diff --git "a/-9AzT4oBgHgl3EQfvf1g/content/tmp_files/load_file.txt" "b/-9AzT4oBgHgl3EQfvf1g/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-9AzT4oBgHgl3EQfvf1g/content/tmp_files/load_file.txt" @@ -0,0 +1,762 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf,len=761 +page_content='Implementation of hyperbolic complex numbers in Julia language Anna V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Korolkova,1, \\ast Migran N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Gevorkyan,1, \\dagger and Dmitry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Kulyabov1, 2, ‡ 1Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation 2Joint Institute for Nuclear Research 6 Joliot-Curie, Dubna, Moscow region, 141980, Russian Federation Background: Hyperbolic complex numbers are used in the description of hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' One of the well-known examples of such spaces is the Minkowski space, which plays a leading role in the problems of the special theory of relativity and electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' However, such numbers are not very common in different programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Purpose: Of interest is the implementation of hyperbolic complex in scientific programming languages, in particular, in the Julia language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Methods: The Julia language is based on the concept of multiple dispatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This concept is an extension of the concept of polymorphism for object-oriented programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' To implement hyperbolic complex numbers, the multiple dispatching approach of the Julia language was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Results: The result is a library that implements hyperbolic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Conclusions: Based on the results of the study, we can conclude that the concept of multiple dispatching in scientific programming languages is convenient and natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Keywords: Julia programming language, multiple dispatch, abstract data types, type conversion, parametric structures, hyperbolic complex numbers I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' INTRODUCTION The Julia programming language [1, 2] is a promising language for scientific computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' At the moment, the Julia language has reached a stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' By design, Julia solves the problem of two languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This problem lies in the fact that for rapid prototyping, data processing and visualization, an interpreted dynamic language or a mathematical package (Python, Matlab, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') is used, and for intensive numerical calculations, the program has to be rewritten in a compiled language with static typing (C/ C++, Fortran).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' An illustration of this problem can be seen in Python, which has gained wide popularity as an interface language-glue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Numerous wrapper libraries were written on it, which used Python code to call C/C++ and Fortran functions from precompiled libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' For example, the well-known library NumPy [3] consists of 51% C code and only 47% Python code (the remaining percentages are divided between C++, Fortran, JavaScript and Unix shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The Julia language combines the flexibility of dynamically typed interpreted languages with the performance of statically typed compiled languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The basic part of the Julia language is very similar to other scientific programming languages, so it does not cause difficulties in mastering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' However, Julia’s core is built around the concept of multiple dispatch [4], which is rare in other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' It is in this mechanism that the essential difference of Julia from other languages lies, and its understanding is essential for the full use of all the advantages of Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Paper structure In the article, the authors paid great attention to illustrating the mechanism of multiple dispatch and other mechanisms that are closely related to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the first part of the article, we give the necessary definitions and illustrate the concept of multiple dispatch with simple examples that allow you to understand the syntax associated with this part of the language and capture the essence of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the second part, we give an example of the implementation of hyperbolic complex numbers in the Julia language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This example allows you to touch not only multiple dispatch, but also the type casting mechanism, the abstract type hierarchy, overloading arithmetic operators, and specifying user-defined data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' \\ast korolkova-av@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru \\dagger gevorkyan-mn@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru ‡ kulyabov-ds@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='01707v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='MS] 4 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' MULTIPLE DISPATCH A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Common definitions Dynamic dispatch is a mechanism that allows you to choose which of the many implementations of a polymorphic function (or operator) should be called in a given case [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In this case, the choice of one or another implementation is carried out at the stage of program execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Multiple dispatch is based on dynamic dispatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In this case, the choice of implementation of a polymorphic function is made based on the type, number, and order of the function’s arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This is how runtime polymorphic dispatch is implemented [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Note also that in addition to the term multiple dispatch, the term multimethod is also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The mechanism of multiple dispatch is similar to the mechanism of overloading functions and operators, implemented, for example, in the C++ language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Function overloading, however, is done exclusively at compile time, while multiple dispatch should work at runtime as well (runtime polymorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Multiple dispatch in Julia To illustrate the mechanism of multiple dispatch, we will give the following code example in the Julia language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y) println("Generic