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minpoly.h
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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 dependent, 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
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);
81
82 // reset the matrix, so that we can use it to find another linear dependence
83 // Note: there is no need to reinitialize 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.
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);
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
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.
137
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
int l
Definition: cfEzgcd.cc:100
int i
Definition: cfEzgcd.cc:132
Variable x
Definition: cfModGcd.cc:4082
int p
Definition: cfModGcd.cc:4078
g
Definition: cfModGcd.cc:4090
CanonicalForm b
Definition: cfModGcd.cc:4103
unsigned long * tmprow
Definition: minpoly.h:74
bool findLinearDependency(unsigned long *newRow, unsigned long *dep)
Definition: minpoly.cc:96
void normalizeTmp(unsigned i)
Definition: minpoly.cc:88
int firstNonzeroEntry(unsigned long *row)
Definition: minpoly.cc:51
unsigned * pivots
Definition: minpoly.h:75
unsigned long n
Definition: minpoly.h:72
unsigned long ** matrix
Definition: minpoly.h:73
void normalizeRow(unsigned long *row, unsigned i)
Definition: minpoly.cc:225
unsigned * nonPivots
Definition: minpoly.h:111
unsigned p
Definition: minpoly.h:107
unsigned rows
Definition: minpoly.h:112
unsigned long ** matrix
Definition: minpoly.h:109
void insertRow(unsigned long *row)
Definition: minpoly.cc:236
unsigned * pivots
Definition: minpoly.h:110
int findLargestNonpivot()
Definition: minpoly.cc:366
void insertMatrix(LinearDependencyMatrix &mat)
Definition: minpoly.cc:331
int firstNonzeroEntry(unsigned long *row)
Definition: minpoly.cc:216
unsigned long n
Definition: minpoly.h:108
int findSmallestNonpivot()
Definition: minpoly.cc:339
return result
Definition: facAbsBiFact.cc:75
fq_nmod_poly_t * vec
Definition: facHensel.cc:108
void mult(unsigned long *result, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:647
void quo(unsigned long *a, unsigned long *q, unsigned long p, int &dega, int degq)
Definition: minpoly.cc:597
unsigned long modularInverse(long long x, long long p)
Definition: minpoly.cc:744
unsigned long * computeMinimalPolynomial(unsigned long **matrix, unsigned n, unsigned long p)
Definition: minpoly.cc:428
static unsigned long multMod(unsigned long a, unsigned long b, unsigned long p)
Definition: minpoly.h:202
#define ULONG64
void vectorMatrixMult(unsigned long *vec, unsigned long **mat, unsigned **nonzeroIndices, unsigned *nonzeroCounts, unsigned long *result, unsigned n, unsigned long p)
Definition: minpoly.cc:393
int gcd(unsigned long *g, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:666
int lcm(unsigned long *l, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:709
void rem(unsigned long *a, unsigned long *q, unsigned long p, int &dega, int degq)
Definition: minpoly.cc:572