source: git/Singular/minpoly.cc @ db83bf

/***********************************************
 * Copyright (C) 2011 Sebastian Jambor         *
 * sebastian@momo.math.rwth-aachen.de          *
 ***********************************************/


#include<cmath>
//#include<iomanip>

#include "minpoly.h"

// TODO: avoid code copying, use subclassing instead!

LinearDependencyMatrix::LinearDependencyMatrix(unsigned n, unsigned long p) {
    this->n = n;
    this->p = p;

    matrix = new unsigned long*[n];
    for (int i = 0; i < n; i++) {
        matrix[i] = new unsigned long[2*n + 1];
    }
    pivots = new unsigned[n];
    tmprow = new unsigned long[2*n + 1];
    rows = 0;
}

LinearDependencyMatrix::~LinearDependencyMatrix() {
    delete[] tmprow;
    delete[] pivots;

    for (int i = 0; i < n; i++) {
        delete[] (matrix[i]);
    }
    delete[] matrix;
}

void LinearDependencyMatrix::resetMatrix() {
    rows = 0;
}

int LinearDependencyMatrix::firstNonzeroEntry(unsigned long *row) {
    for (int i = 0; i < n; i++)
        if (row[i] != 0)
            return i;

    return -1;
}

void LinearDependencyMatrix::reduceTmpRow() {
    for (int i = 0; i < rows; i++) {
        unsigned piv = pivots[i];
        unsigned long x = tmprow[piv] % p;
        // if the corresponding entry in the row is zero, there is nothing to do
        if (x != 0) {
            // subtract tmprow[i] times the i-th row
            for (int j = piv; j < n + rows + 1; j++) {
                unsigned long tmp = matrix[i][j]*x % p;
                tmp = p - tmp;
                tmprow[j] += tmp;
                // We don't normalize here, so remember to do it after all reductions.
                // tmprow[j] %= p;
            }
        }
    }

    // normalize the entries of tmprow.
    for (int i = 0; i < n + rows + 1; i++) {
        tmprow[i] %= p;
    }
}


void LinearDependencyMatrix::normalizeTmp(unsigned i) {
    unsigned long inv = modularInverse(tmprow[i], p);
    tmprow[i] = 1;
    for (int j = i+1; j < 2*n + 1; j++)
        tmprow[j] = (tmprow[j] * inv) % p;
}

bool LinearDependencyMatrix::findLinearDependency(unsigned long* newRow, unsigned long* dep) {
    // Copy newRow to tmprow (we need to add RHS)
    for (int i = 0; i < n; i++) {
        tmprow[i] = newRow[i];
        tmprow[n + i] = 0;
    }
    tmprow[2*n] = 0;
    tmprow[n + rows] = 1;

    reduceTmpRow();

    // Is tmprow reduced to zero? Then we have found a linear dependence.
    // Otherwise, we just add tmprow to the matrix.
    unsigned newpivot = firstNonzeroEntry(tmprow);
    if (newpivot == -1) {
        for (int i = 0; i <=n; i++) {
            dep[i] = tmprow[n+i];
        }

        return true;
    } else {
        normalizeTmp(newpivot);

        for (int i = 0; i < 2*n + 1; i++) {
            matrix[rows][i] = tmprow[i];
        }

        pivots[rows] = newpivot;
        rows++;

        return false;
    }
}

#if 0
std::ostream& operator<<(std::ostream& out, const LinearDependencyMatrix& mat) {
    int width = ((int)log10(mat.p)) + 1;

    for (int i = 0; i < mat.rows; i++) {
        for (int j = 0; j < mat.n; j++) {
            out << std::setw(width) << mat.matrix[i][j] << " ";
        }
        out << "| ";
        for (int j = mat.n; j <= 2*mat.n; j++) {
            out << std::setw(width) << mat.matrix[i][j] << " ";
        }
        out << std::endl;
    }
    out << "-----" << std::endl;
    for (int j = 0; j < mat.n; j++) {
        out << std::setw(width) << mat.tmprow[j] << " ";
    }
    out << "| ";
    for (int j = mat.n; j <= 2*mat.n; j++) {
        out << std::setw(width) << mat.tmprow[j] << " ";
    }
    out << std::endl;