implementation") return x + y end function f(x) println("For single argument") return x end function f(x::Integer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::Integer) println("Implementation for integers") return x + y end function f(x::String,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::String) println("Implementation for strings") return x * " " * y end function f(x::Tuple{Int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Int},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::Tuple{Int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Int}) println("Implementation for tuples of two integer elements") return (x[1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' x[2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y[1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y[2]) end In this example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' we have created five implementations of the f function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' which differ from each other in different signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In terms of the Julia language, this means that one function f now has four different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the first two methods, we did not use type annotations, so the type of the arguments will be determined either at compile time or at run time (as in interpreted languages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' It is also worth noting that Julia uses dynamic JIT compilation (just-in-time), so the compilation stage is not explicitly separated from the execution stage for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The arguments of the following three methods are annotated with types, so they will only be called if the types match the annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the f for strings, the * concatenation operator is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The choice of the multiplication sign instead of the more traditional addition sign + is justified by the creators of the language by the fact that string concatenation is not a commuting operation, so it is more logical to use the multiplication sign for it, rather than the addition sign, which is often used to denote commuting operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The following code snippet illustrates how multiple dispatch works at compile time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The @show macro is used to print out the name of the function and the arguments passed to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' @show f(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 1) @show f(2, 2) 3 @show f(0x2, 0x1) # numbers in hexadecimal system @show f("Text", "line") @show f(3) @show f([1, 2], [3, 4]) @show f((1, 2), (3, 4)) In the first line, we passed real (floating-point) type arguments to the function, so a generic implementation call was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Since the operator + is defined for floating point numbers, the function succeeded and gave the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Methods for integers were called in the second and third lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Note that the Integer type is an abstract type and includes signed and unsigned integers from 1 to 16 bytes in size, defined in the language core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Numbers written in hexadecimal are interpreted by default as unsigned integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The method for strings was called on the fourth line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the fifth line, the method for one argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The sixth line passed two arrays as arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The + operation is defined for arrays, so the function ran without error and returned an element-wise sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the seventh line, the function arguments are tuples consisting of two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Since we defined a method for such a combination of arguments, the function worked correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Generic implementation f(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0 Implementation for integers f(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2) = 4 Implementation for integers f(0x02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 0x01) = 0x03 Implementation for strings f("Text",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' "line") = "Text line" For single argument f(3) = 3 Generic implementation f([1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4]) = [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6] Implementation for tuples of two integer elements f((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4)) = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4) The above examples will work correctly in languages that support function overloading and do not demonstrate the specifics of dynamic dispatching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' since the types of arguments are known at the compilation stage and are available to the translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' To test the work of dynamic method calls, consider the following code: print("Enter an integer:") # Read a string and convert to an integer type @show n = parse(Int32, readline()) if n > 0 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='1 else x = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y = 2 end f(x, y) Here, the types of variable values x and y are not known at compile time, as they depend on what number the user enters during program execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' However, for the case of integer x and y the corresponding method is called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' HYPERBOLIC NUMBERS We will use hyperbolic numbers to illustrate the multiple dispatch capabilities of the Julia language, so we will limit ourselves to the definition and basic arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4 Hyperbolic numbers [8–11], along with elliptic and parabolic numbers, are a generalization of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Hyperbolic numbers can be defined as follows: z = x + jy, j2 = 1, j \\not = \\pm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The quantity j will be called the hyperbolic imaginary unit, and the quantities x and y will be called the real and imaginary parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' For two hyperbolic numbers z1 = x1 + jy1 and z2 = x2 + jy2 the following arithmetic operations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Addition: z1 + z2 = (x1 + x2) + j(y1 + y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Multiplication: z1z2 = (x1x2 + y1y2) + j(x1y2 + x2y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Conjugation: z\\ast = x - jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Inverse number: z - 1 = x x2 + y2 - j y x2 - y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Division: z1 z2 = x1x2 - y1y2 x2 2 - y2 2 + jx1y1 - x1y2 x2 2 - y2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The