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\begin{document}
%===================== ARTICLE HEADER DEFINITION =================
\Article{Rules of publication and recommendations for formatting 
      of articles for the journal ``Numerical methods and programming''}
      {Правила публикации и рекомендации к оформлению статей\break
      в журнале ``Вычислительные методы и программирование''}
    
\Abstract{The text describes the thematic focus of the journal, criteria 
      for selection of articles for publication, rules of submission of 
      articles for publication, the reviewing procedure, interaction of 
      authors with the editors. It also contains a style sheet with samples, 
      as well as rules of bibliographical descriptions both in Russian 
      and English.}
      {Приведено описание тематической направленности журнала,
      критериев отбора контента журнала, порядка направления статей 
      для публикации, процедуры рецензирования, взаимодействия авторов 
      с редакцией. Изложены рекомендации к оформлению статей. Даны примеры
      оформления списков литературы на русском и английском языках.}

\Keywords{numerical methods and programming, 
      journal, 
      publishing rules.}
      {вычислительные методы и программирование, 
      журнал, 
      правила публикации.}

\Acknowledgements{The work was supported by the Russian Foundation for Basic 
      Research (grant No. ??--??--????-а) and Ministry of Science and Higher 
      Education of the Russian Federation (state assignment ????--????--????).}
      {Исследование выполнено при финансовой поддержке РФФИ 
      (грант ??--??--????-а) и Министерства науки и высшего образования РФ 
      (государственное задание ????--????--????).}

\Citation{John J. Doe,
      ``Rules of publication and recommendations for formatting of articles 
      for the journal ``Numerical methods and programming,''
      Numerical Methods and Programming. \textbf{??} (?), 
      \pageref*{firstPage}--\pageref*{LastPage} (2021).
      doi~10.26089/NumMet.v??r???.}
      {Доу Дж. 
      Правила публикации и рекомендации к оформлению статей
      в журнале ``Вычислительные методы и программирование''~//
      Вычислительные методы и программирование. 2021.
      \textbf{??}, \No~?. \pageref*{firstPage}--\pageref*{LastPage}.  
      doi 10.26089/NumMet.v??r???.}


\DOI{10.26089/NumMet.v??r???}% The doi coding redaction
\VOL{??}{?} % The volume and issue coding redaction
\YEAR{2021}
\Received{June 16, 2021} % The received date set redaction
\Accepted{October 13, 2021} % The accepted date set redaction


\Author{John J. Doe}{Джон Дж. Доу}
\FullName{John J. Doe}{Джон Дж. Доу}
% Place of work 1
\Institution{Lomonosov Moscow State University, Research Computing Center}
      {Московский государственный университет им. М. В. Ломоносова,\break 
      Научно-исследовательский вычислительный центр}
\Address{Leninskie Gory, 1, building 4}{Ленинские горы, 1, стр.~4}
\Postcode{119991}
\City{Moscow}{Москва}
\CountryOfResidence{Russia}{Российская Федерация}
% Place of job 2 (example)
%\Institution[2]{Alikhanyan National Science Laboratory (YerPhI)}{Национальная научная лаборатория им.~А.~Алиханяна (ЕрФИ)}
%\Address[2]{Alikhanyan Brothers ulitsa, 2}{ул.~братьев Алиханян, д.~2}
%\Postcode[2]{0036}
%\City[2]{Yerevan}{Ереван}
%\CountryOfResidence[2]{Armenia}{Армения}
% Place of work 3
% ..............

\AcademicDegree{Ph.D.}{к.ф.-м.н.}
\Position{Leading Scientist}{вед. научн. сотр.}
\Orcid{kkkk-llll-mmmm-nnnn}
\Email{journal@num-meth.ru}

\MakeArticleHeader 
%================== END OF ARTICLE HEADER DEFINITION ==============

\pagebreak

\section{General information}

\subsection{Main data}
The founder and publisher of the journal is the Research Computing Center 
of Lomonosov Moscow State University (RCC MSU).

The title of the journal in English: ``Numerical Methods and Programming''.

The title of the journal in Russian: ``Вычислительные методы и программирование''.

The transliterated title of the journal: ``Vychislitel'nye Metody i 
Programmirovanie''.

The journal has been published since 2000.

Frequency: 4 issues per year (since 2006).

The journal is registered with the Federal Agency for Press and Mass Communications 
as a mass medium~--- a scientific electronic periodical (certificate of registration 
No. 77-4356, ISSN 1726-3522).

The journal is included in the Higher Attestation Commission (Russian Federation) 
List of leading peer-reviewed scientific journals and publications in which the 
main scientific results of dissertations for the degree of doctor of science and 
Ph.~D. should be published.

Bibliographic information on published articles is placed in the scientific 
citation database of the RSCI.

The journal is included in the RSCI bibliographic database on the Web of Science 
platform, in the core of the Russian Science Citation Index, and in All-Russian 
Mathematical Portal MathNet.Ru.

Articles published in the journal are assigned doi (since 2015).


\subsection{Aims \& Scope}
The journal publishes works in the following areas and scientific specialties:
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item Methods and algorithms of computational mathematics and their applications 
      for solving scientific and technical problems.

      The publication should contain new results obtained in the field of methods 
      and algorithms for solving problems of computational mathematics, mathematical 
      modeling and computational experiments, test problems and research to determine 
      the applicability of algorithms, parallel software implementation of problems 
      of computational mathematics. Reviews of methods and software are published, 
      containing recommendations for their application.

\item Parallel software and technologies for solving problems of computational 
      mathematics and its scientific and technical applications. 

      Articles related to numerical software targeted at supercomputers and 
      parallel architecture computing systems.

      Researches in the field of theory and technology of designing programs of 
      computational mathematics and its scientific and technical applications, as 
      well as their integration into software systems (development of taxonomy and 
      classifications of problems and methods, disciplines and programming techniques, 
      technological processes for compiling numerical programs for computers with 
      parallel architecture, methods for conducting computational experiments, etc.).
      
      The results of research on the development of tools for automating the design, 
      operation and maintenance of numerical programs and software systems for 
      computers of various architectures (development of problem-oriented languages, 
      means of remote access and organization of interactive modes, data 
      visualization, conversions of numerical programs and display of computational 
      algorithms on computer architecture, means of verification and 
      macro-generation of programs, macro-extensions of algorithmic languages, 
      preprocessors and macro-processors, etc.) are accepted for publication.
\end{enumerate}

\subsection{Language of publications}
Articles are accepted for publication in Russian and English.

\subsection{Publication ethics}
Journal ``Numerical Methods and Programming'' strictly adheres to the international 
publication ethics standards set by the Committee on Publication Ethics 
[\ref{rfa:enlit:COPE1}].

Authors of publications voluntarily undertake to adhere to the following 
ethical standards:
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item The authors are responsible for the accuracy of the research results, 
      the content of the article and for the very fact of its publication.
\item Authors should not submit to the journal materials for publications that are 
      under consideration in another journal, as well as articles published in 
      other journals. Authors should be aware that submitting the same article to 
      different publications is a violation of publication ethics.
\item The authors guarantee that they submit to the editorial board of the journal 
      original articles, previously unpublished anywhere, free from plagiarism and 
      scientific falsifications. If the authors use their own previously published 
      results as a basis for the presented article, then they are obliged to indicate 
      references to these results and specify what new results were obtained in 
      comparison with the previous ones.
\item Authors should refer to works that were important in the study, as well as 
      acknowledge those who have made significant contributions to the work. 
      Borrowed fragments or statements must be made with the obligatory indication 
      of the author and the original source.
\item Authors should not divide a holistic research into several publications in 
      order to increase their number. Significant in terms of volume of work can, 
      in agreement with the editors, be published in several issues of the journal.
\item All co-authors of the manuscript warrant that they have made significant 
      contributions to the research.
\item The authors provide in the article an exhaustive list of sources of financial 
      support for the study and the role of these sources in the study.
\item Authors must adhere to the rules of paper design accepted in the journal.
\item Authors must explicitly agree with the rules for preparing articles for 
      publication, including peer review, scientific and literary editing, and 
      bringing articles to editorial standards accepted within the journal.
\item The authors must express their consent to the transfer to the journal of the 
      right to publish and distribute the article in electronic and paper versions, 
      including the placement of bibliographic information about the article in 
      scientific citation bases, as well as the placement of the full text of the 
      article in the Scientific Electronic Library (elibrary.ru) for free access 
      to all Internet users, regardless of their category and location.
\item If the authors find any inaccuracies in an already published article, then 
      they are obliged to inform the members of the editorial board of the journal 
      and jointly participate in correcting the publication.
\item In case of violation of the rules of publication ethics by the editors, or 
      if there are unresolved disagreements with the editors, the authors have the 
      right to file a complaint with the Scientific Publications Ethics Council 
      of the Association of Scientific Editors and Publishers (ASEP)
\end{enumerate}

