commit de3f8901fe4f769beb4901d8892dd58db40cae75 Author: jowl Date: Sat Nov 29 03:06:29 2025 -0600 test diff --git a/zsplins.aux b/zsplins.aux new file mode 100644 index 0000000..db7865a --- /dev/null +++ b/zsplins.aux @@ -0,0 +1,30 @@ +\relax +\providecommand\babel@aux[2]{} +\@nameuse{bbl@beforestart} +\catcode `"\active +\catcode `<\active +\catcode `>\active +\@nameuse{es@quoting} +\babel@aux{spanish}{} +\@writefile{toc}{\contentsline {section}{\numberline {1}Los Z-splines}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Introducción}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Definición cardinal y construcción básica}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Conservación de momentos}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Reproducción polinómica}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.5}Soporte compacto y regularidad}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.6}Relación con diferencias finitas}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.7}Límite hacia la función sinc}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.8}Z-splines en mallas no uniformes}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.9}Aplicaciones numéricas}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.10}Conclusión}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {2}El Z-spline cúbico}{3}{}\protected@file@percent } +\newlabel{eq:aiz1}{{1}{4}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Z-spline cúbico en los extremos de un intervalo}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {3}Fórmula de integración numérica}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {4}Fórmula de derivación numérica}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {5}Convergencia}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Integración}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Derivación}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Interpolación equidistante en 2D}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Interpolación en 1D con puntos arbitrarios}{4}{}\protected@file@percent } +\gdef \@abspage@last{4} diff --git a/zsplins.fdb_latexmk b/zsplins.fdb_latexmk new file mode 100644 index 0000000..1d107e1 --- /dev/null +++ b/zsplins.fdb_latexmk @@ -0,0 +1,82 @@ +# Fdb version 4 +["pdflatex"] 1764207864.40331 "/home/jowl/Documents/metodos/zsplins.tex" "zsplins.pdf" "zsplins" 1764207864.67622 0 + "/home/jowl/Documents/metodos/zsplins.tex" 1764207864.29063 6662 9e966fbad8b78d3a72ecc84c95172a1b "" + "/usr/share/texlive/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc" 1136849721 2971 def0b6c1f0b107b3b936def894055589 "" + "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx0900.tfm" 1136768653 3584 1a7de6c99457381c64abc1a7c545505f "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1200.tfm" 1136768653 3584 402da0b29eafbad07963b1224b222f18 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1440.tfm" 1136768653 3584 13049b61b922a28b158a38aeff75ee9b "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm0500.tfm" 1136768653 3584 178baad7ffca7f5d3428a83bd7cc64c3 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm0700.tfm" 1136768653 3584 cf973739aac7ab6247f9150296af7954 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm0900.tfm" 1136768653 3584 d3d8ac8b25ca19c0a40b86a5db1e8ccc "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1000.tfm" 1136768653 3584 adb004a0c8e7c46ee66cad73671f37b4 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1200.tfm" 1136768653 3584 f80ddd985bd00e29e9a6047ebd9d4781 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1440.tfm" 1136768653 3584 3169d30142b88a27d4ab0e3468e963a2 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1728.tfm" 1136768653 3584 3c76ccb63eda935a68ba65ba9da29f1a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1246382020 924 