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recompile software manual after hw 7 and hw 7 p6

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Elizabeth Hunt 2023-11-27 15:13:34 -07:00
parent 0981ffa00c
commit f0b420e8cd
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@ -1,4 +1,4 @@
% Created 2023-11-15 Wed 14:43 % Created 2023-11-27 Mon 15:10
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[11pt]{article} \documentclass[11pt]{article}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -30,7 +30,7 @@
\setlength\parindent{0pt} \setlength\parindent{0pt}
\section{Design} \section{Design}
\label{sec:org78303cd} \label{sec:org138ae3c}
The LIZFCM static library (at \url{https://github.com/Simponic/math-4610}) is a successor to my The LIZFCM static library (at \url{https://github.com/Simponic/math-4610}) is a successor to my
attempt at writing codes for the Fundamentals of Computational Mathematics course in Common attempt at writing codes for the Fundamentals of Computational Mathematics course in Common
Lisp, but the effort required to meet the requirement of creating a static library became Lisp, but the effort required to meet the requirement of creating a static library became
@ -47,7 +47,7 @@ the C programming language. I have a couple tenets for its design:
in files, and not individual files per function. in files, and not individual files per function.
\end{itemize} \end{itemize}
\section{Compilation} \section{Compilation}
\label{sec:orgb417494} \label{sec:orge986ff9}
A provided \texttt{Makefile} is added for convencience. It has been tested on an \texttt{arm}-based M1 machine running A provided \texttt{Makefile} is added for convencience. It has been tested on an \texttt{arm}-based M1 machine running
MacOS as well as \texttt{x86} Arch Linux. MacOS as well as \texttt{x86} Arch Linux.
@ -72,11 +72,11 @@ produce an object file:
Which is then bundled into a static library in \texttt{lib/lizfcm.a} and can be linked Which is then bundled into a static library in \texttt{lib/lizfcm.a} and can be linked
in the standard method. in the standard method.
\section{The LIZFCM API} \section{The LIZFCM API}
\label{sec:org2144095} \label{sec:orgd18dc24}
\subsection{Simple Routines} \subsection{Simple Routines}
\label{sec:orgc9edf4b} \label{sec:orgcc14949}
\subsubsection{\texttt{smaceps}} \subsubsection{\texttt{smaceps}}
\label{sec:org449b8ec} \label{sec:orgd908a7a}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{smaceps} \item Name: \texttt{smaceps}
@ -101,7 +101,7 @@ float smaceps() {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{dmaceps}} \subsubsection{\texttt{dmaceps}}
\label{sec:org9a9ac05} \label{sec:org53d7f6f}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{dmaceps} \item Name: \texttt{dmaceps}
@ -126,9 +126,9 @@ double dmaceps() {
} }
\end{verbatim} \end{verbatim}
\subsection{Derivative Routines} \subsection{Derivative Routines}
\label{sec:orgc31ab7b} \label{sec:orgd7542a0}
\subsubsection{\texttt{central\_derivative\_at}} \subsubsection{\texttt{central\_derivative\_at}}
\label{sec:org83dc368} \label{sec:orgf572396}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{central\_derivative\_at} \item Name: \texttt{central\_derivative\_at}
@ -158,7 +158,7 @@ double central_derivative_at(double (*f)(double), double a, double h) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{forward\_derivative\_at}} \subsubsection{\texttt{forward\_derivative\_at}}
\label{sec:orgf1ec748} \label{sec:org4e1fa4a}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{forward\_derivative\_at} \item Name: \texttt{forward\_derivative\_at}
@ -188,7 +188,7 @@ double forward_derivative_at(double (*f)(double), double a, double h) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{backward\_derivative\_at}} \subsubsection{\texttt{backward\_derivative\_at}}
\label{sec:orga2827be} \label{sec:org50d656f}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{backward\_derivative\_at} \item Name: \texttt{backward\_derivative\_at}
