Lectures on Advanced Numerical Analysis by Fritz John

By Fritz John

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One strategy is to avoid divisions when possible, or replace them by multiplications. 1). 5 of February 21, 2010 when dividing two floating-point significands) or remainder (when multiplying two residues modulo n) is needed. We also discuss exact division — when the remainder is known to be zero — and the problem of dividing by a single word. 1 Naive Division In all division algorithms, we assume that divisors are normalized. We say that B := 0n−1 bj β j is normalized when its most significant word bn−1 satisfies bn−1 ≥ β/2.

One important difference is that LSB-division does not need any correction step, since the carries go in the direction opposite to the cancelled bits. 2). 1 Roots Square Root The “paper and pencil” method once taught at school to extract square roots is very similar to “paper and pencil” division. It decomposes an integer m of the form s2 + r, taking two digits of m at a time, and finding one digit of s for each two digits of m. It is based on the following idea. If m = s2 + r is the current decomposition, then taking two more digits of the argument, we have a decomposition of the form 100m + r′ = 100s2 + 100r + r′ with 0 ≤ r′ < 100.

Since v is not needed here, we can simply avoid computing v and x, by removing steps 2 and 7. 5). All known algorithms for subquadratic gcd rely on an extended gcd subroutine which is called recursively, so we discuss the subquadratic extended gcd in the next section. 5 of February 21, 2010 Half Binary GCD, Divide and Conquer GCD Designing a subquadratic integer gcd algorithm that is both mathematically correct and efficient in practice is a challenging problem. A first remark is that, starting from n-bit inputs, there are O(n) terms in the remainder sequence r0 = a, r1 = b, .

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