scholarly journals Handling of Complex Numbers in the CHProgramming Language

1993 ◽  
Vol 2 (3) ◽  
pp. 77-106 ◽  
Author(s):  
Harry H. Cheng
Keyword(s):  
Variable Number ◽  
Language Design ◽  
Complex Numbers ◽  
Computer Language ◽  
Real Numbers ◽  
Mathematical Functions ◽  
Complex Arithmetic ◽  
Complex Functions ◽  
Complex Features ◽  
Branch Cuts

The handling of complex numbers in the CHprogramming language will be described in this paper. Complex is a built-in data type in CH. The I/O, arithmetic and relational operations, and built-in mathematical functions are defined for both regular complex numbers and complex metanumbers of ComplexZero, Complexlnf, and ComplexNaN. Due to polymorphism, the syntax of complex arithmetic and relational operations and built-in mathematical functions are the same as those for real numbers. Besides polymorphism, the built-in mathematical functions are implemented with a variable number of arguments that greatly simplify computations of different branches of multiple-valued complex functions. The valid lvalues related to complex numbers are defined. Rationales for the design of complex features in CHare discussed from language design, implementation, and application points of views. Sample CHprograms show that a computer language that does not distinguish the sign of zeros in complex numbers can also handle the branch cuts of multiple-valued complex functions effectively so long as it is appropriately designed and implemented.

Design and Implementation of High-Level Numerical Analysis Functions Based on Computational Arrays for Applications in Engineering

2000 ◽  
Author(s):  
Harry H. Cheng ◽  
Xudong Hu ◽  
Bin Lin
Keyword(s):  
Numerical Analysis ◽  
Variable Number ◽  
Data Types ◽  
Real Numbers ◽  
Mathematical Functions ◽  
C Language ◽  
Complex Arithmetic ◽  
Design And Implementation ◽  
Complex Functions ◽  
High Level

Abstract This paper presents the design and implementation of high-level numerical analysis functions in CH, a superset of C language developed for the convenience of scientific and engineering computations. In CH, complex number is treated as a built-in data type, so that the syntaxes of complex arithmetic, relational operations, and built-in mathematical functions are the same as those for real numbers. The variable number of arguments is used in the built-in mathematical functions to simplify the computation of different branches of multi-valued complex functions. The computational arrays are introduced to handle the arrays in the numerical computations. Passing arrays of variable length by arrays of deferred-shape and arrays of assumed-shape to functions are discussed. These methods allow the arrays to be passed with their rank, dimensions and data types. A list of high-level numerical functions and two examples of the applications in the scientific and engineering are given in the paper.

scholarly journals Winding Numbers, Unwinding Numbers, and the Lambert W Function

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Complex Number ◽  
Lambert W Function ◽  
Complex Numbers ◽  
Winding Number ◽  
Line Integral ◽  
Complex Functions

AbstractThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

Encircling Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Nineteenth Century ◽  
Real Number ◽  
Traveling Salesman Problem ◽  
Complex Number ◽  
Traveling Salesman ◽  
Complex Numbers ◽  
Real Numbers ◽  
Hamiltonian Graphs ◽  
The World ◽  
The Traveling Salesman Problem

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

Unbounded Positive Definite Functions

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart
Keyword(s):  
Abelian Group ◽  
Positive Measure ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Definite Function ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

Didactic Features of Studying the Real Numbers in the School Course of Mathematics

2017 ◽  
Vol 6 (1) ◽  
pp. 65-70
Author(s):  
Алексеенко ◽  
A. Alekseenko ◽  
Лихачева ◽  
M. Likhacheva
Keyword(s):  
Real Number ◽  
Complex Numbers ◽  
Algebraic Structures ◽  
Real Numbers ◽  
Irrational Numbers ◽  
The Real ◽  
Clear Idea

The article is devoted to the study of the peculiarities of real numbers in the discipline "Algebra and analysis" in the secondary school. The theme of "Real numbers" is not easy to understand and often causes difficulties for students. However, the study of this topic is now being given enough attention and time. The consequence is a lack of understanding of students and school-leavers, what constitutes the real numbers, irrational numbers. At the same time the notion of a real number is required for further successful study of mathematics. To improve the efficiency of studying the topic and form a clear idea about the different numbers offered to add significantly to the material of modern textbooks, increase the number of hours in the study of real numbers, as well as to include in the school course of algebra topics "Complex numbers" and "Algebraic structures".

