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<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>

<span class="Comment">MODULE: mat_ZZ</span>

<span class="Comment">SUMMARY:</span>

<span class="Comment">Defines the class mat_ZZ.</span>

<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>


<span class="PreProc">#include </span><span class="String">&lt;NTL/matrix.h&gt;</span>
<span class="PreProc">#include </span><span class="String">&lt;NTL/vec_vec_ZZ.h&gt;</span>

<span class="Type">typedef</span> Mat&lt;ZZ&gt; mat_ZZ; <span class="Comment">// backward compatibility</span>

<span class="Type">void</span> add(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> mat_ZZ&amp; B);
<span class="Comment">// X = A + B</span>

<span class="Type">void</span> sub(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> mat_ZZ&amp; B);
<span class="Comment">// X = A - B</span>

<span class="Type">void</span> negate(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A);
<span class="Comment">// X = - A</span>

<span class="Type">void</span> mul(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> mat_ZZ&amp; B);
<span class="Comment">// X = A * B</span>

<span class="Type">void</span> mul(vec_ZZ&amp; x, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> vec_ZZ&amp; b);
<span class="Comment">// x = A * b</span>

<span class="Type">void</span> mul(vec_ZZ&amp; x, <span class="Type">const</span> vec_ZZ&amp; a, <span class="Type">const</span> mat_ZZ&amp; B);
<span class="Comment">// x = a * B</span>

<span class="Type">void</span> mul(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> ZZ&amp; b);
<span class="Type">void</span> mul(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">long</span> b);
<span class="Comment">// X = A * b</span>

<span class="Type">void</span> mul(mat_ZZ&amp; X, <span class="Type">const</span> ZZ&amp; a, <span class="Type">const</span> mat_ZZ&amp; B);
<span class="Type">void</span> mul(mat_ZZ&amp; X, <span class="Type">long</span> a, <span class="Type">const</span> mat_ZZ&amp; B);
<span class="Comment">// X = a * B</span>



<span class="Type">void</span> determinant(ZZ&amp; d, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">long</span> deterministic=<span class="Constant">0</span>);
ZZ determinant(<span class="Type">const</span> mat_ZZ&amp; a, <span class="Type">long</span> deterministic=<span class="Constant">0</span>);
<span class="Comment">// d = determinant(A).  If !deterministic, a randomized strategy may</span>
<span class="Comment">// be used that errs with probability at most 2^{-80}.</span>



<span class="Type">void</span> solve(ZZ&amp; d, vec_ZZ&amp; x,
           <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> vec_ZZ&amp; b,
           <span class="Type">long</span> deterministic=<span class="Constant">0</span>)
<span class="Comment">// computes d = determinant(A) and solves x*A = b*d if d != 0; A must</span>
<span class="Comment">// be a square matrix and have compatible dimensions with b.  If</span>
<span class="Comment">// !deterministic, the computation of d may use a randomized strategy</span>
<span class="Comment">// that errs with probability 2^{-80}.</span>



<span class="Type">void</span> solve1(ZZ&amp; d, vec_ZZ&amp; x, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> vec_ZZ&amp; b);
<span class="Comment">// A must be a square matrix.</span>
<span class="Comment">// If A is singular, this routine sets d = 0 and returns.</span>
<span class="Comment">// Otherwise, it computes d, x such that x*A == b*d, </span>
<span class="Comment">// such that d &gt; 0 and minimal.</span>
<span class="Comment">// Note that d is a positive divisor of the determinant,</span>
<span class="Comment">// and is not in general equal to the determinant.</span>
<span class="Comment">// The routine is deterministic, and uses a Hensel lifting strategy.</span>

<span class="Comment">// For backward compatability, there is also a routine called</span>
<span class="Comment">// HenselSolve1 that simply calls solve1.</span>


<span class="Type">void</span> inv(ZZ&amp; d, mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">long</span> deterministic=<span class="Constant">0</span>);
<span class="Comment">// computes d = determinant(A) and solves X*A = I*d if d != 0; A must</span>
<span class="Comment">// be a square matrix.  If !deterministic, the computation of d may</span>
<span class="Comment">// use a randomized strategy that errs with probability 2^{-80}.</span>


<span class="Comment">// NOTE:  See LLL.txt for routines that compute the kernel and</span>
<span class="Comment">// image of an integer matrix.</span>

<span class="Comment">// NOTE: See HNF.txt for a routine that computes Hermite Normal Forms.</span>

<span class="Type">void</span> sqr(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A);
mat_ZZ sqr(<span class="Type">const</span> mat_ZZ&amp; A);
<span class="Comment">// X = A*A   </span>

<span class="Type">void</span> inv(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A);
mat_ZZ inv(<span class="Type">const</span> mat_ZZ&amp; A);
<span class="Comment">// X = A^{-1}; error is raised if |det(A)| != 1.</span>

<span class="Type">void</span> power(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> ZZ&amp; e);
mat_ZZ power(<span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">const</span> ZZ&amp; e);

<span class="Type">void</span> power(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">long</span> e);
mat_ZZ power(<span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">long</span> e);
<span class="Comment">// X = A^e; e may be negative (in which case A must be nonsingular).</span>



