From: Bjørn Rustad Date: Wed, 16 Nov 2011 09:37:33 +0000 (+0100) Subject: Initial commit X-Git-Url: http://git.rustad.me/?a=commitdiff_plain;h=26daad033ca3aebfc65e0b4ddfaea72888e2f38a;p=nummat2 Initial commit --- diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..13d90dd --- /dev/null +++ b/Makefile @@ -0,0 +1,5 @@ +all: + latex rapport.tex + latex rapport.tex + dvips rapport.dvi + ps2pdf rapport.ps diff --git a/egellip.m b/egellip.m new file mode 100644 index 0000000..fccadb1 --- /dev/null +++ b/egellip.m @@ -0,0 +1,51 @@ +function [sn,cn,dn]=egellip(h,phiin,m,ny)%,tol) +%Solve the elliptic integral from 0 to phi + +%if nargin<3, tol = eps; end +%if nargin<2, error('Not enough input arguments.'); end + +a(1) = 1; +c(1) = sqrt(m); +b(1) = sqrt(1-m); + +i = 1; +while abs(c(i)) > eps + i = i + 1; + a(i) = 0.5 * (a(i-1) + b(i-1)); + b(i) = sqrt(a(i-1)* b(i-1)); + c(i) = 0.5 * (a(i-1) - b(i-1)); +end +n=i-1; +phiit(1)=phiin; +for j=1:n + phijf=mod(phiit(j),pi); + nte=(phiit(j)-phijf)/pi; + tempf=atan((b(j)/a(j))*tan(phijf)); + if tempf<0 + tempf=tempf+pi; + end + phiit(j+1)=tempf+nte*pi+phiit(j); +end +%lamtau=phiit(i)/(a(i)*2^(i-1)); +%un= +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +phin(:,i)=[(2.^(n-1)).*a(i)*h*ny+phiit(i);(2.^n)*a(i)*h*ny+phiit(i)]; +while i > 1 + i = i - 1; + phin(:,i) = phin(:,i+1); + %if ~isempty(in) + phin(:,i) = 0.5*(asin(c(i+1).*sin(rem(phin(:,i+1),2*pi))./a(i+1)) + phin(:,i+1)); + %end +end +sn = sin(rem(phin(:,1),2*pi)); +cn = cos(rem(phin(:,1),2*pi)); +dn = sqrt(1 - m .* sn.^2); + +% special case m = 1 +m1 = find(m==1); +sn(m1) = tanh([h*ny/2+phiit(m1)/((2.^(n)).*a(i));h*ny+phiit(i)/((2.^(n)).*a(i))]); +cn(m1) = sech([h*ny/2+phiit(m1)/((2.^(n)).*a(i));h*ny+phiit(i)/((2.^(n)).*a(i))]); +dn(m1) = sech([h*ny/2+phiit(m1)/((2.^(n)).*a(i));h*ny+phiit(i)/((2.^(n)).*a(i))]); + + diff --git a/elerr.kopi.m b/elerr.kopi.m new file mode 100644 index 0000000..81f0da4 --- /dev/null +++ b/elerr.kopi.m @@ -0,0 +1,8 @@ +function [el, e1] = elerr(k, L) + +[il, i1] = elint(k, pi/2, L) + +el = abs(ellipke(k^2) - il) +e1 = abs(ellipke(k^2) - i1) + +end diff --git a/elerr.m b/elerr.m new file mode 100644 index 0000000..98b02f9 --- /dev/null +++ b/elerr.m @@ -0,0 +1,8 @@ +function [el, e1] = elerr(k, L) + +[il, i1] = elint(k, pi/2, L); + +el = abs(ellipke(k^2) - il) +e1 = abs(ellipke(k^2) - i1) + +end diff --git a/elint.kopi.m b/elint.kopi.m new file mode 100644 index 0000000..6487784 --- /dev/null +++ b/elint.kopi.m @@ -0,0 +1,5 @@ +function [il, i1] = elint(k, phi, L) + +f = @(x) 1.0 / sqrt(1 - k^2 * sin(x)^2); + +[il, i1] = romberg(f, 0, phi, L); diff --git a/elint.m b/elint.m new file mode 100644 index 0000000..5af223f --- /dev/null +++ b/elint.m @@ -0,0 +1,5 @@ +function [il, i1] = elint(k, phi, L) + +f = @(x) 1.0 / sqrt(1 - k^2 * sin(x)^2); + +[il, i1] = rombergold(f, 0, phi, L); diff --git a/freewR.m b/freewR.m new file mode 100644 index 0000000..3b30de3 --- /dev/null +++ b/freewR.m @@ -0,0 +1,149 @@ +function momentum=freewR(h,a,b,c,m01,m02,m03) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Solve ( I=diag(A,B,C), Omega = angular velocity, R(t) \in SO(3) ) +% +% m'= m x (inv(I)*m) +% +% Return exact solution for the angular momentum m=m(t_0+h) in one step h, +% with initial condition m(t_0) = m_0 = (m01,m02,m03). +% +% +% Subfunctions: egellip +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +Inert=diag([a,b,c],0); % Inertia tensor +Inert_inv=diag([1/a,1/b,1/c],0); + +OmeganU(:,1)=[m01/a;m02/b;m03/c]; % Variables +omega0=[m01/a;m02/b;m03/c]; % For temporary use + +b1=sqrt(b*(c-b)); +b3=sqrt(b*(b-a)); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% BEGIN +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +T=(a*omega0(1)^2+b*omega0(2)^2+c*omega0(3)^2)/2; +GG=a^2*omega0(1)^2+b^2*omega0(2)^2+c^2*omega0(3)^2; + +a1=sqrt(2*T*c-GG); +a3=sqrt(GG-2*T*a); + +k=b3*a1/(b1*a3); + %***************************************************** + % First case: (b1/a1 >= b3/a3) + %***************************************************** + if k<=1 + ni=b1*a3/(b*sqrt(a*c)); + %************************************************** + % Adjusting the initial value (first case) + % Calculate "phi_0" (phinoll) + %************************************************** + if omega0(1)>=0 + if omega0(2)>=0 % (First quarter) + phinoll=asin(omega0(2)*b1/a1); + if omega0(3)<0 + ni=-ni; + a3=-a3; + end + else %omega0(2)<0 (Fourth quarter) + phinoll=asin(omega0(2)*b1/a1)+2*pi; + if omega0(3)<0 + ni=-ni; + a3=-a3; + end + end + else %omega0(1)<0 + if omega0(2)>=0 % (Second quarter) + phinoll=pi-asin(omega0(2)*b1/a1); + if omega0(3)<0 + ni=-ni; + a3=-a3; + end + else %omega0(2)<0 (Third quarter) + phinoll=2*pi-acos(omega0(1)*sqrt(a*(c-a))/a1); + if omega0(3)<0 + ni=-ni; + a3=-a3; + end + end + end + %************************************************************* + % Integrating omega by means of Jacobi elliptic functions: + % (first case) + %************************************************************* + [Sn,Cn,Dn]=egellip(h,phinoll,k^2,ni); + v=a1/sqrt(a*(c-a))*Cn'; + midt1=v(1,1); + OmeganU(1,1)=v(1,2); + v=a1/b1*Sn'; + midt2=v(1,1); + OmeganU(2,1)=v(1,2); + v=a3/sqrt(c*(c-a))*Dn'; + midt3=v(1,1); + OmeganU(3,1)=v(1,2); + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + else + %***************************************************** + % Second case: (b1/a1 < b3/a3) + %***************************************************** + k=b1*a3/(a1*b3); + ni=a1*b3/(b*sqrt(a*c)); + %***************************************************** + % Adjusting the initial value (second case) + % Calculate "phi_0" (phinoll) + %***************************************************** + if omega0(3)>=0 + if omega0(2)>=0 % (First quarter) + phinoll=asin(omega0(2)*b3/a3); + if omega0(1)<0 + ni=-ni; + a1=-a1; + end + else %omega0(2)<0 (Fourth quarter) + phinoll=asin(omega0(2)*b3/a3)+2*pi; + if omega0(1)<0 + ni=-ni; + a1=-a1; + end + end + else %omega0(3)<0 + if omega0(2)>=0 % (Second quarter) + phinoll=acos(omega0(3)*sqrt(c*(c-a))/a3); + if omega0(1)<0 + ni=-ni; + a1=-a1; + end + else %omega0(2)<0 (Third quarter) + phinoll=2*pi-acos(omega0(3)*sqrt(c*(c-a))/a3); + if omega0(1)<0 + ni=-ni; + a1=-a1; + end + end + end + %************************************************************* + % Integrating omega by means of Jacobi elliptic functions: + % (first case) + %************************************************************* + [Sn,Cn,Dn]=egellip(h,phinoll,k^2,ni); + v=a1/sqrt(a*(c-a))*Dn'; + midt1=v(1,1); + OmeganU(1,1)=v(1,2); + v=a3/b3*Sn'; + midt2=v(1,1); + OmeganU(2,1)=v(1,2); + v=a3/sqrt(c*(c-a))*Cn'; + midt3=v(1,1); + OmeganU(3,1)=v(1,2); + end +%************************************************************* +% Approximation of rotation matrix +%************************************************************* +%midtt=[midt1;midt2;midt3]; +%Rmat=expso3(h*midtt); +momentum=Inert*OmeganU; + diff --git a/main.kopi.m b/main.kopi.m new file mode 100644 index 0000000..0682cb3 --- /dev/null +++ b/main.kopi.m @@ -0,0 +1 @@ +i = elint(0.5, 2, 15) diff --git a/main.m b/main.m new file mode 100644 index 0000000..0682cb3 --- /dev/null +++ b/main.m @@ -0,0 +1 @@ +i = elint(0.5, 2, 15) diff --git a/newtfree.m b/newtfree.m new file mode 100644 index 0000000..6c3c2d5 --- /dev/null +++ b/newtfree.m @@ -0,0 +1,9 @@ +function estimate = newtfree(previous, h, T) + estimate = previous; + F = @(x) x - previous - .25*h*cross(x+previous,T\(x + previous)); + + J = @(x) [1, .25*h*(1/T(2,2)-1/T(3,3))*(previous(3)+x(3)),.25*h*(1/T(2,2)-1/T(3,3))*(previous(2)+x(2));.25*h*(1/T(3,3)-1/T(1,1))*(previous(3)+x(3)), 1,.25*h*(1/T(3,3)-1/T(1,1))*(previous(1)+x(1));.25*h*(1/T(1,1)-1/T(2,2))*(previous(2)+x(2)),.25*h*(1/T(1,1)-1/T(2,2))*(previous(1)+x(1)),1]; + while norm(F(estimate)) > 10*eps + estimate = estimate - J(estimate)\F(estimate); + end +end diff --git a/newton.m b/newton.m new file mode 100644 index 0000000..29eafee --- /dev/null +++ b/newton.m @@ -0,0 +1,17 @@ +function null = newton(u, k, x0, tol) + +x = x0; + +f = @(x) 1/sqrt(1 - k^2 * sin(x)^2); + +G = romberg(k, 0, x, tol/10) - u; +while (abs(G) > tol) + G = romberg(k, 0, x, eps) - u; + Gder = f(x); + xtemp = x - G/Gder; + x = xtemp; +end + +null = x; + +end diff --git a/newton_error.m b/newton_error.m new file mode 100644 index 0000000..e857fc5 --- /dev/null +++ b/newton_error.m @@ -0,0 +1,12 @@ +start = 0; +k = 0.5; + +C = []; +tol = 100000*eps; +for u = linspace(0,100,10) + am = newton(u, k, start, tol); + err = abs(sin(am) - ellipj(u, k^2)); + C = [C; (err / tol)]; +end + +max(C) diff --git a/octave-core b/octave-core new file mode 100644 index 0000000..88618ba Binary files /dev/null and b/octave-core differ diff --git a/rapport.aux b/rapport.aux new file mode 100644 index 0000000..cb63d8d --- /dev/null +++ b/rapport.aux @@ -0,0 +1,37 @@ +\relax +\@writefile{toc}{\contentsline {section}{Part 1}{1}} +\@writefile{toc}{\contentsline {subsection}{Problem 1}{1}} +\@writefile{toc}{\contentsline {subsubsection}{a)}{1}} +\newlabel{eq:sdiv_linear}{{1}{1}} +\newlabel{eq:sdiv_integrated}{{2}{1}} +\newlabel{eq:sdiv_evaluated}{{3}{1}} +\newlabel{eq:s_difference}{{4}{1}} +\newlabel{eq:sder_sum}{{5}{1}} +\newlabel{eq:s_matrix}{{6}{1}} +\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces \sf Data determining our spline polynomials.}}{2}} +\newlabel{tab:data}{{1}{2}} +\newlabel{eq:poly}{{7}{2}} +\@writefile{toc}{\contentsline {subsubsection}{b)}{2}} +\newlabel{eq:g_def}{{8}{2}} +\newlabel{eq:g_der_zero}{{9}{2}} +\newlabel{eq:xbar_error}{{10}{2}} +\newlabel{eq:2s_bound}{{11}{2}} +\newlabel{eq:si_bound}{{12}{2}} +\newlabel{eq:error_bound}{{13}{2}} +\@writefile{toc}{\contentsline {subsection}{Problem 2}{2}} +\@writefile{toc}{\contentsline {subsubsection}{a)}{2}} +\newlabel{eq:integral}{{14}{2}} +\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces \sf Showing the convergence for the Romberg method over the interval $[0, 5]$ with $k = 0.5$ as $L$ increases.}}{3}} +\newlabel{tab:romberg_conv}{{2}{3}} +\newlabel{eq:euler_maclaurin}{{15}{3}} +\@writefile{toc}{\contentsline {subsubsection}{b)}{3}} +\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces \sf Estimated integral for decreasing values of tolerance limit.}}{3}} +\newlabel{tab:tol_test}{{3}{3}} +\@writefile{toc}{\contentsline {subsubsection}{c)}{3}} +\@writefile{toc}{\contentsline {subsection}{Problem 3}{4}} +\@writefile{toc}{\contentsline {subsubsection}{a)}{4}} 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line 712 +Defining \cs_if_eq:cNF on line 713 +Defining \cs_if_eq_p:Nc on line 714 +Defining \cs_if_eq:NcTF on line 715 +Defining \cs_if_eq:NcT on line 716 +Defining \cs_if_eq:NcF on line 717 +Defining \cs_if_eq_p:cc on line 718 +Defining \cs_if_eq:ccTF on line 719 +Defining \cs_if_eq:ccT on line 720 +Defining \cs_if_eq:ccF on line 721 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3expan.sty +Package: l3expan 2009/08/16 v1464 L3 Experimental Argument Expansion module +Defining \l_exp_tl on line 56 +Defining \exp_arg_next:nnn on line 57 +Defining \exp_arg_next_nobrace:nnn on line 60 +Defining \::: on line 63 +Defining \::n on line 64 +Defining \::N on line 67 +Defining \::c on line 70 +Defining \::o on line 73 +Defining \::f on line 76 +Defining \exp_stop_f: on line 81 +Defining \exp_arg:x on line 82 +Defining \::x on line 93 +Defining \::V on line 94 +Defining \::v on line 102 +Defining \exp_args:No on line 129 +Defining \exp_args:NNo on line 130 +Defining \exp_args:NNNo on line 132 +Defining \exp_args:cc on line 135 +Defining \exp_args:NNc on line 136 +Defining \exp_args:Ncc on line 138 +Defining \exp_args:Nccc on line 140 +Defining \exp_args:Nco on line 143 +Defining \exp_not:o on line 192 +Defining \exp_not:d on line 193 +Defining \exp_not:f on line 196 +Defining \exp_not:v on line 199 +Defining \exp_not:V on line 202 +Defining \exp_not:c on line 205 +Defining \cs_generate_variant:Nn on line 206 +Defining \cs_generate_internal_variant:n on line 242 +Defining \cs_generate_internal_variant_aux:n on line 248 +Defining \exp_arg_last_unbraced:nn on line 255 +Defining \::f_unbraced on line 256 +Defining \::o_unbraced on line 260 +Defining \::V_unbraced on line 263 +Defining \::v_unbraced on line 267 +Defining \exp_last_unbraced:Nf on line 273 +Defining \exp_last_unbraced:NV on line 274 +Defining \exp_last_unbraced:Nv on line 275 +Defining \exp_last_unbraced:NcV on line 276 +Defining \exp_last_unbraced:NNo on line 279 +Defining \exp_last_unbraced:NNV on line 282 +Defining \exp_last_unbraced:NNNo on line 285 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3tl.sty +Package: l3tl 2009/09/03 v1505 L3 Experimental Token Lists +Defining \tl_new:Nn on line 56 +Defining \tl_new:cn on line 60 +Defining \exp_args:Ncn on line 60 +Defining \tl_new:Nx on line 61 +Defining \tl_new:N on line 65 +Defining \tl_new:c on line 66 +Defining \tl_use:N on