~/ntl-11.4.2/doc/mat_lzz_p.cpp.html

    /**************************************************************************\

    MODULE: mat_zz_p

    SUMMARY:

    Defines the class mat_zz_p.
    Note that the modulus p need not be a prime, except as indicated below.

    IMPLEMENTATION NOTES:

    Starting with NTL version 9.7.0 (and 9.7.1), many of the routines here have
    been optimized to take better advantage of specific hardware features available
    on 64-bit Intel CPU's.  Currently, the mul, inv, determinant, solve, gauss,
    kernel, and image routines are fastest for p up to 23-bits long (assuming the
    CPU supports AVX instructions).  After that, performance degrades in three
    stages: stage 1: up to 28-bits; stage 2: up to 31-bits; stage 3: 32-bits and
    up.

    For primes up to 23-bits, AVX floating point instructions are used.  After
    that, ordinary integer arithmetic is used.  In a future version, I may exploit
    AVX2 integer instructions to get better stage 2 performance.  And in the more
    distant future, AVX512 instructions will be used, when they become available.

    On older Intel machines, or non-Intel machines that have "long long" support,
    one still gets optimizations corresponding to the three stages above.  On
    32-bit machines, one still gets three stages, just with smaller crossover
    points.

    \**************************************************************************/

    #include <NTL/matrix.h>
    #include "vec_vec_zz_p.h"

    typedef Mat<zz_p> mat_zz_p; // backward compatibility

    void add(mat_zz_p& X, const mat_zz_p& A, const mat_zz_p& B);
    // X = A + B

    void sub(mat_zz_p& X, const mat_zz_p& A, const mat_zz_p& B);
    // X = A - B

    void mul(mat_zz_p& X, const mat_zz_p& A, const mat_zz_p& B);
    // X = A * B

    void mul(vec_zz_p& x, const mat_zz_p& A, const vec_zz_p& b);
    // x = A * b

    void mul(vec_zz_p& x, const vec_zz_p& a, const mat_zz_p& B);
    // x = a * B

    void mul(mat_zz_p& X, const mat_zz_p& A, zz_p b);
    void mul(mat_zz_p& X, const mat_zz_p& A, long b);
    // X = A * b

    void mul(mat_zz_p& X, zz_p a, const mat_zz_p& B);
    void mul(mat_zz_p& X, long a, const mat_zz_p& B);
    // X = a * B

    void transpose(mat_zz_p& X, const mat_zz_p& A);
    mat_zz_p transpose(const mat_zz_p& A);
    // X = transpose of A

    void determinant(zz_p& d, const mat_zz_p& A);
    zz_p determinant(const mat_zz_p& a);
    // d = determinant(A)

    void solve(zz_p& d, vec_zz_p& x, const mat_zz_p& A, const vec_zz_p& b);
    // A is an n x n matrix, b is a length n vector.  Computes d = determinant(A).
    // If d != 0, solves x*A = b (so x and b are treated as a row vectors).

    void solve(zz_p& d, const mat_zz_p& A, vec_zz_p& x, const vec_zz_p& b);
    // A is an n x n matrix, b is a length n vector.  Computes d = determinant(A).
    // If d != 0, solves A*x = b (so x and b are treated as a column vectors).

    void inv(zz_p& d, mat_zz_p& X, const mat_zz_p& A);
    // A is an n x n matrix.  Computes d = determinant(A).  If d != 0,
    // computes X = A^{-1}.

    void inv(mat_zz_p& X, const mat_zz_p& A);
    mat_zz_p inv(const mat_zz_p& A);
    // X = A^{-1}; error is raised if A is  singular

    void power(mat_zz_p& X, const mat_zz_p& A, const ZZ& e);
    mat_zz_p power(const mat_zz_p& A, const ZZ& e);
    void power(mat_zz_p& X, const mat_zz_p& A, long e);
    mat_zz_p power(const mat_zz_p& A, long e);
    // X = A^e; e may be negative (in which case A must be nonsingular).