Responsibilities of reviewers:
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item The main task of the reviewer is an objective and prompt assessment of the 
      correspondence of the submitted articles to the subject of the journal, 
      scientific novelty and the level of the presented results, the quality of 
      preparation of materials; formulating recommendations for the publication 
      of materials, their revision or rejection with respect to the due level of 
      confidentiality, following the generally accepted standards of objectivity 
      COPE, ensuring non-disclosure of information and promptly reporting conflicts 
      of interest.
\item The reviewer in his comments and recommendations should provide single-blind 
      peer review without revealing his identity to the author.
\item Any manuscript submitted for review should be considered a confidential 
      document. It is unacceptable to show it to other reviewers or discuss it with 
      other experts without the prior permission of the editor-in-chief.
\item One of the important tasks of the reviewer's activity is the improvement of 
      materials and manuscripts, the publication of which in the journal is of 
      interest to the scientific community and can make a significant contribution 
      to the development of branches of science within the scope of the journal's 
      topics, through comments and suggestions to the authors.
\item The reviewer makes a significant contribution to the activities of the 
      scientific journal. His peer review serves as the basis for editorial 
      decisions to be made by the editorial board.
\item Taking into account the high rhythm of modern scientific life, the publication 
      of significant scientific results should take place as soon as possible. Based
      on this, the reviewer selected for the evaluation of the work, who believes 
      that the level of his qualifications or the time resources available to him 
      are insufficient for the operational review of the research presented in the 
      scientific work, must notify the editor about this and abandon the review 
      process.
\item Reviews of scientific papers must be objective. The personal criticism of the 
      author by the reviewer is categorically inappropriate. Reviewers are required 
      to express their views clearly, to the point and reasonably.
\item Peer reviewers should identify papers relevant to the research and previously 
      published in peer-reviewed journals that have not been properly cited or cited 
      by the authors.
\item The reviewer is also obliged to draw the attention of the editor to any 
      significant or partial similarity with any other work with which the reviewer 
      is directly familiar.
\item Unpublished materials used in the submitted manuscript should not be used in 
      the reviewer's own research without the written consent of the author.
\item Non-public information obtained during the review must remain confidential 
      and not be used by the reviewer for personal gain.
\end{enumerate}

\smallskip
The editor-in-chief and the editorial board of the journal undertake to follow the 
generally accepted norms of publishing ethics in relation to the authors of articles 
sent for publication:
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item The decision to publish is made based on the results of assessing the 
      reliability of the information presented in the article, scientific novelty, 
      clarity of presentation, research and compliance of the work with the subject 
      of the journal. Gender identification, race, citizenship, social status, 
      political views and other attributes of the authors not related to scientific 
      activity cannot serve as a reason for refusal to publish.
\item All articles submitted to the editorial board undergo single-blind peer review.
      Reviewers must be experts on the subject of the submitted research, they must 
      guarantee an objective assessment of the articles according to the criteria set 
      out in clause~1, in particular, the absence of a conflict of interest, and 
      clearly formulate their conclusions on the submitted articles, including an 
      assessment of the possibility of their publication and recommendations for 
      revision.
\item The editorial board reserve the right to cancel the positive decision to 
      publish the material in the event of serious violations of publication ethics 
      that were missed at the stage of reviewing.
\item The editorial board of the journal ensures the confidentiality of any 
      information about the submitted articles until the moment of their publication. 
      The knowledge gained in the process of editing and reviewing cannot be used for 
      the purpose of obtaining personal gain before the publication of the article.
\item The editorial board does not allow the publication of articles with signs of 
      plagiarism. The presence of borrowing without reference to the source, 
      unofficial citation and paraphrasing, assignment of rights to the results of 
      other people's research are considered plagiarism and are sufficient grounds 
      for rejection of the article.
\item The editorial board takes measures to identify and prevent the publication of 
      articles reflecting research in the course of which there were violations of 
      the law; including, for this purpose, information about such offenses received 
      from external sources is carefully considered.
\item The editorial board of the journal has the right to withdraw an already 
      published article and, if appropriate, return it to the authors for revision, 
      if the publication of the article entailed a violation of someone's rights or 
      generally accepted norms of scientific ethics.
\item The editors are ready to publish corrections, explanations, refutations and 
      apologies related to previously published articles, including in case of 
      receipt of relevant requests from the authors.
\item In case of revealing violations of publication ethics on the part of the 
      authors, the editors ask the authors for clarifications; in case of an 
      unsatisfactory answer, the editorial board has the right to contact the 
      employer of the authors or the competent executive authorities.
\item The editorial board does not leave unanswered the authors' comments regarding 
      both their published works and those accepted for publication.
\item The editorial board of the journal adheres to the established terms of 
      publication if the authors fulfill the necessary requirements for articles.
\item The editor-in-chief may apply to the Association of Scientific Editors and 
      Publishers (ASEP) to resolve ethical disputes between the editorial board and 
      the authors of the articles.
\item The editorial board of the journal is not responsible for any damage caused 
      by the publication of the article.
\end{enumerate}


\subsection{Publication fee} 
For authors, the publication of articles is free.

\subsection{Copyright for the publisher} 
To protect the rights and interests of both parties, the Research Computing Center 
of Lomonosov Moscow State University (RCC MSU), an exclusive license is required, 
which clearly defines our rights and special rights reserved by the authors. 
We ask each author to provide the RCC MSU with this exclusive license after the 
article has passed the reviewing stage and a decision has been made to publish the 
article in our journal.

Thus, RCC MSU agrees to publish the article and has the right to:
\begin{itemize}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item publish, reproduce, distribute, display and store the article;
\item to translate the article into other languages;
\item use the article to create adaptations, reprints and create summaries, 
      excerpts and / or abstracts;
\item create any other derivative works based on the article;
\item include electronic links from the article to third-party materials;
\item license any third party to do any or all of the above;
\item initiate legal proceedings on behalf of the copyright owner if the author(s) 
      believes that a third party is infringing or may infringe the copyright of 
      the article.
\end{itemize}

\subsection{Author permissions} 
Authors are allowed to use their own articles for non-commercial purposes without 
obtaining permission from the Research Computing Center of Moscow State University. 
But for commercial use, we need to be aware of this.

The authors reserve the right to:
\begin{itemize}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item post a PDF file with your article to fellow researchers for educational 
      purposes only;
\item make a reasonable number of copies for personal or non-commercial use;
\item republish part or all of the article in a book or other publication edited 
      by the author (with the exception of several articles in one book or 
      publication);
\item use illustrations and research data in their future works;
\item use the article in course packages at the authors' institution (does not 
      apply if a commercial fee is charged for the compilation or training program);
\item use the work for further research and presentations at meetings;
\item reuse the published manuscript in whole or in part in other works for 
      non-commercial purposes, while the original publication in the journal of 
      the RCC MSU should be marked by means of a note or reference in a format 
      acceptable to the RCC MSU;
\item to use your own materials in other publications, provided that the journal 
      is indicated as the original place of publication and the journal of the 
      RCC MSU ``Numerical Methods and Programming'' is notified in writing and 
      in advance.
\end{itemize}

\subsection{Creative Commons Attribution License (CC-BY)} 
All articles published by RCC MSU in the journal ``Numerical Methods and 
Programming'' may be redistributed under the terms of the Creative Commons 
Attribution License (CC-BY) 4.0.

Thus, anyone can copy, distribute and transmit the article, provided that the 
original article and source are cited correctly.

\subsection{Long-term deep archiving and digital preservation policy} 
All issues of the journal ``Numerical Methods and Programming'' are uploaded article 
by article to the scientific electronic library \url{https://elibrary.ru} and to 
the portal for searching scientific information on mathematics, physics, 
information technology and related sciences \url{http://mathnet.ru}.

Additionally printed obligatory copies of each issue are also transferred to the 
Russian Book Chamber (Legal Deposit).

\section{Recommendations for the content and structure of articles}
\subsection{Publication forms}
The editorial board of the journal accepts for consideration the author's materials, 
drawn up in the form of a review or an original article.

Reviews can cover any modern achievements related to the subject of the journal. 
Reviews should be written in such a way as to assist the reader in assessing 
emerging prospects and opportunities, as well as in finding publications containing 
key information about new developments. In the introductory part of the review, 
it is desirable to provide an explanation of the main tasks, concepts and terms. 
The reader of the review article should understand the need for an in-depth study 
of the problem using the bibliographic references provided for the article, so the 
review should not look like a listing of innovations and references to sources.

Original articles must correspond to the subject of the journal. The novelty and 
scientific significance of the work are checked by reviewers and the 
editor-in-chief. The information in the original article must be complete, 
allowing a specialist in this field to reproduce the results obtained by the authors. 
It is not recommended to present generally known information in the original 
articles. When submitting manuscripts, the authors are fully responsible for 
the fact that the presented research results are performed personally, comply 
with the norms of scientific ethics and are carried out on an open topic.

\subsection{Volumes of published materials}
Review articles can be up to 30 pages per issue. The publication of reviews in 
several issues takes place in agreement with the editor-in-chief. Authors of 
original articles are recommended to be up to 20 pages long.

\subsection{Article title}
Authors should give the article the most specific and informative title, 
if possible, not exceeding 20 words.

\subsection{Annotations}
Review and original articles should be accompanied by annotations giving an idea 
of the essence and novelty of the published results. The abstract should be 
understandable before reading the article and should not contain links to any 
publications. The volume of the annotation should not exceed 10-15 lines. It is 
not recommended to break annotations into paragraphs.


\subsection{UDC codes}
The authors of the article indicate the UDC (only for articles in Russian), 
which the publication corresponds to.

