libroutines

IML package provides efficient routines to solve nonsingular systems of linear equations, certifie

 *
 * Summary:
 *   Compute a mod p row-echelon transform of a mod p input matrix
 *
 * Description:
 *   Given a n x m mod p matrix A, a row-echelon transform of A is a 4-tuple 
 *   (U,P,rp,d) with rp the rank profile of A (the unique and strictly 
 *   increasing list [j1,j2,...jr] of column indices of the row-echelon form 
 *   which contain the pivots), P a permutation matrix such that all r leading
 *   submatrices of (PA)[0..r-1,rp] are nonsingular, U a nonsingular matrix 
 *   such that UPA is in row-echelon form, and d the determinant of 
 *   (PA)[0..r-1,rp].
 *
 *   Generally, it is required that p be a prime, as inverses are needed, but
 *   in some cases it is possible to obtain an echelon transform when p is 
 *   composite. For the cases where the echelon transform cannot be obtained
 *   for p composite, the function returns an error indicating that p is 
 *   composite.
 *
 *   The matrix U is structured, and has last n-r columns equal to the last n-r
 *   columns of the identity matrix, n the row dimension of A.
 *
 *   The first r rows of UPA comprise a basis in echelon form for the row 
 *   space of A, while the last n-r rows of U comprise a basis for the left 
 *   nullspace of PA.
 *
 *   For efficiency, this function does not output an echelon transform 
 *   (U,P,rp,d) directly, but rather the expression sequence (Q,rp,d).
 *   Q, rp, d are the form of arrays and pointers in order to operate inplace,
 *   which require to preallocate spaces and initialize them. Initially, 
 *   Q[i] = i (i=0..n), rp[i] = 0 (i=0..n), and *d = 1. Upon completion, rp[0]
 *   stores the rank r, rp[1..r] stores the rank profile. i<=Q[i]<=n for 
 *   i=1..r. The input Matrix A is modified inplace and used to store U. 
 *   Let A' denote the state of A on completion. Then U is obtained from the
 *   identity matrix by replacing the first r columns with those of A', and P
 *   is obtained from the identity matrix by swapping row i with row Q[i], for
 *   i=1..r in succession.
 *
 *   Parameters flrows, lrows, redflag, eterm control the specific operations
 *   this function will perform. Let (U,P,rp,d) be as constructed above. If 
 *   frows=0, the first r rows of U will not be correct. If lrows=0, the last
 *   n-r rows of U will not be correct. The computation can be up to four 
 *   times faster if these flags are set to 0.
 *
 *   If redflag=1, the row-echelon form is reduced, that is (UPA)[0..r-1,rp] 
 *   will be the identity matrix. If redflag=0, the row-echelon form will not
 *   be reduced, that is (UPA)[1..r,rp] will be upper triangular and U is unit
 *   lower triangular. If frows=0 then redflag has no effect.
 *
 *   If eterm=1, then early termination is triggered if a column of the 
 *   input matrix is discovered that is linearly dependant on the previous
 *   columns. In case of early termination, the third return value d will be 0
 *   and the remaining components of the echelon transform will not be correct.
 *
 * Input:
 *         p: FiniteField, modulus
 *         A: 1-dim Double array length n*m, representation of a n x m input
 *            matrix
 *         n: long, row dimension of A
 *         m: long, column dimension of A
 *     frows: 1/0, 
 *          - if frows = 1, the first r rows of U will be correct
 *          - if frows = 0, the first r rows of U will not be correct
 *     lrows: 1/0,
 *          - if lrows = 1, the last n-r rows of U will be correct
 *          - if lrows = 0, the last n-r rows of U will not be correct
 *   redflag: 1/0,
 *          - if redflag = 1, compute row-echelon form
 *          - if redflag = 0, not compute reow-echelon form
 *     eterm: 1/0,
 *          - if eterm = 1, terminate early if not in full rank
 *          - if eterm = 0, not terminate early
 *         Q: 1-dim long array length n+1, compact representation of 
 *            permutation vector, initially Q[i] = i, 0 <= i <= n
 *        rp: 1-dim long array length n+1, representation of rank profile, 
 *            initially rp[i] = 0, 0 <= i <= n
 *         d: pointer to FiniteField, storing determinant of the matrix, 
 *            initially *d = 1
 *
 * Precondition:
 *   ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
 *
 */




extern Double * mAdjoint (const FiniteField p, Double *A, const long n);