9904cf1d39e9767e7a3622f2a125a565 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1246382020 908 2921f8a10601f252058503cc6570e581 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1246382020 940 75ac932a52f80982a9f8ea75d03a34cf "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmex10.tfm" 1136768653 992 662f679a0b3d2d53c1b94050fdaa3f50 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi12.tfm" 1136768653 1524 4414a8315f39513458b80dfc63bff03a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1136768653 1520 eccf95517727cb11801f4f1aee3a21b4 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1136768653 1288 655e228510b4c2a1abe905c368440826 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm" 1136768653 1124 6c73e740cf17375f03eec0ee63599741 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb" 1248133631 31809 8670ca339bf94e56da1fc21c80635e2a "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm7.pfb" 1248133631 35309 940e81a5b9e04201a07e8b33a3ae6e64 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfbx0900.pfb" 1215737283 143961 201304b65f97bc643fbd0bcf0685d1fe "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfbx1200.pfb" 1215737283 140176 d4962f948b4cc0adf4d3dde77a128c95 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfbx1440.pfb" 1215737283 135942 859a90cad7494a1e79c94baf546d7de5 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfrm0900.pfb" 1215737283 149037 995a6f1e12c1d647b99b1cf55db78699 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfrm1000.pfb" 1215737283 138258 6525c253f16cededa14c7fd0da7f67b2 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfrm1200.pfb" 1215737283 136101 f533469f523533d38317ab5729d00c8a "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfrm1728.pfb" 1215737283 131438 3aa300b3e40e5c8ba7b4e5c6cebc5dd6 "" + "/usr/share/texlive/texmf-dist/tex/generic/babel-spanish/spanish.ldf" 1622145181 26965 3b6d1655ad716f98e788b1fb9f2e87b8 "" + "/usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty" 1676321701 151363 1f5971af3ef874d432e8fb43e0edb71d "" + "/usr/share/texlive/texmf-dist/tex/generic/babel/locale/es/babel-es.ini" 1663444672 4345 33a8cc6f0a7c6afcb84895b2c963a4e5 "" + "/usr/share/texlive/texmf-dist/tex/generic/babel/locale/es/babel-spanish.tex" 1498512262 336 a1c472f674f6f0c25a2c95ad8c7ad024 "" + "/usr/share/texlive/texmf-dist/tex/generic/babel/txtbabel.def" 1674507072 6927 64b53e78feab932ab94f892bb5a5facf "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty" 1644112042 7237 bdd120a32c8fdb4b433cf9ca2e7cd98a "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty" 1572645307 1057 525c2192b5febbd8c1f662c9468335bb "" + "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c "" + "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty" 1359763108 13829 94730e64147574077f8ecfea9bb69af4 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff "" + "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd" 1359763108 961 d02606146ba5601b5645f987c92e6193 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1654720880 2222 78b930a5a6e3dc2ac69b78c2057b94d7 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty" 1654720880 4173 c989ee3ced31418e3593916ab26c793a "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty" 1654720880 88393 1adf6fa3f245270d06e3d4f8910f7fc5 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty" 1654720880 4474 f04cd1cc7bd76eb033e6fb12eb6a0d77 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty" 1654720880 2444 70065bddd85997dc1fd0bb7ae634e5fa "" + "/usr/share/texlive/texmf-dist/tex/latex/base/article.cls" 1754438400 20144 d5ecf0a5140c8d8d8b72cbe86e320eff "" + "/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty" 1754438400 5119 4ce42f43368f652f9c9522d943cce8e4 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty" 1754438400 5048 84b05796b49b69e2d4257d537721c960 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo" 