\subsection{Keywords}
Authors should indicate the set of keywords that best characterize the published 
research results. Keywords are used to automatically search for an article, for 
example in an electronic library. We recommend that authors choose no more than 
10 keywords or phrases for their article.

\subsection{Acknowledgments}
Authors can express their gratitude to persons who have provided significant 
assistance in the work, or indicate sponsorship and grants.


\subsection{Structure of Articles}
Depending on the size of the article, the text of the article can be structured 
into sections (\verb!\section!) and subsections (\verb!\subsection!) (like this 
article) or only into sections.

It is not recommended to give titles of sections and subsections that would span 
more than two lines. 

Subsections can consist of two or more paragraphs\footnote{These ``Guidelines'' 
is not a typical article, therefore it cannot serve as a guide in this matter.}. 
It is recommended for better perception to break the text into paragraphs no more 
than 10--15~lines in size.


\subsection{Bibliography}
The list of references is an integral part of the article and is drawn up directly 
in the text of the article with the indication doi, if the publication is assigned 
a digital identifier. The list of references is formed in the order of reference 
to the sources in the text of the article.

A mandatory requirement is a complete translation of the bibliography into English 
for indexing an article in foreign databases.

The use of separate files with the bib, bbl and others extensions is not allowed.

\subsection{Translation}
Authors of articles published in the journal in English must provide 
translated into Russian:
\begin{itemize}[itemsep=0pt,parsep=0pt,topsep=0pt,partopsep=0pt]
\item surname, name, patronymic, academic degree, academic title of each author;
\item the name, postal addresses of the organizations that are the main places 
      of work of the authors, and their positions in these organizations;
\item article title;
\item annotation text;
\item keywords;
\item acknowledgments.
\end{itemize}
If the translation into Russian of this information by the author is difficult, 
the editors should be notified of that situation so that this work can be done 
by them.


\section{Requirements for the design of articles}

\subsection{Preparation of the article}
The editors accept articles from authors prepared in the \LaTeX{} publishing 
system~[\ref{rfa:enlit:Kotelnikov}--\ref{rfa:enlit:Lvovski}]. 

Authors of articles can use additional style files, for example algorithmx for 
a set of algorithms, etc. Additional style files must be explicitly included in 
the document preamble. Using your own styles is not recommended.

Articles prepared using Microsoft Word or other editors are accepted only in 
exceptional cases by the decision of the editors and will be published with a delay 
due to the translation of the original text into the \LaTeX{} publishing system and 
additional editorial revisions.

\subsection{Typing requirements}
We draw the attention of the authors to the following typing rules:

\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item There should always be a space between the initials and the surname 
(A.A. Ivanov). In the title of the article, spaces are also placed between 
the initials: A. A. Ivanov.

\item The integer part of the number should be separated from the fractional part 
in the same way in the text of the article: either a comma or a period. On graphs 
and diagrams formed as a result of the work of programs, and in program listings, 
the integer part of the number from the fractional part can be separated by a dot.

\item The quotation marks should be in the form of ``legs''.

\item Dates must be typed in the format dd.mm.yyyy.

\item The period is not put after the section headings, table titles and figure 
captions, after a number of abbreviated names: s~--- second, g~--- gram, 
min~--- minute, d~--- day, mil~--- million, bil~--- billion, etc. The period 
is placed at the end of the last sentence of the annotation, after the 
abbreviations: mon.~--- month, etc.

\item Multi-word abbreviations are separated by spaces, for example 760~mmHg. 
Art., with the exception of commonly used ones (i.e.).

%\item Ссылки на рисунки и таблицы отделяются неразрывными пробелами от номера
%      (правильно: fig.~1, table~2, неправильно: рис.1, табл.2).

\item Quotes and brackets are not separated by spaces from the words enclosed in 
      them: (for $U_\text{max}$), (a); and not ( at $U_\text{max}$ ), ( a ).

\item A non-breaking space is placed between the number or paragraph sign and 
      the number: No.~1, §~5.65.
\end{enumerate}

\subsection{Requirements for a set of formulas, theorems, etc.}
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item Needs to be typed in straight type: 
      a)~all numbers are typed in straight type in formulas; 
      b) abbreviations of words, including in indices (for example, $E_\text{ion}$~---
      ionization energy);
      c) abbreviations of mathematical functions: $\sin$, $\cos$, $\lim$, 
      $\text{max}$.

\item For typing excluded formulas, it is recommended to use the {\tt equation} 
      environment with automatic continuous numbering. To suppress numbering, use 
      the {\tt equation*} environment.

\item The style file provides for the placement of summation indices, integration 
      limits, etc. strictly above or below mathematical symbols:
      \begin{equation}
        \sum_{n=0}^{+\infty}=\ldots\ , \qquad \int_{0}^{\pi}\sin^2(x) dx =\ldots\ .
      \end{equation}

\item Formulas that span one or more lines (so-called exclusion formulas) and are 
      part of a sentence must obey the rules of punctuation.

      \COMMENT{
	  \vspace*{2mm}\hrule\vspace*{2mm}
	  Если плоская область $\Omega$ односвязна и на ней задан непрерывно дифференцируемый вектор
	  \begin{equation}
	  \bm{a} = P(x,y)\bm{i}+Q(x,y)\bm{j},   % \bm{} -- полужирный наклонный шрифт в формулах; принятое в журнале написание вектора
	  \end{equation}
	  для которого
	  \begin{equation}
	  \mathop{\mathrm{rot}}\bm{a}=\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\bm{k}=0,
	  \end{equation}
	  то $\bm{a}$ имеет на $\Omega$ потенциал [\ref{rfa:enlit:rot}].
	  \vspace*{2mm}\hrule\vspace*{4mm}
      }

\item Vectors must be typed using the \verb!\bm{}! (bold italic) command, 
      not \verb!\vec{}!.
\item Matrices should be typed in normal mathematical font, for example, 
      matrix $A$ and transposed $A^T$.
\item The words ``Theorem'', ``Collorary'', ``Lemma'', ``Asserrtion'', ``Proposition''
      are placed in the text of the article explicitly and are highlighted in 
      bold, and the corresponding formulations are in italic font.
\item The words ``Proof'', ``Definition'', ``Remark'' are in roman italic style, 
      and the corresponding wording is in roman type.
\item It is recommended to carry out continuous numbering with Arabic numerals 
      for theorems, definitions, etc. 
\end{enumerate}

\subsection{Requirements for a set of listings of programs and algorithms}
It is allowed to reproduce small in volume (no more than 1 page is recommended) 
fragments of programs illustrating important algorithms, programming language tools, 
etc. Text messages and comments in listings must be in English. Examples of listing 
programs and algorithms are given in Appendix~\ref{pril:prog}.

\smallskip
\subsection{Requirements for illustrations}
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item The editors make minimal edits to the illustrations. In case of insufficient 
      quality illustrations, the article is sent back to the authors for revision.
\item The following dimensions of illustrations are recommended: 
      a)~no more than 165 mm wide and no more than 210~mm high (in the width of a 
      text column);
      b)~no more than 80 mm wide and no more than 140 mm high (half the column 
      width); 
      c)~145~mm high and 240~mm wide (illustration expanded $90^\circ$ on the 
      whole page).

\pagebreak
\item Three types of illustrations are accepted: line drawings (graphs, diagrams, 
      graphs, etc.), screenshots of the computer screen (for example, color 
      visualization of the field obtained as a result of mathematical modeling) 
      and photographs.
\item Line drawings is accepted in the form of PNG (raster) files with a resolution 
      of 600 DPI at a 1:1 scale of the figure or in EPS format (vector images). 
      The thickness of the thinnest line in the picture should not be less than 5~px. 
      It is recommended to use black for the main lines. Elements of the picture that 
      are important for understanding should be highlighted in two or three colors, 
      which are contrastingly reproduced in black and white printing. Recommended 
      background color is white.
\item Photos are accepted in formats JPEG, PNG, TIFF with a resolution of 300 DPI. 
      The image in the photo should be sharp, contrasting. Digital image processing 
      should not raise doubts about the authenticity of the photograph.
\item Figures must be in English. It is recommended to designate important elements 
      of the figure with numbers and give a decryption in the signature.
\item The series of illustrations is intended for comparing graphs, assessing the 
      influence of parameters, explaining trends, dynamics, etc. Illustrations 
      included in the series must be of the same size and resolution. Each 
      illustration in the series must be designated with a letter of the English 
      alphabet. Examples of illustrations design are given in Appendix~\ref{pril:ris}.
\item The editors draw your attention to the fact that the preliminary layout 
      made by the authors often leads to a deterioration in the quality of the 
      pictures, for example, due to the incorrectly set resolution when converting 
      a picture to EPS format. The editors ask to provide, along with EPS files, 
      the original ``raw'' files in the formats TIFF, PNG, JPEG without processing. 
      If the picture is a screenshot from the computer screen, then it must be taken 
      at the maximum window size and no longer scaled. Authors can make a copy of 
      the figure in the format and sizes they need and insert it into the text of 
      the article.
\end{enumerate}
   
\subsection{Requirements for the design of tables}
The editors recommend formatting tabular data using the \LaTeX{} publishing system 
(Appendix~\ref{pril:tbl}). Text in table cells must be in English. Highlighting 
table cells with color unnecessarily is not encouraged.

The exception is tables obtained from the Internet or automated data processing 
systems in the form of screenshots.