line 67 +Defining \tl_use:c on line 75 +Defining \tl_show:N on line 76 +Defining \tl_show:c on line 77 +Defining \tl_set:Nn on line 79 +Defining \tl_set:Nx on line 80 +Defining \tl_gset:Nn on line 81 +Defining \tl_gset:Nx on line 82 +Defining \tl_set:NV on line 83 +Defining \tl_set:No on line 83 +Defining \tl_set:Nv on line 83 +Defining \tl_set:Nf on line 83 +Defining \tl_set:cn on line 83 +Defining \tl_set:cV on line 83 +Defining \exp_args:NcV on line 83 +Defining \tl_set:co on line 83 +Defining \tl_set:cx on line 83 +Defining \tl_gset:NV on line 84 +Defining \tl_gset:No on line 84 +Defining \tl_gset:Nv on line 84 +Defining \tl_gset:cn on line 84 +Defining \tl_gset:cx on line 84 +Defining \tl_set_eq:NN on line 85 +Defining \tl_gset_eq:NN on line 86 +Defining \tl_set_eq:Nc on line 87 +Defining \tl_set_eq:cN on line 87 +Defining \tl_set_eq:cc on line 87 +Defining \tl_gset_eq:Nc on line 88 +Defining \tl_gset_eq:cN on line 88 +Defining \tl_gset_eq:cc on line 88 +Defining \tl_clear:N on line 89 +Defining \tl_clear:c on line 90 +Defining \tl_gclear:N on line 91 +Defining \tl_gclear:c on line 92 +Defining \tl_clear_new:N on line 93 +Defining \tl_clear_new:c on line 94 +Defining \tl_gclear_new:N on line 95 +Defining \tl_gclear_new:c on line 96 +Defining \tl_put_left:Nn on line 97 +Defining \tl_put_left:NV on line 101 +Defining \tl_put_left:No on line 104 +Defining \tl_put_left:Nx on line 108 +Defining \tl_gput_left:Nn on line 111 +Defining \tl_gput_left:NV on line 115 +Defining \tl_gput_left:No on line 118 +Defining \tl_gput_left:Nx on line 122 +Defining \tl_put_left:cn on line 125 +Defining \tl_put_left:co on line 125 +Defining \tl_put_left:cV on line 125 +Defining \tl_gput_left:cn on line 126 +Defining \tl_gput_left:co on line 126 +Defining \tl_gput_left:cV on line 127 +Defining \tl_put_right:Nn on line 128 +Defining \tl_gput_right:Nn on line 132 +Defining \tl_put_right:NV on line 136 +Defining \tl_put_right:No on line 139 +Defining \tl_gput_right:NV on line 143 +Defining \tl_gput_right:No on line 146 +Defining \tl_put_right:cn on line 156 +Defining \tl_put_right:co on line 156 +Defining \tl_put_right:cV on line 157 +Defining \tl_gput_right:cn on line 158 +Defining \tl_gput_right:co on line 158 +Defining \tl_gput_right:cV on line 158 +Defining \tl_gset:Nc on line 159 +Defining \tl_set:Nc on line 162 +Defining \c_job_name_tl on line 163 +Defining \c_empty_tl on line 164 +Defining \g_tmpa_tl on line 165 +Defining \g_tmpb_tl on line 166 +Defining \l_testa_tl on line 167 +Defining \l_testb_tl on line 168 +Defining \g_testa_tl on line 169 +Defining \g_testb_tl on line 170 +Defining \l_tmpa_tl on line 171 +Defining \l_tmpb_tl on line 172 +Defining \tl_if_empty_p:c on line 177 +Defining \tl_if_empty:cTF on line 178 +Defining \tl_if_empty:cT on line 179 +Defining \tl_if_empty:cF on line 180 +Defining \tl_if_eq_p:NN on line 183 +Defining \tl_if_eq:NNTF on line 183 +Defining \tl_if_eq:NNT on line 183 +Defining \tl_if_eq:NNF on line 183 +Defining \tl_if_eq_p:Nc on line 184 +Defining \tl_if_eq_p:cN on line 184 +Defining \tl_if_eq_p:cc on line 184 +Defining \tl_if_eq:NcTF on line 185 +Defining \tl_if_eq:cNTF on line 185 +Defining \tl_if_eq:ccTF on line 185 +Defining \tl_if_eq:NcT on line 186 +Defining \tl_if_eq:cNT on line 186 +Defining \tl_if_eq:ccT on line 186 +Defining \tl_if_eq:NcF on line 187 +Defining \tl_if_eq:cNF on line 187 +Defining \tl_if_eq:ccF on line 187 +Defining \tl_if_empty_p:n on line 191 +Defining \tl_if_empty:nTF on line 191 +Defining \tl_if_empty:nT on line 191 +Defining \tl_if_empty:nF on line 191 +Defining \tl_if_empty_p:V on line 192 +Defining \tl_if_empty:VTF on line 193 +Defining \tl_if_empty:VT on line 194 +Defining \tl_if_empty:VF on line 195 +Defining \tl_if_empty_p:o on line 196 +Defining \tl_if_empty:oTF on line 197 +Defining \tl_if_empty:oT on line 198 +Defining \tl_if_empty:oF on line 199 +Defining \tl_if_blank_p:n on line 202 +Defining \tl_if_blank:nTF on line 202 +Defining \tl_if_blank:nT on line 202 +Defining \tl_if_blank:nF on line 202 +Defining \tl_if_blank_p_aux:w on line 203 +Defining \tl_if_blank_p:V on line 206 +Defining \tl_if_blank:VTF on line 207 +Defining \tl_if_blank:VT on line 208 +Defining \tl_if_blank:VF on line 209 +Defining \tl_if_blank_p:o on line 210 +Defining \tl_if_blank:oTF on line 211 +Defining \tl_if_blank:oT on line 212 +Defining \tl_if_blank:oF on line 213 +Defining \tl_compare:xx on line 214 +Defining \tl_compare:nn on line 215 +Defining \tl_compare:nx on line 218 +Defining \tl_compare:xn on line 221 +Defining \tl_compare:nV on line 224 +Defining \tl_compare:no on line 227 +Defining \tl_compare:Vn on line 230 +Defining \tl_compare:on on line 233 +Defining \tl_compare:VV on line 236 +Defining \tl_compare:oo on line 239 +Defining \tl_compare:xV on line 242 +Defining \tl_compare:xo on line 245 +Defining \tl_compare:Vx on line 248 +Defining \tl_compare:ox on line 251 +Defining \tl_if_eq_p:xx on line 265 +Defining \tl_if_eq:xxTF on line 265 +Defining \tl_if_eq:xxT on line 265 +Defining \tl_if_eq:xxF on line 265 +Defining \tl_if_eq_p:nx on line 265 +Defining \tl_if_eq:nxTF on line 265 +Defining \tl_if_eq:nxT on line 265 +Defining \tl_if_eq:nxF on line 265 +Defining \tl_if_eq_p:ox on line 265 +Defining \tl_if_eq:oxTF on line 265 +Defining \tl_if_eq:oxT on line 265 +Defining \tl_if_eq:oxF on line 265 +Defining \tl_if_eq_p:Vx on line 265 +Defining \tl_if_eq:VxTF on line 265 +Defining \tl_if_eq:VxT on line 265 +Defining \tl_if_eq:VxF on line 265 +Defining \tl_if_eq_p:xn on line 266 +Defining \tl_if_eq:xnTF on line 266 +Defining \tl_if_eq:xnT on line 266 +Defining \tl_if_eq:xnF on line 266 +Defining \tl_if_eq_p:nn on line 266 +Defining \tl_if_eq:nnTF on line 266 +Defining \tl_if_eq:nnT on line 266 +Defining \tl_if_eq:nnF on line 266 +Defining \tl_if_eq_p:on on line 266 +Defining \tl_if_eq:onTF on line 266 +Defining \tl_if_eq:onT on line 266 +Defining \tl_if_eq:onF on line 266 +Defining \tl_if_eq_p:Vn on line 266 +Defining \tl_if_eq:VnTF on line 266 +Defining \tl_if_eq:VnT on line 266 +Defining \tl_if_eq:VnF on line 266 +Defining \tl_if_eq_p:xo on line 267 +Defining \tl_if_eq:xoTF on line 267 +Defining \tl_if_eq:xoT on line 267 +Defining \tl_if_eq:xoF on line 267 +Defining \tl_if_eq_p:no on line 267 +Defining \tl_if_eq:noTF on line 267 +Defining \tl_if_eq:noT on line 267 +Defining \tl_if_eq:noF on line 267 +Defining \tl_if_eq_p:oo on line 267 +Defining \tl_if_eq:ooTF on line 267 +Defining \tl_if_eq:ooT on line 267 +Defining \tl_if_eq:ooF on line 267 +Defining \tl_if_eq_p:xV on line 268 +Defining \tl_if_eq:xVTF on line 268 +Defining \tl_if_eq:xVT on line 268 +Defining \tl_if_eq:xVF on line 268 +Defining \tl_if_eq_p:nV on line 268 +Defining \tl_if_eq:nVTF on line 268 +Defining \tl_if_eq:nVT on line 268 +Defining \tl_if_eq:nVF on line 268 +Defining \tl_if_eq_p:VV on line 268 +Defining \tl_if_eq:VVTF on line 268 +Defining \tl_if_eq:VVT on line 268 +Defining \tl_if_eq:VVF on line 268 +Defining \tl_to_lowercase:n on line 369 +Defining \tl_to_uppercase:n on line 370 +Defining \tl_to_str:n on line 371 +Defining \tl_to_str:N on line 372 +Defining \tl_to_str_aux:w on line 374 +Defining \tl_to_str:c on line 375 +Defining \tl_map_function:nN on line 376 +Defining \tl_map_function:NN on line 379 +Defining \tl_map_function_aux:Nn on line 383 +Defining \tl_map_function:cN on line 387 +Defining \exp_args:NcN on line 387 +Defining \tl_map_inline:nn on line 388 +Defining \tl_map_inline:Nn on line 397 +Defining \tl_map_inline:cn on line 406 +Defining \g_tl_inline_level_num on line 407 +Defining \tl_map_variable:nNn on line 408 +Defining \tl_map_variable:NNn on line 411 +Defining \tl_map_variable:cNn on line 412 +Defining \tl_map_variable_aux:Nnn on line 413 +Defining \tl_map_break: on line 418 +Defining \tl_reverse:n on line 419 +Defining \tl_reverse_aux:nN on line 422 +Defining \tl_reverse:V on line 426 +Defining \tl_reverse:o on line 426 +Defining \tl_reverse:N on line 427 +Defining \tl_elt_count:n on line 430 +Defining \tl_elt_count:V on line 435 +Defining \tl_elt_count:o on line 435 +Defining \tl_elt_count:N on line 436 +Defining \tl_set_rescan:Nnn on line 441 +Defining \tl_gset_rescan:Nnn on line 442 +Defining \c_two_ats_with_two_catcodes_tl on line 451 +Defining \tl_set_rescan_aux:NNnn on line 452 +Defining \tl_set_rescan:Nnx on line 468 +Defining \tl_gset_rescan:Nnx on line 478 +Defining \tl_rescan:nn on line 486 +Defining \tl_if_in:NnTF on line 500 +Defining \tl_if_in:NnT on line 500 +Defining \tl_if_in:NnF on line 500 +Defining \tl_if_in:cnTF on line 501 +Defining \tl_if_in:cnT on line 502 +Defining \tl_if_in:cnF on line 503 +Defining \tl_if_in:nnTF on line 509 +Defining \tl_if_in:nnT on line 509 +Defining \tl_if_in:nnF on line 509 +Defining \tl_if_in:VnTF on line 510 +Defining \tl_if_in:VnT on line 511 +Defining \tl_if_in:VnF on line 512 +Defining \tl_if_in:onTF on line 513 +Defining \tl_if_in:onT on line 514 +Defining \tl_if_in:onF on line 515 +Defining \tl_replace_in_aux:NNnn on line 517 +Defining \tl_replace_in:Nnn on line 532 +Defining \tl_replace_in:cnn on line 533 +Defining \exp_args:Ncnn on line 533 +Defining \tl_greplace_in:Nnn on line 534 +Defining \tl_greplace_in:cnn on line 535 +Defining \tl_replace_all_in:Nnn on line 552 +Defining \tl_replace_all_in:cnn on line 554 +Defining \tl_greplace_all_in:Nnn on line 555 +Defining \tl_greplace_all_in:cnn on line 557 +Defining \tl_remove_in:Nn on line 558 +Defining \tl_gremove_in:Nn on line 559 +Defining \tl_remove_in:cn on line 560 +Defining \tl_gremove_in:cn on line 561 +Defining \tl_remove_all_in:Nn on line 562 +Defining \tl_gremove_all_in:Nn on line 565 +Defining \tl_remove_all_in:cn on line 568 +Defining \tl_gremove_all_in:cn on line 569 +Defining \tl_head:n on line 570 +Defining \tl_head_i:n on line 571 +Defining \tl_tail:n on line 572 +Defining \tl_tail:f on line 573 +Defining \tl_head_iii:n on line 574 +Defining \tl_head_iii:f on line 575 +Defining \tl_head:w on line 576 +Defining \tl_head_i:w on line 577 +Defining \tl_tail:w on line 578 +Defining \tl_head_iii:w on line 579 +Defining \tl_head:V on line 580 +Defining \tl_tail:V on line 581 +Defining \tl_if_head_eq_meaning_p:nN on line 585 +Defining \tl_if_head_eq_meaning:nNTF on line 585 +Defining \tl_if_head_eq_meaning:nNT on line 585 +Defining \tl_if_head_eq_meaning:nNF on line 585 +Defining \tl_if_head_eq_charcode_p:nN on line 590 +Defining \tl_if_head_eq_charcode:nNTF on line 590 +Defining \tl_if_head_eq_charcode:nNT on line 590 +Defining \tl_if_head_eq_charcode:nNF on line 590 +Defining \tl_if_head_eq_charcode_p:fN on line 600 +Defining \tl_if_head_eq_charcode:fNTF on line 601 +Defining \tl_if_head_eq_charcode:fNT on line 602 +Defining \tl_if_head_eq_charcode:fNF on line 603 +Defining \tl_if_head_eq_catcode_p:nN on line 608 +Defining \tl_if_head_eq_catcode:nNTF on line 608 +Defining \tl_if_head_eq_catcode:nNT on line 608 +Defining \tl_if_head_eq_catcode:nNF on line 608 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3num.sty +Package: l3num 2009/08/08 v1451 L3 Experimental token numbers +Defining \if_num:w on line 56 +Defining \if_case:w on line 57 +Defining \num_incr:c on line 62 +Defining \num_decr:c on line 63 +Defining \num_gincr:c on line 64 +Defining \num_gdecr:c on line 65 +Defining \num_zero:N on line 66 +Defining \num_gzero:N on line 67 +Defining \num_zero:c on line 68 +Defining \num_gzero:c on line 69 +Defining \num_new:N on line 70 +Defining \num_new:c on line 71 +Defining \num_set:Nn on line 72 +Defining \num_set:cn on line 75 +Defining \num_gset:Nn on line 76 +Defining \num_gset:cn on line 77 +Defining \num_set_eq:NN on line 78 +Defining \num_set_eq:cN on line 79 +Defining \num_set_eq:Nc on line 79 +Defining \num_set_eq:cc on line 79 +Defining \num_gset_eq:NN on line 80 +Defining \num_gset_eq:cN on line 81 +Defining \num_gset_eq:Nc on line 81 +Defining \num_gset_eq:cc on line 81 +Defining \num_add:Nn on line 82 +Defining \num_add:cn on line 83 +Defining \num_gadd:Nn on line 84 +Defining \num_gadd:cn on line 85 +Defining \num_use:N on line 86 +Defining \num_use:c on line 87 +Defining \num_show:N on line 88 +Defining \num_show:c on line 89 +Defining \num_elt_count:n on line 90 +Defining \num_elt_count_prop:Nn on line 91 +Defining \l_tmpa_num on line 92 +Defining \l_tmpb_num on line 93 +Defining \l_tmpc_num on line 94 +Defining \g_tmpa_num on line 95 +Defining \g_tmpb_num on line 96 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3intexpr.sty +Package: l3intexpr 2009/03/20 v1086 L3 Integer Expressions +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3quark.sty +Package: l3quark 2009/05/31 v1365 L3 Experimental Quark Commands +Defining \quark_new:N on line 56 +Defining \q_stop on line 57 +Defining \q_no_value on line 58 +Defining \q_nil on line 59 +Defining \q_error on line 60 +Defining \q_mark on line 61 +Defining \q_recursion_tail