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minpoly.h
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/***********************************************************************************
* Author: Sebastian Jambor, 2011 *
* (C) GPL (e-mail from June 6, 2012, 17:00:31 MESZ) *
* sebastian@momo.math.rwth-aachen.de *
* *
* Implementation of an algorithm to compute the minimal polynomial of a *
* square matrix A \in \F_p^{n \times n}. *
* *
* Let V_1, \dotsc, V_k \in \F_p^{1 \times n} be vectors such that *
* V_1, V_1*A, V_1*A^2, \dotsc, V_2, V_2*A, V_2*A^2, \dotsc *
* generate \F_p^{1 \times n}. *
* Let mpV_i be the monic polynomial of smallest degree such that *
* V_i*mpV_i(A) = 0. *
* Then the minimal polynomial of A is the least common multiple of the mpV_i. *
* *
* *
* The algorithm uses two classes: *
* *
* 1. LinearDependencyMatrix *
* This is used to find a linear dependency between the vectors V, V*A, \ldotsc. *
* To to this, it has an internal n \times (2n + 1) matrix. *
* Every time a new row VA^i is inserted, it is reduced via Gauss' Algorithm, *
* using right hand sides. If VA^i is reduced to zero, then the vectors are *
* linearly dependent, and the dependency can be read of at the right hand sides. *
* *
* Example: Compute the minimal polynomial of A = [[0,1],[1,1]] with V = [1,0] *
* over F_5. *
* Then LinearDependencyMatrix will be: *
* After the first step (i.e., after inserting V = [1,0]): *
* ( 1 0 | 1 0 0 ) *
* After the second step (i.e., after inserting VA = [0,1]): *
* ( 1 0 | 1 0 0 ) *
* ( 0 1 | 0 1 0 ) *
* In the third step, where VA^2 = [1,1] is inserted, the row *
* ( 1 1 | 0 0 1 ) *
* is reduced to *
* ( 0 0 | 4 4 1) *
* Thus VA^2 + 4*VA + 4*V = 0, so mpV = t^2 + 4*t + 4. *
* *
* *
* *
* 2. NewVectorMatrix *
* If one vector V_1 is not enough to compute the minimal polynomial, i.e. the *
* vectors V_1, V_1*A, V_1*A^2, \dotsc don't generate \F_p^{1 \times n}, then *
* we have to find a vector V_2 which is not in the span of the V_1*A^i. *
* This is done with NewVectorMatrix, which simply holds a reduced n \times n *
* matrix, where the rows generate the span of the V_jA^i. *
* To find a vector which is not in the span, simply take the k-th standard *
* vector, where k is not a pivot element of A. *
* *
* *
* For efficiency reasons, the matrix entries in LinearDependencyMatrix *
* and NewVectorMatrix are not initialized to zero. Instead, a variable rows *
* is used to indicate the number of rows which are nonzero (all further *
* rows are regarded as zero rows). Furthermore, the array pivots stores the *
* pivot entries of the rows, i.e., pivots[i] indicates the position of the *
* first non-zero entry in the i-th row, which is normalized to 1. *
***********************************************************************************/
#ifndef MINPOLY_H
#define MINPOLY_H
class NewVectorMatrix;
class LinearDependencyMatrix {
friend class NewVectorMatrix;
private:
unsigned p;
unsigned long n;
unsigned long **matrix;
unsigned long *tmprow;
unsigned *pivots;
unsigned rows;
public:
LinearDependencyMatrix(unsigned n, unsigned long p);
~LinearDependencyMatrix();
// reset the matrix, so that we can use it to find another linear dependence
// Note: there is no need to reinitialize the matrix and vectors!
void resetMatrix();
// return the first nonzero entry in row (only the first n entries are checked,
// regardless of the size, since we will also apply this for rows with
// right hand sides).