\subsection{Requirements for captions of figures and titles of tables and listings}
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item Figures, tables, formulas, program listings, etc. must have consecutive 
      numbering for each type of structural elements of the article. Examples of 
      registration of captions for figures, tables and listings are given in 
      Appendices~\ref{pril:prog}--\ref{pril:tbl}.
\item Signatures should be informative and concise. It is recommended that 
      signatures be worded so that they do not span more than three lines.
\item The caption to a series of illustrations must contain a common name for all 
      figures in the series and a list of captions specifying each illustration 
      (Appendix~\ref{pril:ris}, Fig.~\ref{pril:ris:fig2}). 
\end{enumerate}


\subsection{Requirements for References}
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item The list of references is formed in the order of reference to the sources 
      in the text of the article.
\item Articles published in Internet publications must be provided with an 
      up-to-date link.
\item The preferable version of the link is to indicate the doi of the publication.
\end{enumerate}

An example of the References format is given in Appendix~\ref{pril:refer}.


\subsection{References to figures, formulas, tables, sources}
The editors do not recommend using manually typed links in the text of the article. 
To reduce the likelihood of errors in links in the text of the article, it is 
necessary to use the commands \verb!\label!, which automatically stores the values
of counters for figures, tables, lists in a symbolic variable, and \verb!\ref!,
which creates a link by the name of the variable 
[\ref{rfa:enlit:Kotelnikov}, \ref{rfa:enlit:Lvovski}].


\section{Interaction with the editors}

\subsection{Required documents}
To publish an article, authors must provide:
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\renewcommand{\labelenumi}{\arabic{enumi})}
\item a cover letter of the authors (Appendix~\ref{pril:sopr} in this document) 
      in electronic form and its scanned version with ``live'' signatures;
\pagebreak
\item electronic \LaTeX-version of the article in English, prepared in accordance 
      with the requirements documents for the design of articles, as well as a 
      pdf-file of the article and pictures;
\item a text file containing the following information in Russian and English: 
      a)~surname, name, patronymic, academic degree, academic title, ORCID and 
      e-mail of each of the authors; b)~the names and postal addresses of 
      organizations that are the main places of work of the authors and their 
      positions in these organizations; c)~the title of the article; d)~annotation 
      text; e)~keywords.
\end{enumerate}

\subsection{Transfer of documents to the editorial office}
The author interacts with the editors mainly through the journal's website 
\url{https://num-meth.ru}. After completing the registration form, the author 
(one of the authors) have to:
\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\renewcommand{\labelenumi}{\arabic{enumi})}
\item download through the form all the prepared necessary documents and article 
      components in the form of a ZIP-archive, choosing when downloading the 
      {\tt Article Component = 
      ``ZIP-archive with source texts (.tex and required additional files)''};
\item give a description of the archive, authorship, etc. (it is recommended to 
      do it at the stage of checking the details of the sent file);
\item confirm sending the archive;
\item enter article metadata (title, abstract and keywords required) and information 
      about the author and all co-authors (all information must be entered in 
      Russian and / or in English);
\item confirm the entered data.
\end{enumerate}

The author receives e-mail notifications for all significant events accompanying the 
process of preparing an article for publication. The author should monitor the 
change in the status of the article and, if necessary, respond to the editorial 
board's invitations to enter into a dialogue on the pages with the discussion of 
the material (for example, when communicating with a literary editor).

\subsection{Publication decision, peer review procedure}

\begin{enumerate}[itemsep=0pt,parsep=2pt,topsep=2pt,partopsep=0pt]
\item The editors consider articles only relevant to the subject of the journal. 
      If the article is rejected, the editorial board notifies the author of the 
      decision and does not enter into a discussion with the author of the 
      correspondence of his article to the subject of the journal. 
      The author has the right to publish his work in another edition.
\item An article accepted by the editorial board for review after preliminary 
      selection undergoes a {\it single-blind peer review} procedure approved 
      by the journal.
\item The author receives a review of his article and a notification from the 
      editor-in-chief about the acceptance of the article for publication, about 
      the need for revision or rejection of the article, indicating the reasons.
\end{enumerate}


\subsection{Preparing an article for publication}
The article accepted for publication goes through the stages of literary and 
technical editing. Depending on the quality of the materials presented, the 
author will be in more or less intensive correspondence with the editors. 
Authors are encouraged to respond to letters from editors within one to two 
days so that the publication takes place in the intended issue of the journal.

\subsection{Publishing an article}
Articles from the typeset issue of the magazine are posted on the website, 
the issue is sent to print. The author is notified of the publication.


\vspace*{16mm plus 2mm minus 2mm}

%===========================  REFERENCES =====================================

\REFERENCES

\elitem{rfa:enlit:COPE1}{
Guidelines on Good Publication Practice. Committee on Publication Ethics (COPE).
\url{https://publicationethics.org/files/u7141/1999pdf13.pdf}.
Cited November 12, 2021.}

\elitem{rfa:enlit:Kotelnikov}{
I.~A.~Kotel'nikov and P.~Z.~Chebotaev, 
\textit{\LaTeX\ in Russian} 
(Sibirskiy Khronograph, Novosibirsk, 2004) [in Russian].}

\elitem{rfa:enlit:Samarin}{
M.~Goossens, F.~Mittelbach, and A.~Samarin, 
\textit{The \LaTeX~Companion.} 
(Addision--Wesley Professional, 1994; Mir, Moscow, 1999) 
[in Russian].}

\elitem{rfa:enlit:Lvovski}{
S.~M.~L'vovsky. \textit{Typesetting and Layout in the \LaTeX~System}
(MCNMO Publ., Moscow, 2003) 
[in Russian].}


\elitem{rfa:enlit:rot}{
Ya.~S.~Bugrov and S.~M.~Nikol'sky, 
\textit{Higher Mathematics: Differential Equations. Multiple Integrals. Ranks} 
(Phoenix Publ., Rostov-na-Donu, 1998) 
[in Russian].}

\elitem{rfa:enlit:Antonov}{
A.~S.~Antonov, 
\textit{Parallel Programming Using OpenMP Technology}
(Mosk. Gos. Univ., Moscow, 2009) 
[in Russian].}

\elitem{rfa:enlit:Lapko}{
O.~Lapko, 
\textit{The Makecell Package}. 
\url{https://mirror.truenetwork.ru/CTAN/macros/latex/contrib/makecell/makecell-rus.pdf}.
Cited November 12, 2021 [in Russian].}


\elitem{rfa:enlit:mrow}{%
Multirow~--- Create Tabular Cells Spanning Multiple Rows.
\url{https://www.ctan.org/pkg/multirow}.
Cited November~12, 2021.}


\medskip
\DateEN

\bigskip
\bigskip

\MakeAuthorsInfoEN

%\end{document}
%============================ КОНЕЦ СТАТЬИ ==================================



%============================ ПРИЛОЖЕНИЕ 1 ==================================
\newpage
\appendix{Registration of listings of programs and algorithms}{pril:prog}


\textbf{Listing~\ref{pril:prog:lst1}.~A C/C++ program that takes up the 
full width of the typing strip.} 
We draw the attention of the authors to the following feature of the
\verb!\begin{lstlisting}...\end{lstlisting}! environment:
square brackets \verb![! \verb!]! are used to specify a parameter block, 
and a comma is used to separate parameters. The listing title in English 
is typed as follows:
\begin{verbatim}
caption={An example of the design of a program fragment 
        (with the permission of the author \mbox{[\ref{rfa:enlit:Antonov}]})},
\end{verbatim}
External curly braces in \verb!caption={...}! 
allow you to use a comma in the caption text, if necessary, and the
\verb!\mbox{[\ref{rfa:enlit:Antonov}]}! construct escapes the square brackets
\verb![! \verb!]!, allowing you to use them in the link to reference.
%
% LISTING 1. С-program
%
\begin{lstlisting}[%
      caption={An example of the design of a program fragment (with the permission of the author \mbox{[\ref{rfa:enlit:Antonov}]})},
      label=pril:prog:lst1,
      float=h!, % may be t, h, b, t!, h!, b!
      language=C, 
      inputencoding=utf8, 
      extendedchars=\true, 
      keywordstyle=\color{blue!75!black}, 
      keepspaces=true, 
      commentstyle=\color{green!40!black}, 
      breakatwhitespace=true,
      frame=single, 
      linewidth=169mm,
      xleftmargin=7mm,
      basicstyle=\ttfamily\small,
      numbers=left,
      breaklines=\true, 
      numberstyle=\footnotesize,
      tabsize=8]
// OpenMP technology
// applying the single directive with the nowait option 
#include <stdio.h>

int main (int argc, char *argv[])
{
#pragma omp parallel
{
      printf("Message 1\n");
#pragma omp single nowait
      {
            printf("One thread\n");
      }
      printf("Message 2\n");
}
}
\end{lstlisting}