on line 62 +Defining \q_recursion_stop on line 63 +Defining \quark_if_recursion_tail_stop:n on line 64 +Defining \quark_if_recursion_tail_stop:N on line 70 +Defining \quark_if_recursion_tail_stop:o on line 75 +Defining \quark_if_recursion_tail_stop_do:nn on line 76 +Defining \quark_if_recursion_tail_stop_do:Nn on line 85 +Defining \quark_if_recursion_tail_stop_do:on on line 93 +Defining \exp_args:Non on line 93 +Defining \quark_if_recursion_tail_aux:w on line 94 +Defining \quark_if_no_value_p:N on line 98 +Defining \quark_if_no_value:NTF on line 98 +Defining \quark_if_no_value:NT on line 98 +Defining \quark_if_no_value:NF on line 98 +Defining \quark_if_no_value_p:n on line 118 +Defining \quark_if_no_value:nTF on line 118 +Defining \quark_if_no_value:nT on line 118 +Defining \quark_if_no_value:nF on line 118 +Defining \quark_if_nil_p:N on line 121 +Defining \quark_if_nil:NTF on line 121 +Defining \quark_if_nil:NT on line 121 +Defining \quark_if_nil:NF on line 121 +Defining \quark_if_nil_p:n on line 140 +Defining \quark_if_nil:nTF on line 140 +Defining \quark_if_nil:nT on line 140 +Defining \quark_if_nil:nF on line 140 +Defining \quark_if_nil_p:V on line 141 +Defining \quark_if_nil:VTF on line 142 +Defining \quark_if_nil:VT on line 143 +Defining \quark_if_nil:VF on line 144 +Defining \quark_if_nil_p:o on line 145 +Defining \quark_if_nil:oTF on line 146 +Defining \quark_if_nil:oT on line 147 +Defining \quark_if_nil:oF on line 148 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3seq.sty +Package: l3seq 2009/06/17 v1415 L3 Experimental sequences and stacks +Defining \seq_elt:w on line 56 +Defining \seq_elt_end: on line 57 +Defining \seq_new:N on line 58 +Defining \seq_new:c on line 59 +Defining \seq_clear:N on line 60 +Defining \seq_clear:c on line 61 +Defining \seq_gclear:N on line 62 +Defining \seq_gclear:c on line 63 +Defining \seq_clear_new:N on line 64 +Defining \seq_clear_new:c on line 65 +Defining \seq_gclear_new:N on line 66 +Defining \seq_gclear_new:c on line 67 +Defining \seq_set_eq:NN on line 68 +Defining \seq_set_eq:cN on line 69 +Defining \seq_set_eq:Nc on line 70 +Defining \seq_set_eq:cc on line 71 +Defining \seq_gset_eq:NN on line 72 +Defining \seq_gset_eq:cN on line 73 +Defining \seq_gset_eq:Nc on line 74 +Defining \seq_gset_eq:cc on line 75 +Defining \seq_gconcat:NNN on line 76 +Defining \seq_gconcat:ccc on line 79 +Defining \seq_if_empty_p:N on line 80 +Defining \seq_if_empty:NTF on line 80 +Defining \seq_if_empty:NT on line 80 +Defining \seq_if_empty:NF on line 80 +Defining \seq_if_empty_p:c on line 81 +Defining \seq_if_empty:cTF on line 81 +Defining \seq_if_empty:cT on line 81 +Defining \seq_if_empty:cF on line 81 +Defining \seq_if_empty_err:N on line 82 +Defining \seq_if_in:NnTF on line 94 +Defining \seq_if_in:NnT on line 94 +Defining \seq_if_in:NnF on line 94 +Defining \seq_if_in:cVTF on line 95 +Defining \seq_if_in:coTF on line 95 +Defining \seq_if_in:cnTF on line 95 +Defining \seq_if_in:cxTF on line 95 +Defining \seq_if_in:cVT on line 96 +Defining \seq_if_in:coT on line 96 +Defining \seq_if_in:cnT on line 96 +Defining \seq_if_in:cxT on line 96 +Defining \seq_if_in:cVF on line 97 +Defining \seq_if_in:coF on line 97 +Defining \seq_if_in:cnF on line 97 +Defining \seq_if_in:cxF on line 97 +Defining \seq_get:NN on line 98 +Defining \seq_get_aux:w on line 102 +Defining \seq_get:cN on line 105 +Defining \seq_pop_aux:nnNN on line 106 +Defining \seq_pop_aux:w on line 110 +Defining \seq_show:N on line 115 +Defining \seq_show:c on line 116 +Defining \seq_display:N on line 117 +Defining \seq_display:c on line 131 +Defining \seq_put_aux:Nnn on line 132 +Defining \seq_put_aux:w on line 135 +Defining \seq_put_left:Nn on line 136 +Defining \seq_put_left:NV on line 139 +Defining \seq_put_left:No on line 139 +Defining \seq_put_left:Nx on line 139 +Defining \seq_put_left:cn on line 139 +Defining \seq_put_left:cV on line 139 +Defining \seq_put_left:co on line 139 +Defining \seq_put_right:Nn on line 140 +Defining \seq_put_right:NV on line 142 +Defining \seq_put_right:No on line 142 +Defining \seq_put_right:Nx on line 142 +Defining \seq_put_right:cn on line 142 +Defining \seq_put_right:cV on line 142 +Defining \seq_put_right:co on line 142 +Defining \seq_gput_left:Nn on line 143 +Defining \seq_gput_right:Nn on line 147 +Defining \seq_gput_left:NV on line 151 +Defining \seq_gput_left:No on line 151 +Defining \seq_gput_left:Nx on line 151 +Defining \seq_gput_left:cn on line 151 +Defining \seq_gput_left:cV on line 151 +Defining \seq_gput_left:co on line 151 +Defining \seq_gput_right:NV on line 152 +Defining \seq_gput_right:No on line 152 +Defining \seq_gput_right:Nx on line 152 +Defining \seq_gput_right:cn on line 152 +Defining \seq_gput_right:cV on line 152 +Defining \seq_gput_right:co on line 152 +Defining \seq_gput_right:Nc on line 153 +Defining \seq_map_variable_aux:Nnw on line 154 +Defining \seq_map_variable:NNn on line 160 +Defining \seq_map_variable:cNn on line 164 +Defining \seq_map_break: on line 166 +Defining \seq_map_break:n on line 167 +Defining \seq_map_function:NN on line 168 +Defining \seq_map_function:cN on line 173 +Defining \seq_map_inline:Nn on line 174 +Defining \seq_map_inline:cn on line 179 +Defining \l_seq_remove_seq on line 180 +Defining \seq_remove_duplicates_aux:NN on line 185 +Defining \seq_remove_duplicates_aux:n on line 190 +Defining \seq_remove_duplicates:N on line 191 +Defining \seq_gremove_duplicates:N on line 194 +Defining \seq_push:Nn on line 197 +Defining \seq_push:NV on line 198 +Defining \seq_push:No on line 199 +Defining \seq_push:cn on line 200 +Defining \seq_pop:NN on line 201 +Defining \seq_pop:cN on line 202 +Defining \seq_gpush:Nn on line 203 +Defining \seq_gpop:NN on line 204 +Defining \seq_gpush:NV on line 205 +Defining \seq_gpush:No on line 205 +Defining \seq_gpush:cn on line 205 +Defining \seq_gpush:Nv on line 205 +Defining \seq_gpop:cN on line 206 +Defining \seq_top:NN on line 207 +Defining \seq_top:cN on line 208 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3toks.sty +Package: l3toks 2009/06/14 v1412 L3 Experimental Token Registers +Defining \toks_new:c on line 57 +Defining \toks_use:N on line 58 +Defining \toks_use:c on line 59 +Defining \toks_set:Nn on line 60 +Defining \toks_set:NV on line 61 +Defining \toks_set:Nv on line 64 +Defining \toks_set:No on line 67 +Defining \toks_set:Nf on line 68 +Defining \toks_set:Nx on line 71 +Defining \toks_set:cn on line 71 +Defining \toks_set:cV on line 71 +Defining \toks_set:cv on line 71 +Defining \exp_args:Ncv on line 71 +Defining \toks_set:co on line 71 +Defining \toks_set:cx on line 71 +Defining \toks_set:cf on line 71 +Defining \toks_gset:Nn on line 72 +Defining \toks_gset:NV on line 73 +Defining \toks_gset:No on line 73 +Defining \toks_gset:Nx on line 73 +Defining \toks_gset:cn on line 73 +Defining \toks_gset:cV on line 73 +Defining \toks_gset:co on line 73 +Defining \toks_gset:cx on line 73 +Defining \toks_set_eq:NN on line 74 +Defining \toks_gset_eq:NN on line 75 +Defining \toks_set_eq:Nc on line 76 +Defining \toks_set_eq:cN on line 76 +Defining \toks_set_eq:cc on line 76 +Defining \toks_gset_eq:Nc on line 77 +Defining \toks_gset_eq:cN on line 77 +Defining \toks_gset_eq:cc on line 77 +Defining \toks_clear:N on line 78 +Defining \toks_gclear:N on line 81 +Defining \toks_clear:c on line 85 +Defining \toks_gclear:c on line 86 +Defining \toks_use_clear:N on line 87 +Defining \toks_use_gclear:N on line 90 +Defining \toks_use_clear:c on line 94 +Defining \toks_use_gclear:c on line 95 +Defining \toks_show:N on line 96 +Defining \toks_show:c on line 97 +Defining \toks_put_left:Nn on line 98 +Defining \toks_put_left:NV on line 102 +Defining \toks_put_left:No on line 102 +Defining \toks_put_left:Nx on line 102 +Defining \toks_put_left:cn on line 102 +Defining \toks_put_left:co on line 102 +Defining \toks_put_left:cV on line 102 +Defining \toks_gput_left:Nn on line 103 +Defining \toks_gput_left:NV on line 107 +Defining \toks_gput_left:No on line 107 +Defining \toks_gput_left:Nx on line 107 +Defining \toks_gput_left:cn on line 107 +Defining \toks_gput_left:cV on line 107 +Defining \toks_gput_left:co on line 107 +Defining \toks_put_left_aux:w on line 108 +Defining \toks_put_right:Nn on line 111 +Defining \toks_gput_right:Nn on line 114 +Defining \toks_put_right:NV on line 118 +Defining \toks_put_right:No on line 124 +Defining \toks_put_right:Nx on line 129 +Defining \toks_put_right:cn on line 129 +Defining \toks_put_right:cV on line 129 +Defining \toks_put_right:co on line 129 +Defining \toks_gput_right:NV on line 130 +Defining \toks_gput_right:No on line 130 +Defining \toks_gput_right:Nx on line 130 +Defining \toks_gput_right:cn on line 130 +Defining \toks_gput_right:cV on line 130 +Defining \toks_gput_right:co on line 130 +Defining \toks_put_right:Nf on line 131 +Defining \toks_if_empty_p:N on line 138 +Defining \toks_if_empty:NTF on line 138 +Defining \toks_if_empty:NT on line 138 +Defining \toks_if_empty:NF on line 138 +Defining \toks_if_empty_p:c on line 139 +Defining \toks_if_empty:cTF on line 140 +Defining \toks_if_empty:cT on line 141 +Defining \toks_if_empty:cF on line 142 +Defining \toks_if_eq_p:NN on line 146 +Defining \toks_if_eq:NNTF on line 146 +Defining \toks_if_eq:NNT on line 146 +Defining \toks_if_eq:NNF on line 146 +Defining \toks_if_eq_p:Nc on line 147 +Defining \toks_if_eq_p:cN on line 147 +Defining \toks_if_eq_p:cc on line 147 +Defining \toks_if_eq:NcTF on line 148 +Defining \toks_if_eq:cNTF on line 148 +Defining \toks_if_eq:ccTF on line 148 +Defining \toks_if_eq:NcT on line 149 +Defining \toks_if_eq:cNT on line 149 +Defining \toks_if_eq:ccT on line 149 +Defining \toks_if_eq:NcF on line 150 +Defining \toks_if_eq:cNF on line 150 +Defining \toks_if_eq:ccF on line 150 +\l_tmpb_toks=\toks25 +\l_tmpc_toks=\toks26 +\g_tmpa_toks=\toks27 +\g_tmpb_toks=\toks28 +\g_tmpc_toks=\toks29 +\c_empty_toks=\toks30 +\l_tl_replace_toks=\toks31 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3int.sty +Package: l3int 2009/05/28 v1362 L3 Experimental Integer module +Defining \int_to_roman:w on line 56 +Defining \int_to_number:w on line 57 +Defining \int_advance:w on line 58 +Defining \int_new:N on line 59 +Defining \int_new:c on line 63 +Defining \int_set:Nn on line 64 +Defining \int_gset:Nn on line 66 +Defining \int_set:cn on line 69 +Defining \int_gset:cn on line 70 +Defining \int_incr:N on line 71 +Defining \int_decr:N on line 73 +Defining \int_gincr:N on line 75 +Defining \int_gdecr:N on line 78 +Defining \int_incr:c on line 85 +Defining \int_decr:c on line 86 +Defining \int_gincr:c on line 87 +Defining \int_gdecr:c on line 88 +Defining \int_zero:N on line 89 +Defining \int_zero:c on line 90 +Defining \int_gzero:N on line 91 +Defining \int_gzero:c on line 92 +Defining \int_add:Nn on line 93 +Defining \int_sub:Nn on line 96 +Defining \int_gadd:Nn on line 99 +Defining \int_gsub:Nn on line 102 +Defining \int_add:cn on line 105 +Defining \int_gadd:cn on line 106 +Defining \int_sub:cn on line 107 +Defining \int_gsub:cn on line 108 +Defining \int_use:N on line 109 +Defining \int_use:c on line 110 +Defining \int_show:N on line 111 +Defining \int_show:c on line 112 +Defining \int_to_arabic:n on line 113 +Defining \int_roman_lcuc_mapping:Nnn on line 114 +Defining \int_to_roman:n on line 126 +Defining \int_to_Roman:n on line 130 +Defining \int_to_roman_lcuc:NN on line 134 +Defining \int_alph_default_conversion_rule:n on line 148 +Defining \int_Alph_default_conversion_rule:n on line 157 +Defining \int_to_alph:n on line 166 +Defining \int_to_Alph:n on line 170 +Defining \int_to_symbol:n on line 174 +Defining \int_symbol_math_conversion_rule:n on line 185 +Defining \int_symbol_text_conversion_rule:n on line 198 +Defining \l_tmpa_int on line 211 +\l_tmpa_int=\count110 +Defining \l_tmpb_int on line 212 +\l_tmpb_int=\count111 +Defining \l_tmpc_int on line 213 +\l_tmpc_int=\count112 +Defining \g_tmpa_int on line 214 +\g_tmpa_int=\count113 +Defining \g_tmpb_int on line 215 +\g_tmpb_int=\count114 +Defining \int_const:Nn on line 221 +Defining \c_one on line 234 +Defining \c_two on line 235 +Defining \c_three on line 236 +Defining \c_four on line 237 +Defining \c_five on line 238 +Defining \c_six on line 239 +Defining \c_seven on line 240 +Defining \c_eight on line 241 +Defining \c_nine on line 242 +Defining \c_ten on line 243 +Defining \c_eleven on line 244 +Defining \c_twelve on line 245 +Defining \c_thirteen on line 246 +Defining \c_fourteen on line 247 +Defining \c_fifteen on line 248 +Defining \c_thirty_two on line 250 +Defining \c_hundred_one on line 251 +Defining \c_twohundred_fifty_five on line 252 +Defining \c_twohundred_fifty_six on line 253 +Defining \c_thousand on line 254 +Defining \c_ten_thousand on line 255 +Defining \c_ten_thousand_one on line 256 +Defining \c_ten_thousand_two on line 257 +Defining \c_ten_thousand_three on line 258 +Defining \c_ten_thousand_four on line 259 +Defining \c_twenty_thousand on line 260 +Defining \c_max_int on line 261 +\c_max_int=\count115 +Defining \int_get_sign_and_digits:n on line 262 +Defining \int_get_sign:n on line 265 +Defining \int_get_digits:n on line 268 +Defining \int_get_sign_and_digits_aux:nNNN on line 271 +Defining \int_get_sign_and_digits_aux:oNNN on line 287 +Defining \exp_args:NoNNN on line 287 +Defining \int_convert_from_base_ten_aux:nnn on line 300 +Defining \int_convert_from_base_ten_aux:nfn on line 313 +Defining \exp_args:Nnfn on line 313 +Defining \int_convert_from_base_ten_aux:ffn on line 314 +Defining \exp_args:Nffn on line 314 +Defining \int_convert_to_base_ten_aux:nn on line 333 +Defining \int_convert_to_base_ten_auxi:nnN on line 336 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3prg.sty +Package: l3prg 2009/09/20 v1545 L3 Experimental control structures +Defining \group_align_safe_begin: on line 72 +Defining \group_align_safe_end: on line 74 +Defining \scan_align_safe_stop: on line 75 +Defining \prg_replicate:nn on line 85 +Defining \prg_replicate_aux:N on line 91 +Defining \prg_replicate_first_aux:N on line 94 +Defining \prg_replicate_ on line 97 +Defining \prg_replicate_0:n on line 98 +Defining \prg_replicate_1:n on line 99 +Defining \prg_replicate_2:n on line 100 +Defining \prg_replicate_3:n on line 101 +Defining \prg_replicate_4:n on line 103 +Defining \prg_replicate_5:n on line 105 +Defining \prg_replicate_6:n on line 107 +Defining \prg_replicate_7:n on line 109 +Defining \prg_replicate_8:n on line 111 +Defining \prg_replicate_9:n on line 113 +Defining \prg_replicate_first_0:n on line 115 +Defining \prg_replicate_first_1:n on line 116 +Defining \prg_replicate_first_2:n on line 117 +Defining \prg_replicate_first_3:n on line 118 +Defining \prg_replicate_first_4:n on line 119 +Defining \prg_replicate_first_5:n on line 120 +Defining \prg_replicate_first_6:n on line 121 +Defining \prg_replicate_first_7:n on line 122 +Defining \prg_replicate_first_8:n on line 123 +Defining \prg_replicate_first_9:n on line 124 +Defining \prg_stepwise_function:nnnN on line 125 +Defining \prg_stepwise_function_incr:nnnN on line 131 +Defining \prg_stepwise_function_decr:nnnN on line 140 +Defining \g_prg_inline_level_int on line 149 +\g_prg_inline_level_int=\count116 +Defining \prg_stepwise_inline:nnnn on line 150 +Defining \prg_stepwise_inline_incr:Nnnn on line 160 +Defining \prg_stepwise_inline_decr:Nnnn on line 168 +Defining \prg_stepwise_variable:nnnNn on line 176 +Defining \prg_stepwise_variable_incr:nnnNn on line 182 +Defining \prg_stepwise_variable_decr:nnnNn on line 190 +Defining \bool_new:N on line 198 +Defining \bool_new:c on line 199 +Defining \bool_set_true:N on line 200 +Defining \bool_set_true:c on line 201 +Defining \bool_set_false:N on line 202 +Defining \bool_set_false:c on line 203 +Defining \bool_gset_true:N on line 204 +Defining \bool_gset_true:c on line 205 +Defining \bool_gset_false:N on line 206 +Defining \bool_gset_false:c on line 207 +Defining \bool_set_eq:NN on line 208 +Defining \bool_set_eq:Nc on line 209 +Defining \bool_set_eq:cN on line 210 +Defining \bool_set_eq:cc on line 211 +Defining \bool_gset_eq:NN on line 212 +Defining \bool_gset_eq:Nc on line 213 +Defining \bool_gset_eq:cN on line 214 +Defining \bool_gset_eq:cc on line 215 +Defining \l_tmpa_bool on line 216 +Defining \g_tmpa_bool on line 217 +Defining \bool_if_p:c on line 221 +Defining \bool_if:cTF on line 222 +Defining \bool_if:cT on line 223 +Defining \bool_if:cF on line 224 +Defining \bool_while_do:Nn on line 225 +Defining \bool_while_do:cn on line 228 +Defining \bool_until_do:Nn on line 229 +Defining \bool_until_do:cn on line 232 +Defining \bool_do_while:Nn on line 233 +Defining \bool_do_while:cn on line 236 +Defining \bool_do_until:Nn on line 237 +Defining \bool_do_until:cn on line 240 +Defining \bool_cleanup:N on line 259 +Defining \bool_choose:NN on line 263 +Defining \bool_&_1:w on line 264 +Defining \bool_|_0:w on line 265 +Defining \bool_)_0:w on line 266 +Defining \bool_)_1:w on line 267 +Defining \bool_S_0:w on line 268 +Defining \bool_S_1:w on line 269 +Defining \bool_set:Nn on line 283 +Defining \bool_gset:Nn on line 284 +Defining \bool_set:cn on line 287 +Defining \bool_gset:cn on line 288 +Defining \bool_not_p:n on line 289 +Defining \bool_xor_p:nn on line 290 +Defining \bool_while_do:nn on line 298 +Defining \bool_until_do:nn on line 301 +Defining \bool_do_while:nn on line 304 +Defining \bool_do_until:nn on line 307 +Defining \prg_case_int:nnn on line 310 +Defining \prg_case_int_aux:nnn on line 314 +Defining \prg_case_dim:nnn on line 320 +Defining \prg_case_dim_aux:nnn on line 324 +Defining \prg_case_str:nnn on line 330 +Defining \prg_case_str_aux:nnn on line 334 +Defining \prg_case_tl:Nnn on line 340 +Defining \prg_case_tl_aux:NNn on line 344 +Defining \prg_end_case:nw on line 350 +Defining \prg_define_quicksort:nnn on line 352 +Defining \prg_variable_get_scope:N on line 442 +Defining \prg_variable_get_scope_aux:w on line 442 +Defining \prg_variable_get_type:N on line 459 +Defining \prg_variable_get_type_aux:w on line 459 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3clist.sty +Package: l3clist 2009/08/05 v1441 L3 Experimental comma separated lists +Defining \clist_new:N on line 56 +Defining \clist_new:c on line 57 +Defining \clist_clear:N on line 58 +Defining \clist_clear:c on line 59 +Defining \clist_gclear:N on line 60 +Defining \clist_gclear:c on line 61 +Defining \clist_clear_new:N on line 62 +Defining \clist_clear_new:c on line 63 +Defining \clist_gclear_new:N on line 64 +Defining \clist_gclear_new:c on line 65 +Defining \clist_set_eq:NN on line 66 +Defining \clist_set_eq:cN on line 67 +Defining \clist_set_eq:Nc on line 68 +Defining \clist_set_eq:cc on line 69 +Defining \clist_gset_eq:NN on line 70 +Defining \clist_gset_eq:cN on line 71 +Defining \clist_gset_eq:Nc on line 72 +Defining \clist_gset_eq:cc on line 73 +Defining \clist_set_from_seq_aux:NNN on line 74 +Defining \clist_set_from_seq:NN on line 81 +Defining \clist_gset_from_seq:NN on line 82 +Defining \clist_if_empty_p:N on line 83 +Defining \clist_if_empty:NTF on line 83 +Defining \clist_if_empty:NT on line 83 +Defining \clist_if_empty:NF on line 83 +Defining \clist_if_empty_p:c on line 84 +Defining \clist_if_empty:cTF on line 84 +Defining \clist_if_empty:cT on line 84 +Defining \clist_if_empty:cF on line 84 +Defining \clist_if_empty_err:N on line 85 +Defining \clist_if_eq_p:NN on line 91 +Defining \clist_if_eq:NNTF on line 91 +Defining \clist_if_eq:NNT on line 91 +Defining \clist_if_eq:NNF on line 91 +Defining \clist_if_eq_p:cN on line 92 +Defining \clist_if_eq:cNTF on line 92 +Defining \clist_if_eq:cNT on line 92 +Defining \clist_if_eq:cNF on line 92 +Defining \clist_if_eq_p:Nc on line 93 +Defining \clist_if_eq:NcTF on line 93 +Defining \clist_if_eq:NcT on line 93 +Defining \clist_if_eq:NcF on line 93 +Defining \clist_if_eq_p:cc on line 94 +Defining \clist_if_eq:ccTF on line 94 +Defining \clist_if_eq:ccT on line 94 +Defining \clist_if_eq:ccF on line 94 +Defining \clist_if_in:NnTF on line 101 +Defining \clist_if_in:NnT on line 101 +Defining \clist_if_in:NnF on line 101 +Defining \clist_if_in:NVTF on line 102 +Defining \clist_if_in:NoTF on line 102 +Defining \clist_if_in:cnTF on line 102 +Defining \clist_if_in:cVTF on line 102 +Defining \clist_if_in:coTF on line 102 +Defining \clist_if_in:NVT on line 103 +Defining \clist_if_in:NoT on line 103 +Defining \clist_if_in:cnT on line 103 +Defining \clist_if_in:cVT on line 103 +Defining \clist_if_in:coT on line 103 +Defining \clist_if_in:NVF on line 104 +Defining \clist_if_in:NoF on line 104 +Defining \clist_if_in:cnF on line 104 +Defining \clist_if_in:cVF on line 104 +Defining \clist_if_in:coF on line 104 +Defining \clist_use:N on line 105 +Defining \clist_use:c on line 113 +Defining \clist_get:NN on line 114 +Defining \clist_get_aux:w on line 118 +Defining \clist_get:cN on line 119 +Defining \clist_pop_aux:nnNN on line 120 +Defining \clist_pop_aux:w on line 124 +Defining \clist_pop_auxi:w on line 129 +Defining \clist_show:N on line 130 +Defining \clist_show:c on line 131 +Defining \clist_display:N on line 132 +Defining \clist_display:c on line 146 +Defining \clist_put_aux:NNnnNn on line 147 +Defining \clist_put_left:Nn on line 152 +Defining \clist_put_left:NV on line 155 +Defining \clist_put_left:No on line 155 +Defining \clist_put_left:Nx on line 155 +Defining \clist_put_left:cn on line 155 +Defining \clist_put_left:cV on line 155 +Defining \clist_put_left:co on line 155 +Defining \clist_gput_left:Nn on line 156 +Defining \clist_gput_left:NV on line 159 +Defining \clist_gput_left:No on line 159 +Defining \clist_gput_left:Nx on line 159 +Defining \clist_gput_left:cn on line 159 +Defining \clist_gput_left:cV on line 159 +Defining \clist_gput_left:co on line 159 +Defining \clist_put_right:Nn on line 160 +Defining \clist_put_right:NV on line 163 +Defining \clist_put_right:No on line 163 +Defining \clist_put_right:Nx on line 163 +Defining \clist_put_right:cn on line 163 +Defining \clist_put_right:cV on line 163 +Defining \clist_put_right:co on line 163 +Defining \clist_gput_right:Nn on line 164 +Defining \clist_gput_right:NV on line 167 +Defining \clist_gput_right:No on line 167 +Defining \clist_gput_right:Nx on line 167 +Defining \clist_gput_right:cn on line 167 +Defining \clist_gput_right:cV on line 167 +Defining \clist_gput_right:co on line 167 +Defining \clist_map_function:NN on line 168 +Defining \clist_map_function:cN on line 174 +Defining \clist_map_function:nN on line 175 +Defining \clist_map_function_aux:Nw on line 180 +Defining \clist_map_break: on line 185 +Defining \g_clist_inline_level_int on line 186 +\g_clist_inline_level_int=\count117 +Defining \clist_map_inline:Nn on line 187 +Defining \clist_map_inline:cn on line 198 +Defining \clist_map_inline:nn on line 199 +Defining \clist_map_variable:nNn on line 210 +Defining \clist_map_variable:NNn on line 216 +Defining \clist_map_variable:cNn on line 217 +Defining \exp_args:NcNn on line 217 +Defining \clist_map_variable_aux:Nnw on line 218 +Defining \clist_concat_aux:NNNN on line 223 +Defining \clist_concat:NNN on line 232 +Defining \clist_gconcat:NNN on line 233 +Defining \clist_concat:ccc on line 234 +Defining \clist_gconcat:ccc on line 235 +Defining \l_clist_remove_clist on line 236 +Defining \clist_remove_duplicates_aux:NN on line 241 +Defining \clist_remove_duplicates_aux:n on line 246 +Defining \clist_remove_duplicates:N on line 247 +Defining \clist_gremove_duplicates:N on line 250 +Defining \clist_remove_element:Nn on line 253 +Defining \clist_gremove_element:Nn on line 256 +Defining \clist_remove_element_aux:NNn on line 268 +Defining \clist_remove_element_aux:n on line 269 +Defining \clist_push:Nn on line 270 +Defining \clist_push:NV on line 271 +Defining \clist_push:No on line 272 +Defining \clist_push:cn on line 273 +Defining \clist_pop:NN on line 274 +Defining \clist_pop:cN on line 275 +Defining \clist_gpush:Nn on line 276 +Defining \clist_gpush:NV on line 277 +Defining \clist_gpush:No on line 278 +Defining \clist_gpush:cn on line 279 +Defining \clist_gpop:NN on line 280 +Defining \clist_gpop:cN on line 281 +Defining \clist_top:NN on line 282 +Defining \clist_top:cN on line 283 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3token.sty +Package: l3token 2009/09/22 v1550 L3 Experimental token investigation and manip +ulation +Defining \char_set_catcode:w on line 56 +Defining \char_set_catcode:nn on line 57 +Defining \char_value_catcode:w on line 60 +Defining \char_value_catcode:n on line 61 +Defining \char_show_value_catcode:w on line 64 +Defining \char_show_value_catcode:n on line 67 +Defining \char_make_escape:N on line 70 +Defining \char_make_begin_group:N on line 71 +Defining \char_make_end_group:N on line 72 +Defining \char_make_math_shift:N on line 73 +Defining \char_make_alignment:N on line 74 +Defining \char_make_end_line:N on line 75 +Defining \char_make_parameter:N on line 76 +Defining \char_make_math_superscript:N on line 77 +Defining \char_make_math_subscript:N on line 78 +Defining \char_make_ignore:N on line 79 +Defining \char_make_space:N on line 80 +Defining \char_make_letter:N on line 81 +Defining \char_make_other:N on line 82 +Defining \char_make_active:N on line 83 +Defining \char_make_comment:N on line 84 +Defining \char_make_invalid:N on line 85 +Defining \char_make_escape:n on line 86 +Defining \char_make_begin_group:n on line 87 +Defining \char_make_end_group:n on line 88 +Defining \char_make_math_shift:n on line 89 +Defining \char_make_alignment:n on line 90 +Defining \char_make_end_line:n on line 91 +Defining \char_make_parameter:n on line 92 +Defining \char_make_math_superscript:n on line 93 +Defining \char_make_math_subscript:n