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 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 
66 //#include<iostream>
67 
68 class NewVectorMatrix;
69 
71  friend class NewVectorMatrix;
72 private:
73  unsigned p;
74  unsigned long n;
75  unsigned long **matrix;
76  unsigned long *tmprow;
77  unsigned *pivots;
78  unsigned rows;
79 
80 public:
81  LinearDependencyMatrix(unsigned n, unsigned long p);
83 
84  // reset the matrix, so that we can use it to find another linear dependence
85  // Note: there is no need to reinitalize the matrix and vectors!
86  void resetMatrix();
87 
88 
89  // return the first nonzero entry in row (only the first n entries are checked,
90  // regardless of the size, since we will also apply this for rows with
91  // right hand sides).
92  // If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector)
93  int firstNonzeroEntry(unsigned long *row);
94 
95  void reduceTmpRow();
96 
97  void normalizeTmp(unsigned i);
98 
99  bool findLinearDependency(unsigned long* newRow, unsigned long* dep);
100 
101  //friend std::ostream& operator<<(std::ostream& out, const LinearDependencyMatrix& mat);
102 };
103 
104 
105 // This class is used to find a new vector for the next step in the
106 // minimal polynomial algorithm.
108 private:
109  unsigned p;
110  unsigned long n;
111  unsigned long **matrix;
112  unsigned *pivots;
113  unsigned *nonPivots;
114  unsigned rows;
115 
116 public:
117  NewVectorMatrix(unsigned n, unsigned long p);
119 
120  // return the first nonzero entry in row (only the first n entries are checked,
121  // regardless of the size, since we will also apply this for rows with
122  // right hand sides).
123  // If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector)
124  int firstNonzeroEntry(unsigned long *row);
125 
126 // // let piv be the pivot position of row i. then this method eliminates entry piv of row
127 // void subtractIthRow(unsigned long *row, unsigned i);
128 
129  void normalizeRow(unsigned long *row, unsigned i);
130 
131  void insertRow(unsigned long* row);
132 
133  // insert each row of the matrix
135 
136  // Finds the smallest integer between 0 and n-1, which is not a pivot position.
137  // If no such number exists, return -1.
138  int findSmallestNonpivot();
139 
140  int findLargestNonpivot();
141 };
142 
143 
144 // compute the minimal polynomial of matrix \in \F_p^{n \times n}.
145 // The result is an array of length n + 1, where the i-th entry represents the i-th coefficient
146 // of the minimal polynomial.
147 //
148 // result should be deleted with delete[]
149 unsigned long* computeMinimalPolynomial(unsigned long** matrix, unsigned n, unsigned long p);
150 
151 
152 
153 /////////////////////////////////
154 // auxiliary methods
155 /////////////////////////////////
156 
157 
158 // compute x^(-1) mod p
159 //
160 // NOTE: this uses long long instead of unsigned long, for the XEA to work.
161 // This shouldn't be a problem, since p has to be < 2^31 for the multiplication to work anyway.
162 //
163 // There is no need to distinguish between 32bit and 64bit architectures: On 64bit, long long
164 // is the same as long, and on 32bit, we need long long so that the variables can hold negative values.
165 unsigned long modularInverse(long long x, long long p);
166 
167 void vectorMatrixMult(unsigned long* vec, unsigned long **mat, unsigned **nonzeroIndices, unsigned *nonzeroCounts, unsigned long* result, unsigned n, unsigned long p);
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 mod q.
172 // Method will change dega accordingly.
173 void rem(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq);
174 
175 // a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1,
176 // representing polynomials \sum_i a[i]t^i \in \F_p[t].
177 // After this method, a will be a / q.
178 // Method will change dega accordingly.
179 void quo(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq);
180 
181 
182 // NOTE: since we don't know the size of result (the list can be longer than the degree of the polynomial),
183 // every entry has to be preinitialized to zero!
184 void mult(unsigned long* result, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
185 
186 
187 // g = gcd(a,b).