\textbf{Listing~\ref{pril:prog:lst2}.~Fortran program.} 
The \verb!float! parameter may be useful for authors, which determines the 
placement of the listing as a floating object on the page: 
\verb+t+ and \verb+t!+ set the placement at the top of the page, 
\verb+b+ and \verb+b!+~--- at the bottom, 
and \verb+h+ and \verb+h!+~--- in the place of the text where the listing 
is located. Using parameters with an exclamation mark 
\verb+t!+, \verb+b!+ and \verb+h!+ ensures that the listing is placed on the 
closest possible page.
%
% ЛИСТИНГ 2. ПРОГРАММА НА FORTRAN
%
\begin{lstlisting}[%
      caption={Undistributed parallel loop in FDVMH language (floating object)},
      label={pril:prog:lst2},
      float=b!, % may be t, h, b, t!, h!, b!
      language=Fortran, 
      inputencoding=utf8, 
      extendedchars=\true, 
      keywordstyle=\color{blue!75!black},  
      keepspaces=true, 
      commentstyle=\color{green!40!black}, 
      breakatwhitespace=true,
      frame=single, 
      linewidth=169mm,
      xleftmargin=7mm,
      basicstyle=\ttfamily\small,
      numbers=left,
      breaklines=true, 
      numberstyle=\footnotesize,
      tabsize=2]  
!DVM$ REGION TARGETS(HOST) OUT(A)
!DVM$ PARALLEL(I,J,K) TIE (A(K,J,I)) 
  DO I = Li, Hi
    DO J = Lj, Hj
      DO K = Lk, Hk
        A(K,J,I) = ....
!DVM$ END REGION       
\end{lstlisting}
 
\smallskip
\textbf{Algorithm~\ref{pril:prog:alg1}.~An example of the design of the algorithm.} 
The packages algorithm, algpseudocode, etc., existing in \LaTeXe{}, 
contain various commands and environments that give the algorithms the look 
accepted in scientific literature. The commands in these packages define the 
representation of algorithmic operations: for example, the command \verb!x\gets 10! 
is displayed as $x\leftarrow 10$ and means that the variable $x$ 
is assigned the value~10.

Often, the authors of our journal, due to the lack of a national standard in this 
area, use the original designations of algorithmic operations, therefore the 
editors recommend using the commands defined in the style file of the journal 
for a set of algorithms. Unlike the packages algorithm, algpseudocode and some 
others, our style file defines commands that are responsible for placing 
algorithmic operations, but not for representing them.

Let's consider the main commands that were used in the set of 
Algorithm~\ref{pril:prog:alg1}.
Environment
\begin{verbatim}
\begin{nmAlgorithm}{<symbolic label>}{<English title>}
...
\end{nmAlgorithm}
\end{verbatim}
marks the area for the algorithm. The command
\begin{verbatim}
\algLine[<number of indents>][<symbolic label>]{<algorithmic operation>}
\end{verbatim}
is used to form an algorithm line. The \verb!<number of indents>! and 
\verb!<symbolic label> ! parameters are optional for the \verb!\algLine! command.

For convenience, the style file defines commands for frequently used algorithmic 
operations and constructions: 
\verb!\algINPUT!~--- prints \textbf{input}; 
\verb!\algFOR!~--- \textbf{for};
\verb!\algDO!~--- \textbf{do};
\verb!\algENDFOR!~--- \textbf{end for};
\verb!\algIF!~--- \textbf{if};
\verb!\algTHEN!~--- \textbf{then};
\verb!\algELSE!~--- \textbf{else};
\verb!\algELIF!~--- \textbf{elseif};
\verb!\algENDIF!~--- \textbf{end if};
\verb!\algOUTPUT!~--- \textbf{output};
\verb!\algSTOP!~--- \textbf{stop};
\verb!\algFUNC!~--- \textbf{function};
\verb!\algENDFUNC!~--- \textbf{end function}.
The definition of these commands is primitive:
\begin{center}
\verb!\newcommand{\algELSE}{\textbf{else\ }}!,
\end{center}
and their set can be extended by the author of the article himself, if necessary.

The \verb!\newcommand{\algGOTO[1]{\textbf{goto}~\ref*{#1}}}! command defined in the style file 
takes the symbolic label specified in the \verb!\algLine! command as an argument 
and generates the text \textbf{goto\ }\verb!<line number! \verb!containing! \verb!the label>!.

\begin{nmAlgorithm}{pril:prog:alg1}
    {Function $g$ (calculating the coordinates of the point $v$ by its ordinal $k$)}
     \algLine{\algFUNC $g(k,d,p)$}
     \algLine[1]{$u_{n-1} \ceq \left\lfloor k/\left(d-1\right)^{n-2}\right\rfloor$}
     \algLine[1]{$u_n \ceq u_{n-1}$}
     \algLine[1]{$k\ceq k\mod (d-1)^{n-2}$}
     \algLine[1]{\algFOR $j=\left(n-3\right)\ldots0$ \algDO}
     \algLine[2]{$u_j\ceq \left\lfloor k/(d-1)^j\right\rfloor+1$}
     \algLine[2]{$k\ceq k\mod(d-1)^j$}
     \algLine[1]{\algENDFOR}
     \algLine[1]{\textbf{return} $(v_1,\dots,v_n)$}
     \algLine{\algENDFUNC}
\end{nmAlgorithm}  

\medskip
\textbf{Algorithm~\ref{pril:prog:alg2}.~Comparison of two variants of the algorithm.} 
To make it convenient for the reader to compare variants of an algorithm or to show 
the difference between two similar algorithms, the author needs to place them line 
by line next to each other (see Algorithm~\ref{pril:prog:alg2}).

For a set of algorithms that require matching, you need to use the command
\begin{verbatim}
\algLineDouble[<number of indents>][<symbolic label>]
              {<algorithmic operation, algorithm 1>}{<algorithmic operation, algorithm 2>},
\end{verbatim}
which arranges lines of algorithms in two columns.


\begin{nmAlgorithm}{pril:prog:alg2}
      {Generating validation set points (parameters $d, \rho$)}
\algLineDoubleComment{a) with dublicates}{b) no dublicates}
\algLineDouble[0][a1:l1]{$\varphi := \pi/d$}{$\varphi := \pi/d$}
\algLineDouble[0][a1:l2]{\algFOR $j_{n-1} = 0\ldots (2d-1)$ \algDO}{\algFOR $j_{n-1} = 0\ldots (2d-1)$ \algDO}
\algLineDouble[1]{$\theta := j_{n-1}\varphi$}{$\theta := j_{n-1}\varphi$}
\algLineDouble[1]{\algFOR $j_{n-2} = 0\ldots d$ \algDO}{\algFOR \corr{$j_{n-2} = 1\ldots d-1$} \algDO}
\algLineDouble[2]{$\phi_{n-2} := j_{n-2}\varphi$}{$\phi_{n-2} := j_{n-2}\varphi$}
\algLineDouble[2]{\ldots}{\ldots}
\algLineDouble[2]{\algFOR $j_2 = 0\ldots d$ \algDO}{\algFOR \corr{$j_2 = 1\ldots d-1$} \algDO}
\algLineDouble[3]{$\phi_2 := j_1\varphi$}{$\phi_2 := j_1\varphi$}
\algLineDouble[3]{\algFOR $j_1 = 0\ldots d$ \algDO}{\algFOR \corr{$j_1 = 1\ldots d-1$} \algDO}
\algLineDouble[4]{$\phi_1 := j_1\varphi$}{$\phi_1 := j_1\varphi$}
\algLineDouble[4]{$\varpi := 1$}{$\varpi := 1$}
\algLineDouble[4]{$v_1 := \rho \cos(\phi_1)$}{$v_1 := \rho \cos(\phi_1)$}
\algLineDouble[4]{\algFOR $l=2\ldots n-2$ \algDO}{\algFOR $l=2\ldots n-2$ \algDO}
\algLineDouble[5]{$\varpi := \sin(\phi_{l-1})\varpi$}{$\varpi := \sin(\phi_{l-1})\varpi$}
\algLineDouble[5]{$v_l := \rho\cos(\phi_l)\varpi$}{$v_l := \rho\cos(\phi_l)\varpi$}
\algLineDouble[4]{\algENDFOR }{\algENDFOR }
\algLineDouble[4]{$v_{n-1} := \rho\sin(\theta)\varpi$}{$v_{n-1} := \rho\sin(\theta)\varpi$}
\algLineDouble[4]{$v_n := \rho\cos(\theta)\varpi$ }{$v_n := \rho\cos(\theta)\varpi$ }
\algLineDouble[4]{\algOUTPUT $v$}{\algOUTPUT $v$}
\algLineDouble[3]{\algENDFOR}{\algENDFOR}
\algLineDouble[2]{\algENDFOR}{\algENDFOR}
\algLineDouble[2]{\ldots}{\ldots}
\algLineDouble[1]{\algENDFOR}{\algENDFOR}
\algLineDouble{\algENDFOR}{\algENDFOR}
\algLineDouble{\algSTOP}{\algSTOP}\par
\end{nmAlgorithm}