@ -218,9 +218,9 @@ double backward_derivative_at(double (*f)(double), double a, double h) {
} }
\end{verbatim} \end{verbatim}
\subsection{Vector Routines} \subsection{Vector Routines}
\label{sec:org4bc395e} \label{sec:orgfb4a8e7}
\subsubsection{Vector Arithmetic: \texttt{add\_v, minus\_v}} \subsubsection{Vector Arithmetic: \texttt{add\_v, minus\_v}}
\label{sec:orgcc76baa} \label{sec:org06284bd}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name(s): \texttt{add\_v}, \texttt{minus\_v} \item Name(s): \texttt{add\_v}, \texttt{minus\_v}
@ -250,7 +250,7 @@ Array_double *minus_v(Array_double *v1, Array_double *v2) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{Norms: \texttt{l1\_norm}, \texttt{l2\_norm}, \texttt{linf\_norm}} \subsubsection{Norms: \texttt{l1\_norm}, \texttt{l2\_norm}, \texttt{linf\_norm}}
\label{sec:org015b19a} \label{sec:org2849a93}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name(s): \texttt{l1\_norm}, \texttt{l2\_norm}, \texttt{linf\_norm} \item Name(s): \texttt{l1\_norm}, \texttt{l2\_norm}, \texttt{linf\_norm}
@ -283,7 +283,7 @@ double linf_norm(Array_double *v) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{vector\_distance}} \subsubsection{\texttt{vector\_distance}}
\label{sec:org78137a7} \label{sec:org4274c99}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{vector\_distance} \item Name: \texttt{vector\_distance}
@ -303,7 +303,7 @@ double vector_distance(Array_double *v1, Array_double *v2,
} }
\end{verbatim} \end{verbatim}
\subsubsection{Distances: \texttt{l1\_distance}, \texttt{l2\_distance}, \texttt{linf\_distance}} \subsubsection{Distances: \texttt{l1\_distance}, \texttt{l2\_distance}, \texttt{linf\_distance}}
\label{sec:orgd71d562} \label{sec:orge4d3e3f}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name(s): \texttt{l1\_distance}, \texttt{l2\_distance}, \texttt{linf\_distance} \item Name(s): \texttt{l1\_distance}, \texttt{l2\_distance}, \texttt{linf\_distance}
@ -328,7 +328,7 @@ double linf_distance(Array_double *v1, Array_double *v2) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{sum\_v}} \subsubsection{\texttt{sum\_v}}
\label{sec:orgb188125} \label{sec:org94e6241}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{sum\_v} \item Name: \texttt{sum\_v}
@ -346,7 +346,7 @@ double sum_v(Array_double *v) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{scale\_v}} \subsubsection{\texttt{scale\_v}}
\label{sec:org0a828aa} \label{sec:orgbc8e308}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{scale\_v} \item Name: \texttt{scale\_v}
@ -364,7 +364,7 @@ Array_double *scale_v(Array_double *v, double m) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{free\_vector}} \subsubsection{\texttt{free\_vector}}
\label{sec:orgfff2e8b} \label{sec:org42b8bd4}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{free\_vector} \item Name: \texttt{free\_vector}
@ -381,7 +381,7 @@ void free_vector(Array_double *v) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{add\_element}} \subsubsection{\texttt{add\_element}}
\label{sec:orgf002846} \label{sec:org69937a3}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{add\_element} \item Name: \texttt{add\_element}
@ -400,7 +400,7 @@ Array_double *add_element(Array_double *v, double x) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{slice\_element}} \subsubsection{\texttt{slice\_element}}
\label{sec:org8ef8f62} \label{sec:org9958a86}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{slice\_element} \item Name: \texttt{slice\_element}
@ -418,7 +418,7 @@ Array_double *slice_element(Array_double *v, size_t x) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{copy\_vector}} \subsubsection{\texttt{copy\_vector}}
\label{sec:org6794d79} \label{sec:org6f1daca}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{copy\_vector} \item Name: \texttt{copy\_vector}