    // NOTE: the routines determinant, solve, inv, and power (with negative
    // exponent) all require that the modulus p is prime: during elimination, if a
    // non-zero pivot element does not have an inverse, and error is raised.  The
    // following "relaxed" versions of these routines will also work with prime
    // powers, if the optional parameter relax is true (which is the default).
    // However, note that in these relaxed routines, if a computed determinant
    // value is zero, this may not be the true determinant: all that you can assume
    // is that the true determinant is not invertible mod p. If the parameter
    // relax==false, then these routines behave identically to their "unrelaxed"
    // counterparts.

    void relaxed_determinant(zz_p& d, const mat_zz_p& A, bool relax=true);
    zz_p relaxed_determinant(const mat_zz_p& a, bool relax=true);
    void relaxed_solve(zz_p& d, vec_zz_p& x, const mat_zz_p& A, const vec_zz_p& b, bool relax=true);
    void relaxed_solve(zz_p& d, const mat_zz_p& A, vec_zz_p& x, const vec_zz_p& b, bool relax=true);
    void relaxed_inv(zz_p& d, mat_zz_p& X, const mat_zz_p& A, bool relax=true);
    void relaxed_inv(mat_zz_p& X, const mat_zz_p& A, bool relax=true);
    mat_zz_p relaxed_inv(const mat_zz_p& A, bool relax=true);
    void relaxed_power(mat_zz_p& X, const mat_zz_p& A, const ZZ& e, bool relax=true);
    mat_zz_p relaxed_power(const mat_zz_p& A, const ZZ& e, bool relax=true);
    void relaxed_power(mat_zz_p& X, const mat_zz_p& A, long e, bool relax=true);
    mat_zz_p relaxed_power(const mat_zz_p& A, long e, bool relax=true);

    void sqr(mat_zz_p& X, const mat_zz_p& A);
    mat_zz_p sqr(const mat_zz_p& A);
    // X = A*A

    void ident(mat_zz_p& X, long n);
    mat_zz_p ident_mat_zz_p(long n);
    // X = n x n identity matrix

    long IsIdent(const mat_zz_p& A, long n);
    // test if A is the n x n identity matrix

    void diag(mat_zz_p& X, long n, zz_p d);
    mat_zz_p diag(long n, zz_p d);
    // X = n x n diagonal matrix with d on diagonal

    long IsDiag(const mat_zz_p& A, long n, zz_p d);
    // test if X is an  n x n diagonal matrix with d on diagonal

    void random(mat_zz_p& x, long n, long m);  // x = random n x m matrix
    mat_zz_p random_mat_zz_p(long n, long m);

    long gauss(mat_zz_p& M);
    long gauss(mat_zz_p& M, long w);
    // Performs unitary row operations so as to bring M into row echelon
    // form.  If the optional argument w is supplied, stops when first w
    // columns are in echelon form.  The return value is the rank (or the
    // rank of the first w columns).

    void image(mat_zz_p& X, const mat_zz_p& A);
    // The rows of X are computed as basis of A's row space.  X is is row
    // echelon form

    void kernel(mat_zz_p& X, const mat_zz_p& A);
    // Computes a basis for the kernel of the map x -> x*A. where x is a
    // row vector.

    // NOTE: the gauss, image, and kernel routines all require that
    // the modulus p is prime.

    // miscellaneous:

    void clear(mat_zz_p& a);
    // x = 0 (dimension unchanged)

    long IsZero(const mat_zz_p& a);
    // test if a is the zero matrix (any dimension)

    // operator notation:

    mat_zz_p operator+(const mat_zz_p& a, const mat_zz_p& b);
    mat_zz_p operator-(const mat_zz_p& a, const mat_zz_p& b);
    mat_zz_p operator*(const mat_zz_p& a, const mat_zz_p& b);

    mat_zz_p operator-(const mat_zz_p& a);

    // matrix/scalar multiplication:

    mat_zz_p operator*(const mat_zz_p& a, zz_p b);
    mat_zz_p operator*(const mat_zz_p& a, long b);

    mat_zz_p operator*(zz_p a, const mat_zz_p& b);
    mat_zz_p operator*(long a, const mat_zz_p& b);

    // matrix/vector multiplication:

    vec_zz_p operator*(const mat_zz_p& a, const vec_zz_p& b);

    vec_zz_p operator*(const vec_zz_p& a, const mat_zz_p& b);

    // assignment operator notation:

    mat_zz_p& operator+=(mat_zz_p& x, const mat_zz_p& a);
    mat_zz_p& operator-=(mat_zz_p& x, const mat_zz_p& a);
    mat_zz_p& operator*=(mat_zz_p& x, const mat_zz_p& a);

    mat_zz_p& operator*=(mat_zz_p& x, zz_p a);
    mat_zz_p& operator*=(mat_zz_p& x, long a);

    vec_zz_p& operator*=(vec_zz_p& x, const mat_zz_p& a);