{"id":50,"date":"2018-08-06T18:47:34","date_gmt":"2018-08-06T18:47:34","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/chapter\/latexpage\/"},"modified":"2021-11-26T20:15:33","modified_gmt":"2021-11-26T20:15:33","slug":"latexpage","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/chapter\/latexpage\/","title":{"raw":"latexpage","rendered":"latexpage"},"content":{"raw":"[latexpage]\nAt first, we sample $f(x)$ in the $N$ ($N$ is odd) equidistant points around $x^*$:\n\\[\nf_k = f(x_k),\\: x_k = x^*+kh,\\: k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}\n\\]\nwhere $h$ is some step.\nThen we interpolate points $\\{(x_k,f_k)\\}$ by polynomial\n\\begin{equation}\u00a0\\label{eq:poly}\nP_{N-1}(x)=\\sum_{j=0}^{N-1}{a_jx^j}\n\\end{equation}\nIts coefficients $\\{a_j\\}$ are found as a solution of system of linear equations:\n\\begin{equation}\u00a0\\label{eq:sys}\n\\left\\{ P_{N-1}(x_k) = f_k\\right\\},\\quad k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}\n\\end{equation}\nHere are references to existing equations: (\\ref{eq:poly}), (\\ref{eq:sys}).\nHere is reference to non-existing equation (\\ref{eq:unknown}).\n\n&nbsp;\n\n\\begin{tikzpicture}\n[+preamble]\n\\usepackage{pgfplots}\n\\pgfplotsset{compat=newest}\n[\/preamble]\n\\begin{axis}\n\\addplot3[surf,domain=0:360,samples=40] {cos(x)*cos(y)};\n\\end{axis}\n\\end{tikzpicture}","rendered":"<p>[latexpage]<br \/>\nAt first, we sample $f(x)$ in the $N$ ($N$ is odd) equidistant points around $x^*$:<br \/>\n\\[<br \/>\nf_k = f(x_k),\\: x_k = x^*+kh,\\: k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}<br \/>\n\\]<br \/>\nwhere $h$ is some step.<br \/>\nThen we interpolate points $\\{(x_k,f_k)\\}$ by polynomial<br \/>\n\\begin{equation}\u00a0\\label{eq:poly}<br \/>\nP_{N-1}(x)=\\sum_{j=0}^{N-1}{a_jx^j}<br \/>\n\\end{equation}<br \/>\nIts coefficients $\\{a_j\\}$ are found as a solution of system of linear equations:<br \/>\n\\begin{equation}\u00a0\\label{eq:sys}<br \/>\n\\left\\{ P_{N-1}(x_k) = f_k\\right\\},\\quad k=-\\frac{N-1}{2},\\dots,\\frac{N-1}{2}<br \/>\n\\end{equation}<br \/>\nHere are references to existing equations: (\\ref{eq:poly}), (\\ref{eq:sys}).<br \/>\nHere is reference to non-existing equation (\\ref{eq:unknown}).<\/p>\n<p>&nbsp;<\/p>\n<p>\\begin{tikzpicture}<br \/>\n[+preamble]<br \/>\n\\usepackage{pgfplots}<br \/>\n\\pgfplotsset{compat=newest}<br \/>\n[\/preamble]<br \/>\n\\begin{axis}<br \/>\n\\addplot3[surf,domain=0:360,samples=40] {cos(x)*cos(y)};<br \/>\n\\end{axis}<br \/>\n\\end{tikzpicture}<\/p>\n","protected":false},"author":266,"menu_order":13,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-50","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/pressbooks\/v2\/chapters\/50","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/wp\/v2\/users\/266"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/pressbooks\/v2\/chapters\/50\/revisions"}],"predecessor-version":[{"id":51,"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/pressbooks\/v2\/chapters\/50\/revisions\/51"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/pressbooks\/v2\/chapters\/50\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/wp\/v2\/media?parent=50"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/pressbooks\/v2\/chapter-type?post=50"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/wp\/v2\/contributor?post=50"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/jctestbookdalcin\/wp-json\/wp\/v2\/license?post=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}