Exact Solution Of Linear Equations Research Articles

Algorithms for Inversion Mod $p^k$

This article describes and analyzes all existing algorithms for computing $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) for a prime $p$ p , and also introduces a new algorithm based on the exact solution of linear equations using $p$ p -adic expansions. The algorithm starts with the initial value $c=a^{-1}\pmod {p}$ c = a - 1 ( mod p ) and iteratively computes the digits of the inverse $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) in base $p$ p . The mod 2 version of the algorithm is more efficient than all existing algorithms for small values of $k$ k . Moreover, it stands out as being the only one that works for any $p$ p , any $k$ k , and digit-by-digit. While the new algorithm is asymptotically worse off, it requires the minimal number of arithmetic operations (just a single addition) per step, as compared to all existing algorithms.

Systolic arrays for integer Chinese remaindering

This paper presents time-minimal and processor-time-minimal systolic arrays for computing a process dependence graph corresponding to the mixed-radix conversion algorithm. The arrays are particularly suitable for implementation of algorithms from the applications of residue number systems on programmable systolic/wavefront arrays and in VLSI. Examples of such applications are exact solution of linear equations and computation of pseudo-inverses of matrices.

A parallel algorithm for exact solution of linear equations via congruence technique

We present a parallel algorithm for computing the exact solution of a system of linear equations via the congruence technique. The basic idea of the technique is to convert the original system of equations into a system of congruences modulo various primes, and combine the solutions by the application of the Chinese remainder theorem. The parallel congruence algorithm proposed in this paper requires only local communication among the processors and is particularly suitable for implementation on distributed-memory multiprocessors and systolic computing systems. We have implemented the parallel congruence algorithm on an Intel cube and obtained an efficiency of greater than 90%.

A Comparison of Algorithms for the Exact Solution of Linear Equations

A Comparison of Algorithms for the Exact Solution of Linear Equations

The Exact Solution of Linear Equations with Rational Function Coefficients

The Exact Solution of Linear Equations with Rational Function Coefficients

Remark on Algorithm 406: Exact Solution of Linear Equations Using Residue Arithmetic [F4

Remark on Algorithm 406: Exact Solution of Linear Equations Using Residue Arithmetic [F4

Algorithm 406: exact solution of linear equations using residue arithmetic [F4

Algorithm 406: exact solution of linear equations using residue arithmetic [F4

Exact solutions of linear equations with rational coefficients

Exact solutions of linear equations with rational coefficients

Exact Solutions of Linear Equations with Rational Coefficients by Congruence Techniques

Exact Solutions of Linear Equations with Rational Coefficients by Congruence Techniques

Exact solutions of linear equations with rational coefficients by congruence techniques

Exact solutions of linear equations with rational coefficients by congruence techniques