// If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector)
int firstNonzeroEntry(unsigned long *row);
void reduceTmpRow();
void normalizeTmp(unsigned i);
bool findLinearDependency(unsigned long* newRow, unsigned long* dep);
//friend std::ostream& operator<<(std::ostream& out, const LinearDependencyMatrix& mat);
};
// This class is used to find a new vector for the next step in the
// minimal polynomial algorithm.
class NewVectorMatrix {
private:
unsigned p;
unsigned long n;
unsigned long **matrix;
unsigned *pivots;
unsigned *nonPivots;
unsigned rows;
public:
NewVectorMatrix(unsigned n, unsigned long p);
~NewVectorMatrix();
// return the first nonzero entry in row (only the first n entries are checked,
// regardless of the size, since we will also apply this for rows with
// right hand sides).
// If the first n entries are all zero, return -1 (so this gives a check if row is the zero vector)
int firstNonzeroEntry(unsigned long *row);
// // let piv be the pivot position of row i. then this method eliminates entry piv of row
// void subtractIthRow(unsigned long *row, unsigned i);
void normalizeRow(unsigned long *row, unsigned i);
void insertRow(unsigned long* row);
// insert each row of the matrix
void insertMatrix(LinearDependencyMatrix& mat);
// Finds the smallest integer between 0 and n-1, which is not a pivot position.
// If no such number exists, return -1.
int findSmallestNonpivot();
int findLargestNonpivot();
};
// compute the minimal polynomial of matrix \in \F_p^{n \times n}.
// The result is an array of length n + 1, where the i-th entry represents the i-th coefficient
// of the minimal polynomial.
//
// result should be deleted with delete[]
unsigned long* computeMinimalPolynomial(unsigned long** matrix, unsigned n, unsigned long p);
/////////////////////////////////
// auxiliary methods
/////////////////////////////////
// compute x^(-1) mod p
//
// NOTE: this uses long long instead of unsigned long, for the XEA to work.
// This shouldn't be a problem, since p has to be < 2^31 for the multiplication to work anyway.
//
// There is no need to distinguish between 32bit and 64bit architectures: On 64bit, long long
// is the same as long, and on 32bit, we need long long so that the variables can hold negative values.
unsigned long modularInverse(long long x, long long p);
void vectorMatrixMult(unsigned long* vec, unsigned long **mat, unsigned **nonzeroIndices, unsigned *nonzeroCounts, unsigned long* result, unsigned n, unsigned long p);
// a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1,
// representing polynomials \sum_i a[i]t^i \in \F_p[t].
// After this method, a will be a mod q.
// Method will change dega accordingly.
void rem(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq);
// a is a vector of length at least dega + 1, and q is a vector of length at least degq + 1,
// representing polynomials \sum_i a[i]t^i \in \F_p[t].
// After this method, a will be a / q.
// Method will change dega accordingly.
void quo(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq);
// NOTE: since we don't know the size of result (the list can be longer than the degree of the polynomial),
// every entry has to be preinitialized to zero!
void mult(unsigned long* result, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
// g = gcd(a,b).
// returns deg(g)
//
// NOTE: since we don't know the size of g, every entry has to be preinitialized to zero!
int gcd(unsigned long* g, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
// l = lcm(a,b).
// returns deg(l)
//
// has side effects for a
//
// NOTE: since we don't know the size of l, every entry has to be preinitialized to zero!