scholarly journals ON THE POWER QUANTUM COMPUTATION OVER REAL HILBERT SPACES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350001 ◽  
Author(s):  
MATTHEW McKAGUE
Keyword(s):  
Hilbert Space ◽  
Quantum Computation ◽  
Hilbert Spaces ◽  
Quantum Circuit ◽  
Complex Hilbert Space ◽  
Complexity Classes ◽  
Complex Numbers ◽  
Real Numbers ◽  
Single Qubit ◽  
Quantum Complexity

We consider the power of various quantum complexity classes with the restriction that states and operators are defined over a real, rather than complex, Hilbert space. It is well known that a quantum circuit over the complex numbers can be transformed into a quantum circuit over the real numbers with the addition of a single qubit. This implies that BQP retains its power when restricted to using states and operations over the reals. We show that the same is true for QMA (k), QIP (k), QMIP and QSZK.

scholarly journals Distributed Control for Multiagent Consensus Motions with Nonuniform Time Delays

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Mengji Shi ◽  
Kaiyu Qin
Keyword(s):  
Distributed Control ◽  
Time Delays ◽  
Original System ◽  
Complex Numbers ◽  
Control Problems ◽  
Rectilinear Motion ◽  
State Variables ◽  
Real Numbers ◽  
Numerical Examples ◽  
Delay Margin

This paper solves control problems of agents achieving consensus motions in presence of nonuniform time delays by obtaining the maximal tolerable delay value. Two types of consensus motions are considered: the rectilinear motion and the rotational motion. Unlike former results, this paper has remarkably reduced conservativeness of the consensus conditions provided in such form: for each system, if all the nonuniform time delays are bounded by the maximal tolerable delay value which is referred to as “delay margin,” the system will achieve consensus motion; otherwise, if all the delays exceed the delay margin, the system will be unstable. When discussing the system which is intended to achieve rotational consensus motion, an expanded system whose state variables are real numbers (those of the original system are complex numbers) is introduced, and corresponding consensus condition is given also in the form of delay margin. Numerical examples are provided to illustrate the results.

scholarly journals Cayley-Dickson Construction

2012 ◽  
Vol 20 (4) ◽  
pp. 281-290
Author(s):  
Artur Korniłowicz
Keyword(s):  
Complex Numbers ◽  
Real Numbers ◽  
Basic Properties

Summary Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers, quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties.

scholarly journals Platform-independent Specification and Verification of the Standard Mathematical Square Root Function

2018 ◽  
Vol 25 (6) ◽  
pp. 637-666
Author(s):  
Nikolay V. Shilov ◽  
Dmitry A. Kondratyev ◽  
Igor S. Anureev ◽  
Eugene V. Bodin ◽  
Alexei V. Promsky
Keyword(s):  
Root Function ◽  
Square Root ◽  
Combined Approach ◽  
Real Numbers ◽  
Mathematical Functions ◽  
Square Roots ◽  
Computer Aided ◽  
Platform Independence ◽  
Fix Point ◽  
Specification And Verification

The project “Platform-independent approach to formal specification and verification of standard mathematical functions” is aimed onto the development of incremental combined approach to specification and verification of standard Mathematical functions like sqrt, cos, sin, etc. Platform-independence means that we attempt to design a relatively simple axiomatization of the computer arithmetics in terms of real arithmetics (i.e. the field \(\mathbb{R}\) of real numbers) but do not specify neither base of the computer arithmetics, nor a format of numbers representation. Incrementality means that we start with the most straightforward specification of the simplest case to verify the algorithm in real numbers and finish with a realistic specification and a verification of the algorithm in computer arithmetics. We call our approach combined because we start with manual (pen-and-paper) verification of the algorithm in real numbers, then use this verification as proof-outlines for a manual verification of the algorithm in computer arithmetics, and finish with a computer-aided validation of the manual proofs with a proof-assistant system (to avoid appeals to “obviousness” that are common in human-carried proofs). In the paper, we apply our platform-independent incremental combined approach to specification and verification of the standard Mathematical square root function. Currently a computer-aided validation was carried for correctness (consistency) of our fix-point arithmetics and for the existence of a look-up table with the initial approximations of the square roots for fix-point numbers.

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