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By Arne Storjohann

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This makes the computation of a single unimodular transform for achieving the form more challenging. An additional issue, especially from a complexity theoretic point of view, is that over a PIR an echelon form might not have a unique number of nonzero rows — this is handled by recovering echelon and Hermite forms with minimal numbers of nonzero rows. The primary purpose of this chapter is to establish sundry complexity results in a general setting — the algorithms return a single unimodular transform and are applicable for any input matrix over any PIR (of course, provided we can compute the basic operations).

We define the triangularizing adjoint F of A to be the lower triangular matrix with first i entries in row i equal to the last row of the adjoint of the principal i-th submatrix of A. F and the corresponding adjoint triangularization T of A satsify      F d0 ∗ .. d1 ∗ ∗  .. ··· dn−1     ∗ ∗ .. ∗ ∗ ∗ ∗ A ··· .. ··· ∗ ∗ .. ∗       =   T d1 where di is the principal i-th minor of A (d0 = 1). ∗ d2 ··· ··· .. ∗ ∗ .. 5. THE TRIANGULARIZING ADJOINT 49 We present an algorithm that requires O(nθ ) multiplication and subtractions plus O(n2 log n) exact divisions from R to compute F .

The amount of progress made during each loop iteration is di . The average cost per unit of progress (per row zeroed out) is thus cmdθ−2 . Since i di ≤ r, and the loop terminates when i di = n − 1, the overall running time is as stated. Finally, E can be recovered in the allotted time by computing the product Ir 0 Ui · · · U2 U1 from left to right. 1). Here we return an echelon form (instead of just triangular as they do) and analyse under the tri-paramater model (instead of just n and m). 7. Let A ∈ Rn×m .

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