1754438400 8448 c33a4e1cb35cee9b33c2b21033b73e39 "" + "/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty" 1578002852 41601 9cf6c5257b1bc7af01a58859749dd37a "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty" 1654720880 2671 7e67d78d9b88c845599a85b2d41f2e39 "" + "/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def" 1673989714 30429 213676d4c7327a21d91ddaed900e7b81 "" + "/usr/share/texlive/texmf-dist/web2c/texmf.cnf" 1754438400 40900 887e0dc8cac988a9e9c574af364cf837 "" + "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1764206285.82275 4602016 72a039a3c4aea0fa25b5d35d65138f54 "" + "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1764206009 7753792 26321a032c2bb762dde4f36fe2e26e5f "" + "zsplins.aux" 1764207864.62159 2581 b15dc604252ad4b5d43b3dc777b944be "pdflatex" + "zsplins.tex" 1764207864.29063 6662 9e966fbad8b78d3a72ecc84c95172a1b "" + (generated) + "zsplins.aux" + "zsplins.log" + "zsplins.pdf" + (rewritten before read) diff --git a/zsplins.fls b/zsplins.fls new file mode 100644 index 0000000..0261e30 --- /dev/null +++ b/zsplins.fls @@ -0,0 +1,269 @@ +PWD /home/jowl/Documents/metodos +INPUT /usr/share/texlive/texmf-dist/web2c/texmf.cnf +INPUT /usr/share/texlive/texmf-dist/web2c/texmf.cnf +INPUT /var/lib/texmf/web2c/pdftex/pdflatex.fmt +INPUT /home/jowl/Documents/metodos/zsplins.tex +OUTPUT zsplins.log +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1000.tfm +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/txtbabel.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel-spanish/spanish.ldf +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel-spanish/spanish.ldf +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel-spanish/spanish.ldf +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel-spanish/spanish.ldf +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/locale/es/babel-spanish.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/locale/es/babel-spanish.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/locale/es/babel-spanish.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/locale/es/babel-spanish.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/babel/locale/es/babel-es.ini +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +INPUT ./zsplins.aux +INPUT zsplins.aux +INPUT zsplins.aux +OUTPUT zsplins.aux +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1728.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1200.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr12.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr8.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr6.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi12.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmex10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm0900.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx0900.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1440.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1440.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1200.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm +OUTPUT zsplins.pdf +INPUT /var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map +INPUT /usr/share/texlive/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm0700.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm0500.tfm +INPUT zsplins.aux +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm7.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfbx0900.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfbx1200.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfbx1440.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfrm0900.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfrm1000.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfrm1200.