    return out;
}
#endif


NewVectorMatrix::NewVectorMatrix(unsigned n, unsigned long p) {
    this->n = n;
    this->p = p;

    matrix = new unsigned long*[n];
    for (int i = 0; i < n; i++) {
        matrix[i] = new unsigned long[n];
    }

    pivots = new unsigned[n];
    rows = 0;
}

NewVectorMatrix::~NewVectorMatrix() {
    delete pivots;

    for (int i = 0; i < n; i++) {
        delete[] matrix[i];
    }
    delete matrix;
}

int NewVectorMatrix::firstNonzeroEntry(unsigned long *row) {
    for (int i = 0; i < n; i++)
        if (row[i] != 0)
            return i;

    return -1;
}

//void NewVectorMatrix::subtractIthRow(unsigned long* row, unsigned i) {
//    unsigned piv = pivots[i];
//    unsigned long x = row[piv];
//    for (int j = piv; j < n; j++) {
//        unsigned tmp = matrix[i][j]*x % p;
//        tmp = p - tmp;
//        row[j] += tmp;
//        row[j] %= p;
//    }
//}
//
void NewVectorMatrix::normalizeRow(unsigned long *row, unsigned i) {
    unsigned long inv = modularInverse(row[i], p);
    row[i] = 1;

    for (int j = i+1; j < n; j++) {
        row[j] = (row[j] * inv) % p;
    }
}

void NewVectorMatrix::insertRow(unsigned long* row) {
    for (int i = 0; i < rows; i++) {
        unsigned piv = pivots[i];
        unsigned long x = row[piv] % p;
        // if the corresponding entry in the row is zero, there is nothing to do
        if (x != 0) {
            // subtract row[i] times the i-th row
            for (int j = piv; j < n; j++) {
                unsigned long tmp = matrix[i][j]*x % p;
                tmp = p - tmp;
                row[j] += tmp;
                // We don't normalize here, so remember to do it after all reductions.
                // row[j] %= p;
            }
        }
    }

    // normalize the entries of row.
    for (int i = 0; i < n + rows + 1; i++) {
        row[i] %= p;
    }

    unsigned piv = firstNonzeroEntry(row);

    if (piv != -1) {
        // normalize and insert row into the matrix
        normalizeRow(row, piv);
        for (int i = 0; i < n; i++) {
            matrix[rows][i] = row[i];
        }

        pivots[rows] = piv;
        rows++;
    }
}


void NewVectorMatrix::insertMatrix(LinearDependencyMatrix& mat) {
    // The matrix in LinearDependencyMatrix is already in reduced form.
    // Thus, if the current matrix is empty, we can simply copy the other matrix.
    if (rows == 0) {
        for (int i = 0; i < mat.rows; i++) {
            for (int j = 0; j < n; j++) {
                matrix[i][j] = mat.matrix[i][j];

                rows = mat.rows;
                for (int i = 0; i < rows; i++) {
                    pivots[i] = mat.pivots[i];
                }
            }
        }
    } else {
        for (int i = 0; i < mat.rows; i++) {
            insertRow(mat.matrix[i]);
        }
    }
}

int NewVectorMatrix::findSmallestNonpivot() {
    // This method isn't very efficient, but it is called at most a few times, so efficiency is not important.
    if (rows == n)
        return -1;

    for (int i = 0; i < n; i++) {
        bool isPivot = false;
        for (int j = 0; j < rows; j++) {
            if (pivots[j] == i) {
                isPivot = true;
                break;
            }
        }

        if (!isPivot) {
            return i;
        }
    }
}


void vectorMatrixMult(unsigned long* vec, unsigned long **mat, unsigned long* result, unsigned n, unsigned long p) {
    unsigned long tmp;
    for (int i = 0; i < n; i++) {
        result[i] = 0;
        for (int j = 0; j < n; j++) {
            tmp = vec[j]*mat[j][i] % p;
            result[i] += tmp;
        }
        // We can afford to reduce mod p only after all additions, since p < 2^31, but an unsigned long can store 2^64.
        // Thus the only possibility for an overflow would occurr for matrices with about 2^31 rows.
        result[i] %= p;
    }
}

