Richard P. Brent

The Parallel Evaluation of General Arithmetic Expressions

It is shown that arithmetic expressions with <italic>n</italic> ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log<subscrpt>2</subscrpt><italic>n</italic> + 10(<italic>n</italic> - 1)/<italic>p</italic> using <italic>p</italic> ≥ 1 processors which can independently perform arithmetic operations in unit time. This bound is within a constant factor of the best possible. A sharper result is given for expressions without the division operation, and the question of numerical stability is discussed.

Fast solution of toeplitz systems of equations and computation of Padé approximants

Abstract We present two new algorithms, ADT and MDT, for solving order-n Toeplitz systems of linear equations Tz = b in time O(n log2 n) and space O(n). The fastest algorithms previously known, such as Trenchs algorithm, require time Ω(n 2 ) and require that all principal submatrices of T be nonsingular. Our algorithm ADT requires only that T be nonsingular. Both our algorithms for Toeplitz systems are derived from algorithms for computing entries in the Pade table for a given power series. We prove that entries in the Pade table can be computed by the Extended Euclidean Algorithm. We describe an algorithm EMGCD (Extended Middle Greatest Common Divisor) which is faster than the algorithm HGCD of Aho, Hopcroft and Ullman, although both require time O(n log2 n), and we generalize EMGCD to produce PRSDC (Polynomial Remainder Sequence Divide and Conquer) which produces any iterate in the PRS, not just the middle term, in time O(n log2 n). Applying PRSDC to the polynomials U0(x) = x2n+1 and U1(x) = a0 + a1x + … + a2nx2n gives algorithm AD (Anti-Diagonal), which computes any (m, p) entry along the antidiagonal m + p = 2n of the Pade table for U1 in time O(n log2 n). Our other algorithm, MD (Main-Diagonal), computes any diagonal entry (n, n) in the Pade table for a normal power series, also in time O(n log2 n). MD is related to Schonhages fast continued fraction algorithm. A Toeplitz matrix T is naturally associated with U1, and the (n, n) Pade approximation to U1 gives the first column of T−1. We show how a formula due to Trench can be used to compute the solution z of Tz = b in time O(n log n) from the first row and column of T−1. Thus, the Pade table algorithms AD and MD give O(n log2 n) Toeplitz algorithms ADT and MDT. Trenchs formula breaks down in certain degenerate cases, but in such cases a companion formula, the discrete analog of the Christoffel-Darboux formula, is valid and may be used to compute z in time O(n log2n) via the fast computation (by algorithm AD) of at most four Pade approximants. We also apply our results to obtain new complexity bounds for the solution of banded Toeplitz systems and for BCH decoding via Berlekamps algorithm.

The Solution of Singular-Value and Symmetric Eigenvalue Problems on Multiprocessor Arrays

Parallel Jacobi-like algorithms are presented for computing a singular-value decomposition of an

Fast Multiple-Precision Evaluation of Elementary Functions

m \times n

Fast Algorithms for Manipulating Formal Power Series

matrix

A Fortran Multiple-Precision Arithmetic Package

(m \geqq n)

An algorithm with guaranteed convergence for finding a zero of a function

and an eigenvalue decomposition of an

The Area-Time Complexity of Binary Multiplication

n \times n

Fast training algorithms for multilayer neural nets

symmetric matrix. A linear array of

On the zeros of the Riemann zeta function in the critical strip

O(n)