on line 94 +Defining \char_make_ignore:n on line 95 +Defining \char_make_space:n on line 96 +Defining \char_make_letter:n on line 97 +Defining \char_make_other:n on line 98 +Defining \char_make_active:n on line 99 +Defining \char_make_comment:n on line 100 +Defining \char_make_invalid:n on line 101 +Defining \char_set_mathcode:w on line 102 +Defining \char_set_mathcode:nn on line 103 +Defining \char_gset_mathcode:w on line 106 +Defining \char_gset_mathcode:nn on line 107 +Defining \char_value_mathcode:w on line 110 +Defining \char_value_mathcode:n on line 111 +Defining \char_show_value_mathcode:w on line 114 +Defining \char_show_value_mathcode:n on line 115 +Defining \char_set_lccode:w on line 118 +Defining \char_set_lccode:nn on line 119 +Defining \char_value_lccode:w on line 122 +Defining \char_value_lccode:n on line 123 +Defining \char_show_value_lccode:w on line 125 +Defining \char_show_value_lccode:n on line 126 +Defining \char_set_uccode:w on line 128 +Defining \char_set_uccode:nn on line 129 +Defining \char_value_uccode:w on line 132 +Defining \char_value_uccode:n on line 133 +Defining \char_show_value_uccode:w on line 135 +Defining \char_show_value_uccode:n on line 136 +Defining \char_set_sfcode:w on line 138 +Defining \char_set_sfcode:nn on line 139 +Defining \char_value_sfcode:w on line 142 +Defining \char_value_sfcode:n on line 143 +Defining \char_show_value_sfcode:w on line 146 +Defining \char_show_value_sfcode:n on line 147 +Defining \token_new:Nn on line 150 +Defining \c_group_begin_token on line 151 +Defining \c_group_end_token on line 152 +Defining \c_math_shift_token on line 155 +Defining \c_alignment_tab_token on line 157 +Defining \c_parameter_token on line 158 +Defining \c_math_superscript_token on line 159 +Defining \c_math_subscript_token on line 161 +Defining \c_space_token on line 162 +Defining \c_letter_token on line 163 +Defining \c_other_char_token on line 164 +Defining \token_if_group_begin_p:N on line 171 +Defining \token_if_group_begin:NTF on line 171 +Defining \token_if_group_begin:NT on line 171 +Defining \token_if_group_begin:NF on line 171 +Defining \token_if_group_end_p:N on line 175 +Defining \token_if_group_end:NTF on line 175 +Defining \token_if_group_end:NT on line 175 +Defining \token_if_group_end:NF on line 175 +Defining \token_if_math_shift_p:N on line 179 +Defining \token_if_math_shift:NTF on line 179 +Defining \token_if_math_shift:NT on line 179 +Defining \token_if_math_shift:NF on line 179 +Defining \token_if_alignment_tab_p:N on line 183 +Defining \token_if_alignment_tab:NTF on line 183 +Defining \token_if_alignment_tab:NT on line 183 +Defining \token_if_alignment_tab:NF on line 183 +Defining \token_if_parameter_p:N on line 187 +Defining \token_if_parameter:NTF on line 187 +Defining \token_if_parameter:NT on line 187 +Defining \token_if_parameter:NF on line 187 +Defining \token_if_math_superscript_p:N on line 191 +Defining \token_if_math_superscript:NTF on line 191 +Defining \token_if_math_superscript:NT on line 191 +Defining \token_if_math_superscript:NF on line 191 +Defining \token_if_math_subscript_p:N on line 195 +Defining \token_if_math_subscript:NTF on line 195 +Defining \token_if_math_subscript:NT on line 195 +Defining \token_if_math_subscript:NF on line 195 +Defining \token_if_space_p:N on line 199 +Defining \token_if_space:NTF on line 199 +Defining \token_if_space:NT on line 199 +Defining \token_if_space:NF on line 199 +Defining \token_if_letter_p:N on line 203 +Defining \token_if_letter:NTF on line 203 +Defining \token_if_letter:NT on line 203 +Defining \token_if_letter:NF on line 203 +Defining \token_if_other_char_p:N on line 207 +Defining \token_if_other_char:NTF on line 207 +Defining \token_if_other_char:NT on line 207 +Defining \token_if_other_char:NF on line 207 +Defining \token_if_active_char_p:N on line 211 +Defining \token_if_active_char:NTF on line 211 +Defining \token_if_active_char:NT on line 211 +Defining \token_if_active_char:NF on line 211 +Defining \token_if_eq_meaning_p:NN on line 215 +Defining \token_if_eq_meaning:NNTF on line 215 +Defining \token_if_eq_meaning:NNT on line 215 +Defining \token_if_eq_meaning:NNF on line 215 +Defining \token_if_eq_catcode_p:NN on line 219 +Defining \token_if_eq_catcode:NNTF on line 219 +Defining \token_if_eq_catcode:NNT on line 219 +Defining \token_if_eq_catcode:NNF on line 219 +Defining \token_if_eq_charcode_p:NN on line 223 +Defining \token_if_eq_charcode:NNTF on line 223 +Defining \token_if_eq_charcode:NNT on line 223 +Defining \token_if_eq_charcode:NNF on line 223 +Defining \token_if_macro_p:N on line 226 +Defining \token_if_macro:NTF on line 226 +Defining \token_if_macro:NT on line 226 +Defining \token_if_macro:NF on line 226 +Defining \token_if_macro_p_aux:w on line 227 +Defining \token_if_cs_p:N on line 233 +Defining \token_if_cs:NTF on line 233 +Defining \token_if_cs:NT on line 233 +Defining \token_if_cs:NF on line 233 +Defining \token_if_expandable_p:N on line 241 +Defining \token_if_expandable:NTF on line 241 +Defining \token_if_expandable:NT on line 241 +Defining \token_if_expandable:NF on line 241 +Defining \token_if_chardef_p:N on line 333 +Defining \token_if_chardef:NTF on line 333 +Defining \token_if_chardef:NT on line 333 +Defining \token_if_chardef:NF on line 333 +Defining \token_if_chardef_aux:w on line 333 +Defining \token_if_mathchardef_p:N on line 333 +Defining \token_if_mathchardef:NTF on line 333 +Defining \token_if_mathchardef:NT on line 333 +Defining \token_if_mathchardef:NF on line 333 +Defining \token_if_mathchardef_aux:w on line 333 +Defining \token_if_int_register_p:N on line 333 +Defining \token_if_int_register:NTF on line 333 +Defining \token_if_int_register:NT on line 333 +Defining \token_if_int_register:NF on line 333 +Defining \token_if_int_register_aux:w on line 333 +Defining \token_if_skip_register_p:N on line 333 +Defining \token_if_skip_register:NTF on line 333 +Defining \token_if_skip_register:NT on line 333 +Defining \token_if_skip_register:NF on line 333 +Defining \token_if_skip_register_aux:w on line 333 +Defining \token_if_dim_register_p:N on line 333 +Defining \token_if_dim_register:NTF on line 333 +Defining \token_if_dim_register:NT on line 333 +Defining \token_if_dim_register:NF on line 333 +Defining \token_if_dim_register_aux:w on line 333 +Defining \token_if_toks_register_p:N on line 333 +Defining \token_if_toks_register:NTF on line 333 +Defining \token_if_toks_register:NT on line 333 +Defining \token_if_toks_register:NF on line 333 +Defining \token_if_toks_register_aux:w on line 333 +Defining \token_if_protected_macro_p:N on line 333 +Defining \token_if_protected_macro:NTF on line 333 +Defining \token_if_protected_macro:NT on line 333 +Defining \token_if_protected_macro:NF on line 333 +Defining \token_if_protected_macro_aux:w on line 333 +Defining \token_if_long_macro_p:N on line 333 +Defining \token_if_long_macro:NTF on line 333 +Defining \token_if_long_macro:NT on line 333 +Defining \token_if_long_macro:NF on line 333 +Defining \token_if_long_macro_aux:w on line 333 +Defining \token_if_protected_long_macro_p:N on line 333 +Defining \token_if_protected_long_macro:NTF on line 333 +Defining \token_if_protected_long_macro:NT on line 333 +Defining \token_if_protected_long_macro:NF on line 333 +Defining \token_if_protected_long_macro_aux:w on line 333 +Defining \token_get_prefix_arg_replacement_aux:w on line 364 +Defining \token_get_prefix_spec:N on line 364 +Defining \token_get_arg_spec:N on line 364 +Defining \token_get_replacement_spec:N on line 364 +Defining \token_if_primitive_p:N on line 383 +Defining \token_if_primitive:NTF on line 383 +Defining \token_if_primitive:NT on line 383 +Defining \token_if_primitive:NF on line 383 +Defining \token_if_primitive_p_aux:N on line 384 +Defining \l_peek_token on line 405 +Defining \g_peek_token on line 406 +Defining \l_peek_search_token on line 407 +Defining \peek_after:NN on line 408 +Defining \peek_gafter:NN on line 409 +Defining \l_peek_true_tl on line 412 +Defining \l_peek_false_tl on line 413 +Defining \peek_tmp:w on line 414 +Defining \l_peek_search_tl on line 415 +Defining \peek_token_generic:NNTF on line 416 +Defining \peek_token_generic:NNT on line 424 +Defining \peek_token_generic:NNF on line 427 +Defining \peek_token_remove_generic:NNTF on line 430 +Defining \peek_token_remove_generic:NNT on line 439 +Defining \peek_token_remove_generic:NNF on line 442 +Defining \l_peek_true_aux_tl on line 445 +Defining \c_peek_true_remove_next_tl on line 448 +Defining \peek_execute_branches_meaning: on line 449 +Defining \peek_execute_branches_catcode: on line 456 +Defining \peek_execute_branches_charcode: on line 463 +Defining \peek_execute_branches_charcode_aux:NN on line 472 +Defining \peek_def_aux:nnnn on line 480 +Defining \peek_def_aux_ii:nnnnn on line 485 +Defining \peek_meaning:NTF on line 498 +Defining \peek_meaning:NT on line 498 +Defining \peek_meaning:NF on line 498 +Defining \peek_meaning_ignore_spaces:NTF on line 503 +Defining \peek_meaning_ignore_spaces:NT on line 503 +Defining \peek_meaning_ignore_spaces:NF on line 503 +Defining \peek_meaning_remove:NTF on line 508 +Defining \peek_meaning_remove:NT on line 508 +Defining \peek_meaning_remove:NF on line 508 +Defining \peek_meaning_remove_ignore_spaces:NTF on line 513 +Defining \peek_meaning_remove_ignore_spaces:NT on line 513 +Defining \peek_meaning_remove_ignore_spaces:NF on line 513 +Defining \peek_catcode:NTF on line 518 +Defining \peek_catcode:NT on line 518 +Defining \peek_catcode:NF on line 518 +Defining \peek_catcode_ignore_spaces:NTF on line 523 +Defining \peek_catcode_ignore_spaces:NT on line 523 +Defining \peek_catcode_ignore_spaces:NF on line 523 +Defining \peek_catcode_remove:NTF on line 528 +Defining \peek_catcode_remove:NT on line 528 +Defining \peek_catcode_remove:NF on line 528 +Defining \peek_catcode_remove_ignore_spaces:NTF on line 533 +Defining \peek_catcode_remove_ignore_spaces:NT on line 533 +Defining \peek_catcode_remove_ignore_spaces:NF on line 533 +Defining \peek_charcode:NTF on line 538 +Defining \peek_charcode:NT on line 538 +Defining \peek_charcode:NF on line 538 +Defining \peek_charcode_ignore_spaces:NTF on line 543 +Defining \peek_charcode_ignore_spaces:NT on line 543 +Defining \peek_charcode_ignore_spaces:NF on line 543 +Defining \peek_charcode_remove:NTF on line 548 +Defining \peek_charcode_remove:NT on line 548 +Defining \peek_charcode_remove:NF on line 548 +Defining \peek_charcode_remove_ignore_spaces:NTF on line 553 +Defining \peek_charcode_remove_ignore_spaces:NT on line 553 +Defining \peek_charcode_remove_ignore_spaces:NF on line 553 +Defining \peek_ignore_spaces_aux: on line 554 +Defining \peek_ignore_spaces_execute_branches: on line 557 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3io.sty +Package: l3io 2009/05/31 v1369 L3 Experimental i/o module +Defining \iow_new:c on line 57 +Defining \iow_open:Nn on line 58 +Defining \iow_open:cn on line 62 +Defining \iow_close:N on line 63 +Defining \c_iow_term_stream on line 64 +Defining \c_ior_term_stream on line 65 +Defining \c_iow_log_stream on line 66 +Defining \c_ior_log_stream on line 67 +Defining \iow_now:Nx on line 68 +Defining \iow_now:Nn on line 69 +Defining \iow_log:n on line 73 +Defining \iow_term:n on line 75 +Defining \iow_now_when_avail:Nn on line 76 +Defining \iow_now_when_avail:cn on line 79 +Defining \iow_now_buffer_safe_aux:w on line 80 +Defining \iow_now_buffer_safe:Nx on line 83 +Defining \iow_now_buffer_safe:Nn on line 86 +Defining \iow_shipout_x:Nx on line 90 +Defining \iow_shipout:Nn on line 91 +Defining \iow_shipout:Nx on line 94 +Defining \iow_newline: on line 95 +Defining \iow_space: on line 96 +Defining \iow_char:N on line 97 +Defining \c_iow_comment_char on line 98 +Defining \c_iow_lbrace_char on line 99 +Defining \c_iow_rbrace_char on line 100 +Defining \ior_open:Nn on line 103 +Defining \if_eof:w on line 107 +Defining \ior_if_eof_p:N on line 110 +Defining \ior_if_eof:NTF on line 110 +Defining \ior_if_eof:NT on line 110 +Defining \ior_if_eof:NF on line 110 +Defining \ior_to:NN on line 111 +Defining \ior_gto:NN on line 114 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3prop.sty +Package: l3prop 2009/09/16 v1536 L3 Experimental Property Lists +Defining \q_prop on line 56 +Defining \prop_new:N on line 57 +Defining \prop_new:c on line 58 +Defining \prop_clear:N on line 59 +Defining \prop_clear:c on line 60 +Defining \prop_gclear:N on line 61 +Defining \prop_gclear:c on line 62 +Defining \prop_set_eq:NN on line 63 +Defining \prop_set_eq:Nc on line 64 +Defining \prop_set_eq:cN on line 65 +Defining \prop_set_eq:cc on line 66 +Defining \prop_gset_eq:NN on line 67 +Defining \prop_gset_eq:Nc on line 68 +Defining \prop_gset_eq:cN on line 69 +Defining \prop_gset_eq:cc on line 70 +Defining \prop_show:N on line 71 +Defining \prop_show:c