diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux index cd33fb04..f14144b0 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux @@ -21,14 +21,21 @@ \@writefile{lot}{\addvspace {10\p@ }} \@writefile{toc}{\contentsline {section}{\numberline {1.1}Problem}{2}{section.1.1}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {1.2}Theoretical introduction}{3}{section.1.2}\protected@file@percent } -\@writefile{toc}{\contentsline {chapter}{\numberline {2}Determine trajectory of the motion}{4}{chapter.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}Linear Least Squares Problem}{3}{subsection.1.2.1}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.2}Gram matrix and QR decomposition}{3}{subsection.1.2.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.3}Gram-Schmidt algorithm}{4}{subsection.1.2.3}\protected@file@percent } +\@writefile{toc}{\contentsline {chapter}{\numberline {2}Determine trajectory of the motion}{6}{chapter.2}\protected@file@percent } \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} -\@writefile{toc}{\contentsline {section}{\numberline {2.1}a) Runge-Kutta method of $4^{th}$ order and Adams PC}{4}{section.2.1}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.1}Problem}{4}{subsection.2.1.1}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.2}Theoretical Introdution}{5}{subsection.2.1.2}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\numberline {2.2}b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}{5}{section.2.2}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.1}Problem}{5}{subsection.2.2.1}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.2}Theoretical Introdution}{5}{subsection.2.2.2}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {2.1}a) Runge-Kutta method of $4^{th}$ order and Adams PC}{6}{section.2.1}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.1}Problem}{6}{subsection.2.1.1}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.2}Theoretical Introdution}{6}{subsection.2.1.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsubsection}{Runge-Kutta method of order 4}{7}{section*.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.1.3}Adams PC method}{7}{subsection.2.1.3}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {2.2}b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}{8}{section.2.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.1}Problem}{8}{subsection.2.2.1}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.2}Theoretical Introdution}{8}{subsection.2.2.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsubsection}{Error estimation using step doubling-approach}{8}{section*.3}\protected@file@percent } +\@writefile{toc}{\contentsline {subsubsection}{Actual correction of the step size}{10}{section*.4}\protected@file@percent } \bibcite{texbook}{1} -\gdef \@abspage@last{7} +\gdef \@abspage@last{13} diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fdb_latexmk b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fdb_latexmk index d49feb38..ca918b5e 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fdb_latexmk +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fdb_latexmk @@ -1,16 +1,18 @@ # Fdb version 3 -["pdflatex"] 1641089981 "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex" "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf" "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME" 1641089982 +["pdflatex"] 1641343564 "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex" "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf" "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME" 1641343565 "/etc/texmf/web2c/texmf.cnf" 1624878795 475 c0e671620eb5563b2130f56340a5fde8 "" "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/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/ecbx1728.tfm" 1136768653 3584 ad49a18d8515beef6d92d4d3f197d0fd "" "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx2488.tfm" 1136768653 3584 0181dbc4d429c3ba4e30feba37b5df96 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm0600.tfm" 1136768653 3584 291a5713401683441e0a8c8f4417b17b "" "/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/jknappen/ec/ecrm2074.tfm" 1136768653 3584 8e2870ec7aa9776f59654942b0923f51 "" "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm2488.tfm" 1136768653 3584 406ad7b70d9a41f7833f92b6313150c8 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecss0600.tfm" 1136768653 3584 29c1411d292c531bfea559d981c42321 "" "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecti1200.tfm" 1136768653 3072 8b5a64dc91775463bc95e2d818524028 "" "/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 "" @@ -41,10 +43,13 @@ "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmtt12.tfm" 1136768653 772 9a936b7f5e2ff0557fce0f62822f0bbf "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm" 1136768653 768 d7b9a2629a0c353102ad947dc9221d49 "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1248133631 32080 340ef9bf63678554ee606688e7b5339d "" + "/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/cmmi12.pfb" 1248133631 36741 fa121aac0049305630cf160b86157ee4 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb" 1248133631 37166 8ab3487cbe3ab49ebce74c29ea2418db "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1248133631 35469 70d41d2b9ea31d5d813066df7c99281c "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb" 1248133631 32722 d7379af29a190c3f453aba36302ff5a9 "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb" 1248133631 32362 179c33bbf43f19adbb3825bb4e36e57a "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb" 1248133631 32734 69e00a6b65cedb993666e42eedb3d48f "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 "" "/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/cmsy8.pfb" 1248133631 32626 4f5c1b83753b1dd3a97d1b399a005b4b "" @@ -83,11 +88,14 @@ "/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty" 1601675358 4947 8cb7717f0cc771eca0fda15160c7fee9 "" "/usr/share/texlive/texmf-dist/tex/latex/base/report.cls" 1601675358 23204 74c91ecbcc47161218f25d9d0651c0f7 "" "/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo" 1601675358 8450 6fd3588c0e9d06f6f56c6cf4f7246466 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/t1cmss.fd" 1580683321 2622 2e7b4cea91736cafdafba1c0db58bc1c "" "/usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty" 1581112666 2821 2c0928feafd5527387e29a1af774d030 "" "/usr/share/texlive/texmf-dist/tex/latex/bigfoot/bigfoot.sty" 1440967810 43914 273a3a42cdc1e0e69e94929e58fcd49d "" "/usr/share/texlive/texmf-dist/tex/latex/bigfoot/perpage.sty" 1414965027 5707 8a111e2f4c8f511ad622494ac2fe3f6d "" "/usr/share/texlive/texmf-dist/tex/latex/bigfoot/suffix.sty" 1414965027 3090 223874a03a08cecb9fb79bd86e5dcf97 "" + "/usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty" 1399239813 4378 f429f0da968c278653359293040a8f52 "" "/usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1579991033 13886 d1306dcf79a944f6988e688c1785f9ce "" + "/usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty" 1504561519 9004 ad3bdc4b26a2ac7912e82bc12a76a39c "" "/usr/share/texlive/texmf-dist/tex/latex/filecontents/filecontents.sty" 1569102703 3408 71173360dc73c4a3f80bb0bc7b926ba0 "" "/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/color.cfg" 1459978653 1213 620bba36b25224fa9b7e1ccb4ecb76fd "" "/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg" 1465944070 1224 978390e9c2234eab29404bc21b268d1e "" @@ -126,6 +134,7 @@ "/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty" 1580683321 10216 5efd55f2010055e7b7875afd6a75be82 "" "/usr/share/texlive/texmf-dist/tex/latex/translator/translator-basic-dictionary-English.dict" 1596662134 3535 7dc96051305a7e943219126c49c44cd6 "" "/usr/share/texlive/texmf-dist/tex/latex/translator/translator.sty" 1596662134 8765 3368c4a5a4b5466261cafb836195e916 "" + "/usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty" 1253232110 1380 971a51b00a14503ddf754cab24c3f209 "" "/usr/share/texlive/texmf-dist/tex/latex/url/url.sty" 1388531844 12796 8edb7d69a20b857904dd0ea757c14ec9 "" "/usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty" 1463002160 55589 34128738f682d033422ca125f82e5d62 "" "/usr/share/texlive/texmf-dist/web2c/texmf.cnf" 1613593815 38841 799d1dd9682a55ce442e10c99777ecc1 "" @@ -137,17 +146,18 @@ "/usr/share/texmf/fonts/type1/public/cm-super/sfrm1200.pfb" 1624878812 136101 f533469f523533d38317ab5729d00c8a "" "/usr/share/texmf/fonts/type1/public/cm-super/sfrm1440.pfb" 1624878812 131078 d96015a2fa5c350129e933ca070b2484 "" "/usr/share/texmf/fonts/type1/public/cm-super/sfrm2074.pfb" 1624878812 131290 ea265c7de37664eae04a6f91a1f7a51f "" + "/usr/share/texmf/fonts/type1/public/cm-super/sfss0600.pfb" 1624878812 99931 c4bf5cbd82bc527fffa6bd4047ab5d6a "" "/usr/share/texmf/fonts/type1/public/cm-super/sfti1200.pfb" 1624878812 198221 ca5aa71411090ef358a6cc78b7458365 "" "/usr/share/texmf/web2c/texmf.cnf" 1613593815 38841 799d1dd9682a55ce442e10c99777ecc1 "" "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1624878842 5160710 ecf427ae8fa19139d8691f526e47bb9b "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1624878876 2570852 ef2a47e64eb64912b01ccaedc81044b8 "" - "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux" 1641089981 2219 3dff7fc0859351b10fd12fb9bf91ebe8 "pdflatex" - "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out" 1641089981 777 5d77f6c54e5123a329f82b6e7183390f "pdflatex" - "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex" 1641089980 3437 59b84c536ae7b6b60e448a42543a481c "" - "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc" 1641089981 950 ba0618c32d78135c7e142c5be958acfd "pdflatex" + "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux" 1641343565 3137 e1a9f64d6b4f309c2cfc8a653381e1b0 "pdflatex" + "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out" 1641343565 1090 eb8f27c1fc516a185703a5fa58130694 "pdflatex" + "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex" 1641343564 11711 e9f6bfd65c19b6444d6ee3c680488a9e "" + "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc" 1641343565 1580 4ae1a23b3059e50d1811bf0e771d41fe "pdflatex" (generated) - "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.log" - "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out" "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux" "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf" + "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out" "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc" + "ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.log" diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fls b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fls index 8e3db05b..dac3a506 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fls +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fls @@ -22,6 +22,50 @@ INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size12.clo INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size12.clo INPUT /usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr12.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty @@ -209,16 +253,6 @@ INPUT /usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse-generic.tex INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty INPUT /usr/share/texlive/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty @@ -838,9 +872,16 @@ INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1200.tfm OUTPUT ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1728.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1728.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1440.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmbx6.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm0600.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/t1cmss.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/t1cmss.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/t1cmss.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/t1cmss.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecss0600.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr17.