Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computation

Abstract

Blackbox algorithms for linear algebra problems start with projection of the sequence of powers of a matrix to a sequence of vectors (Lanczos), a sequence of scalars (Wiedemann) or a sequence of smaller matrices (block methods). Such algorithms usually depend on the minimal polynomial of the resulting sequence being that of the given matrix. Here exact formulas are given for the probability that this occurs. They are based on the generalized Jordan normal form (direct sum of companion matrices of the elementary divisors) of the matrix. Sharp bounds follow from this for matrices of unknown elementary divisors. The bounds are valid for all finite field sizes and show that a small blocking factor can give high probability of success for all cardinalities and matrix dimensions.

Probabilistic analysis of block wiedemann for leading invariant factors

“…This makes precise previous observations and estimates regarding block size. The results in this paper are an extension of our previous work in which we presented formulas for P q,b,1 (A) and P q,b,1 (n) [Harrison et al, 2016].…”

“…It will show that with a blocksize modestly larger than r the probability of preserving r invariant factors is quite high, even for small fields. The construction and formula generalize Theorem 20 from [Harrison et al, 2016] which obtained a similar bound for preserving the minimal polynomial. To develop the formula, we begin with the following properties derived from Theorems 7 and 14 to compute the probability for the leading Jordan block and the Schur complement to induct on the remaining blocks.…”

“…, where M is nonsingular, and U i,e and V 1,i are the rightmost and topmost blocks of U i and V i respectively [Harrison et al, 2016]. Because V 1,i is uniformly random and M is nonsingular, M V 1,i is uniformly random, and therefore, P q,b,r (A) = P q,b,r (⊕ t i=1 C f ).…”

“…The probability that the sum of t outer products has rank r can be computed with the following recurrence [Harrison et al, 2016].…”

Quantum Walks on Generalized Quadrangles

“…See [6] for analysis of Wiedemann's algorithm. Since the minimal polynomial is squarefree, S + (U(X) 3 ) and S + (U(Y ) 3 ) are both diagonalizable over each finite field for which the computation was done.…”