on line 72 +Defining \prop_display:N on line 73 +Defining \prop_display:c on line 89 +Defining \prop_split_aux:Nnn on line 90 +Defining \prop_get:NnN on line 96 +Defining \prop_get_aux:w on line 99 +Defining \prop_get:NVN on line 100 +Defining \exp_args:NNVN on line 100 +Defining \prop_get:cnN on line 100 +Defining \exp_args:NcnN on line 100 +Defining \prop_get:cVN on line 100 +Defining \exp_args:NcVN on line 100 +Defining \prop_gget:NnN on line 101 +Defining \prop_gget_aux:w on line 103 +Defining \prop_gget:NVN on line 104 +Defining \prop_gget:cnN on line 104 +Defining \prop_gget:cVN on line 104 +Defining \prop_get_gdel:NnN on line 105 +Defining \prop_get_del_aux:w on line 107 +Defining \prop_put:Nnn on line 113 +Defining \prop_gput:Nnn on line 119 +Defining \prop_put_aux:w on line 125 +Defining \prop_put:NnV on line 133 +Defining \exp_args:NNnV on line 133 +Defining \prop_put:NVn on line 133 +Defining \exp_args:NNVn on line 133 +Defining \prop_put:NVV on line 133 +Defining \exp_args:NNVV on line 133 +Defining \prop_put:cnn on line 133 +Defining \prop_gput:NnV on line 134 +Defining \prop_gput:Nno on line 134 +Defining \prop_gput:Nnx on line 134 +Defining \prop_gput:Nox on line 134 +Defining \prop_gput:cnn on line 134 +Defining \prop_gput:ccx on line 134 +Defining \prop_del:Nn on line 135 +Defining \prop_gdel:Nn on line 137 +Defining \prop_del_aux:w on line 139 +Defining \prop_del:NV on line 146 +Defining \prop_gdel:NV on line 147 +Defining \prop_gput_if_new:Nnn on line 148 +Defining \prop_put_if_new_aux:w on line 150 +Defining \prop_if_empty_p:N on line 152 +Defining \prop_if_empty:NTF on line 152 +Defining \prop_if_empty:NT on line 152 +Defining \prop_if_empty:NF on line 152 +Defining \prop_if_empty_p:c on line 153 +Defining \prop_if_empty:cTF on line 153 +Defining \prop_if_empty:cT on line 153 +Defining \prop_if_empty:cF on line 153 +Defining \prop_if_eq_p:NN on line 154 +Defining \prop_if_eq:NNTF on line 154 +Defining \prop_if_eq:NNT on line 154 +Defining \prop_if_eq:NNF on line 154 +Defining \prop_if_eq_p:cN on line 155 +Defining \prop_if_eq:cNTF on line 155 +Defining \prop_if_eq:cNT on line 155 +Defining \prop_if_eq:cNF on line 155 +Defining \prop_if_eq_p:Nc on line 156 +Defining \prop_if_eq:NcTF on line 156 +Defining \prop_if_eq:NcT on line 156 +Defining \prop_if_eq:NcF on line 156 +Defining \prop_if_eq_p:cc on line 157 +Defining \prop_if_eq:ccTF on line 157 +Defining \prop_if_eq:ccT on line 157 +Defining \prop_if_eq:ccF on line 157 +Defining \prop_if_in:NnTF on line 160 +Defining \prop_if_in:NnT on line 160 +Defining \prop_if_in:NnF on line 160 +Defining \prop_if_in_aux:w on line 161 +Defining \prop_if_in:NVTF on line 164 +Defining \prop_if_in:NoTF on line 164 +Defining \prop_if_in:cnTF on line 164 +Defining \prop_if_in:ccTF on line 164 +Defining \prop_if_in:NVT on line 165 +Defining \prop_if_in:NoT on line 165 +Defining \prop_if_in:cnT on line 165 +Defining \prop_if_in:ccT on line 165 +Defining \prop_if_in:NVF on line 166 +Defining \prop_if_in:NoF on line 166 +Defining \prop_if_in:cnF on line 166 +Defining \prop_if_in:ccF on line 166 +Defining \prop_map_function:cN on line 178 +Defining \prop_map_function:Nc on line 178 +Defining \prop_map_function:cc on line 178 +Defining \g_prop_inline_level_num on line 179 +Defining \prop_map_inline:Nn on line 180 +Defining \prop_map_inline:cn on line 188 +Defining \prop_map_break: on line 189 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3msg.sty +Package: l3msg 2009/09/01 v1502 L3 Experimental LaTeX Messages module +Defining \c_msg_fatal_tl on line 58 +Defining \c_msg_error_tl on line 59 +Defining \c_msg_warning_tl on line 60 +Defining \c_msg_info_tl on line 61 +Defining \c_msg_fatal_text_tl on line 64 +Defining \c_msg_help_text_tl on line 67 +Defining \c_msg_kernel_bug_text_tl on line 70 +Defining \c_msg_kernel_bug_more_text_tl on line 76 +Defining \c_msg_no_info_text_tl on line 80 +Defining \c_msg_return_text_tl on line 86 +Defining \c_msg_hide_tl + on line 94 +Defining \c_msg_on_line_tl on line 95 +Defining \c_msg_text_prefix_tl on line 96 +Defining \c_msg_more_text_prefix_tl on line 97 +Defining \c_msg_code_prefix_tl on line 98 +Defining \l_msg_class_tl on line 99 +Defining \l_msg_current_class_tl on line 100 +Defining \l_msg_names_clist on line 101 +\l_msg_redirect_classes_prop=\toks32 +\l_msg_redirect_names_prop=\toks33 +Defining \l_msg_redirect_classes_clist on line 104 +Defining \msg_line_number: on line 107 +Defining \msg_line_context: on line 113 +Defining \msg_newline: on line 114 +Defining \msg_two_newlines: on line 115 +Defining \msg_space: on line 116 +Defining \msg_two_spaces: on line 117 +Defining \msg_four_spaces: on line 118 +Defining \msg_generic_new:nnnn on line 119 +Defining \msg_generic_new:nnn on line 123 +Defining \msg_generic_new:nn on line 127 +Defining \msg_generic_set:nnnn on line 131 +Defining \msg_generic_set:nnn on line 136 +Defining \msg_generic_set:nn on line 141 +Defining \msg_generic_set_clist:n on line 146 +Defining \msg_generic_set_text:nn on line 154 +Defining \msg_generic_set_more_text:nnn on line 163 +Defining \msg_generic_set_code:nnnn on line 164 +Defining \msg_generic_set_code:nn on line 180 +Defining \msg_direct_interrupt:xxxxn on line 207 +Defining \msg_direct_interrupt_aux:n on line 214 +Defining \msg_direct_log:xx on line 221 +Defining \msg_direct_term:xx on line 228 +Defining \msg_new:nnnnn on line 229 +Defining \msg_new:nnnn on line 232 +Defining \msg_new:nnn on line 235 +Defining \msg_set:nnnnn on line 238 +Defining \msg_set:nnnn on line 241 +Defining \msg_set:nnn on line 244 +Defining \msg_class_new:nn on line 247 +Defining \msg_class_set:nn on line 263 +Defining \msg_use:nnnnxx on line 286 +Defining \msg_use_code: on line 287 +Defining \msg_use_loop:n on line 288 +Defining \msg_use_aux:nnn on line 297 +Defining \msg_use_aux:nn on line 308 +Defining \msg_use_loop_check:nn on line 316 +Defining \msg_fatal:nnxx on line 317 +\l_msg_redirect_fatal_prop=\toks34 +Defining \msg_error:nnxx on line 328 +\l_msg_redirect_error_prop=\toks35 +Defining \msg_warning:nnxx on line 349 +\l_msg_redirect_warning_prop=\toks36 +Defining \msg_info:nnxx on line 356 +\l_msg_redirect_info_prop=\toks37 +Defining \msg_log:nnxx on line 363 +\l_msg_redirect_log_prop=\toks38 +Defining \msg_trace:nnxx on line 369 +\l_msg_redirect_trace_prop=\toks39 +Defining \msg_none:nnxx on line 375 +\l_msg_redirect_none_prop=\toks40 +Defining \msg_redirect_class:nn on line 378 +Defining \msg_redirect_module:nnn on line 381 +Defining \msg_redirect_name:nnn on line 384 +Defining \msg_kernel_new:nnnn on line 385 +Defining \msg_kernel_new:nnn on line 388 +Defining \msg_kernel_new:nn on line 391 +Defining \msg_kernel_set:nnnn on line 394 +Defining \msg_kernel_set:nnn on line 397 +Defining \msg_kernel_set:nn on line 400 +Defining \msg_kernel_classes_new:n on line 410 +Defining \msg_kernel_fatal:nxx on line 421 +Defining \msg_kernel_fatal:nx on line 422 +Defining \msg_kernel_fatal:n on line 422 +Defining \msg_kernel_error:nxx on line 443 +Defining \msg_kernel_error:nx on line 444 +Defining \msg_kernel_error:n on line 444 +Defining \msg_kernel_warning:nxx on line 445 +Defining \msg_kernel_warning:nx on line 448 +Defining \msg_kernel_warning:n on line 448 +Defining \msg_kernel_info:nxx on line 449 +Defining \msg_kernel_info:nx on line 452 +Defining \msg_kernel_info:n on line 452 +Defining \msg_text > LaTeX/coding-bug:nn on line 453 +Defining \msg_text > LaTeX/message-unknown:nn on line 459 +Defining \msg_text > LaTeX/message-class-unknown:nn on line 467 +Defining \msg_text > LaTeX/message-loop:nn on line 474 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3skip.sty +Package: l3skip 2009/05/28 v1362 L3 Experimental skip registers +Defining \skip_new:N on line 56 +Defining \skip_new:c on line 60 +Defining \skip_set:Nn on line 61 +Defining \skip_gset:Nn on line 64 +Defining \skip_set:cn on line 68 +Defining \skip_gset:cn on line 69 +Defining \skip_zero:N on line 70 +Defining \skip_gzero:N on line 73 +Defining \skip_zero:c on line 77 +Defining \skip_gzero:c on line 78 +Defining \skip_add:Nn on line 79 +Defining \skip_add:cn on line 82 +Defining \skip_sub:Nn on line 83 +Defining \skip_gadd:Nn on line 86 +Defining \skip_gadd:cn on line 90 +Defining \skip_gsub:Nn on line 91 +Defining \skip_horizontal:N on line 95 +Defining \skip_horizontal:c on line 96 +Defining \skip_horizontal:n on line 97 +Defining \skip_vertical:N on line 98 +Defining \skip_vertical:c on line 99 +Defining \skip_vertical:n on line 100 +Defining \skip_use:N on line 101 +Defining \skip_use:c on line 102 +Defining \skip_show:N on line 103 +Defining \skip_show:c on line 104 +Defining \skip_eval:n on line 105 +Defining \l_tmpa_skip on line 108 +\l_tmpa_skip=\skip46 +Defining \l_tmpb_skip on line 109 +\l_tmpb_skip=\skip47 +Defining \l_tmpc_skip on line 110 +\l_tmpc_skip=\skip48 +Defining \g_tmpa_skip on line 111 +\g_tmpa_skip=\skip49 +Defining \g_tmpb_skip on line 112 +\g_tmpb_skip=\skip50 +Defining \skip_if_infinite_glue_p:n on line 120 +Defining \skip_if_infinite_glue:nTF on line 120 +Defining \skip_if_infinite_glue:nT on line 120 +Defining \skip_if_infinite_glue:nF on line 120 +Defining \skip_split_finite_else_action:nnNN on line 121 +Defining \dim_new:N on line 133 +Defining \dim_new:c on line 137 +Defining \dim_set:Nn on line 138 +Defining \dim_set:cn on line 139 +Defining \dim_set:Nc on line 139 +Defining \dim_gset:Nn on line 140 +Defining \dim_gset:cn on line 141 +Defining \dim_gset:Nc on line 141 +Defining \dim_gset:cc on line 141 +Defining \dim_zero:N on line 142 +Defining \dim_zero:c on line 143 +Defining \dim_gzero:N on line 144 +Defining \dim_gzero:c on line 145 +Defining \dim_add:Nn on line 146 +Defining \dim_add:cn on line 149 +Defining \dim_add:Nc on line 149 +Defining \dim_gadd:Nn on line 150 +Defining \dim_gadd:cn on line 151 +Defining \dim_sub:Nn on line 152 +Defining \dim_sub:cn on line 153 +Defining \dim_sub:Nc on line 153 +Defining \dim_gsub:Nn on line 154 +Defining \dim_gsub:cn on line 155 +Defining \dim_use:N on line 156 +Defining \dim_use:c on line 157 +Defining \dim_show:N on line 158 +Defining \dim_show:c on line 159 +Defining \l_tmpa_dim on line 160 +\l_tmpa_dim=\dimen122 +Defining \l_tmpb_dim on line 161 +\l_tmpb_dim=\dimen123 +Defining \l_tmpc_dim on line 162 +\l_tmpc_dim=\dimen124 +Defining \l_tmpd_dim on line 163 +\l_tmpd_dim=\dimen125 +Defining \g_tmpa_dim on line 164 +\g_tmpa_dim=\dimen126 +Defining \g_tmpb_dim on line 165 +\g_tmpb_dim=\dimen127 +Defining \c_zero_dim on line 166 +Defining \c_max_dim on line 167 +Defining \dim_eval:n on line 168 +Defining \if_dim:w on line 169 +Defining \dim_compare_p:nNn on line 173 +Defining \dim_compare:nNnTF on line 173 +Defining \dim_compare:nNnT on line 173 +Defining \dim_compare:nNnF on line 173 +Defining \dim_while_do:nNnn on line 174 +Defining \dim_until_do:nNnn on line 177 +Defining \dim_do_while:nNnn on line 180 +Defining \dim_do_until:nNnn on line 183 +Defining \muskip_new:N on line 186 +Defining \muskip_set:Nn on line 190 +Defining \muskip_gset:Nn on line 191 +Defining \muskip_add:Nn on line 192 +Defining \muskip_gadd:Nn on line 193 +Defining \muskip_sub:Nn on line 194 +Defining \muskip_gsub:Nn on line 195 +Defining \muskip_use:N on line 196 +)) +(/usr/share/texmf-texlive/tex/latex/expl3/l3box.sty +Package: l3box 2009/06/01 v1375 L3 Experimental Box module +Defining \box_new:N on line 56 +Defining \box_new:c on line 60 +Defining \if_hbox:N on line 61 +Defining \if_vbox:N on line 62 +Defining \if_box_empty:N on line 63 +Defining \box_if_horizontal_p:N on line 66 +Defining \box_if_horizontal:NTF on line 66 +Defining \box_if_horizontal:NT on line 66 +Defining \box_if_horizontal:NF on line 66 +Defining \box_if_vertical_p:N on line 69 +Defining \box_if_vertical:NTF on line 69 +Defining \box_if_vertical:NT on line 69 +Defining \box_if_vertical:NF on line 69 +Defining \box_if_horizontal_p:c on line 70 +Defining \box_if_horizontal:cTF on line 71 +Defining \box_if_horizontal:cT on line 72 +Defining \box_if_horizontal:cF on line 73 +Defining \box_if_vertical_p:c on line 74 +Defining \box_if_vertical:cTF on line 75 +Defining \box_if_vertical:cT on line 76 +Defining \box_if_vertical:cF on line 77 +Defining \box_if_empty_p:N on line 80 +Defining \box_if_empty:NTF on line 80 +Defining \box_if_empty:NT on line 80 +Defining \box_if_empty:NF on line 80 +Defining \box_if_empty_p:c on line 81 +Defining \box_if_empty:cTF on line 82 +Defining \box_if_empty:cT on line 83 +Defining \box_if_empty:cF on line 84 +Defining \box_set_eq:NN on line 85 +Defining \box_set_eq:cN on line 86 +Defining \box_set_eq:Nc on line 86 +Defining \box_set_eq:cc on line 86 +Defining \box_gset_eq:NN on line 87 +Defining \box_gset_eq:cN on line 88 +Defining \box_gset_eq:Nc on line 88 +Defining \box_gset_eq:cc on line 88 +Defining \l_last_box