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi12.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm @@ -851,17 +892,19 @@ INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmss17.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmtt12.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/msbm10.tfm -INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecbx1440.tfm INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecti1200.tfm INPUT ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux INPUT ./ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out INPUT ./ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out INPUT /usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi12.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.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/cmsy8.pfb @@ -873,4 +916,5 @@ INPUT /usr/share/texmf/fonts/type1/public/cm-super/sfbx2488.pfb INPUT /usr/share/texmf/fonts/type1/public/cm-super/sfrm1200.pfb INPUT /usr/share/texmf/fonts/type1/public/cm-super/sfrm1440.pfb INPUT /usr/share/texmf/fonts/type1/public/cm-super/sfrm2074.pfb +INPUT /usr/share/texmf/fonts/type1/public/cm-super/sfss0600.pfb INPUT /usr/share/texmf/fonts/type1/public/cm-super/sfti1200.pfb diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.log b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.log index 9acbd9f9..c3596405 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.log +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.6.28) 2 JAN 2022 03:19 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.6.28) 5 JAN 2022 01:46 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -23,39 +23,62 @@ File: size12.clo 2020/04/10 v1.4m Standard LaTeX file (size option) \abovecaptionskip=\skip47 \belowcaptionskip=\skip48 \bibindent=\dimen138 -) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty +) (/usr/share/texlive/texmf-dist/tex/latex/eqparbox/eqparbox.sty +Package: eqparbox 2017/09/03 v4.1 Create equal-widthed boxes +\eqp@tempdima=\skip49 +\eqp@tempdimb=\skip50 + (/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +Package: array 2020/10/01 v2.5c Tabular extension package (FMi) +\col@sep=\dimen139 +\ar@mcellbox=\box47 +\extrarowheight=\dimen140 +\NC@list=\toks15 +\extratabsurround=\skip51 +\backup@length=\skip52 +\ar@cellbox=\box48 +) +\eqp@tabular@box=\box49 +\eqp@list@box=\box50 +\eqp@list@indent=\skip53 + (/usr/share/texlive/texmf-dist/tex/latex/environ/environ.sty +Package: environ 2014/05/04 v0.3 A new way to define environments + (/usr/share/texlive/texmf-dist/tex/latex/trimspaces/trimspaces.sty +Package: trimspaces 2009/09/17 v1.1 Trim spaces around a token list +) +\@envbody=\toks16 +)) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mathtools.sty Package: mathtools 2020/03/24 v1.24 mathematical typesetting tools (/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty Package: keyval 2014/10/28 v1.15 key=value parser (DPC) -\KV@toks@=\toks15 +\KV@toks@=\toks17 ) (/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty Package: calc 2017/05/25 v4.3 Infix arithmetic (KKT,FJ) \calc@Acount=\count186 \calc@Bcount=\count187 -\calc@Adimen=\dimen139 -\calc@Bdimen=\dimen140 -\calc@Askip=\skip49 -\calc@Bskip=\skip50 +\calc@Adimen=\dimen141 +\calc@Bdimen=\dimen142 +\calc@Askip=\skip54 +\calc@Bskip=\skip55 LaTeX Info: Redefining \setlength on input line 80. LaTeX Info: Redefining \addtolength on input line 81. \calc@Ccount=\count188 -\calc@Cskip=\skip51 +\calc@Cskip=\skip56 ) (/usr/share/texlive/texmf-dist/tex/latex/mathtools/mhsetup.sty Package: mhsetup 2017/03/31 v1.3 programming setup (MH) ) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty Package: amsmath 2020/09/23 v2.17i AMS math features -\@mathmargin=\skip52 +\@mathmargin=\skip57 For additional information on amsmath, use the `?' option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty Package: amstext 2000/06/29 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=\toks16 -\ex@=\dimen141 +\@emptytoks=\toks18 +\ex@=\dimen143 )) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty Package: amsbsy 1999/11/29 v1.2d Bold Symbols -\pmbraise@=\dimen142 +\pmbraise@=\dimen144 ) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty Package: amsopn 2016/03/08 v2.02 operator names ) @@ -69,9 +92,9 @@ LaTeX Info: Redefining \overline on input line 399. 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@=\box47 -\strutbox@=\box48 -\big@size=\dimen143 +\Mathstrutbox@=\box51 +\strutbox@=\box52 +\big@size=\dimen145 LaTeX Font Info: Redeclaring font encoding OML on input line 743. LaTeX Font Info: Redeclaring font encoding OMS on input line 744. \macc@depth=\count194 @@ -79,21 +102,21 @@ LaTeX Font Info: Redeclaring font encoding OMS on input line 744. \dotsspace@=\muskip16 \c@parentequation=\count196 \dspbrk@lvl=\count197 -\tag@help=\toks17 +\tag@help=\toks19 \row@=\count198 \column@=\count199 \maxfields@=\count266 -\andhelp@=\toks18 -\eqnshift@=\dimen144 -\alignsep@=\dimen145 -\tagshift@=\dimen146 -\tagwidth@=\dimen147 -\totwidth@=\dimen148 -\lineht@=\dimen149 -\@envbody=\toks19 -\multlinegap=\skip53 -\multlinetaggap=\skip54 -\mathdisplay@stack=\toks20 +\andhelp@=\toks20 +\eqnshift@=\dimen146 +\alignsep@=\dimen147 +\tagshift@=\dimen148 +\tagwidth@=\dimen149 +\totwidth@=\dimen150 +\lineht@=\dimen151 +\@envbody=\toks21 +\multlinegap=\skip58 +\multlinetaggap=\skip59 +\mathdisplay@stack=\toks22 LaTeX Info: Redefining \[ on input line 2923. LaTeX Info: Redefining \] on input line 2924. ) @@ -102,16 +125,16 @@ LaTeX Info: Thecontrolsequence`\)'isalreadyrobust on input line 130. LaTeX Info: Thecontrolsequence`\['isalreadyrobust on input line 130. LaTeX Info: Thecontrolsequence`\]'isalreadyrobust on input line 130. \g_MT_multlinerow_int=\count267 -\l_MT_multwidth_dim=\dimen150 -\origjot=\skip55 -\l_MT_shortvdotswithinadjustabove_dim=\dimen151 -\l_MT_shortvdotswithinadjustbelow_dim=\dimen152 -\l_MT_above_intertext_sep=\dimen153 -\l_MT_below_intertext_sep=\dimen154 -\l_MT_above_shortintertext_sep=\dimen155 -\l_MT_below_shortintertext_sep=\dimen156 -\xmathstrut@box=\box49 -\xmathstrut@dim=\dimen157 +\l_MT_multwidth_dim=\dimen152 +\origjot=\skip60 +\l_MT_shortvdotswithinadjustabove_dim=\dimen153 +\l_MT_shortvdotswithinadjustbelow_dim=\dimen154 +\l_MT_above_intertext_sep=\dimen155 +\l_MT_below_intertext_sep=\dimen156 +\l_MT_above_shortintertext_sep=\dimen157 +\l_MT_below_shortintertext_sep=\dimen158 +\xmathstrut@box=\box53 +\xmathstrut@dim=\dimen159 ) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support \symAMSa=\mathgroup4 @@ -125,8 +148,8 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold' ) Package: xstring 2019/02/06 v1.83 String manipulations (CT) ) (/usr/share/texlive/texmf-dist/tex/generic/systeme/systeme.tex -\SYS_systemecode=\toks21 -\SYS_systempreamble=\toks22 +\SYS_systemecode=\toks23 +\SYS_systempreamble=\toks24 \SYSeqnum=\count270 ) Package: systeme 2020/05/03 v0.34 Mise en forme de systemes d'equations (CT) @@ -135,25 +158,15 @@ Package: expl3 2021-01-09 L3 programming layer (loader) (/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def File: l3backend-pdftex.def 2020-01-29 L3 backend support: PDF output (pdfTeX) \l__color_backend_stack_int=\count271 -\l__pdf_internal_box=\box50 +\l__pdf_internal_box=\box54 )) (/usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty Package: xparse 2020-10-27 L3 Experimental document command parser (/usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse-generic.tex)) Package: siunitx 2020/02/25 v2.8b A comprehensive (SI) units package - (/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty -Package: array 2020/10/01 v2.5c Tabular extension package (FMi) -\col@sep=\dimen158 -\ar@mcellbox=\box51 -\extrarowheight=\dimen159 -\NC@list=\toks23 -\extratabsurround=\skip56 -\backup@length=\skip57 -\ar@cellbox=\box52 -) -(/usr/share/texlive/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty + (/usr/share/texlive/texmf-dist/tex/latex/l3packages/l3keys2e/l3keys2e.sty Package: l3keys2e 2020-10-27 LaTeX2e option processing using LaTeX3 keys ) -\l__siunitx_tmp_box=\box53 +\l__siunitx_tmp_box=\box55 \l__siunitx_tmp_dim=\dimen160 \l__siunitx_tmp_int=\count272 \l__siunitx_number_mantissa_length_int=\count273 @@ -165,8 +178,8 @@ Package: l3keys2e 2020-10-27 LaTeX2e option processing using LaTeX3 keys \l__siunitx_process_integer_min_int=\count279 \l__siunitx_process_precision_int=\count280 \l__siunitx_group_min_int=\count281 -\l__siunitx_angle_marker_box=\box54 -\l__siunitx_angle_unit_box=\box55 +\l__siunitx_angle_marker_box=\box56 +\l__siunitx_angle_unit_box=\box57 \l__siunitx_angle_marker_dim=\dimen161 \l__siunitx_angle_unit_dim=\dimen162 \l__siunitx_unit_int=\count282 @@ -176,8 +189,8 @@ Package: l3keys2e 2020-10-27 LaTeX2e option processing using LaTeX3 keys \l__siunitx_unit_prefix_base_int=\count286 \l__siunitx_unit_prefix_gram_int=\count287 \l__siunitx_number_product_int=\count288 -\c__siunitx_one_fill_skip=\skip58 -\l__siunitx_table_unit_align_skip=\skip59 +\c__siunitx_one_fill_skip=\skip61 +\l__siunitx_table_unit_align_skip=\skip62 \l__siunitx_table_exponent_dim=\dimen163 \l__siunitx_table_integer_dim=\dimen164 \l__siunitx_table_mantissa_dim=\dimen165 @@ -187,13 +200,14 @@ Package: l3keys2e 2020-10-27 LaTeX2e option processing using LaTeX3 keys \l__siunitx_table_fill_pre_dim=\dimen169 \l__siunitx_table_fill_post_dim=\dimen170 \l__siunitx_table_fill_mid_dim=\dimen171 -\l__siunitx_table_pre_box=\box56 -\l__siunitx_table_post_box=\box57 -\l__siunitx_table_mantissa_box=\box58 -\l__siunitx_table_result_box=\box59 -\l__siunitx_table_number_align_skip=\skip60 -\l__siunitx_table_text_align_skip=\skip61 - (/usr/share/texlive/texmf-dist/tex/latex/translator/translator.sty +\l__siunitx_table_pre_box=\box58 +\l__siunitx_table_post_box=\box59 +\l__siunitx_table_mantissa_box=\box60 +\l__siunitx_table_result_box=\box61 +\l__siunitx_table_number_align_skip=\skip63 +\l__siunitx_table_text_align_skip=\skip64 + +(/usr/share/texlive/texmf-dist/tex/latex/translator/translator.sty Package: translator 2020-08-03 v1.12c Easy translation of strings in LaTeX )) (/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty Package: fontenc 2020/08/10 v2.0s Standard LaTeX package @@ -229,12 +243,12 @@ Package: kvoptions 2020-10-07 v3.14 Key value format for package options (HO) \@linkdim=\dimen172 \Hy@linkcounter=\count289 \Hy@pagecounter=\count290 - -(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def + (/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def File: pd1enc.def 2020-05-15 v7.00e Hyperref: PDFDocEncoding definition (HO) Now handling font encoding PD1 ... ... no UTF-8 mapping file for font encoding PD1 -) (/usr/share/texlive/texmf-dist/tex/generic/intcalc/intcalc.sty +) +(/usr/share/texlive/texmf-dist/tex/generic/intcalc/intcalc.sty Package: intcalc 2019/12/15 v1.3 Expandable calculations with integers (HO) ) (/usr/share/texlive/texmf-dist/tex/generic/etexcmds/etexcmds.sty Package: etexcmds 2019/12/15 v1.7 Avoid name clashes with e-TeX commands (HO) @@ -295,7 +309,7 @@ Package: uniquecounter 2019/12/15 v1.4 Provide unlimited unique counter (HO) ) Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 286. ) -\Hy@SectionHShift=\skip62 +\Hy@SectionHShift=\skip65 ) (/usr/share/texlive/texmf-dist/tex/latex/bigfoot/bigfoot.sty Package: bigfoot 2015/08/30 2.1 makes footnotes work @@ -325,12 +339,12 @@ Package: perpage 2014/10/25 2.0 Reset/sort counters per page \FN@slave=\marks2 \FN@color=\marks3 \FN@outervsize=\dimen177 -\FN@vsize=\skip63 -\FN@insertions=\box60 -\FN@output=\toks24 -\FN@tempbox=\box61 +\FN@vsize=\skip66 +\FN@insertions=\box62 +\FN@output=\toks25 +\FN@tempbox=\box63 \FN@savebox=\insert252 -\FN@topmarkbox=\box62 +\FN@topmarkbox=\box64 \FN@outputflag=\count306 \FN@myvsize=\dimen178 \bigfoottolerance=\count307 @@ -358,8 +372,8 @@ Package xcolor Info: Model `Gray' substituted by `gray' on input line 1370. Package xcolor Info: Model `wave' substituted by `hsb' on input line 1371. ) (/usr/share/texlive/texmf-dist/tex/latex/listings/listings.sty \lst@mode=\count308 -\lst@gtempboxa=\box63 -\lst@token=\toks25 +\lst@gtempboxa=\box65 +\lst@token=\toks26 \lst@length=\count309 \lst@currlwidth=\dimen179 \lst@column=\count310 @@ -373,7 +387,7 @@ Package xcolor Info: Model `wave' substituted by `hsb' on input line 1371. File: lstmisc.sty 2020/03/24 1.8d (Carsten Heinz) \c@lstnumber=\count314 \lst@skipnumbers=\count315 -\lst@framebox=\box64 +\lst@framebox=\box66 ) (/usr/share/texlive/texmf-dist/tex/latex/listings/listings.cfg File: listings.cfg 2020/03/24 1.8d listings configuration )) @@ -381,10 +395,10 @@ Package: listings 2020/03/24 1.8d (Carsten Heinz) \netBracketCount@mlpr=\count316 \blkLvl@mlpr=\count317 \blkLvlAtClassdef@mlpr=\count318 -\emHeight@mlpr=\skip64 -\jayDepth@mlpr=\skip65 -\sectionRuleOffset@mlpr=\skip66 -\toks@mlpr=\toks26 +\emHeight@mlpr=\skip67 +\jayDepth@mlpr=\skip68 +\sectionRuleOffset@mlpr=\skip69 +\toks@mlpr=\toks27 ) (/usr/share/texlive/texmf-dist/tex/latex/filecontents/filecontents.sty Package: filecontents 2019/09/20 v1.5 Create an external file from within a LaTeX document @@ -407,33 +421,33 @@ Package graphics Info: Driver file: pdftex.def on input line 105. \Gin@req@height=\dimen183 \Gin@req@width=\dimen184 ) -Package hyperref Info: Option `colorlinks' set `true' on input line 21. -Package Listings Info: Made " a short reference for \lstinline on input line 32. +Package hyperref Info: Option `colorlinks' set `true' on input line 23. +Package Listings Info: Made " a short reference for \lstinline on input line 34. (./ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux) \openout1 = `ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux'. -LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 41. -LaTeX Font Info: ... okay on input line 41. -LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 41. -LaTeX Font Info: ... okay on input line 41. -LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 41. -LaTeX Font Info: ... okay on input line 41. -LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 41. -LaTeX Font Info: ... okay on input line 41. -LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 41. -LaTeX Font Info: ... okay on input line 41. -LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 41. -LaTeX Font Info: ... okay on input line 41. -LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 41. -LaTeX Font Info: ... okay on input line 41. -LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 41. -LaTeX Font Info: ... okay on input line 41. +LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 46. +LaTeX Font Info: ... okay on input line 46. +LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 46. +LaTeX Font Info: ... okay on input line 46. +LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 46. +LaTeX Font Info: ... okay on input line 46. +LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 46. +LaTeX Font Info: ... okay on input line 46. +LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 46. +LaTeX Font Info: ... okay on input line 46. +LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 46. +LaTeX Font Info: ... okay on input line 46. +LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 46. +LaTeX Font Info: ... okay on input line 46. +LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 46. +LaTeX Font Info: ... okay on input line 46. (/usr/share/texlive/texmf-dist/tex/latex/siunitx/siunitx-abbreviations.cfg File: siunitx-abbreviations.cfg 2017/11/26 v2.7k siunitx: Abbreviated units ) (/usr/share/texlive/texmf-dist/tex/latex/translator/translator-basic-dictionary-English.dict Dictionary: translator-basic-dictionary, Language: English ) -Package hyperref Info: Link coloring ON on input line 41. +Package hyperref Info: Link coloring ON on input line 46. (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty Package: nameref 2019/09/16 v2.46 Cross-referencing by name of section (/usr/share/texlive/texmf-dist/tex/latex/refcount/refcount.sty @@ -443,28 +457,28 @@ Package: gettitlestring 2019/12/15 v1.6 Cleanup title references (HO) ) \c@section@level=\count319 ) -LaTeX Info: Redefining \ref on input line 41. -LaTeX Info: Redefining \pageref on input line 41. -LaTeX Info: Redefining \nameref on input line 41. +LaTeX Info: Redefining \ref on input line 46. +LaTeX Info: Redefining \pageref on input line 46. +LaTeX Info: Redefining \nameref on input line 46. (./ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out) (./ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out) \@outlinefile=\write3 \openout3 = `ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out'. \footinsdefault=\insert251 -\FN@cache251=\box65 +\FN@cache251=\box67 (/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] \scratchcounter=\count320 \scratchdimen=\dimen185 -\scratchbox=\box66 +\scratchbox=\box68 \nofMPsegments=\count321 \nofMParguments=\count322 -\everyMPshowfont=\toks27 +\everyMPshowfont=\toks28 \MPscratchCnt=\count323 \MPscratchDim=\dimen186 \MPnumerator=\count324 \makeMPintoPDFobject=\count325 -\everyMPtoPDFconversion=\toks28 +\everyMPtoPDFconversion=\toks29 ) (/usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 485. @@ -472,11 +486,11 @@ Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 48 File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live )) \c@lstlisting=\count326 -LaTeX Font Info: Trying to load font information for U+msa on input line 41. +LaTeX Font Info: Trying to load font information for U+msa on input line 46. (/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 41. +LaTeX Font Info: Trying to load font information for U+msb on input line 46. (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd File: umsb.fd 2013/01/14 v3.01 AMS symbols B ) [1 @@ -488,61 +502,71 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B pdfTeX warning (ext4): destination with the same identifier (name{page.1}) has been already used, duplicate ignored \relax -l.45 \chapter +l.50 \chapter {Determine polynomial function fitting experimental data} [1 ] Chapter 1. [2 -] [3] +] +LaTeX Font Info: Trying to load font information for T1+cmss on input line 102. + (/usr/share/texlive/texmf-dist/tex/latex/base/t1cmss.fd +File: t1cmss.fd 2019/12/16 v2.5j Standard LaTeX font definitions +) [3] [4] [5] Chapter 2. