/**************************************************************************\
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 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
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);