#if 0
void printVec(unsigned long* vec, int n) {
    for (int i = 0; i < n; i++) {
        std::cout << vec[i] << ", ";
    }

    std::cout << std::endl;
}
#endif


unsigned long* computeMinimalPolynomial(unsigned long** matrix, unsigned n, unsigned long p) {
    LinearDependencyMatrix lindepmat(n, p);
    NewVectorMatrix newvectormat(n, p);

    unsigned long* result = new unsigned long[n+1];
    unsigned long* mpvec = new unsigned long[n+1];
    unsigned long* tmp = new unsigned long[n+1];

    // initialize result = 1
    for (int i = 0; i <= n; i++) {
        result[i] = 0;
    }
    result[0] = 1;

    int degresult = 0;


    int i = 0;

    unsigned long* vec = new unsigned long[n];
    unsigned long* vecnew = new unsigned long[n];

    while (i != -1) {
        for (int j = 0; j < n; j++) {
            vec[j] = 0;
        }
        vec[i] = 1;

        lindepmat.resetMatrix();

        while (true) {
            bool ld = lindepmat.findLinearDependency(vec, mpvec);

            if (ld) {
                break;
            }

            vectorMatrixMult(vec, matrix, vecnew, n, p);
            unsigned long* swap = vec;
            vec = vecnew;
            vecnew = swap;
        }


        unsigned degmpvec = n;
        while (mpvec[degmpvec] == 0) {
            degmpvec--;
        }

        // if the dimension is already maximal, i.e., the minimal polynomial of vec has the highest possible degree,
        // then we are done.
        if (degmpvec == n) {
            unsigned long* swap = result;
            result = mpvec;
            mpvec = swap;
            i = -1;
        } else {
            // otherwise, we have to compute the lcm of mpvec and prior result

            // tmp = zeropol
            for (int j = 0; j <= n; j++) {
                tmp[j] = 0;
            }
            degresult  = lcm(tmp, result, mpvec, p, degresult, degmpvec);
            unsigned long* swap = result;
            result = tmp;
            tmp = swap;

            if (degresult == n) {
                // minimal polynomial has highest possible degree, so we are done.
                i = -1;
            } else {
                newvectormat.insertMatrix(lindepmat);
                i = newvectormat.findSmallestNonpivot();
            }
        }
    }

    // TODO: take lcms of the different monomials!

    delete[] vecnew;
    delete[] vec;
    delete[] tmp;
    delete[] mpvec;

    return result;
}


void rem(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq) {
    while (degq <= dega) {
        unsigned d = dega - degq;
        long factor = a[dega]*modularInverse(q[degq], p) % p;
        for (int i = degq; i >= 0; i--) {
            long tmp = p - (factor*q[i] % p);
            a[d + i] += tmp;
            a[d + i] %= p;
        }

        while (a[dega] == 0 && dega >= 0) {
            dega--;
        }
    }
}


void quo(unsigned long* a, unsigned long* q, unsigned long p, int & dega, int degq) {
    unsigned degres = dega - degq;
    unsigned long* result = new unsigned long[degres + 1];

    while (degq <= dega) {
        unsigned d = dega - degq;
        long inv = modularInverse(q[degq], p);
        result[d] = a[dega]*inv % p;
        for (int i = degq; i >= 0; i--) {
            long tmp = p - (result[d]*q[i] % p);
            a[d + i] += tmp;
            a[d + i] %= p;
        }

        while (a[dega] == 0 && dega >= 0) {
            dega--;
        }
    }

    // copy result into a
    for (int i = 0; i <= degres; i++) {
        a[i] = result[i];
    }
    // set all following entries of a to zero
    for (int i = degres + 1; i <= degq + degres; i++) {
        a[i] = 0;
    }

    dega = degres;

    delete[] result;
}


void mult(unsigned long* result, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb) {
    // NOTE: we assume that every entry in result is preinitialized to zero!