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 92. +(hyperref) removing `math shift' on input line 160. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `superscript' on input line 92. +(hyperref) removing `superscript' on input line 160. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 92. +(hyperref) removing `math shift' on input line 160. + +[6 + +] + +Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): +(hyperref) removing `math shift' on input line 198. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 104. +(hyperref) removing `superscript' on input line 198. Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `superscript' on input line 104. +(hyperref) removing `math shift' on input line 198. +[7] +Underfull \hbox (badness 10000) in paragraph at lines 211--217 -Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): -(hyperref) removing `math shift' on input line 104. + [] -[4 - -] [5] [6 +[8] [9] [10] [11] [12 ] (./ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux) Package rerunfilecheck Info: File `ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out' has not changed. -(rerunfilecheck) Checksum: 5D77F6C54E5123A329F82B6E7183390F;777. +(rerunfilecheck) Checksum: EB8F27C1FC516A185703A5FA58130694;1090. ) Here is how much of TeX's memory you used: - 11994 strings out of 479304 - 212612 string characters out of 5869779 - 522720 words of memory out of 5000000 - 28948 multiletter control sequences out of 15000+600000 - 424697 words of font info for 91 fonts, out of 8000000 for 9000 + 12143 strings out of 479304 + 215350 string characters out of 5869779 + 533293 words of memory out of 5000000 + 29066 multiletter control sequences out of 15000+600000 + 426235 words of font info for 93 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 - 81i,7n,88p,298b,753s stack positions out of 5000i,500n,10000p,200000b,80000s -{/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc} -Output written on ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf (7 pages, 206214 bytes). + 81i,8n,88p,327b,753s stack positions out of 5000i,500n,10000p,200000b,80000s +{/usr/share/texmf/fonts/enc/dvips/cm-super/cm-super-t1.enc} +Output written on ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf (13 pages, 261703 bytes). PDF statistics: - 167 PDF objects out of 1000 (max. 8388607) - 139 compressed objects within 2 object streams - 20 named destinations out of 1000 (max. 500000) - 81 words of extra memory for PDF output out of 10000 (max. 10000000) + 234 PDF objects out of 1000 (max. 8388607) + 196 compressed objects within 2 object streams + 34 named destinations out of 1000 (max. 500000) + 113 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out index e569c60a..ad7d29e9 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out @@ -1,10 +1,14 @@ \BOOKMARK [0][-]{chapter.1}{Determine polynomial function fitting experimental data}{}% 1 \BOOKMARK [1][-]{section.1.1}{Problem}{chapter.1}% 2 \BOOKMARK [1][-]{section.1.2}{Theoretical introduction}{chapter.1}% 3 -\BOOKMARK [0][-]{chapter.2}{Determine trajectory of the motion}{}% 4 -\BOOKMARK [1][-]{section.2.1}{a\) Runge-Kutta method of 4th order and Adams PC}{chapter.2}% 5 -\BOOKMARK [2][-]{subsection.2.1.1}{Problem}{section.2.1}% 6 -\BOOKMARK [2][-]{subsection.2.1.2}{Theoretical Introdution}{section.2.1}% 7 -\BOOKMARK [1][-]{section.2.2}{b\) Runge-Kutta method of 4th order with variable step size automatically adjusted}{chapter.2}% 8 -\BOOKMARK [2][-]{subsection.2.2.1}{Problem}{section.2.2}% 9 -\BOOKMARK [2][-]{subsection.2.2.2}{Theoretical Introdution}{section.2.2}% 10 +\BOOKMARK [2][-]{subsection.1.2.1}{Linear Least Squares Problem}{section.1.2}% 4 +\BOOKMARK [2][-]{subsection.1.2.2}{Gram matrix and QR decomposition}{section.1.2}% 5 +\BOOKMARK [2][-]{subsection.1.2.3}{Gram-Schmidt algorithm}{section.1.2}% 6 +\BOOKMARK [0][-]{chapter.2}{Determine trajectory of the motion}{}% 7 +\BOOKMARK [1][-]{section.2.1}{a\) Runge-Kutta method of 4th order and Adams PC}{chapter.2}% 8 +\BOOKMARK [2][-]{subsection.2.1.1}{Problem}{section.2.1}% 9 +\BOOKMARK [2][-]{subsection.2.1.2}{Theoretical Introdution}{section.2.1}% 10 +\BOOKMARK [2][-]{subsection.2.1.3}{Adams PC method}{section.2.1}% 11 +\BOOKMARK [1][-]{section.2.2}{b\) Runge-Kutta method of 4th order with variable step size automatically adjusted}{chapter.2}% 12 +\BOOKMARK [2][-]{subsection.2.2.1}{Problem}{section.2.2}% 13 +\BOOKMARK [2][-]{subsection.2.2.2}{Theoretical Introdution}{section.2.2}% 14 diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf index f555f772..c735aa63 100644 Binary files a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf and b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf differ diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.synctex.gz b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.synctex.gz index 187bffda..afb7331d 100644 Binary files a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.synctex.gz and b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.synctex.gz differ diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex index 9fd61852..5693ea98 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex @@ -1,5 +1,7 @@ \documentclass[12pt]{report} + +\usepackage{eqparbox} \usepackage{mathtools} \usepackage{amsmath} \usepackage{amsfonts} @@ -38,6 +40,9 @@ linkto=all, mlshowsectionrules = true, } +\newcommand\myeq{\mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily def}}}{=}}} + + \begin{document} \maketitle \tableofcontents @@ -80,15 +85,78 @@ We have to determine polynomial function $ y = f(x) $ that best fits this data. We will use least-square approximation using system of normal equation with QR factorization. \section{Theoretical introduction} +\subsection{Linear Least Squares Problem} We have polynomial function: \[ y(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x^1 + a_0 \] where n - degree of the polynomial we use to approximate function \\ Given matrix $\mathbf{A} \in \mathbb{R}^{m \times n} \quad (m > n)$ and vector $\mathbf{y} \in \mathbb{R}^m$ \\ We need to find vector \^{x} such that: \[ \forall x \in \mathbb{R}^n \quad|| \mathbf{y} - \mathbf{A} \hat{x} ||_2 \leq || \mathbf{y} - \mathbf{A}x ||_2 \] +$\mathbf{A}\hat{x}$ is the vector of values of the approximating function at each data point and $y$ is the vector of values of data points. \\ +We have to minimize the Euclidean norm of difference between approximation and data points. \\ +To obtain $\hat{x}$ we will calculate the derrivative of error function with respect to $b_n$ and equate it to 0. This way we will get system of equations caled \textbf{normal equations} which have unique solution in form of vector $\hat{x}$ + +\subsection{Gram matrix and QR decomposition} +We will use Gram's matrix since it's condition number tends to be pretty high which improves the accuracy of our solution. \\ +Gram's matrix is a matrix of set of normal equations. The simplest way to present normal equations is by using formula: +\[ \langle \phi_i, \phi_k \rangle \myeq \sum_{j=0}^N \phi_i(x_j)fi_k(x_j) \] +Which then leads to following matrix \textbf{A}: + +\[ +A = \begin{bmatrix} +\phi_0(x_0) & \phi_1(x_0) & \dots & \phi_n(x_0) \\ +\phi_0(x_1) & \phi_1(x_1) & \dots & \phi_n(x_1) \\ +\phi_0(x_2) & \phi_1(x_2) & \dots & \phi_n(x_2) \\ +\vdots & \vdots & \vdots & \vdots \\ +\phi_0(x_N) & \phi_1(x_N) & \dots & \phi_n(x_N) \\ +\end{bmatrix} +\] + +In our case we define matrix \textbf{A} as: +\[ +A = \begin{bmatrix} +1 & x_0 & (x_0)^2 & \dots & (x_0)^n \\ +1 & x_1 & (x_1)^2 & \dots & (x_1)^n\\ +1 & x_2 & (x_2)^2 & \dots & (x_2)^n\\ +\vdots & \vdots & \vdots & \vdots \\ +1 & x_N & (x_N)^2 & \dots & (x_N)^n\\ +\end{bmatrix} +\] + +Where $x_N$ is the x-axis position for the N-th data point. Then we can express system of natural equations using A in this way: +\[ A^Ty = (A^TA)\hat{x} \] + +With condition number of A equal to square root of condition number of Gram's matrix which gives us small errors. It is also usefull for QR factoization since we can rewrite it as follows: +\[ (R^TQ^T)(QR)\hat{x} = (R^TQ^T)y \] +Since determinant of $R \neq 0$ and $Q^TQ = I$ we can simplify aforementioned equation to: +\[ R\hat{x} = Q^Ty \] +And to optimize even further ahead insetead of using QR decomposition we can use faster $\tilde{Q}\tilde{R}$ decomposition: +\[ \tilde{R}\hat{x} = (\tilde{Q}^T\tilde{Q})^{-1}(\tilde{Q}^T) \] +This is faster since $\tilde{R}$ is an upper triangular matrix, easily solved using back-substitution. + +\subsection{Gram-Schmidt algorithm} +We will use Gram-Schmidt algorithm for $\tilde{Q}\tilde{R}$ decomposition. +Gram-Schmidt algorithm takes non-orthogonal function basis denoted as: +\[ \phi_0, \dots, \phi_n \] +and orthogonalizes it into orthogonal function basis denoted as: +\[ \psi_0, \dots, \psi_n \] +Standard Gram-Schmid algorithm looks like this: +\[ \psi_0 = \phi_0 \] +\[ \psi_1 = \phi_1 - \frac{\langle \psi_0, \phi_1 \rangle}{\langle \psi_0, \psi_0 \rangle} \psi_0 \] +\[ \psi_n = \phi_n - \sum_{j = 0}^{i-1} \frac{\langle \psi_j, \phi_i \rangle}{\langle \psi_j, \psi_j \rangle} \psi_j \] +Where $ i = 2, \dots, n $ +Then we get a very well conditioned set of normal equations: +\[ a_0\langle \psi_0, \psi_0 \rangle = \langle f, \psi_0 \rangle \] +\[ a_1\langle \psi_1, \psi_1 \rangle = \langle f, \psi_1 \rangle \] +\[ \vdots \] +\[ a_n\langle \psi_n, \psi_n \rangle = \langle f, \psi_n \rangle \] + + \chapter{Determine trajectory of the motion} + + \section{a) Runge-Kutta method of $4^{th}$ order and Adams PC} \subsection{Problem} We are given following equations: @@ -100,6 +168,32 @@ $ x_1(0) = 8; x_2(0) = 9 $ In this section we will use Runge-Kutta method of $4^{th}$ order and Adams PC with different step-sizes until we find an optimal constant step size - when the decrease of the step size does not influence the solution significantly. \subsection{Theoretical Introdution} +Determining the trajectory of the motion in this example can be mathematically described as solving a system of first order ordinary differential equations with given initical conditions. That is why we use Runge-Kutta method of order 4 which deals with exactly this kind of problem. +\subsubsection{Runge-Kutta method of order 4} +Any single step method are defined by the following formula, where h is a fixed step-size: +\[ y_{n+1} = y_n + \Phi_f(x_n, y_n, h) \] +where: +\[ x_n = x_0 + nh \] +\[ y(x_0) = y_0 = y_a \] +where $y_a$ is given. +Runge-Kutta method of order 4 - RK4 method looks like this: +\[ y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2+2k_3+k_4) \] +\[ k_1 = f(x_n, y_n) \] +\[ k_3 = f(x_n + \frac{1}{2}h, y_n + \frac{1}{2}hk_1) \] +\[ k_2 = f(x_n + \frac{1}{2}h, y_n + \frac{1}{2}hk_2) \] +\[ k_4 = f(x_n + h, y_n + hk_3) \] + + +\subsection{Adams PC method} +Adams method with $P_kEC_kE$ algorithm has the following form: \\ +P: \[ y_n^{[0]} = y_{n-1} + h\sum_{j=1}^k \beta_jf_{n-j} \] +E: \[ f_n^{[0]} = f(x_n, y_n^{[0]}) \] +C: \[ y_n = y_{n-1} + h \sum_{j=1}^k \beta_j^* f_{n-j} + h \beta_0^* f_n^{[0]}\] +E: \[ f_n = f(x_n, y_n) \] + +When compared to explicit and implicit adams method PC metho has smaller absolute stability intervals than implicit method, while having significant advantage in calculations cost. + + \section{b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted} \subsection{Problem} @@ -110,6 +204,69 @@ And we have to determine the trajectory of the motion on interval $[0, 15]$ with $ x_1(0) = 8; x_2(0) = 9 $ In this section we will use Runge-Kutta method of $4^{th}$ order with step size automatically adjusted by the algorithm, with error estimation made according to the step-doubling rule. \subsection{Theoretical Introdution} +Since Runge-Kutta method was explained in previous task I will focus on step-doubling rule. +Everytime we increase step size we receive less accurate results at the benefit of less calculations cost, this means that we need to have reasonable approach to setting step size to maximise accuracy and minimize calculations cost. We will use step-doubling rule in this example. + +\subsubsection{Error estimation using step doubling-approach} +For every step of size $h$ we perform two additional steps of size $\frac{h}{2}$. We denote: \\ +$y_n^{1}$ as a new point obtained using the original step-size $h$ \\ +$y_n^{2}$ as a new point obtained using steps of the size $\frac{h}{2}$ \\ +$r^{(1)}$ - approximation error affter the single step $h$ \\ +$r^{(2)}$ - summed approximation errors after the two smaller steps of length $\frac{h}{2}$ each. \\ +We have: \\ +\newpage +After a single step: +\[ y(x_n + h) = y_n^{(1)} + \frac{r_n^{p+1}(0)}{(p+1)!