/* 
 * Calling Sequence:
 *   Adj <-- mAdjoint(p, A, n)
 *
 * Summary:
 *   Compute the adjoint of a mod p square matrix
 *  
 * Description:
 *   Given a n x n mod p matrix A, the function computes adjoint of A. Input
 *   A is not modified upon completion.
 *
 * Input:
 *   p: FiniteField, prime modulus
 *      if p is a composite number, the routine will still work if no error 
 *      message is returned
 *   A: 1-dim Double array length n*n, representation of a n x n mod p matrix.
 *      The entries of A are casted from integers
 *   n: long, dimension of A
 *
 * Return:
 *   1-dim Double matrix length n*n, repesentation of a n x n mod p matrix,
 *   adjoint of A
 *
 * Precondition:
 *   n*(p-1)^2 <= 2^53-1 = 9007199254740991
 *
 */


extern long mBasis (const FiniteField p, Double *A, const long n, 
  		    const long m, const long basis, const long nullsp, 
		    Double **B, Double **N);

/* 
 * Calling Sequence:
 *   r/-1 <-- mBasis(p, A, n, m, basis, nullsp, B, N)
 *
 * Summary:
 *   Compute a basis for the rowspace and/or a basis for the left nullspace 
 *   of a mod p matrix
 *  
 * Description:
 *   Given a n x m mod p matrix A, the function computes a basis for the
 *   rowspace B and/or a basis for the left nullspace N of A. Row vectors in 
 *   the r x m matrix B consist of basis of A, where r is the rank of A in
 *   Z/pZ. If r is zero, then B will be NULL. Row vectors in the n-r x n
 *   matrix N consist of the left nullspace of A. N will be NULL if A is full
 *   rank.
 *
 *   The pointers are passed into argument lists to store the computed basis 
 *   and nullspace. Upon completion, the rank r will be returned. The 
 *   parameters basis and nullsp control whether to compute basis and/or
 *   nullspace. If set basis and nullsp in the way that both basis and 
 *   nullspace will not be computed, an error message will be printed and 
 *   instead of rank r, -1 will be returned.
 *  
 * Input:
 *        p: FiniteField, prime modulus
 *           if p is a composite number, the routine will still work if no 
 *           error message is returned
 *        A: 1-dim Double array length n*m, representation of a n x m mod p 
 *           matrix. The entries of A are casted from integers
 *        n: long, row dimension of A
 *        m: long, column dimension of A
 *    basis: 1/0, flag to indicate whether to compute basis for rowspace or 
 *           not
 *         - basis = 1, compute the basis
 *         - basis = 0, not compute the basis
 *   nullsp: 1/0, flag to indicate whether to compute basis for left nullspace
 *           or not
 *         - nullsp = 1, compute the nullspace
 *         - nullsp = 0, not compute the nullspace
 *
 * Output:
 *   B: pointer to (Double *), if basis = 1, *B will be a 1-dim r*m Double
 *      array, representing the r x m basis matrix. If basis = 1 and r = 0, 
 *      *B = NULL
 *  
 *   N: pointer to (Double *), if nullsp = 1, *N will be a 1-dim (n-r)*n Double
 *      array, representing the n-r x n nullspace matrix. If nullsp = 1 and
 *      r = n, *N = NULL.
 *
 * Return:
 *   - if basis and/or nullsp are set to be 1, then return the rank r of A 
 *   - if both basis and nullsp are set to be 0, then return -1
 *
 * Precondition:
 *   n*(p-1)^2 <= 2^53-1 = 9007199254740991
 *
 * Note:
 *   - In case basis = 0, nullsp = 1, A will be destroyed inplace. Otherwise,
 *     A will not be changed.
 *   - Space of B and/or N will be allocated in the function
 *
 */



extern long mDeterminant (const FiniteField p, Double *A, const long n);