implementation of hyperbolic numbers is in many respects similar to the implementation of complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Operators +, -, * must be overloaded, and /, root extraction, exponentiation, elementary math functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' At the same time, for the purposes of illustrating the mechanism of operation of multiple dispatching, it is arithmetic operations that are of primary interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This is due to the fact that elementary functions take only one argument, and it is enough to define only one method for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the case of arithmetic operators, it is necessary to provide combinations of arguments of different numeric types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' So, for example, it should be possible to add a hyperbolic number to an integer, rational, irrational number, which automatically affects not only multiple dispatch, but also type casting mechanisms, an abstract type hierarchy, and default constructor overloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Therefore, we will confine ourselves to examples of the implementation of precisely arithmetic operations and that’s all, without touching on the more mathematically complex calculations of various elementary functions of a hyperbolic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Note that in addition to the term hyperbolic numbers, there are also terms in the literature: double numbers, split complex numbers, perplex numbers, hyperbolic numbers [8, 12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' IMPLEMENTATION OF HYPERBOLIC NUMBERS IN JULIA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Declaring a Data Structure The implementation of hyperbolic numbers in Julia was based on the code for complex numbers available in the official Julia repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' We also used the developments obtained in the implementation of parabolic complex numbers [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' New type Hyperbolic defined with an immutable structure: struct Hyperbolic{T<:Real} <: Number "Real part" re::T "Imaginary part" jm::T end The structure is simple and contains only two fields of parametric type T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This requires that the type T was a subtype of the abstract type Real (syntax T<:Real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The type Hyperbolic is a subtype of the abstract type Number (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Thus, hyperbolic numbers are built into an already existing hierarchy of numeric types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' After the structure is defined, a new object of type Hyperbolic can be created by calling the default constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' So, for example, the number h = 1 + j3 is given as follows: h = Hyperbolic{Float64}(1, 3) After creation, you can access the fields of the structure as h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm, but an attempt changing the value of a field of an already existing object will result in an error, since structs are immutable entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' h = Hyperbolic(1, 3) 5 Number Hyperbolic Complex Real Integer Signed Int8 Int16 Int32 Int64 Int128 Bool Unsigned UInt8 UInt16 UInt32 UInt64 UInt128 Rational AbstractFloat Float16 Float32 Float64 Legend: Abstract type Primitive type Structure Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Location of Hyperbolic Numbers in Julia’s Type Hierarchy However, if the argument types are different, then the default constructor will not be able to implicitly cast and create a new object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In this case, you must explicitly specify the parametric type # Float64 и Int64 h = Hyperbolic(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 3) # Error h = Hyperbolic{Float64}(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 3) # Correct B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Additional constructors The default constructor is a normal function whose name is the same as the type name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' By creating additional methods for this function, you can create additional constructors to handle various special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' So, for example, in order not to specify a parametric type every time, you should add a new constructor of the following form: """Constructor №2""" function Hyperbolic(x::Real, y::Real) return Hyperbolic(promote(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end The promote function casts the arguments passed to it to a common type and returns the result as a tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Postfix operator .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' unpacks the tuple and passes its elements as arguments to the constructor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The language core defines casting rules for all subtypes of the Real abstract type, so now the constructor will work correctly for any combination of arguments, as long as the T<:Real rule is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' For example, the following code will work correctly: # Rational и Float64 h = Hyperbolic(1//3, pi) >> Hyperbolic{Float64}(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='141592653589793) We passed a rational number (type Rational) and a built-in global constant (number \\pi ) of type Float64 to the constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' After that, the type casting rule worked and both arguments were cast to the type Float64 as more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Declaring two more additional constructors will allow you to specify hyperbolic numbers with zero imaginary part: """Constructor №3""" function Hyperbolic{T}(x::Real) where {T<:Real} return Hyperbolic{T}(x, 0) end """Constructor №4""" function Hyperbolic(x::Real) return Hyperbolic(promote(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end Constructor number 3 is a parametric function that is declared using the where construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The T is a subtype of the abstract type Real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Constructor number 4 works similarly to constructor number 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Two more constructors will allow you to pass other hyperbolic numbers as an argument to the constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6 """Constructor №5""" function Hyperbolic{T}(h::Hyperbolic) where {T<:Real} Hyperbolic{T}(h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) end """Constructor №6""" function Hyperbolic(h::Hyperbolic) return Hyperbolic(promote(h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end For more convenience, you can also create a separate constant for the imaginary cost j: const jm = Hyperbolic(0, 1) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Data printing To be able to print hyperbolic type values in a compact and readable form, you should