MonteCarloLevel: RngIntElt Default: 0
Proof: BoolElt Default: true
pAdic: BoolElt Default: true
Divisor: RngIntElt Default: 0
Given a square matrix A over the ring R, return the determinant of A as an element of R. R may be any commutative ring. The determinant of the 0 x 0 matrix over R is defined to be R!1.If the coefficient ring is the integer ring Z or the rational field Q then a modular algorithm based on that of Abbott et al. [ABM99] is used, which first computes a divisor d of the determinant D using a fast p-adic nullspace computation, and then computes the quotient D/d by computing the determinant D modulo enough small primes to cover the Hadamard bound divided by d. This always yields a correct answer.
If the parameter MonteCarloLevel is set to a small positive integer s, then a probabilistic Monte-Carlo modular technique is used. Rather than using sufficient primes to cover the Hadamard bound divided by the divisor d, this version of the algorithm terminates when the constructed residue remains constant for s steps. The probability of this being wrong is non-zero but extremely small, even if s is only 1 or 2. If the level is set to 0, then the normal deterministic algorithm is used. Setting the parameter Proof to false is equivalent to setting MonteCarloLevel to 2.
If the coefficient ring is Z and the parameter Divisor is set to an integer d, then d must be a known exact divisor of the determinant (the sign does not matter), and the algorithm may be sped up because of this knowledge.
Given a square matrix A over the ring R, return the trace of A as an element of R, which is simply the sum of the diagonal elements of A.
Given square matrices A and B over the ring R, with the same size, return the trace of A.B as an element of R. This is in general much faster than the call Trace(A*B).
Given an m x n matrix A over a ring R, return the rank of A. This is defined to be the largest r such that there exists a non-zero r x r subdeterminant of A, so r≤m and r ≤n. The rank may have to be obtained by computing the Smith form or echelon form of A, and this computation may be quite expensive over some rings.
The determinant of the submatrix of M (which must be square) formed by removing the i-th row and j-th column.
The determinant of the submatrix of M given by the row indices in I and the column indices in J.
Returns a sequence of all the r by r minors of the matrix M.
The appropriate cofactor of M, equal to ( - 1)i + j times the corresponding minor.
Returns a sequence of all the cofactors of the matrix M.
Returns a sequence of all the r by r cofactors of the matrix M.
Let M be an anti-symmetric square matrix. Then its determinant is always a square and a particular square-root of this, which can be described by a universal polynomial in its entries, is called the Pfaffian of M. The first function returns this. The second function returns the Pfaffian of the submatrix of M described by the indices in I and J. The third function returns the sequence of Pfaffians of the (n choose r) principal r by r submatrices of M (n= the number of rows of M).These are primarily convenience functions and are computed naively by Pfaffian row-expansion.