on line 89 +Defining \box_set_to_last:N on line 90 +Defining \box_set_to_last:c on line 91 +Defining \box_gset_to_last:N on line 92 +Defining \box_gset_to_last:c on line 93 +Defining \box_move_left:nn on line 94 +Defining \box_move_right:nn on line 95 +Defining \box_move_up:nn on line 96 +Defining \box_move_down:nn on line 97 +Defining \box_clear:N on line 98 +Defining \box_clear:c on line 99 +Defining \box_gclear:N on line 100 +Defining \box_gclear:c on line 101 +Defining \box_ht:N on line 102 +Defining \box_dp:N on line 103 +Defining \box_wd:N on line 104 +Defining \box_ht:c on line 105 +Defining \box_dp:c on line 106 +Defining \box_wd:c on line 107 +Defining \box_use_clear:N on line 108 +Defining \box_use_clear:c on line 109 +Defining \box_use:N on line 110 +Defining \box_use:c on line 111 +Defining \box_show:c on line 113 +Defining \l_tmpa_box on line 115 +Defining \l_tmpb_box on line 116 +\l_tmpb_box=\box30 +Defining \vbox:n on line 117 +Defining \vbox_set:Nn on line 118 +Defining \vbox_set:cn on line 119 +Defining \vbox_gset:Nn on line 120 +Defining \vbox_gset:cn on line 121 +Defining \vbox_set_to_ht:Nnn on line 122 +Defining \vbox_set_to_ht:cnn on line 125 +Defining \vbox_gset_to_ht:Nnn on line 126 +Defining \vbox_gset_to_ht:cnn on line 127 +Defining \vbox_gset_to_ht:ccn on line 127 +Defining \exp_args:Nccn on line 127 +Defining \vbox_set_inline_begin:N on line 128 +Defining \vbox_set_inline_end: on line 130 +Defining \vbox_gset_inline_begin:N on line 131 +Defining \vbox_gset_inline_end: on line 133 +Defining \vbox_to_ht:nn on line 134 +Defining \vbox_to_zero:n on line 135 +Defining \vbox_set_split_to_ht:NNn on line 136 +Defining \vbox_unpack:N on line 139 +Defining \vbox_unpack:c on line 140 +Defining \vbox_unpack_clear:N on line 141 +Defining \vbox_unpack_clear:c on line 142 +Defining \hbox:n on line 143 +Defining \hbox_set:Nn on line 144 +Defining \hbox_set:cn on line 145 +Defining \hbox_gset:Nn on line 146 +Defining \hbox_gset:cn on line 147 +Defining \hbox_set_to_wd:Nnn on line 148 +Defining \hbox_set_to_wd:cnn on line 151 +Defining \hbox_gset_to_wd:Nnn on line 152 +Defining \hbox_gset_to_wd:cnn on line 153 +Defining \hbox_set_inline_begin:N on line 154 +Defining \hbox_set_inline_begin:c on line 157 +Defining \hbox_set_inline_end: on line 158 +Defining \hbox_gset_inline_begin:N on line 159 +Defining \hbox_gset_inline_begin:c on line 162 +Defining \hbox_gset_inline_end: on line 163 +Defining \hbox_to_wd:nn on line 164 +Defining \hbox_to_zero:n on line 165 +Defining \hbox_unpack:N on line 166 +Defining \hbox_unpack:c on line 167 +Defining \hbox_unpack_clear:N on line 168 +Defining \hbox_unpack_clear:c on line 169 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3keyval.sty +Package: l3keyval 2009/07/13 v1424 L3 Experimental keyval processing +Defining \l_KV_tmpa_tl on line 56 +Defining \l_KV_tmpb_tl on line 57 +Defining \c_KV_single_equal_sign_tl on line 58 +\l_KV_parse_toks=\toks41 +\l_KV_currkey_toks=\toks42 +\l_KV_currval_toks=\toks43 +Defining \l_KV_remove_one_level_of_braces_bool on line 62 +Defining \KV_sanitize_outerlevel_active_equals:N on line 76 +Defining \KV_sanitize_outerlevel_active_commas:N on line 76 +Defining \KV_remove_surrounding_spaces:nw on line 79 +Defining \KV_remove_surrounding_spaces_auxi:w on line 82 +Defining \KV_remove_surrounding_spaces_auxii:w on line 85 +Defining \KV_add_value_element:w on line 86 +Defining \KV_set_key_element:w on line 94 +Defining \KV_add_element_aux:w on line 109 +Defining \KV_parse_sanitize_aux:n on line 110 +Defining \KV_parse_no_sanitize_aux:n on line 121 +Defining \KV_split_key_value_current:w on line 139 +Defining \KV_split_key_value_space_removal:w on line 140 +Defining \KV_split_key_value_space_removal_detect_error:wTF on line 157 +Defining \KV_val_preserve_braces:NnN on line 160 +Defining \KV_split_key_value_space_removal_aux:w on line 161 +Defining \KV_split_key_value_no_space_removal:w on line 187 +Defining \KV_key_no_value_elt:n on line 201 +Defining \KV_key_value_elt:nn on line 202 +Defining \KV_parse_no_space_removal_no_sanitize:n on line 203 +Defining \KV_parse_space_removal_sanitize:n on line 207 +Defining \KV_parse_space_removal_no_sanitize:n on line 211 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3keys.sty +Package: l3keys 2009/09/17 v1542 L3 Experimental key-value support +Defining \c_keys_root_tl on line 56 +Defining \c_keys_properties_root_tl on line 57 +Defining \c_keys_value_forbidden_tl on line 58 +Defining \c_keys_value_required_tl on line 59 +Defining \l_keys_choice_int on line 60 +\l_keys_choice_int=\count118 +Defining \l_keys_choice_tl on line 61 +Defining \l_keys_choice_code_tl on line 62 +Defining \l_keys_key_tl on line 63 +Defining \l_keys_path_tl on line 64 +Defining \l_keys_property_tl on line 65 +Defining \l_keys_module_tl on line 66 +Defining \l_keys_nesting_seq on line 67 +Defining \l_keys_nesting_tl on line 68 +Defining \l_keys_no_value_bool on line 69 +\l_keys_value_toks=\toks44 +Defining \keys_bool_set:NN on line 86 +Defining \keys_choice_code_store:x on line 89 +Defining \keys_choice_find:n on line 94 +Defining \keys_choice_make: on line 103 +Defining \keys_choices_generate:n on line 117 +Defining \keys_choices_generate_aux:n on line 126 +Defining \keys_cmd_set:nn on line 131 +Defining \keys_cmd_set:nx on line 136 +Defining \keys_cmd_set_aux:n on line 140 +Defining \keys_default_set:n on line 143 +Defining \keys_default_set:V on line 144 +Defining \keys_define:nn on line 157 +Defining \keys_define_elt:n on line 161 +Defining \keys_define_elt:nn on line 165 +Defining \keys_define_elt_aux:nn on line 176 +Defining \keys_define_key:n on line 191 +Defining \keys_execute: on line 196 +Defining \keys_execute_unknown: on line 201 +Defining \keys_execute_aux:nn on line 209 +Defining \keys_if_value_requirement:nTF on line 217 +Defining \keys_meta_make:n on line 226 +Defining \keys_meta_make:x on line 227 +Defining \keys_property_find:n on line 235 +Defining \keys_property_find_aux:n on line 238 +Defining \keys_property_find_aux:w on line 239 +Defining \keys_property_new:nn on line 252 +Defining \keys_property_undefine:n on line 255 +Defining \keys_set:nn on line 268 +Defining \keys_set:nV on line 269 +Defining \keys_set:nv on line 269 +Defining \exp_args:Nnv on line 269 +Defining \keys_set_elt:n on line 273 +Defining \keys_set_elt:nn on line 277 +Defining \keys_set_elt_aux:nn on line 291 +Defining \keys_set_elt_aux: on line 303 +Defining \keys_show:nn on line 306 +Defining \keys_tmp:w on line 307 +Defining \keys_value_or_default:n on line 317 +Defining \keys_value_requirement:n on line 321 +Defining \keys_variable_set:NnNN on line 329 +Defining \keys_properties.bool_set:N on line 332 +Defining \keys_properties.bool_gset:N on line 335 +Defining \keys_properties.choice: on line 338 +Defining \keys_properties.choice_code:n on line 341 +Defining \keys_properties.choice_code:x on line 344 +Defining \keys_properties.code:n on line 347 +Defining \keys_properties.code:x on line 350 +Defining \keys_properties.default:n on line 353 +Defining \keys_properties.default:V on line 356 +Defining \keys_properties.dim_set:N on line 359 +Defining \keys_properties.dim_gset:N on line 362 +Defining \keys_properties.generate_choices:n on line 365 +Defining \keys_properties.int_set:N on line 368 +Defining \keys_properties.int_gset:N on line 371 +Defining \keys_properties.meta:n on line 374 +Defining \keys_properties.meta:x on line 377 +Defining \keys_properties.skip_set:N on line 380 +Defining \keys_properties.skip_gset:N on line 383 +Defining \keys_properties.tl_set:N on line 386 +Defining \keys_properties.tl_set_x:N on line 389 +Defining \keys_properties.tl_gset:N on line 392 +Defining \keys_properties.tl_gset_x:N on line 395 +Defining \keys_properties.value_forbidden: on line 398 +Defining \keys_properties.value_required: on line 401 +Defining \msg_text > LaTeX/key-choice-unknown:nn on line 402 +Defining \msg_text > LaTeX/key-initial-without-code:nn on line 406 +Defining \msg_text > LaTeX/key-unknown:nn on line 410 +Defining \msg_text > LaTeX/key-value-forbidden:nn on line 413 +Defining \msg_text > LaTeX/key-value-required:nn on line 417 +Defining \msg_text > LaTeX/key-no-property:nn on line 421 +Defining \msg_text > LaTeX/key-no-set-function:nn on line 424 +Defining \msg_text > LaTeX/key-property-unknown:nn on line 434 +Defining \msg_text > LaTeX/key-property-value-required:nn on line 437 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3precom.sty +Package: l3precom 2009/08/08 v1453 L3 Experimental precompilation module +\c_cs_dump_stream=\write3 +Defining \g_cs_dump_name_tl on line 60 +Defining \g_cs_dump_seq on line 61 +Defining \cs_record_name:N on line 62 +Defining \cs_record_name:c on line 67 +Defining \cs_dump: on line 68 +Defining \cs_load_dump:n on line 94 +Defining \g_gen_sym_num on line 103 +Defining \g_ggen_sym_num on line 104 +Defining \cs_gen_sym:N on line 105 +Defining \cs_ggen_sym:N on line 110 +Defining \g_cs_trace_seq on line 115 +Defining \g_register_trace_seq on line 116 +Defining \dumpLaTeXstate on line 123 +) +(/usr/share/texmf-texlive/tex/latex/expl3/l3calc.sty +Package: l3calc 2009/05/28 v1362 L3 Experimental calc module +Defining \l_calc_expression_tl on line 56 +Defining \g_calc_A_register on line 57 +Defining \l_calc_B_register on line 58 +Defining \l_calc_current_type_int on line 59 +\l_calc_current_type_int=\count119 +Defining \g_calc_A_int on line 60 +\g_calc_A_int=\count120 +Defining \l_calc_B_int on line 61 +\l_calc_B_int=\count121 +Defining \l_calc_C_int on line 62 +\l_calc_C_int=\count122 +Defining \g_calc_A_dim on line 63 +\g_calc_A_dim=\dimen128 +Defining \l_calc_B_dim on line 64 +\l_calc_B_dim=\dimen129 +Defining \l_calc_C_dim on line 65 +\l_calc_C_dim=\dimen130 +Defining \g_calc_A_skip on line 66 +\g_calc_A_skip=\skip51 +Defining \l_calc_B_skip on line 67 +\l_calc_B_skip=\skip52 +Defining \l_calc_C_skip on line 68 +\l_calc_C_skip=\skip53 +Defining \g_calc_A_muskip on line 69 +\g_calc_A_muskip=\muskip11 +Defining \l_calc_B_muskip on line 70 +\l_calc_B_muskip=\muskip12 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written on rapport.dvi (4 pages, 19104 bytes). diff --git a/rapport.tex b/rapport.tex new file mode 100644 index 0000000..daab39c --- /dev/null +++ b/rapport.tex @@ -0,0 +1,356 @@ +% Fag: Numerisk matematikk +% Project 1 +% Utført av: Bjørn, Mats og Edvard + +\documentclass[12pt, a4paper]{article} +\usepackage[latin1]{inputenc} +\usepackage{listings} +\usepackage{graphicx} +\usepackage{color} +\usepackage{amssymb, amsmath} +\setcounter{secnumdepth}{0} +\setlength{\textheight}{240mm} +\setlength{\textwidth}{150mm} +\usepackage[hmargin=1cm,vmargin=2cm]{geometry} +\usepackage{xfrac} + +\definecolor{orange}{rgb}{1,0.5,0} + +\lstset{ % + language=Matlab, % the language of the code + basicstyle=\footnotesize, % the size of the fonts that are used for the code + numbers=left, % where to put the line-numbers + numberstyle=\footnotesize, % the size of the fonts that are used for the line-numbers + stepnumber=1, % the step between two line-numbers. If it's 1, each line + % will be numbered + numbersep=5pt, % how far the line-numbers are from the code + backgroundcolor=\color{white}, % choose the background color. You must add \usepackage{color} + tabsize=4, % sets default tabsize to 2 spaces + captionpos=b, % sets the caption-position to bottom + breaklines=true, % sets automatic line breaking + keywordstyle=\color{orange}\bfseries, + commentstyle=\color[rgb]{0,0,0.9} +} + +\begin{document} + +\title{Semester Project 2 in TMA4215} +\author{ + Candidate numbers: 10012, 10023, 10056 +} + +\maketitle + +\abstract{ +In this project we look at quadratic splines for interpolating a set of data +points. We also use the Romberg method for numerical evaluation of an integral. +In the end we use a mid-point Runge-Kutta method for solving a differential +equation. +} + +\section{Part 1} +\subsection{Problem 1} +\subsubsection{a)} +Since we are going to make a quadratic spline, we know that the derivative of +ths spline will be a function consisting of interconnected line segments. If we +call the spline $s$ and write $s_i$ for the value of the spline in data point +$i$, we know that +\begin{equation} + \label{eq:sdiv_linear} + s'(x) = s_i' + \frac{s_{i+1}' - s_i'}{x_{i+1} - x_i} \left(x-x_i\right). +\end{equation} +Integrating gives +\begin{equation} + \label{eq:sdiv_integrated} + s(x) = s_i'x + \frac{s_{i+1}' - s_i'}{2\left( x_{i+1} - x_i \right)} \left(x-x_i\right)^2 + C_i, +\end{equation} +and evaluating in the point $x_i$ gives +\begin{equation} + \label{eq:sdiv_evaluated} + s(x_i) = s_i'x_i + C_i = f_i. +\end{equation} +The value of our spline in the data points, $s_i$ should be equal to the +function value in the data points, $f_i$. Using \eqref{eq:sdiv_integrated} and +\eqref{eq:sdiv_evaluated} to write the difference between two succeeding points +we get +\begin{equation} + \label{eq:s_difference} + s_{i+1} - s_i = \frac{1}{2} s_i' \left( x_{i+1} - x_i \right) + \frac{1}{2} + s_{i+1}' \left(x_{i+1} - x_i\right). +\end{equation} +Moving the derivatives to the left side and simplifying leads to the following +set of equations: +\begin{equation} + \label{eq:sder_sum} + s_i' + s_{i+1}' = \frac{2\left(f_{i+1} - f_i\right)}{x_{i+1} - x_i}. +\end{equation} +Together with the extra condition that $s'(x_0) = 0$, we can write this as the +matrix equation +\begin{equation} + \label{eq:s_matrix} + \begin{bmatrix} +1 & 0 & 0 & \hdots & 0 \\ +1 & 1 & 0 & & \\ +0 & 1 & 1 & & \\ +\vdots & & & \ddots & \\ +0 & & & 1 & 1 + \end{bmatrix} + \begin{bmatrix} +s_0' \\ +s_1' \\ +s_2' \\ +\vdots \\ +s_n' + \end{bmatrix} + = + \begin{bmatrix} +0 \\ +\frac{2\left(f_1 - f_0\right)}{x_1 - x_0} \\ +\frac{2\left(f_2 - f_1\right)}{x_2 - x_1} \\ +\frac{2\left(f_3 - f_2\right)}{x_3 - x_2} \\ +\vdots \\ +\frac{2\left(f_n - f_{n-1}\right)}{x_n - x_{n-1}} + \end{bmatrix} +\end{equation} +Inserting the interpolation data we obtain the derivatives of our spline, and +table \ref{tab:data} shows what our spline has to satisfy. +\begin{table}[h] + \caption{\sf Data determining our spline polynomials.