    for (int i = 0; i <= dega; i++) {
        for (int j = 0; j <= degb; j++) {
            result[i + j] += (a[i]*b[j] % p);
            result[i + j] %= p;
        }
    }
}


int gcd(unsigned long* g, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb) {
    unsigned long* tmp1 = new unsigned long[dega + 1];
    unsigned long* tmp2 = new unsigned long[degb + 1];
    for (int i = 0; i <= dega; i++) {
        tmp1[i] = a[i];
    }
    for (int i = 0; i <= degb; i++) {
        tmp2[i] = b[i];
    }
    int degtmp1 = dega;
    int degtmp2 = degb;

    unsigned long* swappol;
    int swapdeg;

    while (degtmp2 >= 0) {
        rem(tmp1, tmp2, p, degtmp1, degtmp2);
        swappol = tmp1;
        tmp1 = tmp2;
        tmp2 = swappol;

        swapdeg = degtmp1;
        degtmp1 = degtmp2;
        degtmp2 = swapdeg;
    }

    for (int i = 0; i <= degtmp1; i++) {
        g[i] = tmp1[i];
    }

    delete[] tmp1;
    delete[] tmp2;

    return degtmp1;
}


int lcm(unsigned long* l, unsigned long* a, unsigned long* b, unsigned long p, int dega, int degb) {
    unsigned long* g = new unsigned long[dega + 1];
    // initialize entries of g to zero
    for (int i = 0; i <= dega; i++) {
        g[i] = 0;
    }

    int degg = gcd(g, a, b, p, dega, degb);

    if (degg > 0) {
        // non-trivial gcd, so compute a = (a/g)
        quo(a, g, p, dega, degg);
    }
    mult(l, a, b, p, dega, degb);

    // normalize
    if (l[dega + degb + 1] != 1) {
        unsigned long inv = modularInverse(l[dega + degb], p);
        for (int i = 0; i <= dega + degb; i++) {
            l[i] *= inv;
            l[i] %= p;
        }
    }

    return dega + degb;
}


// compute x^(-1) mod p
long modularInverse(long x, long p) {
    long u1 = 1;
    long u2 = 0;
    long u3 = x;
    long v1 = 0;
    long v2 = 1;
    long v3 = p;

    long q, t1, t2, t3;
    while (v3 != 0) {
        q = u3 / v3;
        t1 = u1 - q * v1;
        t2 = u2 - q * v2;
        t3 = u3 - q * v3;
        u1 = v1;
        u2 = v2;
        u3 = v3;
        v1 = t1;
        v2 = t2;
        v3 = t3;
    }

    if (u1 < 0) {
        u1 += p;
    }

    return u1;
}
/*
int main() {
//    unsigned long p = 5;
//    unsigned n = 5;
//
//    unsigned long tmp[5][5] = {
//        {4, 2, 2, 0, 0},
//        {1, 0, 4, 0, 0},
//        {1, 4, 0, 0, 0},
//        {0, 0, 0, 0, 2},
//        {0, 0, 0, 1, 2}
//    };
//
//    unsigned long** mat = new unsigned long*[n];
//
//    for (int i = 0; i < n; i++) {
//        mat[i] = new unsigned long[n];
//        for (int j = 0; j < n; j++) {
//            mat[i][j] = tmp[i][j];
//        }
//    }

    unsigned long p = 2147483647;
    unsigned n = 144;