add the appropriate methods to the show function from the Base module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='show(io::IO, h::Hyperbolic) print(io, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, "+", h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm, "j") end Function show is used when printing data to the console, in particular, it is called by the println and macro @show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The code and output listings below will assume that the show method has been added for hyperbolic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Type casting Before proceeding to the implementation of methods for arithmetic operations with hyperbolic numbers, it is necessary to define the rules for type casting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' To do this, create a new method for the function promote_rule from the Base module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='promote_rule(::Type{Hyperbolic{T}}, ::Type{S}) where {T<:Real, S<:Real} return Hyperbolic{promote_type(T, S)} end function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='promote_rule(::Type{Hyperbolic{T}}, ::Type{Hyperbolic{S}}) where {T<:Real, S<:Real} \\lhook → return Hyperbolic{promote_type(T, S)} end As arguments in promote_rule parametric types are specified, which should be cast to one enclosing type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In our case, this is possible if one of the types is a subtype of Real, then the enclosing type is Hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' After adding methods for promote_rule, it becomes possible to use functions promote, promote_type and convert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' >>h = Hyperbolic(1 // 2) >>promote(h, 1) (1//2+0//1j, 1//1+0//1j) >>promote_type(Hyperbolic{Int64}, Float32) Hyperbolic{Float32} The first function is already familiar to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The second allows you to infer the enclosing type not of specific variable values, but of the types themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A type in Julia is an object of the first kind (type DataType) and can be assigned to other variables, passed as function arguments, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Function convert allows you to convert the type specific value, for example: >>convert(Hyperbolic, 1) 1+0j After adding methods for type casting, you can start adding methods for arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A feature of Julia is the implementation of arithmetic operations not in the form of operators, but in the form of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' For example, the following calls are correct: 7 >>+(1,2) 3 >>+(1,2,3,4) 10 >>+((i for i in 1:10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') # числа от 1 до 10 55 In this regard, adding methods for arithmetic operations is no different from the corresponding process for other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Adding methods for unary operations + and - is carried out as follows: Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :+(h::Hyperbolic) = Hyperbolic(+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :-(h::Hyperbolic) = Hyperbolic(-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, -h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) This is an abbreviated function declaration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Similarly, methods are added for binary addition, subtraction, multiplication, and division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Here is the code for addition and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' # Binary + and * function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :+(x::Hyperbolic, y::Hyperbolic) xx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re yy = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm Hyperbolic(xx, yy) end function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :*(x::Hyperbolic, y::Hyperbolic) xx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm yy = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='je * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re return Hyperbolic(xx, yy) end V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' CONCLUSION We examined the mechanism of multiple dispatch underlying the Julia language, using the example of the implemen- tation of hyperbolic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This example allowed us to touch upon such concepts of the language as the hierarchy of data types, composite data types, type casting mechanisms, function overloading (creating new methods for functions in terms of the Julia language), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ACKNOWLEDGMENTS This paper has been supported by the RUDN University Strategic Academic Leadership Program.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Kulyabov, Approaches to the implementation of generalized complex numbers in the julia language, in: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Kulyabov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Samouylov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Sevastianov (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ), Workshop on information technology and scientific computing in the framework of the X International Conference Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems (ITTMM-2020), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2639 of CEUR Workshop Proceedings, Aachen, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 141–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' URL http://ceur-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='org/Vol-2639/paper-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='pdf Реализация гиперболических комплексных чисел на языке Julia А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Королькова,1, \\ast М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Н.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Геворкян,1, \\dagger and Д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Кулябов1, 2, ‡ 1Российский университет дружбы народов, 117198, Москва, ул.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Миклухо-Маклая, д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6 2Объединённый институт ядерных исследований, ул.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Жолио-Кюри 6, Дубна, Московская область, Россия, 141980 Предпосылки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Гиперболические комплексные числа применяются при описании гиперболи- ческих пространств.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Одним из известных примером таких пространств является пространство Минковского, играющее ведущее значение в задачах частной теории относительности, электро- динамики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Однако такие числа не очень распространены в разных языках программирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Цель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Представляет интерес реализация гиперболических комплексных в языках научного программирования, в частности, в языке Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Методы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В основе языка Julia лежит концепция множественной диспетчеризации (multiple dispatch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Эта концепция является расширением концепции полиморфизма для объектно-ориентированных языков программирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Для реализации гиперболических комплексных чисел использован подход множественной дис- петчеризацию языка Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Результаты.