<span class="Type">void</span> ident(mat_ZZ&amp; X, <span class="Type">long</span> n);
mat_ZZ ident_mat_ZZ(<span class="Type">long</span> n);
<span class="Comment">// X = n x n identity matrix</span>

<span class="Type">long</span> IsIdent(<span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">long</span> n);
<span class="Comment">// test if A is the n x n identity matrix</span>

<span class="Type">void</span> diag(mat_ZZ&amp; X, <span class="Type">long</span> n, <span class="Type">const</span> ZZ&amp; d);
mat_ZZ diag(<span class="Type">long</span> n, <span class="Type">const</span> ZZ&amp; d);
<span class="Comment">// X = n x n diagonal matrix with d on diagonal</span>

<span class="Type">long</span> IsDiag(<span class="Type">const</span> mat_ZZ&amp; A, <span class="Type">long</span> n, <span class="Type">const</span> ZZ&amp; d);
<span class="Comment">// test if X is an  n x n diagonal matrix with d on diagonal</span>


<span class="Type">void</span> transpose(mat_ZZ&amp; X, <span class="Type">const</span> mat_ZZ&amp; A);
mat_ZZ transpose(<span class="Type">const</span> mat_ZZ&amp; A);
<span class="Comment">// X = transpose of A</span>


<span class="Type">long</span> CRT(mat_ZZ&amp; a, ZZ&amp; prod, <span class="Type">const</span> mat_zz_p&amp; A);
<span class="Comment">// Incremental Chinese Remaindering: If p is the current zz_p modulus with</span>
<span class="Comment">// (p, prod) = 1; Computes a' such that a' = a mod prod and a' = A mod p,</span>
<span class="Comment">// with coefficients in the interval (-p*prod/2, p*prod/2]; </span>
<span class="Comment">// Sets a := a', prod := p*prod, and returns 1 if a's value changed.</span>



<span class="Comment">// miscellaneous:</span>

<span class="Type">void</span> clear(mat_ZZ&amp; a);
<span class="Comment">// x = 0 (dimension unchanged)</span>

<span class="Type">long</span> IsZero(<span class="Type">const</span> mat_ZZ&amp; a);
<span class="Comment">// test if a is the zero matrix (any dimension)</span>


<span class="Comment">// operator notation:</span>

mat_ZZ <span class="Statement">operator</span>+(<span class="Type">const</span> mat_ZZ&amp; a, <span class="Type">const</span> mat_ZZ&amp; b);
mat_ZZ <span class="Statement">operator</span>-(<span class="Type">const</span> mat_ZZ&amp; a, <span class="Type">const</span> mat_ZZ&amp; b);
mat_ZZ <span class="Statement">operator</span>*(<span class="Type">const</span> mat_ZZ&amp; a, <span class="Type">const</span> mat_ZZ&amp; b);

mat_ZZ <span class="Statement">operator</span>-(<span class="Type">const</span> mat_ZZ&amp; a);


<span class="Comment">// matrix/scalar multiplication:</span>

mat_ZZ <span class="Statement">operator</span>*(<span class="Type">const</span> mat_ZZ&amp; a, <span class="Type">const</span> ZZ&amp; b);
mat_ZZ <span class="Statement">operator</span>*(<span class="Type">const</span> mat_ZZ&amp; a, <span class="Type">long</span> b);

mat_ZZ <span class="Statement">operator</span>*(<span class="Type">const</span> ZZ&amp; a, <span class="Type">const</span> mat_ZZ&amp; b);
mat_ZZ <span class="Statement">operator</span>*(<span class="Type">long</span> a, <span class="Type">const</span> mat_ZZ&amp; b);

<span class="Comment">// matrix/vector multiplication:</span>

vec_ZZ <span class="Statement">operator</span>*(<span class="Type">const</span> mat_ZZ&amp; a, <span class="Type">const</span> vec_ZZ&amp; b);

vec_ZZ <span class="Statement">operator</span>*(<span class="Type">const</span> vec_ZZ&amp; a, <span class="Type">const</span> mat_ZZ&amp; b);



<span class="Comment">// assignment operator notation:</span>

mat_ZZ&amp; <span class="Statement">operator</span>+=(mat_ZZ&amp; x, <span class="Type">const</span> mat_ZZ&amp; a);
mat_ZZ&amp; <span class="Statement">operator</span>-=(mat_ZZ&amp; x, <span class="Type">const</span> mat_ZZ&amp; a);
mat_ZZ&amp; <span class="Statement">operator</span>*=(mat_ZZ&amp; x, <span class="Type">const</span> mat_ZZ&amp; a);

mat_ZZ&amp; <span class="Statement">operator</span>*=(mat_ZZ&amp; x, <span class="Type">const</span> ZZ&amp; a);
mat_ZZ&amp; <span class="Statement">operator</span>*=(mat_ZZ&amp; x, <span class="Type">long</span> a);

vec_ZZ&amp; <span class="Statement">operator</span>*=(vec_ZZ&amp; x, <span class="Type">const</span> mat_ZZ&amp; a);


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