\medskip
\textbf{Algorithm~\ref{pril:prog:alg3}.~An example of combining the text of the article 
and the algorithm.} 
Below is an example of the design of a small algorithm next to the text of 
the article. For this, the environment is used 
\begin{verbatim}
\begin{wrapfigure}{r}{80mm}
      ...
\end{wrapfigure}.
\end{verbatim}
%
% ПРИМЕР "ОБТЕКАНИЯ" АЛГОРИТМА ТЕКСТОМ
%
\medskip
\hrule
\bigskip

\begin{wrapfigure}{r}{80mm}
      \begin{nmAlgorithm}{pril:prog:alg3}            
            {Validation of the solution $\widetilde{x}$ to the LP problem}{\ }
            \algLine{\algINPUT $n,A,b,c,d,\rho,\varepsilon,\widetilde{x}$}
            \algLine{$\varphi \ceq \pi/d$}
            \algLine{\algFOR $k=0\ldots 2d(d-1)^{n-2}-1$ \algDO}
            \algLine[1]{$v \ceq g(k,d,\rho)$}
            \algLine[1]{$Av \leqslant b \And \langle c,v\rangle  > \langle c,\widetilde{x}\rangle + \varepsilon$ \textbf{goto} 9}
            \algLine{\algENDFOR}
            \algLine{\algOUTPUT ``Solution is correct''}
            \algLine{\algGOTO{a3:l10}}
            \algLine{\algOUTPUT ``Solution is incorrect''}
            \algLine[0][a3:l10]{\algSTOP}
      \end{nmAlgorithm}
\end{wrapfigure}

In steps~11--18, spherical coordinates are converted to Cartesian coordinates 
using formulas (2). Using variable bounds in the headers of the loops in 
steps 2, 4,\,\dots, 7, 9, it is easy to calculate that Algorithm 1a generates 
$2d{(d + 1)^{n - 2}}$ points. The disadvantage of Algorithm 1a is that it 
generates some points with repetitions. The performed computational experiment 
showed that for dimension $n = 4$ and the number of parallels $d = 5$, 
Algorithm 1a generates 189 duplicates with a total number of points equal to 360, 
which is more than 50\%. Duplicates are generated in iterations when 
${\phi _i} = 0$ or ${\phi _i} = \pi$, which corresponds to the values 
${j_i} = 0$ and ${j_i} = d$ ($i = 1, \ldots , n-2$). This is due to the fact 
that in this case one of the factors $\sin(\phi _i)$ in (2) will be equal to zero 
and, therefore, variations of other factors will no longer be able to 
change the value of the corresponding coordinate. The solution to the problem 
of duplicates without major reworking of Algorithm 1a is achieved by changing 
the lower and upper bounds of the variables in the headers\ldots


\bigskip
\hrule

%
% КОНЕЦ ПРИМЕРА "ОБТЕКАНИЯ" АЛГОРИТМА ТЕКСТОМ
%
    
      



%============================ ПРИЛОЖЕНИЕ 2 ==================================
\newpage\MakeVerbDense
\appendix{Design of illustrations}{pril:ris}

\textbf{Figure~\ref{pril:ris:fig1}. Inserting a picture that takes 
up half the typesetting width into text.} It is recommended to ``wrap around'' 
the pictures, the width of which does not exceed 80~mm. This requires
use the environment
\begin{verbatim}
\begin{wrapfigure}{r}{<linear size>}
      ...
\end{wrapfigure}.
\end{verbatim}
The required parameter \verb!{r|l}! specifies the position of the picture at 
the right or left edges of the text. Linear size can be 
specified both in absolute units, for example \verb!{80mm}!, and in relative 
units~--- \verb!{0.5\linewidth}!.
%
% ПРИМЕР 1. ВСТАВКА РИСУНКОВ
%
\medskip
\hrule
\bigskip

\begin{wrapfigure}{r}{0.54\linewidth}
\centering
\includegraphics[scale=0.86]{./pic/sokol_Fig-1}
\caption{Construction of points of the validation set $V$ on\\ 
      the three-dimensional sphere for $d=5$}
\label{pril:ris:fig1}
\end{wrapfigure}

Let a linear programming problem be given in the Euclidean space ${\mathbb{R}^n}$
\begin{equation*}
\bar x = \arg \max \left\{ {\left\langle {c,x} \right\rangle \mid {Ax \leqslant b,x \in {\mathbb{R}^n}} } \right\},\eqno(1)
\end{equation*}
where $c$ is the vector of the objective function coefficients. Here and below, 
$\langle{\cdot, \cdot}\rangle$ denotes the scalar product of two vectors. We set
$M = \left\{ {x \in {\mathbb{R}^n} \mid {Ax \leqslant b}} \right\}$ is the set of 
feasible points of problem (1). By definition, the set $M$ is convex. Throughout 
what follows, we will assume that $M$ is a non-empty, bounded (and, therefore, 
closed) set, that is, problem~(1) has at least one solution. 
Let $\tilde x \in {\mathbb{R}^n}$ be an approximate solution to problem~(1) 
obtained using some LP-solver and requiring certification.

The idea of the VaLiPro validation method described in this work is to construct 
a finite set of points~$V$ covering a hypersphere $S$ of small (compared to the 
size of the polyhedron $M$) radius $\rho$ centered at the point of the being 
certified solution~$\tilde x$:
\begin{equation*}
V \subset S = \bigl\{ x \in \mathbb{R}^n \bigm\vert {{\left\| {x - \tilde x} \right\|}^2 = \rho ^2} \bigr\}.
\end{equation*}
\smallskip
\hrule
\bigskip
%
% ПРИМЕР 1. КОНЕЦ
%

\textbf{Figure~\ref{pril:ris:fig2}. Inserting a series of figures.}
We draw the attention of the authors to the problem of integrating formulas and 
fonts of the publishing system \LaTeXe{} into figures created in graphic editors 
or in any other way. In the overwhelming majority of cases, authors try to place 
on their drawings formulas and inscriptions that look unnatural due to the fact 
that they were typed in fonts that are not included in the \LaTeXe{} publishing 
system. 

Let's consider the most common case of inserting a series of figures into the text. 
We assume that the author has chosen the optimal size of the pictures and there are 
no inscriptions on the pictures.

1.~Creates a floating object using the environment
\begin{verbatim}
\begin{figure}{t!}
      ...
\end{figure}.
\end{verbatim}
Mandatory parameters \verb+{t}+ and \verb+{t!}+ specify the placement at the top 
of the page, \verb+{b}+ and \verb+{b!}+~--- at the bottom, 
and \verb+{h}+ and \verb+{h!}+~--- where the floating object is located.

\pagebreak
2. Inside the floating object create an environment \verb!{picture}!
\begin{verbatim}
\begin{figure}{t!}
   \begin{picture}(170,65)
      ...
   \end{picture}
\end{figure}.
\end{verbatim}
The size of the area reserved for drawings is indicated in parentheses in millimeters.
In this example, the area is 170~mm wide and 65~mm high. Note that the size of the 
drawings themselves slightly less than 65~mm in height.

3. Using the command \verb!\put(<x-координата>, <y-координата>){<объект>}!, both the 
figures themselves and any inscriptions are placed in the desired place in the 
reserved area. 

For example, the command
\begin{verbatim}
      \put(11.3,7.5){\includegraphics[width=70mm]{./pic/Bel_sol1r.eps}}
\end{verbatim}
loads the \verb!Bel_sol1r.eps! picture from the \verb!./pic/!,
directory, proportionally scales it to a width of 70~mm and places it inside the 
area reserved in the \verb!{picture}! environment so that the coordinates of the 
lower left corner of the picture become 11.3~mm along the horizontal axis and 
7.5~mm along the vertical axis.

The \verb!\put(35,62){$u(x_1, x_2)$}! command displays the inscription 
$u(x_1, x_2)$ over the picture.

The \verb!\put(x,y)! command can take the \verb!{picture}! environment as an output 
object:
\begin{verbatim}
      \put(x0,y0){\begin{picture}(0,0) ... \end{picture}}.
\end{verbatim}
Such use of nested environments \verb!{picture}! makes sense if the coordinates of 
the second picture and inscriptions on it differ from the coordinates of the first 
picture by the vector $(x0,y0)$. The size of the area reserved by the 
\verb!{picture}! nested environment is immaterial and can be \verb!(0,0)!.

The following commands display inscriptions over the image:
\begin{verbatim}
\put(75,43){\rotatebox{90}{Legend}} 
\end{verbatim}

4. After the \verb!{picture}! environment, place captions to figures using the 
\verb!\caption!:
\begin{verbatim} 
\begin{figure}{t!}
      \begin{picture}(170,60)
      ...
      \end{picture}
      \caption{<signature text>}
\end{figure}.
\end{verbatim}
%
% ПРИМЕР 2. ВСТАВКА РИСУНКОВ
%
\begin{figure}[t!]
\begin{picture}(170,65)\small
      \put(11.3,7.5){\includegraphics[width=70mm]{./pic/Bel_sol1r}}
      \put(35,62){$u(x_1, x_2)$}
      \put(6,55.5){$1.00$}
      \put(7,44){$0.50$}
      \put(8.3,34){$0.00$}
      \put(9,31){$0.50$}
      \put(15.7,20.5){$0.00$}
      \put(20.5,9){$-0.50$}
      \put(13,15){$x_2$}
      \put(31,6){$-0.50$}
      \put(55,10){$0.00$}
      \put(68,9){$x_1$}
      \put(77,14){$0.50$}
      \put(43,0){a)}
      \put(85,0){\begin{picture}(0,0)
            \put(11.3,7.5){\includegraphics[width=70mm]{./pic/Bel_sol2r}}
            \put(25,62){$u(x_1, x_2)$}
            \put(5.5,49.5){$1.00$}
            \put(7,39){$0.50$}
            \put(8.3,30){$0.00$}
            \put(9.5,27){$1.00$}
            \put(19,18.5){$0.50$}
            \put(28.5,9){$0.00$}
            \put(16,13){$x_2$}
            \put(39.6,6){$0.00$}
            \put(58,11.5){$0.50$}
            \put(69,10){$x_1$}
            \put(75,17){$1.00$}
            \put(43,0){b)}
      \end{picture}}
      % надпись, повернутая на 90 градусов
      \put(79,43){\rotatebox{90}{Legend}} % на английском языке
\end{picture}
\caption{Exact solution: a) in example~1.1 at $\beta = 1000$; b) in example~1.2 at $\beta = 10^{-4}$}
\label{pril:ris:fig2}
\end{figure}
%
% ПРИМЕР 2. КОНЕЦ
%

