Large Integer Arithmetic Research Articles

A revision of the subtract-with-borrow random number generators

The most popular and widely used subtract-with-borrow generator, also known as RANLUX, is reimplemented as a linear congruential generator using large integer arithmetic with the modulus size of 576 bits. Modern computers, as well as the specific structure of the modulus inferred from RANLUX, allow for the development of a fast modular multiplication — the core of the procedure. This was previously believed to be slow and have too high cost in terms of computing resources. Our tests show a significant gain in generation speed which is comparable with other fast, high quality random number generators. An additional feature is the fast skipping of generator states leading to a seeding scheme which guarantees the uniqueness of random number sequences. Program summary/New version program summaryProgram Title: RANLUX++Licensing provisions: GPLv3Programming language: C++, C, Assembler

An algorithm to solve integer linear systems exactly using numerical methods

In this paper, we present a new algorithm for the exact solutions of linear systems with integer coefficients using numerical methods. It terminates with the correct answer in well-conditioned cases or quickly aborts in ill-conditioned cases. Success of this algorithm on a linear equation requires that the linear system must be sufficiently well-conditioned for the numeric linear algebra method being used to compute a solution with sufficient accuracy. Our method is to find an initial approximate solution by using a numerical method, then amplify the approximate solution by a scalar, and adjust the amplified solution and corresponding residual to integers so that they can be computed without large integer arithmetic involved and can be stored exactly. Then we repeat these steps to refine the solution until sufficient accuracy is achieved, and finally reconstruct the rational solution. Our approximating, amplifying, and adjusting idea enables us to compute the solutions without involving high precision software floating point operations in the whole procedure or large integer arithmetic except at the final rational reconstruction step. We will expose the theoretical cost and show some experimental results.

Discrete weighted transforms and large-integer arithmetic

It is well known that Discrete Fourier Transform (DFT) techniques may be used to multiply large integers. We introduce the concept of Discrete Weighted Transforms (DWTs) which, in certain situations, substantially improve the speed of multiplication by obviating costly zero-padding of digits. In particular, when arithmetic is to be performed modulo Fermat Numbers 2 2 m + 1 {2^{{2^m}}} + 1 , or Mersenne Numbers 2 q − 1 {2^q} - 1 , weighted transforms effectively reduce FFT run lengths. We indicate how these ideas can be applied to enhance known algorithms for general multiplication, division, and factorization of large integers.

Discrete Weighted Transforms and Large-Integer Arithmetic

Discrete Weighted Transforms and Large-Integer Arithmetic

A canonical simplifier for trigonometric expressions in the kinematic equation

We present a new canonical simplifier for a class of trigonometric expressions that occur in the kinematic equation. The expressions consist of rational numbers and the function symbols +, -, *, /, sin, and cos. Variables are not allowed and the arguments of sin and cos are limited to rational multiples of π. We present an efficient algorithm which computes a simple normal form in the real domain, in contrast to the known methods which either produce unnecessarily complicated normal forms or are inefficient due to large integer arithmetic. The simplifier has successfully been applied in the field of robot kinematics.