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В результате получена библиотека, реализующая гиперболические числа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Выводы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' По результатам исследования можно сделать вывод об удобстве и естественности концепции множественной диспетчеризации в языках научного программирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Keywords: язык программирования Julia, множественная диспетчеризация, абстрактные типы данных, конвертация типов, параметрические структуры, гиперболические комплексные числа I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ВВЕДЕНИЕ Язык программирования Julia [1, 2] — это перспективный язык, предназначенный для научных вычислений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В настоящий момент язык Julia достиг стабильного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' По замыслу разработчиков Julia решает проблему двух языков.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Данная проблема заключается в том, что для быстрого прототипирования, обработки данных и визуализации используется интерпретируемый динамический язык или математический пакет (Python, Matlab и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ), а для интенсивных численных расчётов программу приходится переписывать на компилируемом языке со статической типизацией (C/C++, Fortran).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Иллюстрацию данной проблемы можно увидеть на примере языка Python, который приобрел широкую попу- лярность в качестве интерфейсного «языка-клея».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' На нем было написано большое количество библиотек-обёрток, которые использовали Python-код для вызова C/C++ и Fortran функций из предварительно скомпилированных библиотек.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, известная библиотека NumPy [3] на 51% состоит из кода на языке Си и лишь на 47% из кода на языке Python (оставшиеся проценты делят между собой C++, Fortran, JavaScript и Unix shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Язык Julia совмещает в себе гибкости интерпретируемых языков с динамической типизацией и производитель- ность компилируемых языков со статической типизацией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Базовая часть языка Julia крайне схожа с другими языками научного программирования поэтому не вызывает трудности при освоении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Однако ядро Julia построено вокруг концепцию множественной диспетчеризации (multiple dispatch) [4], которая редко встречается в других языках.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Именно в этом механизме лежит существенное отличие Julia от других языков и его понимание существенно для полноценного использования всех преимуществ Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Структура статьи В статье авторы уделили большое внимание иллюстрации механизма множественной диспетчеризации и других механизмов, которые близко с ней связаны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В первой части статьи мы даем необходимые определения и иллюстрируем концепцию множественной диспетчеризации на простых примерах, позволяющих понять синтаксис, связанный с этой частью языка и \\ast korolkova-av@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru \\dagger gevorkyan-mn@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru ‡ kulyabov-ds@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='01707v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='MS] 4 Jan 2023 2 уловить суть данного подхода.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Во второй части мы приводим пример реализации гиперболических комплексных чисел на языке Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Данный пример позволяет затронуть не только множественную диспетчеризацию, но и механизм приведения типов, иерархию абстрактных типов, перегрузку арифметических операторов и задание пользовательских типов данных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' МНОЖЕСТВЕННАЯ ДИСПЕТЧЕРИЗАЦИЯ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Общие определения Динамическая диспетчеризация (dynamic dispatch) — это механизм, который позволяет выбрать какую из множества реализаций полиморфной функции (или оператора) следует вызвать в данном конкретном случае [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' При этом выбор той или иной реализации осуществляется на стадии выполнения программы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Множественная диспетчеризация основывается на динамической диспетчеризации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В этом случае выбор реализации полиморф- ной функции делается исходя из типа, количества и порядка следования аргументов функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Таким образом реализуется полиморфизм времени выполнения (runtime polymorphic dispatch) [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Заметим также, что кроме термина «множественная диспетчеризация», также употребляется термин мультиметод.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Механизм множественной диспетчеризации похож на механизм перегрузки функций и операторов, реали- зованный, например, в языке C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Перегрузка функций, однако, осуществляется исключительно на стадии компиляции, тогда как множественная диспетчеризация должна работать также и на стадии выполнения программы (полиморфизм времени выполнения).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Множественная диспетчеризация в Julia Для иллюстрации механизма множественной диспетчеризации приведём следующий пример кода на языке Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y) println("Общая реализация") return x + y end function f(x) println("Для одного аргумента") return x end function f(x::Integer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::Integer) println("Реализация для целых чисел") return x + y end function f(x::String,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::String) println("Реализация для строк") return x * " " * y end function f(x::Tuple{Int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Int},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::Tuple{Int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Int}) println("Реализация для кортежей из двух целочисленных элементов") return (x[1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' x[2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y[1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y[2]) end В данном примере мы создали пять реализаций функции f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' которые отличаются друг от друга разными сигнатурами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В терминах языка Julia это означает, что у одной функции f теперь существует четыре разных метода.