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/cm-super/sfrm1728.pfb diff --git a/zsplins.log b/zsplins.log new file mode 100644 index 0000000..97861a2 --- /dev/null +++ b/zsplins.log @@ -0,0 +1,234 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023) (preloaded format=pdflatex 2025.11.26) 26 NOV 2025 19:44 +entering extended mode + restricted \write18 enabled. + file:line:error style messages enabled. + %&-line parsing enabled. +**/home/jowl/Documents/metodos/zsplins.tex +(/home/jowl/Documents/metodos/zsplins.tex +LaTeX2e <2022-11-01> patch level 1 +L3 programming layer <2023-02-22> (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls +Document Class: article 2022/07/02 v1.4n Standard LaTeX document class +(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +File: size10.clo 2022/07/02 v1.4n Standard LaTeX file (size option) +) +\c@part=\count185 +\c@section=\count186 +\c@subsection=\count187 +\c@subsubsection=\count188 +\c@paragraph=\count189 +\c@subparagraph=\count190 +\c@figure=\count191 +\c@table=\count192 +\abovecaptionskip=\skip48 +\belowcaptionskip=\skip49 +\bibindent=\dimen140 +) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +Package: inputenc 2021/02/14 v1.3d Input encoding file +\inpenc@prehook=\toks16 +\inpenc@posthook=\toks17 +) (/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +Package: fontenc 2021/04/29 v2.0v Standard LaTeX package +) (/usr/share/texlive/texmf-dist/tex/generic/babel/babel.sty +Package: babel 2023/02/13 3.86 The Babel package +\babel@savecnt=\count193 +\U@D=\dimen141 +\l@unhyphenated=\language85 + (/usr/share/texlive/texmf-dist/tex/generic/babel/txtbabel.def) +\bbl@readstream=\read2 +\bbl@dirlevel=\count194 + (/usr/share/texlive/texmf-dist/tex/generic/babel-spanish/spanish.ldf +Language: spanish.ldf 2021/05/27 v5.0q Spanish support from the babel system +\es@quottoks=\toks18 +\es@quotdepth=\count195 +Package babel Info: Making " an active character on input line 570. +Package babel Info: Making . an active character on input line 675. +Package babel Info: Making < an active character on input line 722. +Package babel Info: Making > an active character on input line 722. +)) (/usr/share/texlive/texmf-dist/tex/generic/babel/locale/es/babel-spanish.tex +Package babel Info: Importing font and identification data for spanish +(babel) from babel-es.ini. Reported on input line 11. +) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +Package: amsmath 2022/04/08 v2.17n AMS math features +\@mathmargin=\skip50 + +For additional information on amsmath, use the `?' option. +(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +Package: amstext 2021/08/26 v2.01 AMS text + (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +File: amsgen.sty 1999/11/30 v2.0 generic functions +\@emptytoks=\toks19 +\ex@=\dimen142 +)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +Package: amsbsy 1999/11/29 v1.2d Bold Symbols +\pmbraise@=\dimen143 +) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +Package: amsopn 2022/04/08 v2.04 operator names +) +\inf@bad=\count196 +LaTeX Info: Redefining \frac on input line 234. +\uproot@=\count197 +\leftroot@=\count198 +LaTeX Info: Redefining \overline on input line 399. +LaTeX Info: Redefining \colon on input line 410. +\classnum@=\count199 +\DOTSCASE@=\count266 +LaTeX Info: Redefining \ldots on input line 496. +LaTeX Info: Redefining \dots on input line 499. +LaTeX Info: Redefining \cdots on input line 620. +\Mathstrutbox@=\box51 +\strutbox@=\box52 +LaTeX Info: Redefining \big on input line 722. +LaTeX Info: Redefining \Big on input line 723. +LaTeX Info: Redefining \bigg on input line 724. +LaTeX Info: Redefining \Bigg on input line 725. +\big@size=\dimen144 +LaTeX Font Info: Redeclaring font encoding OML on input line 743. +LaTeX Font Info: Redeclaring font encoding OMS on input line 744. +\macc@depth=\count267 +LaTeX Info: Redefining \bmod on input line 905. +LaTeX Info: Redefining \pmod on input line 910. +LaTeX Info: Redefining \smash on input line 940. +LaTeX Info: Redefining \relbar on input line 970. +LaTeX Info: Redefining \Relbar on input line 971. +\c@MaxMatrixCols=\count268 +\dotsspace@=\muskip16 +\c@parentequation=\count269 +\dspbrk@lvl=\count270 +\tag@help=\toks20 +\row@=\count271 +\column@=\count272 +\maxfields@=\count273 +\andhelp@=\toks21 +\eqnshift@=\dimen145 +\alignsep@=\dimen146 +\tagshift@=\dimen147 +\tagwidth@=\dimen148 +\totwidth@=\dimen149 +\lineht@=\dimen150 +\@envbody=\toks22 +\multlinegap=\skip51 +\multlinetaggap=\skip52 +\mathdisplay@stack=\toks23 +LaTeX Info: Redefining \[ on input line 2953. +LaTeX Info: Redefining \] on input line 2954. +) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +Package: amssymb 2013/01/14 v3.01 AMS font symbols + (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support +\symAMSa=\mathgroup4 +\symAMSb=\mathgroup5 +LaTeX Font Info: Redeclaring math symbol \hbar on input line 98. +LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' +(Font) U/euf/m/n --> U/euf/b/n on input line 106. +)) (/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +Package: geometry 2020/01/02 v5.9 Page Geometry + (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty +Package: keyval 2022/05/29 v1.15 key=value parser (DPC) +\KV@toks@=\toks24 +) (/usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +Package: ifvtex 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead. + (/usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +Package: iftex 2022/02/03 v1.0f TeX engine tests +)) +\Gm@cnth=\count274 +\Gm@cntv=\count275 +\c@Gm@tempcnt=\count276 +\Gm@bindingoffset=\dimen151 +\Gm@wd@mp=\dimen152 +\Gm@odd@mp=\dimen153 +\Gm@even@mp=\dimen154 +\Gm@layoutwidth=\dimen155 +\Gm@layoutheight=\dimen156 +\Gm@layouthoffset=\dimen157 +\Gm@layoutvoffset=\dimen158 +\Gm@dimlist=\toks25 +) (/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +File: l3backend-pdftex.def 2023-01-16 L3 backend support: PDF output (pdfTeX) +\l__color_backend_stack_int=\count277 +\l__pdf_internal_box=\box53 +) (./zsplins.aux +LaTeX Info: Redefining \. on input line 8. +LaTeX Info: Redefining \% on input line 8. +) +\openout1 = `zsplins.aux'. + +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 16. +LaTeX Font Info: ... okay on input line 16. +LaTeX Info: Redefining \. on input line 16. +LaTeX Info: Redefining \% on input line 16. + +*geometry* driver: auto-detecting +*geometry* detected driver: pdftex +*geometry* verbose mode - [ preamble ] result: +* driver: pdftex +* paper: +* layout: +* layoutoffset:(h,v)=(0.0pt,0.0pt) +* modes: +* h-part:(L,W,R)=(92.14519pt, 430.00462pt, 92.14519pt) +* v-part:(T,H,B)=(95.39737pt, 556.47656pt, 143.09605pt) +* \paperwidth=614.295pt +* \paperheight=794.96999pt +* \textwidth=430.00462pt +* \textheight=556.47656pt +* \oddsidemargin=19.8752pt +* \evensidemargin=19.8752pt +* \topmargin=-13.87262pt +* \headheight=12.0pt +* \headsep=25.0pt +* \topskip=10.0pt +* \footskip=30.0pt +* \marginparwidth=65.0pt +* \marginparsep=11.0pt +* \columnsep=10.0pt +* \skip\footins=9.0pt plus 4.0pt minus 2.0pt +* \hoffset=0.0pt +* \voffset=0.0pt +* \mag=1000 +* \@twocolumnfalse +* \@twosidefalse +* \@mparswitchfalse +* \@reversemarginfalse +* (1in=72.27pt=25.4mm, 1cm=28.453pt) + +LaTeX Font Info: Trying to load font information for U+msa on input line 18. +(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2013/01/14 v3.01 AMS symbols A +) +LaTeX Font Info: Trying to load font information for U+msb on input line 18. + (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2013/01/14 v3.01 AMS symbols B +) [1 + +{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}{/usr/share/texlive/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc}] [2] [3] [4] (./zsplins.aux +LaTeX Info: Redefining \. on input line 8. +LaTeX Info: Redefining \% on input line 8. +) ) +Here is how much of TeX's memory you