                                 *
50 *                                                                                 *
51 * For efficiency reasons, the matrix entries in LinearDependencyMatrix            *
52 * and NewVectorMatrix are not initialized to zero. Instead, a variable rows       *
53 * is used to indicate the number of rows which are nonzero (all further           *
54 * rows are regarded as zero rows). Furthermore, the array pivots stores the       *
55 * pivot entries of the rows, i.e., pivots[i] indicates the position of the        *
56 * first non-zero entry in the i-th row, which is normalized to 1.                 *
57 *                                                                                 *
58 *                                                                                 *
59 * To use:                                                                         *
60 * Call the method computeMinimalPolynomial(...)                                   *
61 ***********************************************************************************/
62
63
64
65
66#ifndef MINPOLY_H
67#define MINPOLY_H
68
69#include<iostream>
70
71class NewVectorMatrix;
72
73class LinearDependencyMatrix {
74    friend class NewVectorMatrix;
75private:
76    unsigned p;
77    unsigned long n;
78    unsigned long **matrix;
79    unsigned long *tmprow;
80    unsigned *pivots;
81    unsigned rows;
82
83public:
84    LinearDependencyMatrix(unsigned n, unsigned long p);
85    ~LinearDependencyMatrix();
86
87    // reset the matrix, so that we can use it to find another linear dependence
88    // Note: there is no need to reinitalize the matrix and vectors!
89    void resetMatrix();
90
91
92    // return the first nonzero entry in row (only the first n entries are checked,
93    // regardless of the size, since we will also apply this for rows with
94    // right hand sides).
95    // If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector)
96    int firstNonzeroEntry(unsigned long *row);
97
98    void reduceTmpRow();
99
100    void normalizeTmp(unsigned i);
101
102    bool findLinearDependency(unsigned long* newRow, unsigned long* dep);
103
104    friend std::ostream& operator<<(std::ostream& out, const LinearDependencyMatrix& mat);
105};
106
107
108// This class is used to find a new vector for the next step in the
109// minimal polynomial algorithm.
110class NewVectorMatrix {
111private:
112    unsigned p;
113    unsigned long n;
114    unsigned long **matrix;
115    unsigned *pivots;
116    unsigned rows;
117
118public:
119    NewVectorMatrix(unsigned n, unsigned long p);
120    ~NewVectorMatrix();
121
122    // return the first nonzero entry in row (only the first n entries are checked,
123    // regardless of the size, since we will also apply this for rows with
124    // right hand sides).
125    // If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector)
126    int firstNonzeroEntry(unsigned long *row);
127
128    void normalizeRow(unsigned long *row, unsigned i);
129
130    void insertRow(unsigned long* row);
131
132    // insert each row of the matrix
133    void insertMatrix(LinearDependencyMatrix& mat);
134
135    // Finds the smallest integer between 0 and n-1, which is not a pivot position.
136    // If no such number exists, return -1.
137    int findSmallestNonpivot();
138};
139
140
141// compute the minimal polynomial of matrix \in \F_p^{n \times n}.
142// The result is an array of length n + 1, where the i-th entry represents the i-th coefficient
143// of the minimal polynomial.
144//
145// result should be deleted with delete[]
146unsigned long* computeMinimalPolynomial(unsigned long** matrix, unsigned n, unsigned long p);
147
148
149
150/////////////////////////////////
151// auxiliary methods
152/////////////////////////////////
153
154
155// compute x^(-1) mod p
156//
157// NOTE: this uses long instead of unsigned long, for the XEA to work.
158// This shouldn't be a problem, since p has to be < 2^31 for the multiplication to work anyway.
159long modularInverse(long x, long p);
160
161void vectorMatrixMult(unsigned long* vec, unsigned long **mat, unsigned long* result, unsigned n, unsigned long p);
162
163// a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1,
164// representing polynomials \sum_i a[i]t^i \in \F_p[t].
165// After this method, a will be a mod q.
166// Method will change dega accordingly.
167void rem(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq);
168
169// a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1,
170// representing polynomials \sum_i a[i]t^i \in \F_p[t].
171// After this method, a will be a / q.
172// Method will change dega accordingly.
173void quo(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq);
174
175
176// NOTE: since we don't know the size of result (the list can be longer than the degree of the polynomial),
177// every entry has to be preinitialized to zero!
178void mult(unsigned long* result, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
179
180
181// g = gcd(a,b).
182// returns deg(g)
183//
184// NOTE: since we don't know the size of g, every entry has to be preinitialized to zero!
185int gcd(unsigned long* g, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
186
187// l = lcm(a,b).
188// returns deg(l)
189//
190// has side effects for a
191//
192// NOTE: since we don't know the size of l, every entry has to be preinitialized to zero!
193int lcm(unsigned long* l, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
194
195#endif // MINPOLY_H
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