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В первых двух методах мы не использовали аннотаций типов, поэтому тип аргументов будет определен или на стадии компиляции или на стадии выполнения программы (как в интерпретируемых языках).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Стоит также отметит, что Julia использует динамическую JIT-компиляцию (just-in-time), поэтому стадия компиляции от стадии выполнения отделена для пользователя не явным образом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 3 Аргументы трех следующих методов аннотированы типами, поэтому будут вызываться только в случае совпадения типов с аннотациями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В методе f для строк используется оператор конкатенации *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Выбор знака умножения * вместо более традиционного знака сложения + обосновывается создателями языка тем, что конкатенация строк операция не коммутирующая, поэтому более логично использовать для нее знак умножения, а не сложения, которым чаще все принято обозначать коммутирующие операции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Следующий фрагмент кода иллюстрирует работу множественной диспетчеризации на стадии компиляции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Макрос @show служит для распечатки имени функции и переданных ей аргументов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' @show f(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 1) @show f(2, 2) @show f(0x2, 0x1) # числа в шестнадцатеричной системе @show f("Строка", "текста") @show f(3) @show f([1, 2], [3, 4]) @show f((1, 2), (3, 4)) В первой строке мы передали функции аргументы вещественного типа (с плавающей точкой), поэтому был осуществлен вызов общей реализации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так как для чисел с плавающей точкой определен ��ператор +, то функция выполнилась успешно и дала правильный результат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Во второй и третей строках были вызваны методы для целых чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Заметим, что тип Integer является абстрактным типом и включает в себя знаковые и беззнаковые целые числа размером от 1 до 16 байт, определённые в ядре языка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Числа, записанные в шестнадцатеричной системе счисления интерпретируются по умолчанию как беззнаковые целые.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В четвертой строке был вызван метод для строк.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В пятой строке метод для одного аргумента.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В шестой строке в качестве аргументов переданы два массива.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Операция + определена для массивов, поэтому функция выполнилась без ошибок и вернула поэлементную сумму.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В седьмой строке аргументами функции являются кортежи, состоящие из двух целых чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так как нами был определен метод для такой комбинации аргументов – функция отработала корректно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Общая реализация f(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0 Реализация для целых чисел f(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2) = 4 Реализация для целых чисел f(0x02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 0x01) = 0x03 Реализация для строк f("Строка",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' "текста") = "Строка текста" Для одного аргумента f(3) = 3 Общая реализация f([1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4]) = [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6] Реализация для кортежей из двух целочисленных элементов f((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4)) = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4) Приведённые примеры корректно сработают и в языках,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' поддерживающих перегрузку функций и не демон- стрируют специфику динамической диспетчеризации,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' так как типы аргументов известны на стадии компиляции и доступны транслятору.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Для проверки работы именно динамического вызова методов рассмотрим следующий код: print("Введите целое число:") # Считываем строку и конвертируем в целый тип @show n = parse(Int32, readline()) if n > 0 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='1 else x = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y = 2 end f(x, y) 4 Здесь типы значений переменных x и y не известны на стадии компиляции, так как зависят от того, какое число введёт пользователь во время выполнения программы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Тем не менее, для случая целочисленных x и y вызывается соответствующий метод.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ГИПЕРБОЛИЧЕСКИЕ ЧИСЛА Мы будем использовать гиперболические числа для иллюстрации возможностей множественной диспетчериза- ции языка Julia, поэтому ограничимся лишь определением и основными арифметическими операциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Гиперболические числа [8–11], наряду с эллиптическими и параболическими числами, являются обобщением комплексных чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Гиперболические числа можно определить следующим образом: z = x + jy, j2 = 1, j \\not = \\pm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Величину j будем называть гиперболической мнимой единицей, а величины x и y действительной и мнимой частями соответственно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Для двух гиперболических чисел z1 = x1 + jy1 и z2 = x2 + jy2 выполняются следующие арифметические операции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Сложение: z1 + z2 = (x1 + x2) + j(y1 + y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Умножение: z1z2 = (x1x2 + y1y2) + j(x1y2 + x2y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Сопряжение: z\\ast = x - jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Обратное число: z - 1 = x x2 + y2 - j y x2 - y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Деление: z1 z2 = x1x2 - y1y2 x2 2 - y2 2 + jx1y1 - x1y2 x2 2 - y2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Реализация гиперболических чисел во многом аналогична реализации комплексных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Необходимо перегрузить операторы +, -, * и /, функции извлечения корня, возведения в степень, элементарные математические функции и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' При этом для целей иллюстрации механизма работы множественной диспетчеризации основной интерес представляют именно арифметические операции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Это обусловлено тем, что элементарные функции принимают только один аргумент и для них достаточно определить только один метод.