188 // returns deg(g)
189 //
190 // NOTE: since we don't know the size of g, every entry has to be preinitialized to zero!
191 int gcd(unsigned long* g, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
192 
193 // l = lcm(a,b).
194 // returns deg(l)
195 //
196 // has side effects for a
197 //
198 // NOTE: since we don't know the size of l, every entry has to be preinitialized to zero!
199 int lcm(unsigned long* l, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
200 
201 
202 // method suggested by Hans Schoenemann to multiply elements in finite fields
203 // on 32bit and 64bit machines
204 static inline unsigned long multMod(unsigned long a, unsigned long b, unsigned long p)
205 {
206 #if SIZEOF_LONG == 4
207 #define ULONG64 (unsigned long long)
208 #else
209 #define ULONG64 (unsigned long)
210 #endif
211  return (unsigned long)((ULONG64 a)*(ULONG64 b) % (ULONG64 p));
212 }
213 
214 #endif // MINPOLY_H
unsigned * pivots
Definition: minpoly.h:112
const poly a
Definition: syzextra.cc:212
LinearDependencyMatrix(unsigned n, unsigned long p)
Definition: minpoly.cc:21
#define ULONG64
return P p
Definition: myNF.cc:203
unsigned long * tmprow
Definition: minpoly.h:76
unsigned long modularInverse(long long x, long long p)
Definition: minpoly.cc:746
g
Definition: cfModGcd.cc:4031
void normalizeTmp(unsigned i)
Definition: minpoly.cc:90
unsigned long ** matrix
Definition: minpoly.h:75
static unsigned long multMod(unsigned long a, unsigned long b, unsigned long p)
Definition: minpoly.h:204
void vectorMatrixMult(unsigned long *vec, unsigned long **mat, unsigned **nonzeroIndices, unsigned *nonzeroCounts, unsigned long *result, unsigned n, unsigned long p)
Definition: minpoly.cc:395
void mult(unsigned long *result, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:649
fq_nmod_poly_t * vec
Definition: facHensel.cc:103
bool findLinearDependency(unsigned long *newRow, unsigned long *dep)
Definition: minpoly.cc:98
void rem(unsigned long *a, unsigned long *q, unsigned long p, int &dega, int degq)
Definition: minpoly.cc:574
unsigned * pivots
Definition: minpoly.h:77
void normalizeRow(unsigned long *row, unsigned i)
Definition: minpoly.cc:227
unsigned long n
Definition: minpoly.h:110
void insertMatrix(LinearDependencyMatrix &mat)
Definition: minpoly.cc:333
int i
Definition: cfEzgcd.cc:123
unsigned long n
Definition: minpoly.h:74
NewVectorMatrix(unsigned n, unsigned long p)
Definition: minpoly.cc:183
unsigned * nonPivots
Definition: minpoly.h:113
Variable x
Definition: cfModGcd.cc:4023
int gcd(unsigned long *g, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:668
unsigned rows
Definition: minpoly.h:114
void insertRow(unsigned long *row)
Definition: minpoly.cc:238
void quo(unsigned long *a, unsigned long *q, unsigned long p, int &dega, int degq)
Definition: minpoly.cc:599
unsigned long ** matrix
Definition: minpoly.h:111
unsigned long * computeMinimalPolynomial(unsigned long **matrix, unsigned n, unsigned long p)
Definition: minpoly.cc:430
unsigned p
Definition: minpoly.h:109
int firstNonzeroEntry(unsigned long *row)
Definition: minpoly.cc:218
int findLargestNonpivot()
Definition: minpoly.cc:368
const poly b
Definition: syzextra.cc:213
int findSmallestNonpivot()
Definition: minpoly.cc:341
int l
Definition: cfEzgcd.cc:94
return result
Definition: facAbsBiFact.cc:76
int lcm(unsigned long *l, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:711
int firstNonzeroEntry(unsigned long *row)
Definition: minpoly.cc:53