int lcm(unsigned long* l, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb);
// method suggested by Hans Schoenemann to multiply elements in finite fields
// on 32bit and 64bit machines
static inline unsigned long multMod(unsigned long a, unsigned long b, unsigned long p)
{
#if SIZEOF_LONG == 4
#define ULONG64 (unsigned long long)
#else
#define ULONG64 (unsigned long)
#endif
return (unsigned long)((ULONG64 a)*(ULONG64 b) % (ULONG64 p));
}
#endif // MINPOLY_H
l
int l
Definition cfEzgcd.cc:100
i
int i
Definition cfEzgcd.cc:132
x
Variable x
Definition cfModGcd.cc:4090
p
int p
Definition cfModGcd.cc:4086
g
g
Definition cfModGcd.cc:4098
b
CanonicalForm b
Definition cfModGcd.cc:4111
LinearDependencyMatrix
Definition minpoly.h:68
LinearDependencyMatrix::tmprow
unsigned long * tmprow
Definition minpoly.h:74
LinearDependencyMatrix::findLinearDependency
bool findLinearDependency(unsigned long *newRow, unsigned long *dep)
Definition minpoly.cc:96
LinearDependencyMatrix::normalizeTmp
void normalizeTmp(unsigned i)
Definition minpoly.cc:88
LinearDependencyMatrix::firstNonzeroEntry
int firstNonzeroEntry(unsigned long *row)
Definition minpoly.cc:51
LinearDependencyMatrix::p
unsigned p
Definition minpoly.h:71
LinearDependencyMatrix::LinearDependencyMatrix
LinearDependencyMatrix(unsigned n, unsigned long p)
Definition minpoly.cc:19
LinearDependencyMatrix::reduceTmpRow
void reduceTmpRow()
Definition minpoly.cc:60
LinearDependencyMatrix::resetMatrix
void resetMatrix()
Definition minpoly.cc:46
LinearDependencyMatrix::pivots
unsigned * pivots
Definition minpoly.h:75
LinearDependencyMatrix::n
unsigned long n
Definition minpoly.h:72
LinearDependencyMatrix::matrix
unsigned long ** matrix
Definition minpoly.h:73
LinearDependencyMatrix::rows
unsigned rows
Definition minpoly.h:76
LinearDependencyMatrix::~LinearDependencyMatrix
~LinearDependencyMatrix()
Definition minpoly.cc:34
LinearDependencyMatrix::NewVectorMatrix
friend class NewVectorMatrix
Definition minpoly.h:69
NewVectorMatrix
Definition minpoly.h:105
NewVectorMatrix::normalizeRow
void normalizeRow(unsigned long *row, unsigned i)
Definition minpoly.cc:225
NewVectorMatrix::~NewVectorMatrix
~NewVectorMatrix()
Definition minpoly.cc:204
NewVectorMatrix::nonPivots
unsigned * nonPivots
Definition minpoly.h:111
NewVectorMatrix::p
unsigned p
Definition minpoly.h:107
NewVectorMatrix::NewVectorMatrix
NewVectorMatrix(unsigned n, unsigned long p)
Definition minpoly.cc:181
NewVectorMatrix::rows
unsigned rows
Definition minpoly.h:112
NewVectorMatrix::matrix
unsigned long ** matrix
Definition minpoly.h:109
NewVectorMatrix::insertRow
void insertRow(unsigned long *row)
Definition minpoly.cc:236
NewVectorMatrix::pivots
unsigned * pivots
Definition minpoly.h:110
NewVectorMatrix::findLargestNonpivot
int findLargestNonpivot()
Definition minpoly.cc:366
NewVectorMatrix::insertMatrix
void insertMatrix(LinearDependencyMatrix &mat)
Definition minpoly.cc:331
NewVectorMatrix::firstNonzeroEntry
int firstNonzeroEntry(unsigned long *row)
Definition minpoly.cc:216
NewVectorMatrix::n
unsigned long n
Definition minpoly.h:108
NewVectorMatrix::findSmallestNonpivot
int findSmallestNonpivot()
Definition minpoly.cc:339
result
return result
Definition facAbsBiFact.cc:76
vec
fq_nmod_poly_t * vec
Definition facHensel.cc:108
matrix
ip_smatrix * matrix
Definition matpol.h:43
mult
void mult(unsigned long *result, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition minpoly.cc:647
quo
void quo(unsigned long *a, unsigned long *q, unsigned long p, int &dega, int degq)
Definition minpoly.cc:597
modularInverse
unsigned long modularInverse(long long x, long long p)
Definition minpoly.cc:744
computeMinimalPolynomial
unsigned long * computeMinimalPolynomial(unsigned long **matrix, unsigned n, unsigned long p)
Definition minpoly.cc:428
multMod
static unsigned long multMod(unsigned long a, unsigned long b, unsigned long p)
Definition minpoly.h:202
ULONG64
#define ULONG64
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
gcd
int gcd(unsigned long *g, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition minpoly.cc:666
lcm
int lcm(unsigned long *l, unsigned long *a, unsigned long *b, unsigned long p, int dega, int degb)
Definition minpoly.cc:709
rem
void rem(unsigned long *a, unsigned long *q, unsigned long p, int &dega, int degq)
Definition minpoly.cc:572