@ -437,7 +437,7 @@ Array_double *copy_vector(Array_double *v) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{format\_vector\_into}} \subsubsection{\texttt{format\_vector\_into}}
\label{sec:orgaaea3a7} \label{sec:orgb326fc6}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{format\_vector\_into} \item Name: \texttt{format\_vector\_into}
@ -466,9 +466,9 @@ void format_vector_into(Array_double *v, char *s) {
} }
\end{verbatim} \end{verbatim}
\subsection{Matrix Routines} \subsection{Matrix Routines}
\label{sec:org7b74a8b} \label{sec:org8bc3f25}
\subsubsection{\texttt{lu\_decomp}} \subsubsection{\texttt{lu\_decomp}}
\label{sec:orgb5ebd91} \label{sec:org0d25547}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{lu\_decomp} \item Name: \texttt{lu\_decomp}
@ -528,7 +528,7 @@ Matrix_double **lu_decomp(Matrix_double *m) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{bsubst}} \subsubsection{\texttt{bsubst}}
\label{sec:org4b2bdc3} \label{sec:orge607e22}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{bsubst} \item Name: \texttt{bsubst}
@ -553,7 +553,7 @@ Array_double *bsubst(Matrix_double *u, Array_double *b) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fsubst}} \subsubsection{\texttt{fsubst}}
\label{sec:orgf6f799e} \label{sec:org3da184d}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{fsubst} \item Name: \texttt{fsubst}
@ -580,7 +580,7 @@ Array_double *fsubst(Matrix_double *l, Array_double *b) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{solve\_matrix\_lu\_bsubst}} \subsubsection{\texttt{solve\_matrix\_lu\_bsubst}}
\label{sec:org789acbf} \label{sec:orgf2bb0ea}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -616,7 +616,7 @@ Array_double *solve_matrix_lu_bsubst(Matrix_double *m, Array_double *b) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{gaussian\_elimination}} \subsubsection{\texttt{gaussian\_elimination}}
\label{sec:orge5cbe95} \label{sec:org6a7faa2}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -671,7 +671,7 @@ Matrix_double *gaussian_elimination(Matrix_double *m) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{solve\_matrix\_gaussian}} \subsubsection{\texttt{solve\_matrix\_gaussian}}
\label{sec:org9c2b7c3} \label{sec:org5379c90}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -704,7 +704,7 @@ Array_double *solve_matrix_gaussian(Matrix_double *m, Array_double *b) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{m\_dot\_v}} \subsubsection{\texttt{m\_dot\_v}}
\label{sec:org4c184b5} \label{sec:org333640a}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -725,7 +725,7 @@ Array_double *m_dot_v(Matrix_double *m, Array_double *v) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{put\_identity\_diagonal}} \subsubsection{\texttt{put\_identity\_diagonal}}
\label{sec:org45882fa} \label{sec:orgdf99950}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -743,7 +743,7 @@ Matrix_double *put_identity_diagonal(Matrix_double *m) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{slice\_column}} \subsubsection{\texttt{slice\_column}}
\label{sec:org520c709} \label{sec:org90e8eb5}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -766,7 +766,7 @@ Matrix_double *slice_column(Matrix_double *m, size_t x) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{add\_column}} \subsubsection{\texttt{add\_column}}
\label{sec:org84191b6} \label{sec:org7f9f9a5}
\begin{itemize} \begin{itemize}
\item Author: Elizabet Hunt \item Author: Elizabet Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -789,7 +789,7 @@ Matrix_double *add_column(Matrix_double *m, Array_double *v) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{copy\_matrix}} \subsubsection{\texttt{copy\_matrix}}
\label{sec:orgb84b548} \label{sec:orgcd0ab1c}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -808,7 +808,7 @@ Matrix_double *copy_matrix(Matrix_double *m) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{free\_matrix}} \subsubsection{\texttt{free\_matrix}}