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В случае же арифметических операторов необходимо предусмотреть комбинации аргументов разных числовых типов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, должна иметься возможность сложения гиперболического числа с целым, рациональны, иррациональным числом, что автоматически затрагивает не только множественную диспетчеризацию, но и механизмы приведения типов, иерархию абстрактных типов и перегрузку конструктора по умолчанию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Поэтому мы ограничимся примерами реализации именно арифметических операций и все, не затронув более сложные в математическом плане вычисления разнообразных элементарных функций от гиперболического числа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Отметим, что кроме термина гиперболические числа, в литературе встречаются также термины: двойные числа, расщепленные комплексные числа, комплексные числа гиперболического типа (double numbers, split complex numbers, perplex numbers, hyperbolic numbers) [8, 12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' РЕАЛИЗАЦИЯ ГИПЕРБОЛИЧЕСКИХ ЧИСЕЛ В JULIA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Объявление структуры данных При реализации гиперболических чисел в Julia за основу был взят код для комплексных чисел, доступный в официальном репозитории Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Также использовались наработки, полученные при реализации параболических комплексных чисел [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Новый тип Hyperbolic определяется с помощью неизменяемой структуры: struct Hyperbolic{T<:Real} <: Number "Real part" re::T "Imaginary part" jm::T end 5 Number Hyperbolic Complex Real Integer Signed Int8 Int16 Int32 Int64 Int128 Bool Unsigned UInt8 UInt16 UInt32 UInt64 UInt128 Rational AbstractFloat Float16 Float32 Float64 Легенда: Абстрактный тип Примитивный тип Структура Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Местоположение гиперболических чисел в иерархии типов Julia Структура проста и содержит всего два поля параметрического типа T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' При этом требуется, чтобы тип T был подтипом абстрактного типа Real (синтаксис T<:Real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Сам тип Hyperbolic является подтипом абстрактного типа Number (см рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Таким образом гиперболические числа встраиваются в уже существующую иерархию числовых типов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' После определения структуры новый объект типа Hyperbolic можно создать путем вызова конструктора по умолчанию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, число h = 1 + j3 задается следующим образом: h = Hyperbolic{Float64}(1, 3) После создания можно обращаться к полям структуры как h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re и h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm, но попытка изменения значения поля уже существующего объекта приведёт к ошибке, так как структуры являются неизменяемыми сущностями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Если оба аргумента конструктора имеют один и тот же тип T, то его можно явно не указывать в фигурных скобках, так как он будет выведен автоматически из типа передаваемых аргументов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' h = Hyperbolic(1, 3) Однако, если типы аргументов отличаются, то конструктор по умолчанию не сможет осуществить неявное приведение типов и создать новый объект.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В этом случае необходимо явно указывать параметрический тип # Float64 и Int64 h = Hyperbolic(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 3) # Error h = Hyperbolic{Float64}(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 3) # Correct B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Дополните��ьные конструкторы Конструктор по умолчанию представляет собой обычную функцию, имя которой совпадает с именем типа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Со- здавая дополнительные методы для этой функции можно создать дополнительные конструкторы для обработки различных частных случаев.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, чтобы не указывать всякий раз параметрический тип, следует добавить новый конструктор следующего вида: """Constructor №2""" function Hyperbolic(x::Real, y::Real) return Hyperbolic(promote(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end Функция promote осуществляет приведение типов переданных ей аргументов к общему типу и возвращает результат в виде кортежа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Постфиксный оператор .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' распаковывает картеж и передает его элементы в виде аргументов в функцию-конструктор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В ядре языка определены правила приведения для всех подтипов абстракт- ного типа Real, поэтому теперь конструктор будет корректно работать для любой комбинации аргументов, главное чтобы выполнялось правило T<:Real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Например, следующий код сработает корректно: # Rational и Float64 h = Hyperbolic(1//3, pi) >> Hyperbolic{Float64}(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='141592653589793) Мы передали в конструктор рациональное число (тип Rational) и встроенную глобальную константу (число \\pi ) типа Float64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' После чего сработало правило приведения типов и оба аргументы были приведены к типу Float64 как к более общему.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6 Объявление еще двух дополнительных конструкторов позволит задавать гиперболические числа с нулевой мнимой частью: """Constructor №3""" function Hyperbolic{T}(x::Real) where {T<:Real} return Hyperbolic{T}(x, 0) end """Constructor №4""" function Hyperbolic(x::Real) return Hyperbolic(promote(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end Конструктор №3 является параметрической функцией, которая объявляется с использованием конструк- ции where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Параметр T является подтипом абстрактного типа Real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Конструктор №4 работает аналогично конструктору №2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Ещё два конструктора позволят передавать в качестве аргумента конструктора другие гиперболические числа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' """Constructor №5""" function Hyperbolic{T}(h::Hyperbolic) where {T<:Real} Hyperbolic{T}(h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) end """Constructor №6""" function Hyperbolic(h::Hyperbolic) return Hyperbolic(promote(h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end Для большего удобства также можно создать отдельную константу для мнимой единицы j: const jm = Hyperbolic(0, 1) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Вывод данных Для возможности распечатывать значения гиперболического типа в компактном и читаемом виде, следует добавить соответствующие методы для функции show из модуля Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='show(io::IO, h::Hyperbolic) print(io, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, "+", h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm, "j") end Функция show используется при распечатке данных в консоль, в частности