used: + 3977 strings out of 476041 + 59858 string characters out of 5793173 + 1861388 words of memory out of 6000000 + 24417 multiletter control sequences out of 15000+600000 + 526320 words of font info for 68 fonts, out of 8000000 for 9000 + 1137 hyphenation exceptions out of 8191 + 57i,9n,62p,294b,365s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on zsplins.pdf (4 pages, 210763 bytes). +PDF statistics: + 98 PDF objects out of 1000 (max. 8388607) + 59 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 1 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/zsplins.pdf b/zsplins.pdf new file mode 100644 index 0000000..2638391 Binary files /dev/null and b/zsplins.pdf differ diff --git a/zsplins.synctex.gz b/zsplins.synctex.gz new file mode 100644 index 0000000..e725782 Binary files /dev/null and b/zsplins.synctex.gz differ diff --git a/zsplins.tex b/zsplins.tex new file mode 100644 index 0000000..6bb9c2d --- /dev/null +++ b/zsplins.tex @@ -0,0 +1,213 @@ +\documentclass{article} + +% --- PAQUETES ESENCIALES --- +\usepackage[utf8]{inputenc} % Para reconocer tildes y ñ directamente +\usepackage[T1]{fontenc} +\usepackage[spanish]{babel} % Configuración para español (fechas, silabeo) +\usepackage{amsmath} % Matemáticas avanzadas (\text, \operatorname, etc.) +\usepackage{amssymb} % Símbolos matemáticos (\mathbb) +\usepackage{geometry} % (Opcional) Para mejorar los márgenes +% --------------------------- + +\title{Proyecto 1: Métodos numéricos del Z-spline cúbico} +\author{Reyes Garcia Joel} +\date{\today} + +\begin{document} + +\maketitle + +\begin{abstract} +En este trabajo se presentan los métodos numéricos asociados a... +\end{abstract} + +\section{Los Z-splines} + +\subsection{Introducción} + +Los Z-splines constituyen una familia de splines cardinales de soporte +compacto que preservan momentos discretos y poseen alto orden de +regularidad. La idea central es construir una base spline +cuya definición incorpora explícitamente operadores de diferencias +finitas derivados de desarrollos de Taylor. Su creador, +Julián T. Becerra-Sagredo, desarrolló esta teoría en dos trabajos clave +(2003 y 2020), en los cuales se presentan tanto la construcción como +sus propiedades analíticas y aplicaciones. + +\subsection{Definición cardinal y construcción básica} + +Sea $\{x_j\} = \{j\}$ una malla equiespaciada. +Los Z-splines se definen mediante un núcleo cardinal $Z_m(x)$ tal que la +interpolación de un conjunto de datos $\{y_j\}$ está dada por + +\[ +f_m(x) = \sum_{j\in\mathbb{Z}} y_j \, Z_m(x - j). +\] + +Cada función base $Z_m$ es un polinomio por tramos de grado $2m-1$ y con +regularidad $C^{m-1}$. El parámetro $m$ determina el orden del spline. + +La construcción impone condiciones de interpolación entre nodos y +conservación de momentos discretos. Para cada intervalo +$[j,j+1]$, el polinomio está determinado por un conjunto de condiciones +de tipo Hermite–Birkhoff derivadas de operadores de diferencias finitas. + +\subsection{Conservación de momentos} + +Una de las propiedades fundamentales es que las Z-splines preservan los +primeros $2m-1$ momentos discretos. Sea + +\[ +M_k = \sum_j y_j j^k, \qquad k=0,\dots,2m-2, +\] + +entonces la interpolante satisface + +\[ +\int_{-\infty}^{\infty} x^k f_m(x) \, dx = M_k, +\] + +lo que asegura que la interpolación no altera estos invariantes +(discretos o físicos). + +\subsection{Reproducción polinómica} + +El artículo demuestra que las Z-splines reproducen exactamente +polinomios hasta grado $2m-2$: + +\[ +p(x) = x^n, \qquad n \le 2m-2 \quad \Rightarrow \quad +f_m(x) = p(x). +\] + +Esto implica un orden de aproximación + +\[ +\| f - f_m \| = \mathcal{O}(h^{\,2m-1}), +\] + +cuando $f$ es suficientemente regular. + +\subsection{Soporte compacto y regularidad} + +Cada $Z_m$ tiene soporte compacto mínimo: + +\[ +Z_m(x) = 0 \qquad \text{si } |x| > m, +\] + +y