}h^{p+1} + O(h^{p+2}) \] +After a double step: +\[ y(x_n+h) \simeq y_n^{(2)} + 2\frac{r_n^{p+1}(0)}{(p+1)!}(\frac{h}{2})^{p+1} + O(h^{p+2}) \] + +We evaluate unknown coefficient from the first equation $ \frac{r_n^{p+1}(0)}{(p+1)!} $ and insert it into second one: +\[ y(x_n+h) = y_n^{(2)} + \frac{h^{p+1}}{2^p}\frac{(x_n+h) - y_n^{(1)}}{h^{p+1} + O(h^{p+2})} \] +We continue further and receive: +\[ y(x_n + h)(1-\frac{1}{2^p}) = y_n^{(2)}(1-\frac{1}{2^p}) + \frac{y_n^{2}}{2^p} - \frac{y_n^{(1)}}{2^p} + O(h^{p+2}) \] +We multiply by $\frac{2^p}{2^p - 1}$ and get: +\begin{equation} +y(x_n+h) = y_n^{(2)} + \frac{ y_n^{(2)} - y_n^{(1)} }{ 2^p - 1 } + O(h^{p+2}) +\end{equation} +We also obtain for first equation in a similar way: + \[ y(x_n+h) = y_n^{(1)} + 2^p\frac{y_n^{(2)} - y_n^{(1)}}{2^p - 1} + O(h^{p+2}) \] +Assuming we take the main part of the error we get error estimate: +\[ \delta_n(h) = \frac{2^p}{2^p-1}(y_n^{(2)} - y_n^{(1)}) \] +But from the equation (2.1) we can treat expression: +\[ \delta_n(2 \times \frac{h}{2}) = \frac{ y_n^{(2)} - y_n^{(1)}}{2^p - 1} \] +as en estimate of the error of two consecutive steps of size $\frac{h}{2}$ +\subsubsection{Actual correction of the step size} +In general formula for main part of the approximation error where: +\\ $h$ - length of a step +\[ \gamma = \frac{r_n^{(p+1)}(0)}{(p+1)!} \] +Looks like this: +\[ \delta_n(h) = \gamma h^{p+1} \] +If we change step-size to $\alpha h$ then we get: +\[ \delta_n(\alpha h) = \gamma (\alpha h)^{p+1} \] +Then: +\[ \delta_n(\alpha h) = \alpha^{p+1} \delta_n (h) \] +We assume tolerance $\epsilon$: +\[ |\delta_n(\alpha h)| = \epsilon \] +We get: +\[ \alpha^{p+1}|\delta_n(h)| = \epsilon \] +Coefficient $\alpha$ for the step-size correction can be evaluated in such a way: +\[ \alpha = (\frac{\epsilon}{|\delta_n (h)|})^{\frac{1}{p+1}} \] + +This method is also correct for RK4 method when $\delta_n(h) = \delta_n(2 \times \frac{h}{2}) $ +We should also take into considertaion lack of accuracy we do so by using safety factor $s$: +\[ h_{n+1} = s \alpha h_n \] +where $ s < 1 $ +For RK4 we use $ s \approx 0.9 $ \\ +We define accuracy parameters in such a way: +\[ \epsilon = |y_n| \epsilon_r + \epsilon_a \] +where $\epsilon_r$ is relative tolerance and $\epsilon_a$ is absolute tolerance. \\ +In case of set of more than on differential equations we must use worst case approach so the equations that need smallest step-size define step-size for all other equations. +\newpage +For RK methods we have following parameters: +\[ \delta_n(h)_i = \frac{(y_i)_n^{(2)} - (y_i)_n^{(1)}}{2^p - 1} \] +\[ \epsilon_i = |(y_i)_n^{(2)}| \epsilon_r + \epsilon_a \] +\[ \alpha = \min_{1 \leq i \leq m} (\frac{\epsilon_i}{|\delta_n(h)_i}|)^{\frac{1}{p+1}} \] + \begin{thebibliography}{9} \bibitem{texbook} diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc index 227ca940..57edbde9 100644 --- a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc @@ -1,10 +1,17 @@ \contentsline {chapter}{\numberline {1}Determine polynomial function fitting experimental data}{2}{chapter.1}% \contentsline {section}{\numberline {1.1}Problem}{2}{section.1.1}% \contentsline {section}{\numberline {1.2}Theoretical introduction}{3}{section.1.2}% -\contentsline {chapter}{\numberline {2}Determine trajectory of the motion}{4}{chapter.2}% -\contentsline {section}{\numberline {2.1}a) Runge-Kutta method of $4^{th}$ order and Adams PC}{4}{section.2.1}% -\contentsline {subsection}{\numberline {2.1.1}Problem}{4}{subsection.2.1.1}% -\contentsline {subsection}{\numberline {2.1.2}Theoretical Introdution}{5}{subsection.2.1.2}% -\contentsline {section}{\numberline {2.2}b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}{5}{section.2.2}% -\contentsline {subsection}{\numberline {2.2.1}Problem}{5}{subsection.2.2.1}% -\contentsline {subsection}{\numberline {2.2.2}Theoretical Introdution}{5}{subsection.2.2.2}% +\contentsline {subsection}{\numberline {1.2.1}Linear Least Squares Problem}{3}{subsection.1.2.1}% +\contentsline {subsection}{\numberline {1.2.2}Gram matrix and QR decomposition}{3}{subsection.1.2.2}% +\contentsline {subsection}{\numberline {1.2.3}Gram-Schmidt algorithm}{4}{subsection.1.2.3}% +\contentsline {chapter}{\numberline {2}Determine trajectory of the motion}{6}{chapter.2}% +\contentsline {section}{\numberline {2.1}a) Runge-Kutta method of $4^{th}$ order and Adams PC}{6}{section.2.1}% +\contentsline {subsection}{\numberline {2.1.1}Problem}{6}{subsection.2.1.1}% +\contentsline {subsection}{\numberline {2.1.2}Theoretical Introdution}{6}{subsection.2.1.2}% +\contentsline {subsubsection}{Runge-Kutta method of order 4}{7}{section*.2}% +\contentsline {subsection}{\numberline {2.1.3}Adams PC method}{7}{subsection.2.1.3}% +\contentsline {section}{\numberline {2.2}b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}{8}{section.2.2}% +\contentsline {subsection}{\numberline {2.2.1}Problem}{8}{subsection.2.2.1}% +\contentsline {subsection}{\numberline {2.2.2}Theoretical Introdution}{8}{subsection.2.2.2}% +\contentsline {subsubsection}{Error estimation using step doubling-approach}{8}{section*.3}% +\contentsline {subsubsection}{Actual correction of the step size}{10}{section*.4}%