%============================ ПРИЛОЖЕНИЕ 3 ==================================
\newpage
\appendix{Design of tables}{pril:tbl}

\textbf{Table~\ref{pril:tbl:tab1}. Inserting text in two languages into table cells.}
Presentation of information in tabular form is often used in the design of articles. 
In simple cases, it is enough to use the well-known \verb!{array}! and \verb!{tabular}! 
environments~[\ref{rfa:enlit:Kotelnikov}, \ref{rfa:enlit:Lvovski}]. 
The \verb!{tabularx}!~[\ref{rfa:enlit:Kotelnikov}, \ref{rfa:enlit:Samarin}] 
environment and the \verb!\makecell!~[\ref{rfa:enlit:Lapko}] command are more 
convenient to use.

Consider the key points in the design of the Table~\ref{pril:tbl:tab1}. 
The table header is formed by the \verb!\caption! command. Table~\ref{pril:tbl:tab1}, 
there are ten columns, but the number of columns in the first row of the table is 
reduced to four by combining three columns into one command 
\verb!\multicolumn{3}{c|}{}!. The table is set by the \verb!{tabularx}! environment 
with parameters:
\begin{verbatim}
\begin{tabularx}{170mm}{|>{\centering\arraybackslash}X|
      *{3}{>{\centering\arraybackslash}p{11mm}|}
      *{3}{>{\centering\arraybackslash}p{14mm}|}
      *{3}{>{\centering\arraybackslash}p{11mm}|}},
\end{verbatim}
where the table width is set equal to the width of the stripe (\verb!{170mm}!), 
the first column has an undefined width (\verb!X!), due to which the entire table is 
justified in width during compilation, the width of the remaining columns is set 
explicitly in the corresponding specifiers.

To place text on two lines inside a cell, we recommend using the 
\verb!\makecell{}! command, in which the \verb!\\! command saves its line breaking 
function. Horizontal snapping is specified in the \verb!\makecell[]{}! command using 
an optional argument in square brackets: 
\verb![t]!~--- snapping to the top line, 
\verb![b]!~--- to the bottom line, 
\verb![c]!~--- to the center of the block. 


%
% ПРИМЕР 1. ВСТАВКА ТАБЛИЦЫ
%
\begin{table}[h!]\small
\caption{Times in seconds of Fortran programs, NPB 3.3 class D}\label{pril:tbl:tab1}
\begin{tabularx}{170mm}{|>{\centering\arraybackslash}X|
      *{3}{>{\centering\arraybackslash}p{11mm}|}
      *{3}{>{\centering\arraybackslash}p{14mm}|}
      *{3}{>{\centering\arraybackslash}p{11mm}|}}
      \hline
      & \multicolumn{3}{c|}{\makecell{MPI\\ programs}} 
      & \multicolumn{3}{c|}{\makecell{Converted\\ MPI programs}}  
      & \multicolumn{3}{c|}{\makecell{MPI programs +\\ \textbf{FDVMH}}} \\
      \cline{2-10}
      & BT & CG & EP & BT & CG & EP & BT & CG & EP \\
      \hline
      \makecell[t]{1 node} & 665.1 & 397.5 & 93.68 & 785.29 & 376.8 & 83.34 & 63.3 & 80.99 & 0.62 \\
      \hline
      \makecell[c]{2 nodes} & 361.6 & 209.6 & 46.53 & 428.07 & 229.61 & 42.06 & 50.3 & 42.6  & 0.38 \\
      \hline
      \makecell[b]{4 nodes} & 196.8 & 91.3  & 23.3  & 232.36 & 96.67  & 25.16 & 45.5 & 34.3  & 0.17 \\
      \hline
\end{tabularx}
\end{table}
%
% ПРИМЕР 1. КОНЕЦ
%
  
\textbf{Table~\ref{pril:tbl:tab2}. Combining several vertically spaced cells into one.} 
The following table shows the use of the command
\begin{verbatim}
            \multirow[<vpos>]{<rows>}[<bigstruts>]{<width>}[<vmove>]{<text>}.
\end{verbatim}
The example uses only the required arguments 
\verb!{<rows>}!, \verb!{<width>}!, and \verb!{<text>}! [\ref{rfa:enlit:mrow}].


%
% ПРИМЕР 2. ВСТАВКА ТАБЛИЦЫ
%
\begin{table}[h!]
\caption{Results of numerical experiments in example~1.1}
\label{pril:tbl:tab2}
\small
\renewcommand{\tabcolsep}{2pt}
\begin{tabularx}{170mm}{|%
            >{\centering\arraybackslash}p{5mm} |
            >{\centering\arraybackslash}p{16mm} ||
            >{\centering\arraybackslash}p{21mm} |
            >{\centering\arraybackslash}p{12mm} |
            >{\centering\arraybackslash}p{21mm} |
            >{\centering\arraybackslash}p{12mm} ||
            >{\centering\arraybackslash}p{21mm} |
            >{\centering\arraybackslash}p{12mm} |
            >{\centering\arraybackslash}p{21mm} |
            >{\centering\arraybackslash} X |
      }
      \hline
      \multirow{3}{*}{$K$} &  
      \multirow{3}{*}{$N$$\times$$N$} & 
      \multicolumn{2}{c|}{\makecell{FDM}} & 
      \multicolumn{2}{c||}{\makecell{LSC method}} & 
      \multicolumn{2}{c|}{\makecell{FDM}} & 
      \multicolumn{2}{c|}{\makecell{LSC method}} \\
      \cline{3-10}
      &
      &
      $\|E_a\|_{\infty}$\rl &
      $R$ &
      $\|E_a\|_{\infty}$ &
      $R$ &
      $\|E_a\|_{\infty}$ &
      $R$ &
      $\|E_a\|_{\infty}$ &
      $R$ \\
      \cline{3-10}
      & & \multicolumn{4}{c||}{$\beta = 100$} & \multicolumn{4}{c|}{$\beta = 1000$} \\
      \hline
      \multirow{4}{*}{$2$} & 10$\times$10 & $3.843 \cdot 10^{-1}$ & --- 	 & $2.92 \cdot 10^{-1}$ & ---    & $19.978$ 			 & --- 	  & $2.23$               & --- \\
      & 20$\times$20 & $6.771 \cdot 10^{-2}$ & $2.50$ & $5.85 \cdot 10^{-2}$ & $2.31$ & $3.11$ 			     & $2.68$ & $7.74 \cdot 10^{-1}$ & $1.52$\\
      & 40$\times$40 & $1.591 \cdot 10^{-2}$ & $2.09$ & $1.38 \cdot 10^{-2}$ & $2.08$ & $1.959 \cdot 10^{-1}$ & $3.99$ & $1.64 \cdot 10^{-1}$ & $2.23$\\
      & 80$\times$80 & $3.921 \cdot 10^{-3}$ & $2.02$ & $3.45 \cdot 10^{-3}$ & $2.00$ & $4.104 \cdot 10^{-2}$ & $2.25$ & $3.55 \cdot 10^{-2}$ & $2.20$\\
      \hline
      \multirow{4}{*}{$6$}	
      & 10$\times$10 & $4.942 \cdot 10^{-3}$ & ---    & $7.11 \cdot 10^{-4}$ & ---    & $7.985$ 			    & --- 	 & $2.46 \cdot 10^{-1}$ & ---  \\
      & 20$\times$20 & $3.594 \cdot 10^{-5}$ & $7.10$ & $7.30 \cdot 10^{-6}$ & $6.60$ & $1.13 \cdot 10^{-1}$  & $6.14$ & $1.71 \cdot 10^{-2}$ & $3.84$\\
      & 40$\times$40 & $4.635 \cdot 10^{-7}$ & $6.28$ & $8.65 \cdot 10^{-8}$ & $6.39$ & $7.624 \cdot 10^{-4}$ & $7.21$ & $1.44 \cdot 10^{-4}$ & $6.89$\\
      & 80$\times$80 & $7.110 \cdot 10^{-9}$ & $6.03$ & $1.23 \cdot 10^{-9}$ & $6.13$ & $8.193 \cdot 10^{-6}$ & $6.54$ & $1.61 \cdot 10^{-6}$ & $6.48$\\
      \hline	
\COMMENT{
      \multirow{4}{*}{$10$} 
      & 10$\times$10 & $3.036 \cdot 10^{-3}$ & --- 	 & $7.37 \cdot 10^{-7}$  & ---     & $2.936$               & --- 	   & $1.51 \cdot 10^{-2}$  & ---\\
      & 20$\times$20 & $9.894 \cdot 10^{-7}$ & $11.58$ & $4.81 \cdot 10^{-10}$ & $10.58$ & $3.086 \cdot 10^{-1}$ & $3.25$  & $8.76 \cdot 10^{-5}$  & $7.42$\\
      & 40$\times$40 & $8.473 \cdot 10^{-10}$& $10.19$ & $3.54 \cdot 10^{-13}$ & $10.40$ & $1.559 \cdot 10^{-4}$ & $10.95$ & $5.93 \cdot 10^{-8}$  & $10.52$\\
      & 80$\times$80 & $8.407 \cdot 10^{-13}$& $9.98$  & $7.52 \cdot 10^{-14}$ & $2.23$  & $8.78  \cdot 10^{-8}$ & $10.79$ & $7.75 \cdot 10^{-11}$ & $9.57$\\
      \hline	
      \multirow{5}{*}{$18$} 
      & 5$\times$5   & ---  & --- 	 & $1.74 \cdot 10^{-8}$  & ---     & ---				   & --- 	 & $1.03$  & ---\\
      & 10$\times$10 & $8.388 \cdot 10^{-2}$ & --- 	 & $1.43 \cdot 10^{-13}$ & $13.57$ & $154283$              & --- 	 & $2.91 \cdot 10^{-5}$ & $15.11$\\
      & 20$\times$20 & $2.844 \cdot 10^{-7}$ & $18.17$ & $3.04 \cdot 10^{-14}$ & $2.23$  & $307.82$              & $8.97$  & $1.08 \cdot 10^{-9}$ & $14.71$\\
      & 40$\times$40 & $1.826 \cdot 10^{-12}$& $17.25$ & ---					 &  ---    & $1.19  \cdot 10^{-3}$ & $17.98$ & $4.33 \cdot 10^{-13}$	 & $11.28$\\
      & 80$\times$80 & $2.992 \cdot 10^{-12}$& $-0.71$ & ---					 &  ---    & $3.92  \cdot 10^{-9}$ & $18.21$ & ---					 &  ---\\
      \hline	
}% COMMENT
\end{tabularx}
\end{table}
%
% ПРИМЕР 2. КОНЕЦ
%
              