regularidad + +\[ +Z_m \in C^{m-1}. +\] + +Esto permite su uso eficiente en contextos numéricos, especialmente en +métodos de remapeo o mallas adaptativas. + +\subsection{Relación con diferencias finitas} + +Una contribución teórica clave es que los coeficientes de los polinomios +que forman $Z_m$ coinciden con los coeficientes de operadores de +diferencias finitas de orden elevado. En particular, se demuestra que +para derivadas, + +\[ +f_m^{(k)}(x_j) = \sum_{\ell=-m}^m c^{(k)}_{\ell} \, y_{j+\ell}, +\] + +donde los $c^{(k)}_\ell$ son coeficientes de diferencias finitas +centradas obtenidos por expansión de Taylor. + +\subsection{Límite hacia la función sinc} + +A medida que $m \to \infty$, las Z-splines convergen hacia la función +ideal de reconstrucción cardinal: + +\[ +Z_m(x) \longrightarrow \operatorname{sinc}(x) += \frac{\sin(\pi x)}{\pi x}. +\] + +Esto establece un vínculo entre splines compactos y teoría de muestreo. + +\subsection{Z-splines en mallas no uniformes} + +Los artículos extienden la teoría a mallas arbitrarias +$\{x_j\}$ no equidistantes. En ese caso, el núcleo cardinal ya no es +trasladable, y se construyen polinomios locales mediante matrices de +diferencias finitas adaptadas a la geometría de los nodos. + +En regiones de frontera se emplean splines ``one-sided'', lo cual permite +resolver la falta de simetría. + +\subsection{Aplicaciones numéricas} + +Las Z-splines son especialmente útiles en contextos donde es esencial la +conservación de cantidades físicas discretas: + +\begin{itemize} + \item métodos de remapeo ALE y semi-Lagrangianos, + \item interpolaciones conservativas en dinámica de fluidos, + \item regularización de funciones delta, + \item remallado adaptativo, + \item filtrado numérico (denoising) preservando momentos. +\end{itemize} + +En estas aplicaciones, la preservación de momentos juega un papel +fundamental para mantener estabilidad física del sistema. + +\subsection{Conclusión} + +Los Z-splines constituyen una herramienta moderna de +interpolación de gran interés tanto teórico como práctico, especialmente +en contextos de simulación numérica donde la preservación de invariantes +es esencial. + + +\section{El Z-spline cúbico} + +La función base Z-spline cúbica para datos a intervalos arbitrarios $a_{1}, a_{2}, a_{3}$ y $a_{4}$ +está dada por: + +\begin{equation} \label{eq:aiz1} +\widetilde{Z}_{1} (x) = \left\{ +\begin{array}{lcl} +0 & & x < -a_{1}-a_{2} \\ [0.3cm] + +\left( \frac{a_{2}+a_{1}}{a_{1}} \right) + \left( +\frac{3a_{2}+a_{1}}{a_{2}a_{1}} \right) x & & \\ [0.1cm] +\quad + \frac{3a_{2}+2a_{1}}{a_{2}a_{1}(a_{2}+a_{1})} +x^2 + \frac{1}{a_{2}a_{1}(a_{2}+a_{1})}x^{3} & & -a_{1}-a_{2} \leq x \leq -a_{2}, \\ [0.3cm] + +1- \left( \frac{1}{a_{3}}-\frac{1}{a_{2}} \right) x & & \\ [0.1cm] +\quad - \frac{a_{3}+2(a_{2}+a_{1})}{a_{3}a_{2}(a_{2}+a_{1})} x^2 +- \frac{a_{3}+a_{2}+a_{1}}{a_{3}a_{2}^2(a_{2}+a_{1})} x^3 & & -a_{2} \leq x \leq 0, \\ [0.3cm] + +1+ \left( \frac{1}{a_{2}}-\frac{1}{a_{3}} \right) x & & \\ [0.1cm] +\quad - \frac{a_{2}+2(a_{3}+a_{4})}{a_{2}a_{3}(a_{3}+a_{4})} x^2 ++ \frac{a_{2}+a_{3}+a_{4}}{a_{2}a_{3}^2(a_{3}+a_{4})} x^3 & & 0 \leq x \leq a_{3}, \\ [0.3cm] + +\left( \frac{a_{3}+a_{4}}{a_{4}} \right) - \left( +\frac{3a_{3}+a_{4}}{a_{3}a_{4}} \right) x & & \\ [0.1cm] +\quad + \frac{3a_{3}+2a_{4}}{a_{3}a_{4}(a_{3}+a_{4})} x^2 +- \frac{1}{a_{3}a_{4}(a_{3}+a_{4})}x^{3} & & a_{3} \leq x \leq a_{3}+a_{4}, \\ [0.3cm] + +0 & & x > a_{3}+a_{4}. +\end{array} +\right. +\end{equation} + + +\subsection{Z-spline cúbico en los extremos de un intervalo} + +\section{Fórmula de integración numérica} + +\section{Fórmula de derivación numérica} +\section{Convergencia} +\subsection{Integración} +\subsection{Derivación} +\subsection{Interpolación equidistante en 2D} +\subsection{Interpolación en 1D con puntos arbitrarios} + +\end{document} \ No newline at end of file