Related papers: Algorithm to Compute Squares of 1st N Natural Numb…
It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$…
This paper offers a matrix-free first-order numerical method to solve large-scale conic optimization problems. Solving systems of linear equations pose the most computationally challenging part in both first-order and second-order numerical…
We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the…
An algorithm for computing /pi(N) is presented.It is shown that using a symmetry of natural numbers we can easily compute /pi(N).This method relies on the fact that counting the number of odd composites not exceeding N suffices to calculate…
A method of determining two factors of an odd integer without need of multiplication or division operation in iterative portion of computation is presented. It is feasible for an implementing algorithm to use only integer addition and…
The main topic of this contribution is the problem of counting square-free numbers not exceeding $n$. Before this work we were able to do it in time (Comparing to the Big-O notation, Soft-O ($\softO$) ignores logarithmic factors)…
In this paper, new schemes for a squarer, multiplier and divider of complex numbers are proposed. Traditional structural solutions for each of these operations require the presence some number of general-purpose binary multipliers. The…
We consider the following "partition and sum" operation on a natural number: Treating the number as a long string of digits insert several plus signs in between some of the digits and carry out the indicated sum. This results in a smaller…
In this note we consider optimised circuits for implementing Shor's quantum factoring algorithm. First I give a circuit for which none of the about 2n qubits need to be initialised (though we still have to make the usual 2n measurements…
This paper shows that, for matrix multiplications and convolutions, it is possible to asymptotically replace each real multiplication with a single squaring operation. Similarly, a single complex multiplication can be replaced with 3…
We show that all perfect odd integer squares not divisible by 3, can be usefully written as sqrt(N) = a + 18p, where the constant a is determined by the basic properties of N. The equation can be solved deterministically by an efficient…
The unit cost model is both convenient and largely realistic for describing integer decision algorithms over (+,*). Additional operations like division with remainder or bitwise conjunction, although equally supported by computing hardware,…
In this paper, we propose a framework based on sum-of-squares programming to design iterative first-order optimization algorithms for smooth and strongly convex problems. Our starting point is to develop a polynomial matrix inequality as a…
Today's PCs can directly manipulate numbers not longer than 64 bits because the size of the CPU registers and the data-path are limited. Consequently, arithmetic operations such as addition, can only be performed on numbers of that length.…
Given a list of N numbers, the maximum can be computed in N iterations. During these N iterations, the maximum gets updated on average as many times as the Nth harmonic number. We first use this fact to approximate the Nth harmonic number…
We offer multiplication method for factoring big natural numbers which extends the group of the Fermat's and Lehman's factorization algorithms and has run-time complexity $O(n^{1/3})$. This paper is argued the finiteness of proposed…
Matrix multiplication consumes a large fraction of the time taken in many machine-learning algorithms. Thus, accelerator chips that perform matrix multiplication faster than conventional processors or even GPU's are of increasing interest.…
We present fast and highly parallelized versions of Shor's algorithm. With a sizable quantum computer it would then be possible to factor numbers with millions of digits. The main algorithm presented here uses FFT-based fast integer…
Multiplying matrices is among the most fundamental and compute-intensive operations in machine learning. Consequently, there has been significant work on efficiently approximating matrix multiplies. We introduce a learning-based algorithm…
Consider the fundamental problem of drawing a simple random sample of size k without replacement from [n] := {1, . . . , n}. Although a number of classical algorithms exist for this problem, we construct algorithms that are even simpler,…