\label{sec:org0de0d86} \label{sec:orgfece6dc}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -826,7 +826,7 @@ void free_matrix(Matrix_double *m) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{format\_matrix\_into}} \subsubsection{\texttt{format\_matrix\_into}}
\label{sec:orgf8ba876} \label{sec:org3d91cb0}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{format\_matrix\_into} \item Name: \texttt{format\_matrix\_into}
@ -853,9 +853,9 @@ void format_matrix_into(Matrix_double *m, char *s) {
} }
\end{verbatim} \end{verbatim}
\subsection{Root Finding Methods} \subsection{Root Finding Methods}
\label{sec:org8f80c14} \label{sec:orgc546ba0}
\subsubsection{\texttt{find\_ivt\_range}} \subsubsection{\texttt{find\_ivt\_range}}
\label{sec:orgf7fc734} \label{sec:orgf0457e8}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{find\_ivt\_range} \item Name: \texttt{find\_ivt\_range}
@ -887,7 +887,7 @@ Array_double *find_ivt_range(double (*f)(double), double start_x, double delta,
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{bisect\_find\_root}} \subsubsection{\texttt{bisect\_find\_root}}
\label{sec:orgcf0f46b} \label{sec:orge2c8a6a}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name(s): \texttt{bisect\_find\_root} \item Name(s): \texttt{bisect\_find\_root}
@ -918,7 +918,7 @@ Array_double *bisect_find_root(double (*f)(double), double a, double b,
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{bisect\_find\_root\_with\_error\_assumption}} \subsubsection{\texttt{bisect\_find\_root\_with\_error\_assumption}}
\label{sec:org64e4346} \label{sec:org0f219ab}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{bisect\_find\_root\_with\_error\_assumption} \item Name: \texttt{bisect\_find\_root\_with\_error\_assumption}
@ -946,7 +946,7 @@ double bisect_find_root_with_error_assumption(double (*f)(double), double a,
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fixed\_point\_iteration\_method}} \subsubsection{\texttt{fixed\_point\_iteration\_method}}
\label{sec:orge6f6ba8} \label{sec:orgd9fe8e6}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{fixed\_point\_iteration\_method} \item Name: \texttt{fixed\_point\_iteration\_method}
@ -977,7 +977,7 @@ double fixed_point_iteration_method(double (*f)(double), double (*g)(double),
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fixed\_point\_newton\_method}} \subsubsection{\texttt{fixed\_point\_newton\_method}}
\label{sec:org9c22d8f} \label{sec:org4cf044d}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{fixed\_point\_newton\_method} \item Name: \texttt{fixed\_point\_newton\_method}
@ -1005,7 +1005,7 @@ double fixed_point_newton_method(double (*f)(double), double (*fprime)(double),
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fixed\_point\_secant\_method}} \subsubsection{\texttt{fixed\_point\_secant\_method}}
\label{sec:org446d473} \label{sec:orgc5cfd8b}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{fixed\_point\_secant\_method} \item Name: \texttt{fixed\_point\_secant\_method}
@ -1032,7 +1032,7 @@ double fixed_point_secant_method(double (*f)(double), double x_0, double x_1,
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fixed\_point\_secant\_bisection\_method}} \subsubsection{\texttt{fixed\_point\_secant\_bisection\_method}}
\label{sec:orgade170f} \label{sec:orgf3fe7ad}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{fixed\_point\_secant\_method} \item Name: \texttt{fixed\_point\_secant\_method}
@ -1083,9 +1083,9 @@ double fixed_point_secant_bisection_method(double (*f)(double), double x_0,
} }
\end{verbatim} \end{verbatim}
\subsection{Linear Routines} \subsection{Linear Routines}
\label{sec:orgc389980} \label{sec:org8b9d4db}
\subsubsection{\texttt{least\_squares\_lin\_reg}} \subsubsection{\texttt{least\_squares\_lin\_reg}}
\label{sec:org850d9f6} \label{sec:orgca5faa9}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{least\_squares\_lin\_reg} \item Name: \texttt{least\_squares\_lin\_reg}