      


%============================ ПРИЛОЖЕНИЕ 5 ==================================
\newpage
\appendix{Design of References}{pril:refer}
\setcounter{enlitcnt}{0}

\medskip
Books:

\elitem{i8}{%
V.~V.~Voevodin and Vl.~V.~Voevodin, {\it The Parallel Computing}
(BHV-Petersburg, St. Petersburg, 2002) [in Russian].}

\elitem{i13}{%
R.~P.~Fedorenko, \textit{Introduction to Computational Physics} 
(Moscow Inst. Phys. Technol., Moscow, 1994) [in Russian].}

\elitem{i3}{%
L.~D.~Landau and E.~M.~Lifshitz, 
{\it Course of Theoretical Physics}, Vol.~1: {\it Mechanics}
(Nauka, Moscow, 1973; Pergamon, Oxford, 1977).}


\medskip
Articles:

\elitem{i5}{%
A.~S.~Antonov, D.~A.~Nikitenko, and V.~V.~Voevodin, 
``Algo500~--- A New Approach to the Joint Analysis of Algorithms and Computers,’’ 
Lobachevskii J. Math. \textbf{41}~(8), 1435--1443 (2020).
\doi{10.1134/S1995080220080041}.
}

\elitem{i1}{%
V.~I.~Isaev and V.~P.~Shapeev, ``High-Accuracy Versions of the Collocations 
and Least Squares Method for the Numerical Solution of the Navier--Stokes 
Equations,’’
Zh. Vychisl. Mat. Mat. Fiz. {\bf 50} (10), 1758--1770 (2010) 
 [Comput. Math. Math. Phys. {\bf 50} (10), 1670--1681 (2010).
\doi{10.1134/S0965542510100040}].
}

\elitem{i2}{%
E.~V.~Vorozhtsov and V.~P.~Shapeev, ``On the Efficiency of Combining Different 
Methods for Acceleration of Iterations at the Solution of PDEs by the Method of 
Collocations and Least Residuals,'' Appl. Math. Comput. {\bf 363} (2019). 
\doi{10.1016/j.amc.2019.124644}. 
}

\elitem{i8}{%
V.~P.~Shapeev, S.~K.~Golushko, V.~A.~Belyaev, et al., 
``New Versions of the Least-Squares Collocation Method for Solving 
Differential and Integral Equations,'’
  J. Phys.: Conf. Ser. {\bf 1715} (2021), 1--8.
\doi{10.1088/1742-6596/1715/1/012031}. 
}

\elitem{i15}{%
A.~V.~Panyukov and V.~V.~Gorbik, ``Using Massively Parallel Computations for Absolutely Precise Solution of the Linear Programming Problems,’’
  Avtom. Telemekh., No.~2, 73--88 (2012)
  [Autom. Rem. Contr. \textbf{73}~(2), 276--290 (2012). 
  \doi{10.1134/S0005117912020063}].
}


\medskip
Conference proceedings:

\elitem{i24}{%
V.~Bakhtin, A.~Kolganov, V.~Krukov, et al.,
``Dynamic Tuning Methods of DVMH-Programs for Clusters with Accelerators,’’
in {\it Proc. Int. Conf. on Russian Supercomputing Days, Moscow, Russia,
September~28--29, 2015}
(Mosk. Gos. Univ., Moscow, 2015), pp.~257--269.
}

\elitem{i23}{%
J.~Doerfert, K.~Streit, S.~Hack, and Z.~Benaissa,
``Polly’s Polyhedral Scheduling in the Presence of Reductions,''
  in {\it Proc. 5th Int. Workshop on Polyhedral Compilation Techniques (IMPACT 2015),
  Amsterdam, The Netherlands, January~19, 2015},
\url{https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1054.1804&rep=rep1&type=pdf}. 
Cited October~15, 2021.
}

\medskip
Internet resources:

\elitem{i6}{%
OpenMP Compilers \& Tools. 
\url{https://www.openmp.org/resources/openmp-compilers-tools/}. 
Cited October~15, 2021.
}

\elitem{i15}{%
NAS Parallel Benchmarks. 
\url{https://www.nas.nasa.gov/publications/npb.html}.
Cited October 15, 2021.
}

\elitem{iii}{%
NVidia CUDA Zone. 
\url{https://developer.nvidia.com/cuda-zone}. 
Cited November 15, 2020.}


\medskip
Other resources:

\elitem{i21}{%
L.~B.~Sokolinsky, {\it Parallel Algorithmic Skeleton BSF},
  Certificate of RF Registration of Computer Program No.~2 020 661 344.
Date of Registration: September 22, 2020. 
}





%============================ ПРИЛОЖЕНИЕ 6 ==================================
\newpage\nolinenumbers
%
% ПРИЛОЖЕНИЕ 4
% используйте как шаблон письма в редакцию
%
\thispagestyle{empty}
%
% Следующие две строки нужно убрать
%
\appendix{Cover letter}{pril:sopr}

\vspace*{3mm}

{\large
\hfill\parbox[t]{85mm}{
      To the deputy editor-in-chief\\ 
      of the scientific journal\\
      ``Numerical Methods and Programming''\\
      Doctor of Physical and Mathematical Sciences,\\
      A. V. Smirnov
}
\vspace*{9mm}
\begin{center}
COVER LETTER
\end{center}

\vspace*{3mm}

We ask you to publish the article ``<TITLE OF ARTICLE>'' in the journal 
``Numerical Methods and Programming''. 

We provide the following information about each author of the article.

1.~Surname, name, patronymic in full (written in English).

2.~The main place of work in full, indicating the full postal address and URL of the organization's website (with translation into English).

3.~Position at the main place of work (with translation into English).

4.~Academic degree and academic title (with translation into English).

5.~Office, home and mobile phone numbers.

6.~E-mail.

7.~ORCID (required; if not already received, visit \url{https://orcid.org/register}).

8.~Number and title of Code of State Categories Scientific and Technical Information 
(ГРНТИ/CSCSTI) heading (only for articles in Russian) and Organization for Economic 
Co-operation and Development (OECD) code (for articles in English) for the submitted 
article.

9.~Desirable: links to authors' profiles in elibrary.ru (if exists) for 
simplification and correctness of article linking.

\vspace*{2mm}
Current correspondence on the publication of the article should be conducted with 
<place here name of contact person>.

\vspace*{2mm}
The authors agree with the rules for preparing an article for publication, including 
peer review, scientific and literary editing, and bringing the article to editorial 
standards accepted within the journal.

The authors are familiar with the Publication Ethics of the journal and agree with 
its provisions.

The authors also agree to transfer to the journal their right to publish and 
distribute the article in electronic and paper versions, including the placement 
of bibliographic information about the article in the Russian Science Citation 
Index (RSCI) and other scientific citation bases and to post full texts of articles 
in the Scientific electronic library (elibrary.ru) and the portal 
http://www.mathnet.ru for free access to all Internet users regardless of their 
category and location.


\vspace*{8mm}
<Date>\hfill\parbox[t]{77mm}{\hrulefill~<Sign \& Author Name 1>}

\vspace*{8mm}

\hfill\parbox[t]{77mm}{\hrulefill~<Sign \& Author Name 2>}

}%\large



\end{document}