    unsigned long tmp[144][144] = {
        { 675404679,  229969202,  2043258051,  216211357,  1477018606,  262035292,  1238185201,  435546435,  1676840408,  569764551,  2054060931,  1532785340,  1457180050,  1796715512,  174490674,  353680680,  611990038,  1518663662,  1121048639,  1606061363,  241589639,  974320457,  498680823,  1941401535,  1220735521,  1184097691,  432129432,  1840441318,  2128392102,  1641711326,  564594829,  1054998227,  1766260889,  2032769309,  1784198938,  1565682907,  919118918,  2013474666,  1381346629,  721427183,  632729103,  198283955,  1465572480,  1835621413,  1806225960,  654832333,  1419918559,  1104401876,  1817114492,  1265761615,  206058687,  39984355,  1721271367,  1593702828,  450778316,  1516483668,  1517771444,  933944570,  204699028,  1008598420,  1607178098,  2010187045,  48940036,  1830218438,  918121738,  1065078103,  1745505967,  487449558,  397953698,  668658033,  405082609,  1338582332,  98120140,  502769790,  1910313730,  149887431,  51797432,  1121994256,  849712142,  1767705575,  1068487735,  308103054,  577736468,  272560152,  1529134312,  2116085457,  1742262556,  520841687,  2044222268,  70020898,  1597466390,  1115776764,  1361778128,  1006313653,  2005848800,  1479207546,  742754273,  1266820293,  1263819461,  998016414,  744269980,  692793104,  1482930079,  162917395,  583832157,  584134967,  20241179,  1779562579,  1760041266,  324634378,  1247577427,  72614237,  835460172,  1755609857,  1144805770,  54391225,  1466477071,  758359386,  128420243,  1374058397,  2002875633,  1914039055,  982546257,  1663813011,  259779940,  1615396466,  1421924476,  644551086,  172497626,  1215179715,  1795454028,  1058716673,  837131709,  1558144290,  1673115315,  310533731,  1425175896,  851106753,  365425953,  420700709,  807284530,  1561393777,  609416170,  2041479232,  },
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        { 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483645,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483646,  0,  0,  0,  0,  0,  0,  0,  2147483646,  2,  2,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483644,  0,  0,  0,  0,  2147483640,  0,  0,  3,  0,  0,  0,  2147483641,  0,  14,  2147483644,  3,  0,  1073741821,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483614,  0,  0,  6,  0,  2147483640,  28,  2147483641,  21,  0,  0,  2147483644,  0,  2147483645,  2147483617,  2,  14,  21,  12,  1073741815,  6,  2147483644,  1073741821,  1073741825,  },
        { 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  106,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  53,  0,  0,  0,  0,  0,  2147483641,  0,  53,  2147483541,  2147483541,  0,  0,  0,  0,  0,  0,  0,  0,  0,  159,  0,  0,  0,  0,  20,  0,  0,  2147483488,  0,  0,  0,  20,  1,  2147483607,  159,  2147483488,  0,  1073741956,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483329,  0,  20,  0,  2147483642,  2147483587,  0,  0,  159,  0,  100,  20,  2147483518,  2147483607,  2147483587,  66,  1073741926,  2147483329,  159,  1073741956,  1073741744,  },
        { 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  28,  0,  0,  0,  0,  0,  0,  0,  2147483624,  33,  6,  0,  0,  0,  0,  0,  },
        { 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1789569726,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  357913951,  0,  0,  0,  0,  0,  0,  0,  357913951,  357913921,  1431655745,  0,  0,  0,  0,  0,  0,  0,  0,  0,  357913957,  0,  0,  0,  0,  357913951,  0,  0,  357913926,  0,  0,  0,  357913951,  0,  1431655745,  24,  1073741794,  0,  178957001,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1789569637,  0,  0,  357913957,  357913949,  357913928,  1073741801,  0,  357913951,  0,  0,  357913926,  0,  1431655779,  1342177309,  0,  1789569726,  357913951,  357913921,  1431655745,  1073741794,  0,  178957001,  2147483603,  984263375,  715827913,  894784835,  },
        { 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483641,  0,  0,  0,  0,  0,  0,  0,  2147483642,  0,  12,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483617,  0,  0,  0,  0,  2147483641,  28,  2147483641,  18,  0,  0,  0,  0,  0,  2147483618,  0,  12,  3,  18,  12,  2147483641,  },
        { 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483641,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483614,  0,  0,  0,  0,  0,  0,  0,  2147483614,  1,  66,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2147483488,  0,  0,  0,  0,  2147483614,  0,  2147483642,  99,  0,  0,  0,  0,  2147483641,  2147483614,  2147483624,  66,  2147483488,  99,  66,  2147483617,  },
        { 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  28,  0,  0,  0,  0,  0,  0,  0,  2147483624,  0,  33,  6,  0,  }
    };

    unsigned long** mat = new unsigned long*[n];

    for (int i = 0; i < n; i++) {
        mat[i] = new unsigned long[n];
        for (int j = 0; j < n; j++) {
            mat[i][j] = tmp[i][j];
        }
    }


    // now mat is a 144 \times 144 matrix

    // compute the minimal polynomial 99 times without printing
    for (int i = 0; i < 99; i++) {
        delete[] computeMinimalPolynomial(mat, n, p);
    }

    // do it one last time with printing
    unsigned long* res = computeMinimalPolynomial(mat, n, p);
    for (int i = 0; i <= n; i++) {
        if (res[i] != 0) {
            std::cout << " + " << res[i] << "*t^" << i;
        }
    }
    std::cout << std::endl;

    delete[] res;

    return 0;
}
*/