/*
 * Calling Sequence:
 *   det <-- mDeterminant(p, A, n)
 * 
 * Summary:
 *   Compute the determinant of a square mod p matrix
 *
 * Input:
 *   p: FiniteField, prime modulus
 *      if p is a composite number, the routine will still work if no error 
 *      message is returned
 *   A: 1-dim Double array length n*n, representation of a n x n mod p matrix.
 *      The entries of A are casted from integers
 *   n: long, dimension of A
 *
 * Output:
 *   det(A) mod p, the determinant of square matrix A
 *
 * Precondition:
 *   ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
 *
 * Note:
 *   A is destroyed inplace
 *
 */


extern long mInverse (const FiniteField p, Double *A, const long n);

/*
 * Calling Sequence:
 *   1/0 <-- mInverse(p, A, n)
 * 
 * Summary:
 *   Certified compute the inverse of a mod p matrix inplace
 *
 * Description:
 *   Given a n x n mod p matrix A, the function computes A^(-1) mod p 
 *   inplace in case A is a nonsingular matrix in Z/Zp. If the inverse does
 *   not exist, the function returns 0.
 *
 *   A will be destroyed at the end in both cases. If the inverse exists, A is
 *   inplaced by its inverse. Otherwise, the inplaced A is not the inverse.
 *
 * Input:
 *   p: FiniteField, prime modulus
 *      if p is a composite number, the routine will still work if no error 
 *      message is returned
 *   A: 1-dim Double array length n*n, representation of a n x n mod p matrix.
 *      The entries of A are casted from integers
 *   n: long, dimension of A
 *
 * Return: 
 *   - 1, if A^(-1) mod p exists
 *   - 0, if A^(-1) mod p does not exist
 *
 * Precondition:
 *   ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
 *
 * Note:
 *   A is destroyed inplace
 *
 */



extern long mRank (const FiniteField p, Double *A, const long n, const long m);

/*
 * Calling Sequence:
 *   r <-- mRank(p, A, n, m)
 *
 * Summary:
 *   Compute the rank of a mod p matrix
 *
 * Input:
 *   p: FiniteField, prime modulus
 *      if p is a composite number, the routine will still work if no 
 *      error message is returned
 *   A: 1-dim Double array length n*m, representation of a n x m mod p 
 *      matrix. The entries of A are casted from integers
 *   n: long, row dimension of A
 *   m: long, column dimension of A
 *   
 * Return:
 *   r: long, rank of matrix A
 *
 * Precondition:
 *   ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
 *
 * Note:
 *   A is destroyed inplace
 *
 */



extern long * mRankProfile (const FiniteField p, Double *A, 
                            const long n, const long m);

/*
 * Calling Sequence:
 *   rp <-- mRankProfile(p, A, n, m)
 *
 * Summary:
 *   Compute the rank profile of a mod p matrix
 *
 * Input:
 *   p: FiniteField, prime modulus
 *      if p is a composite number, the routine will still work if no 
 *      error message is returned
 *   A: 1-dim Double array length n*m, representation of a n x m mod p 
 *      matrix. The entries of A are casted from integers
 *   n: long, row dimension of A
 *   m: long, column dimension of A
 *   
 * Return:
 *   rp: 1-dim long array length n+1, where
 *     - rp[0] is the rank of matrix A
 *     - rp[1..r] is the rank profile of matrix A
 *
 * Precondition:
 *   ceil(n/2)*(p-1)^2+(p-1) <= 2^53-1 = 9007199254740991 (n >= 2)
 *
 * Note:
 *   A is destroyed inplace
 *
 */