ее вызывают функция println и макрос @show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В приведенных далее листингах кода и результатов его работы будет предполагаться, что добавлен метод show для гиперболических чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Приведение типов Прежде чем переходить к реализации методов для арифметических операций с гиперболическими числами, необходимо определить правила приведения типов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Для этого следует создать новый метод для функции promote_rule из модуля Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='promote_rule(::Type{Hyperbolic{T}}, ::Type{S}) where {T<:Real, S<:Real} return Hyperbolic{promote_type(T, S)} end function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='promote_rule(::Type{Hyperbolic{T}}, ::Type{Hyperbolic{S}}) where {T<:Real, S<:Real} \\lhook → return Hyperbolic{promote_type(T, S)} end В качестве аргументов в promote_rule указываются параметрические типы, которые следует привести к одному объемлющему типу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В нашем случае это возможно, если один из типов является подтипом Real, тогда объемлющим типом будет тип Hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 7 После добавления методов для promote_rule становится возможным использовать функции promote, promote_type и convert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' >>h = Hyperbolic(1 // 2) >>promote(h, 1) (1//2+0//1j, 1//1+0//1j) >>promote_type(Hyperbolic{Int64}, Float32) Hyperbolic{Float32} Первая функция нам уже знакома.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Вторая же позволяет выводить объемлющий тип не конкретных значений переменных, а самих типов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Тип в Julia является объектом первого рода (тип DataType) и его можно присваивать другим переменным, передавать в качестве аргументов функции и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Функция convert позволяет преобразовать тип конкретного значения, например: >>convert(Hyperbolic, 1) 1+0j E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Арифметические операции над гиперболическими числами После добавления методов для приведения типов, можно приступить к добавлению методов для арифметиче- ских операций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Особенностью Julia является реализация арифметических операций не в виде операторов, а в виде функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, корректны следующие вызовы: >>+(1,2) 3 >>+(1,2,3,4) 10 >>+((i for i in 1:10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') # числа от 1 до 10 55 В связи с этим, добавление методов для арифметических операций ничем не отличается от соответствующего процесса для других функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Добавление методов для унарных операций + и - осуществляется следующим образом: Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :+(h::Hyperbolic) = Hyperbolic(+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :-(h::Hyperbolic) = Hyperbolic(-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, -h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) Здесь используется сокращенная запись объявления фу��кции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Аналогично добавляются методы для бинарного сложения, вычитания, умножения и деления.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Приведем здесь код для сложения и умножения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' # Binary + and * function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :+(x::Hyperbolic, y::Hyperbolic) xx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re yy = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm Hyperbolic(xx, yy) end function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :*(x::Hyperbolic, y::Hyperbolic) xx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm yy = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='je * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re return Hyperbolic(xx, yy) end V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ЗАКЛЮЧЕНИЕ Мы рассмотрели механизм множественной диспетчеризации, лежащий в основе языка Julia, на примере реализации гиперболических чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Данный пример позволил затронуть такие понятия языка как иерархия типов данных, составные типы данных, механизмы приведения типов, перегрузка функций (создание новых методов для функций в терминах языка Julia) и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 8 БЛАГОДАРНОСТИ Публикация выполнена при поддержке Программы стратегического академического лидерства РУДН.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [1] Bezanson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Edelman A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Karpinski S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Shah V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Julia: A fresh approach to numerical computing // SIAM Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 65--98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [2] Gevorkyan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Kulyabov D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Sevastyanov L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Review of Julia programming language for scientific computing // The 6th International Conference "Distributed Computing and Grid-technologies in Science and Education".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [3] Oliphant T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Guide to NumPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- 2nd edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- CreateSpace Independent Publishing Platform, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- ISBN: 978- 1517300074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [4] Zappa Nardelli F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Belyakova J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Pelenitsyn A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Chung B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Bezanson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Vitek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Julia subtyping: a rational reconstruction // Proceedings of the ACM on Programming Languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' OOPSLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1--27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [5] Driesen K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', H\\"olzle U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Vitek J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=" Message Dispatch on 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' High-Performance Symbolic-Numerics via Multiple Dispatch // ACM Communications in Computer Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 3.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Kulyabov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Samouylov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Sevastianov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2639 of 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