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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 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 class NewVectorMatrix;
67 
69  friend class NewVectorMatrix;
70 private:
71  unsigned p;
72  unsigned long n;
73  unsigned long **matrix;
74  unsigned long *tmprow;
75  unsigned *pivots;
76  unsigned rows;
77 
78 public:
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 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.
105 class NewVectorMatrix {
106 private:
107  unsigned p;
108  unsigned long n;
109  unsigned long **matrix;
110  unsigned *pivots;
111  unsigned *nonPivots;
112  unsigned rows;
113 
114 public:
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.
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[]
147 unsigned 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.
163 unsigned long modularInverse(long long x, long long p);
164 
165 void 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.
171 void 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.
177 void 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!
182 void 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!
189 int 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!
197 int 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
202 static 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
ip_smatrix
Definition: matpol.h:13
x
Variable x
Definition: cfModGcd.cc:4023
NewVectorMatrix::insertMatrix
void insertMatrix(LinearDependencyMatrix &mat)
Definition: minpoly.cc:331
result
return result
Definition: facAbsBiFact.cc:76
LinearDependencyMatrix::rows
unsigned rows
Definition: minpoly.h:75
NewVectorMatrix::normalizeRow
void normalizeRow(unsigned long *row, unsigned i)
Definition: minpoly.cc:225
NewVectorMatrix::matrix
unsigned long ** matrix
Definition: minpoly.h:108
ULONG64
#define ULONG64
LinearDependencyMatrix::resetMatrix
void resetMatrix()
Definition: minpoly.cc:46
NewVectorMatrix::NewVectorMatrix
NewVectorMatrix(unsigned n, unsigned long p)
Definition: minpoly.cc:181
g
g
Definition: cfModGcd.cc:4031
NewVectorMatrix::firstNonzeroEntry
int firstNonzeroEntry(unsigned long *row)
Definition: minpoly.cc:216
LinearDependencyMatrix::pivots
unsigned * pivots
Definition: minpoly.h:74
multMod
static unsigned long multMod(unsigned long a, unsigned long b, unsigned long p)
Definition: minpoly.h:201
NewVectorMatrix::insertRow
void insertRow(unsigned long *row)
Definition: minpoly.cc:236
LinearDependencyMatrix::normalizeTmp
void normalizeTmp(unsigned i)
Definition: minpoly.cc:88
b
CanonicalForm b
Definition: cfModGcd.cc:4044
modularInverse
unsigned long modularInverse(long long x, long long p)
Definition: minpoly.cc:744
NewVectorMatrix::pivots
unsigned * pivots
Definition: minpoly.h:109
vectorMatrixMult
void vectorMatrixMult(unsigned long *vec, unsigned long **mat, unsigned **nonzeroIndices, unsigned *nonzeroCounts, unsigned long *result, unsigned n, unsigned long p)
Definition: minpoly.cc:393
LinearDependencyMatrix::findLinearDependency
bool findLinearDependency(unsigned long *newRow, unsigned long *dep)
Definition: minpoly.cc:96
NewVectorMatrix::p
unsigned p
Definition: minpoly.h:106
LinearDependencyMatrix::~LinearDependencyMatrix
~LinearDependencyMatrix()
Definition: minpoly.cc:34
LinearDependencyMatrix
Definition: minpoly.h:67
mult
void mult(unsigned long *result, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:647
i
int i
Definition: cfEzgcd.cc:125
LinearDependencyMatrix::tmprow
unsigned long * tmprow
Definition: minpoly.h:73
rem
void rem(unsigned long *a, unsigned long *q, unsigned long p, int &dega, int degq)
Definition: minpoly.cc:572
LinearDependencyMatrix::matrix
unsigned long ** matrix
Definition: minpoly.h:72
LinearDependencyMatrix::LinearDependencyMatrix
LinearDependencyMatrix(unsigned n, unsigned long p)
Definition: minpoly.cc:19
NewVectorMatrix::findSmallestNonpivot
int findSmallestNonpivot()
Definition: minpoly.cc:339
NewVectorMatrix::~NewVectorMatrix
~NewVectorMatrix()
Definition: minpoly.cc:204
NewVectorMatrix::n
unsigned long n
Definition: minpoly.h:107
NewVectorMatrix::findLargestNonpivot
int findLargestNonpivot()
Definition: minpoly.cc:366
NewVectorMatrix::rows
unsigned rows
Definition: minpoly.h:111
NewVectorMatrix
Definition: minpoly.h:104
gcd
int gcd(unsigned long *g, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:666
LinearDependencyMatrix::reduceTmpRow
void reduceTmpRow()
Definition: minpoly.cc:60
LinearDependencyMatrix::n
unsigned long n
Definition: minpoly.h:71
quo
void quo(unsigned long *a, unsigned long *q, unsigned long p, int &dega, int degq)
Definition: minpoly.cc:597
NewVectorMatrix::nonPivots
unsigned * nonPivots
Definition: minpoly.h:110
l
int l
Definition: cfEzgcd.cc:93
computeMinimalPolynomial
unsigned long * computeMinimalPolynomial(unsigned long **matrix, unsigned n, unsigned long p)
Definition: minpoly.cc:428
LinearDependencyMatrix::firstNonzeroEntry
int firstNonzeroEntry(unsigned long *row)
Definition: minpoly.cc:51
p
int p
Definition: cfModGcd.cc:4019
lcm
int lcm(unsigned long *l, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition: minpoly.cc:709
LinearDependencyMatrix::p
unsigned p
Definition: minpoly.h:70
vec
fq_nmod_poly_t * vec
Definition: facHensel.cc:103