add lecture 10

lecture_10/.ipynb_checkpoints/lecture_10-checkpoint.ipynb

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+{
+ "cells": [],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

lecture_10/GaussNaive.m

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+function [x,Aug] = GaussNaive(A,y)
+% GaussNaive: naive Gauss elimination
+%   x = GaussNaive(A,b): Gauss elimination without pivoting.
+% input:
+%   A = coefficient matrix
+%   y = right hand side vector
+% output:
+%   x = solution vector
+[m,n] = size(A);
+if m~=n, error('Matrix A must be square'); end
+nb = n+1;
+Aug = [A y];
+% forward elimination
+for k = 1:n-1
+  for i = k+1:n
+    factor = Aug(i,k)/Aug(k,k);
+    Aug(i,k:nb) = Aug(i,k:nb)-factor*Aug(k,k:nb);
+  end
+end
+% back substitution
+x = zeros(n,1);
+x(n) = Aug(n,nb)/Aug(n,n);
+for i = n-1:-1:1
+  x(i) = (Aug(i,nb)-Aug(i,i+1:n)*x(i+1:n))/Aug(i,i);
+end

lecture_10/GaussPivot.m

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+function [x,Aug,npivots] = GaussPivot(A,b)
+% GaussPivot: Gauss elimination pivoting
+%   x = GaussPivot(A,b): Gauss elimination with pivoting.
+% input:
+%   A = coefficient matrix
+%   b = right hand side vector
+% output:
+%   x = solution vector
+[m,n]=size(A);
+if m~=n, error('Matrix A must be square'); end
+nb=n+1;
+Aug=[A b];
+npivots=0; % initially no pivots used
+% forward elimination
+for k = 1:n-1
+  % partial pivoting
+  [big,i]=max(abs(Aug(k:n,k)));
+  ipr=i+k-1;
+  if ipr~=k
+    npivots=npivots+1; % if the max is not the current index ipr, pivot count
+    Aug([k,ipr],:)=Aug([ipr,k],:);
+  end
+  for i = k+1:n
+    factor=Aug(i,k)/Aug(k,k);
+    Aug(i,k:nb)=Aug(i,k:nb)-factor*Aug(k,k:nb);
+  end
+end
+% back substitution
+x=zeros(n,1);
+x(n)=Aug(n,nb)/Aug(n,n);
+for i = n-1:-1:1
+  x(i)=(Aug(i,nb)-Aug(i,i+1:n)*x(i+1:n))/Aug(i,i);
+end

lecture_10/Tridiag.m

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+function x = Tridiag(e,f,g,r)
+% Tridiag: Tridiagonal equation solver banded system
+%   x = Tridiag(e,f,g,r): Tridiagonal system solver.
+% input:
+%   e = subdiagonal vector
+%   f = diagonal vector
+%   g = superdiagonal vector
+%   r = right hand side vector
+% output:
+%   x = solution vector
+n=length(f);
+% forward elimination
+for k = 2:n
+  factor = e(k)/f(k-1);
+  f(k) = f(k) - factor*g(k-1);
+  r(k) = r(k) - factor*r(k-1);
+end
+% back substitution
+x(n) = r(n)/f(n);
+for k = n-1:-1:1
+  x(k) = (r(k)-g(k)*x(k+1))/f(k);
+end

lecture_10/lecture_10.aux

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+\relax 
+\providecommand\hyper@newdestlabel[2]{}
+\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
+\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
+\global\let\oldcontentsline\contentsline
+\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
+\global\let\oldnewlabel\newlabel
+\gdef\newlabel#1#2{\newlabelxx{#1}#2}
+\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
+\AtEndDocument{\ifx\hyper@anchor\@undefined
+\let\contentsline\oldcontentsline
+\let\newlabel\oldnewlabel
+\fi}
+\fi}
+\global\let\hyper@last\relax 
+\gdef\HyperFirstAtBeginDocument#1{#1}
+\providecommand\HyField@AuxAddToFields[1]{}
+\providecommand\HyField@AuxAddToCoFields[2]{}
+\providecommand \oddpage@label [2]{}
+\@writefile{toc}{\contentsline {section}{\numberline {1}Gauss Elimination}{1}{section.1}}
+\newlabel{gauss-elimination}{{1}{1}{Gauss Elimination}{section.1}{}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.0.1}Solving sets of equations with matrix operations}{1}{subsubsection.1.0.1}}
+\newlabel{solving-sets-of-equations-with-matrix-operations}{{1.0.1}{1}{Solving sets of equations with matrix operations}{subsubsection.1.0.1}{}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Gauss elimination}{4}{subsection.1.1}}
+\newlabel{gauss-elimination}{{1.1}{4}{Gauss elimination}{subsection.1.1}{}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.1.1}Solving sets of equations systematically}{4}{subsubsection.1.1.1}}
+\newlabel{solving-sets-of-equations-systematically}{{1.1.1}{4}{Solving sets of equations systematically}{subsubsection.1.1.1}{}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Springs-masses\relax }}{4}{figure.caption.1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Automate Gauss Elimination}{6}{subsection.1.2}}
+\newlabel{automate-gauss-elimination}{{1.2}{6}{Automate Gauss Elimination}{subsection.1.2}{}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}Problem (Diagonal element is zero)}{6}{subsection.1.3}}
+\newlabel{problem-diagonal-element-is-zero}{{1.3}{6}{Problem (Diagonal element is zero)}{subsection.1.3}{}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.3.1}Spring-Mass System again}{8}{subsubsection.1.3.1}}
+\newlabel{spring-mass-system-again}{{1.3.1}{8}{Spring-Mass System again}{subsubsection.1.3.1}{}}
+\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Springs-masses\relax }}{9}{figure.caption.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Tridiagonal matrix}{9}{subsection.1.4}}
+\newlabel{tridiagonal-matrix}{{1.4}{9}{Tridiagonal matrix}{subsection.1.4}{}}
+\gdef \LT@i {\LT@entry 
+    {2}{67.64778pt}\LT@entry 
+    {1}{261.39424pt}}

lecture_10/lecture_10.blg

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