} + \begin{center} + \begin{tabular}{c c c} + \hline + $x$ & $s(x)$ & $s'(x)$ \\ + \hline + 0 & 1 & 0 \\ + 0.5 & 2 & 4 \\ + 1 & 0.5 & -10 \\ + \hline + \end{tabular} + \end{center} + \label{tab:data} +\end{table} +From table \ref{tab:data}, we easily derive the two quadratic polynomials: +\begin{equation} + \label{eq:poly} + \begin{split} + p_1 &= 4x^2 + 1 \\ + p_2 &= -14x^2 + 18x - \frac{7}{2} + \end{split} +\end{equation} + +\subsubsection{b)} +To find an error bound we define +\begin{equation} + \label{eq:g_def} + g(x) = f(x) - s(x) - \frac{x-x_{j-1}}{\bar{x} - x_{j-1}}\left( f(\bar{x}) - + s(\bar{x}) \right) +\end{equation} +on the interval $I_j = [x_{j-1}, x_j]$ where $\bar{x} \in I_j$. Since this +function is $0$ in both $x_{j-1}$ and $\bar{x}$, we know that $\exists \xi \in +\left( x_{j-1}, \bar{x} \right)$ such that +\begin{equation} + \label{eq:g_der_zero} + 0 = g'(\xi) = f'(\xi) - s'(\xi) - \left(f(\bar{x}) - s(\bar{x})\right) + \frac{1}{\bar{x} - x_{j-i}}, +\end{equation} +which gives +\begin{equation} + \label{eq:xbar_error} + \left|f(\bar{x}) - s(\bar{x})\right| = \left(\bar{x} - x_{j-1}\right) + \left|f'(\xi) - s'(\xi)\right| \leq h\left(\|f'\|_{\infty} + + \max_i|s_i'|\right). +\end{equation} +If we choose $i$ such that $|s_i'| = \max_l|s_l'|$, we derive an upper bound for +$|s_i'|$ using equation \eqref{eq:s_difference} by looking at +\begin{equation} + \label{eq:2s_bound} + \begin{split} + 2s_{i+1}' \leq s_i' + s_{i+1}' &= \frac{2(f_{i+1} - f_i)}{h} \\ + &= 2f'(\eta) \\ + &\leq 2\|f'\|_{\infty} + \end{split} +\end{equation} +which simplifies to +\begin{equation} + \label{eq:si_bound} + \begin{split} + |s_i'| &= |2f'(\eta) - s_{i+1}'| \\ + &\leq 2|f'(\eta)| + |s_{i+1}'| \\ + &\leq 3|f'(\eta)| \leq 3\|f'\|_{\infty} + \end{split} +\end{equation} +Inserting into equation \eqref{eq:xbar_error} gives an upper bound for the error +\begin{equation} + \label{eq:error_bound} + \left|f(x) - s(x)\right| \leq 4h \|f'\|_{\infty}, \forall x \in + [a,b]. +\end{equation} + +\subsection{Problem 2} +\subsubsection{a)} +In this problem we are looking at how to use the Romberg method to evaluate the +following integral: +\begin{equation} + \label{eq:integral} + \varphi = \int{0}{\varphi} \frac{\mathrm{d}\theta}{\sqrt{1 - k^2 + \sin{\theta}^2}}. +\end{equation} + +Our code for integrating functions using the Romberg method is shown in the file +\emph{romberg.m}. Table \ref{tab:romberg_conv} shows the convergence of the +Romberg method applied to the elliptic integral when $L$ increases. +\begin{table}[h!] + \caption{\sf Showing the convergence for the Romberg method over the interval +$[0, 5]$ with $k = 0.5$ as $L$ increases.} + \begin{center} + \begin{tabular}{c l c} + \hline + $L$ & Romberg \\ + \hline + 1 & 5.34 \\ + 2 & 5.27 \\ + 3 & 5.40 \\ + 5 & 5.3879 \\ + 7 & 5.3878724 \\ + 10 & 5.38787242753557 \\ + \hline + quad() & 5.38787242753557 \\ + \hline + \end{tabular} + \end{center} + \label{tab:romberg_conv} +\end{table} + +For the complete elliptic integral of the first kind, we compare the error in +the Romberg method with the error in the trapezoid method. We observe that as +$L$ increases, the error in the trapezoid method decreases much faster than the +error in the Romberg method. For example with $k = 0.5$ and $L = 4$ the error in +the trapezoid method is $2.2e-16$ while the error in the Romberg method is +$1.5e-06$. + +The small error in the trapezoid method can be explained using the +Euler-Maclaurin formula which gives the error in terms of the derivatives of the +function in the end points: +\begin{equation} + \label{eq:euler_maclaurin} + S - I = \sum_{k=2}^p \frac{B_k}{k!} \left( f^{(k-1)}(n) - f^{(k-1)}(0) \right) + + R. +\end{equation} +One can easily derive that the all the odd derivatives of the function we are +integrating, \eqref{eq:integral}, are periodic and equal to zero in +$n \sfrac{\pi}{2}$. This, combined with the fact that all the odd Bernoulli +numbers are zero except for $B_1$, makes the error in \eqref{eq:euler_maclaurin} +disappear Q=EQE=WQR=)WQR)=. + +\subsubsection{b)} +The file \emph{romberg.m} shows the use of the given estimate for the error. +Although the accuracy increases when we decrease the given tolerance value, it +does not match the number of correct decimals. This is shown in table +\ref{tab:tol_test} where we have integrated with $k = 0.5$ from $0$ to $5$. +\begin{table}[h!] + \caption{\sf Estimated integral for decreasing values of tolerance limit.} + \begin{center} + \begin{tabular}{c l c} + \hline + Tolerance & Romberg \\ + \hline + $10^{-1}$ & 5.28 \\ + $10^{-3}$ & 5.39 \\ + $10^{-5}$ & 5.38786 \\ + $10^{-7}$ & 5.38786 \\ + $10^{-9}$ & 5.38787243 \\ + $10^{-11}$ & 5.38787242753 \\ + \hline + quad() & 5.38787242753557 \\ + \hline + \end{tabular} + \end{center} + \label{tab:tol_test} +\end{table} + +\subsubsection{c)} +The Octave file \emph{newton.m} shows an implementation of Newton's method to +solve $\varphi = am(u,k)$ which is the inverse of the elliptic integral. As the +Newton method is used to find the zero, we stop when the function is no further +away from zero than the given tolerance. + +We ran this method on 100 values between -1 and 1 and compared it with the +Octave function \emph{ellipj}. This gave an error of $C\cdot TOL$ where $C$ had +a maximum value of 5. + +When choosing larger values for $|u|$ our code does not work as well. We have to +increase our tolerance to get it to return something at all, and the error in +the results varies a lot. + +\subsection{Problem 3} +\subsubsection{a)} +The mid-point Runge-Kutta method is implemented in the Octave file +\emph{PENISPENIS.m}. This gives the relation between two succeeding function +values on the following form: +\begin{equation} + \label{eq:runge_kutta} + \mathbf{m}_{n+1} - \mathbf{m}_n = hf \left(t + \frac{1}{2} h, \frac{1}{2} +(\mathbf{m}_{n+1} + \mathbf{m}_n) \right). +\end{equation} +Since this is an implicit method we use Newtons method to estimate $y_{n+1}$ for +each step of Runge-Kutta. Rearranging so that we have something that can be +solved by Newton gives +\begin{equation} + \label{eq:runge_kutta_newton} + F(\mathbf{m}_{n+1}) = \mathbf{m}_{n+1} - \mathbf{m}_n - h +\mathbf{\bar{m}}_{n+1} + \times \left(T^{-1} \mathbf{\bar{m}}_{n+1}\right) = 0, +\end{equation} +where $\mathbf{\bar{m}}_{n+1} = \frac{1}{2} \left(\mathbf{m}_{n} + +\mathbf{m}_{n+1}\right)$. + +While $\mathbf{m}_n$ is a known constant calculated earlier, the +$\mathbf{m}_{n+1}$ is a three-dimensional vector of unknowns. We need the +Jacobian to use the Newton method: +\begin{equation} + \label{eq:jacobian} + J(\mathbf{x}) = + \begin{bmatrix} +1 & +\frac{h}{2} \left(\frac{1}{I_2} - \frac{1}{I_3}\right) x_3 & +\frac{h}{2} \left(\frac{1}{I_2} - \frac{1}{I_3}\right) x_2 \\ +\frac{h}{2} \left(\frac{1}{I_3} - \frac{1}{I_1}\right) x_3 & +1 & +\frac{h}{2} \left(\frac{1}{I_3} - \frac{1}{I_1}\right) x_1 \\ +\frac{h}{2} \left(\frac{1}{I_1} - \frac{1}{I_2}\right) x_2 & +\frac{h}{2} \left(\frac{1}{I_1} - \frac{1}{I_2}\right) x_1 & +1 \\ + \end{bmatrix}. +\end{equation} +The Newton iteration is now given by +\begin{equation} + \mathbf{m}_{n+1}^{k+1} = \mathbf{m}_{n+1}^k - + \frac{F(\mathbf{m}_{n+1}^k)}{J(\mathbf{\bar{m}}_{n+1}^k)}, +\end{equation} +where we iterate over $k$ until we are pleased with our new approximation for +$\mathbf{m}_{n+1}$. + +To find the order of our method we consider our means of solving the ODE with +step sizes $\frac{1}{2^k}$ where $k = 1, 2, ..., 5$. For each $k$ we compute the +values of our Runge-Kutta method and the provided elliptic method in the +interval zero to one. The values of the Runge-Kutta method is subtracted from +the elliptic method, and the norms of these give a sequence of errors. ***. The +elements in the sequence are now on the form $C (\frac{1}{2^k})^p$ where $p$ is +the order of the method. We now divide the first element by the second, the +second by the third and so on. Taking the base-two-logarithm of this gives us +$p$. This is found numerically to be $2$ in the Octave function +\emph{freerigid.m}. + +HER MÅ VI SETTE INN EN TABELL SOM SAMMENLIGNER LØSNINGENE + +\subsubsection{b)} +We chose to use the Octave-forge function \emph{ode45} as a comparison against +the other methods. With the time interval $[0, 50]$, the Octave script +\emph{rigidbodysimulator.m} produced the plot showed in figure +\ref{fig:ode_comparison}. PLX INSERT IGUURE. EXPLAIN ANALYZE. We observe that +the \emph{ode45}-method, which is a Runge-Kutta-method of order (4, 5), has an +error that often grows faster than the others. + +\subsubsection{c)} + +\end{document} diff --git a/rigidbodysimulator.m b/rigidbodysimulator.m new file mode 100644 index 0000000..4885802 --- /dev/null +++ b/rigidbodysimulator.m @@ -0,0 +1,33 @@ +start = [1;1;1]; +interval = [0, 30]; +h = 1; +tspan = [0:h:30]; +T = diag([3, 4, 5]); +f = @(t,x) cross(x,T\x); + +midpoint = []; +midpoint(1,:) = start; +elliptic = []; +elliptic(1,:) = start; +ode = []; +ode(1,:) = start; +t = 0; +i = 2; +while t < interval(2) + midpoint(i,:) = newtfree(midpoint(i-1,:)', h, T); + elliptic(i,:) = freewR(h, T(1,1), T(2,2), T(3,3), elliptic(i-1,1), elliptic(i-1,2), elliptic(i-1,3)); + t = t + h; + i = i + 1; +end +ode = ode45(f, interval, start'); + +clf; +hold off; +figure(1); +hold on; +plot3(midpoint(:,1), midpoint(:,2), midpoint(:,3), 'r-'); +hold on; +plot3(elliptic(:,1), elliptic(:,2), elliptic(:,3), 'b-'); +hold on; +plot3(ode.y(:,1), ode.y(:,2), ode.y(:,3), 'g-'); + diff --git a/romberg.m b/romberg.m new file mode 100644 index 0000000..3d903d2 --- /dev/null +++ b/romberg.m @@ -0,0 +1,30 @@ +function integral = romberg(k, a, b, tol) + +f = @(x) 1/sqrt(1 - k^2 * sin(x)^2); + +R(1,1) = 0.5 * (b - a) * (f(a) + f(b)); +E(1,1) = 0; + +row = 2; +while 1 + hn = (b - a)/(2.0^(row-1)); + R(row,1) = 0.5 * R(row-1,1); + s = 0; + for k = [1:2^(row-2)] + s += f(a + (2*k-1)*hn); + end + R(row,1) += hn * s; + E(row,1) = 0; + + for col = [2:row] + R(row,col) = ((4^(col-1))*R(row,col-1) - R(row-1,col-1)) / (4**(col-1) - 1); + %est = R(row,col) - R(row,col-1); + est = (1.0/(4.0^col - 1)) * (R(row,col-1) - R(row-1,col-1)); + E(row,col) = abs(est); + if abs(est) < tol + integral = R(row,col); + return + end + end + row++; +end diff --git a/romberg.py b/romberg.py new file mode 100644 index 0000000..51e3d97 --- /dev/null +++ b/romberg.py @@ -0,0 +1,26 @@ +import numpy as np + +def romberg(f, a, b, tol): + + R = np.zeros((tol, tol)) + + R[0][0] = 0.5 * (b - a) * (f(a) + f(b)) + for i in xrange(1, tol): + hn = float(b - a)/float(2**i) + R[i][0] = 0.5 * R[i-1][0] + R[i][0] += hn * sum([f(a + (2*k-1)*hn) for k in xrange(1, 2**(i-1)+1)]) + print hn + print sum([f(a + (2*k-1)*hn) for k in xrange(1, 2**(i-1)+1)]) + + for i in xrange(1, tol): + for row in xrange(0, tol - i): + R[row][i] = ((4**i)*R[row+1][i-1] - R[row][i-1]) / (4**i - 1) + + return R + +def main(): + + print romberg(lambda x: np.sqrt(x), 0, 5, 6) + +if __name__ == '__main__': + main() diff --git a/rombergold.m b/rombergold.m new file mode 100644 index 0000000..4b3a26d --- /dev/null +++ b/rombergold.m @@ -0,0 +1,24 @@ +function [rll, rl0] = rombergold(f, a, b, L) + +R(1,1) = 0.5 * (b - a) * (f(a) + f(b)); + +for i = [2:L] + hn = (b - a)/(2.0^(i-1)); + R(i,1) = 0.5 * R(i-1,1); + s = 0; + for k = [1:2^(i-2)] + s += f(a + (2*k-1)*hn); + end + R(i,1) += hn * s; +end + +for i = [2:L] + for row = [i:L] + R(row,i) = ((4^(i-1))*R(row,i-1) - R(row-1,i-1)) / (4**(i-1) - 1); + end +end + +rll = R(L,L); +rl0 = R(L,1); + +end