@ -1115,9 +1115,9 @@ Line *least_squares_lin_reg(Array_double *x, Array_double *y) {
} }
\end{verbatim} \end{verbatim}
\subsection{Eigen-Adjacent} \subsection{Eigen-Adjacent}
\label{sec:org6bea1aa} \label{sec:orgb053100}
\subsubsection{\texttt{dominant\_eigenvalue}} \subsubsection{\texttt{dominant\_eigenvalue}}
\label{sec:org0e70920} \label{sec:orgc345d49}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{dominant\_eigenvalue} \item Name: \texttt{dominant\_eigenvalue}
@ -1142,6 +1142,10 @@ double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance,
while (error >= tolerance && (--iter) > 0) { while (error >= tolerance && (--iter) > 0) {
Array_double *eigenvector_2 = m_dot_v(m, eigenvector_1); Array_double *eigenvector_2 = m_dot_v(m, eigenvector_1);
Array_double *normalized_eigenvector_2 =
scale_v(eigenvector_2, 1.0 / linf_norm(eigenvector_2));
free_vector(eigenvector_2);
eigenvector_2 = normalized_eigenvector_2;
Array_double *mx = m_dot_v(m, eigenvector_2); Array_double *mx = m_dot_v(m, eigenvector_2);
double new_lambda = double new_lambda =
@ -1156,8 +1160,115 @@ double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance,
return lambda; return lambda;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{shift\_inverse\_power\_eigenvalue}}
\label{sec:org7bb6f14}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{least\_dominant\_eigenvalue}
\item Location: \texttt{src/eigen.c}
\item Input: a pointer to an invertible matrix \texttt{m}, an initial eigenvector guess \texttt{v} (that is non
zero or orthogonal to an eigenvector with the dominant eigenvalue), a \texttt{shift} to act as the
shifted \(\delta\), and \texttt{tolerance} and \texttt{max\_iterations} that act as stop conditions.
\item Output: the eigenvalue closest to \texttt{shift} with the lowest magnitude closest to 0, approximated
with the Inverse Power Iteration Method
\end{itemize}
\begin{verbatim}
double shift_inverse_power_eigenvalue(Matrix_double *m, Array_double *v,
double shift, double tolerance,
size_t max_iterations) {
assert(m->rows == m->cols);
assert(m->rows == v->size);
Matrix_double *m_c = copy_matrix(m);
for (size_t y = 0; y < m_c->rows; ++y)
m_c->data[y]->data[y] = m_c->data[y]->data[y] - shift;
double error = tolerance;
size_t iter = max_iterations;
double lambda = shift;
Array_double *eigenvector_1 = copy_vector(v);
while (error >= tolerance && (--iter) > 0) {
Array_double *eigenvector_2 = solve_matrix_lu_bsubst(m_c, eigenvector_1);
Array_double *normalized_eigenvector_2 =
scale_v(eigenvector_2, 1.0 / linf_norm(eigenvector_2));
free_vector(eigenvector_2);
Array_double *mx = m_dot_v(m, normalized_eigenvector_2);
double new_lambda =
v_dot_v(mx, normalized_eigenvector_2) /
v_dot_v(normalized_eigenvector_2, normalized_eigenvector_2);
error = fabs(new_lambda - lambda);
lambda = new_lambda;
free_vector(eigenvector_1);
eigenvector_1 = normalized_eigenvector_2;
}
return lambda;
}
\end{verbatim}
\subsubsection{\texttt{least\_dominant\_eigenvalue}}
\label{sec:orgdef7c62}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{least\_dominant\_eigenvalue}
\item Location: \texttt{src/eigen.c}
\item Input: a pointer to an invertible matrix \texttt{m}, an initial eigenvector guess \texttt{v} (that is non
zero or orthogonal to an eigenvector with the dominant eigenvalue), a \texttt{tolerance} and
\texttt{max\_iterations} that act as stop conditions.
\item Output: the least dominant eigenvalue with the lowest magnitude closest to 0, approximated
with the Inverse Power Iteration Method.
\end{itemize}
\begin{verbatim}
double least_dominant_eigenvalue(Matrix_double *m, Array_double *v,
double tolerance, size_t max_iterations) {
return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations);
}
\end{verbatim}
\subsubsection{\texttt{partition\_find\_eigenvalues}}
\label{sec:orgc68645a}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{partition\_find\_eigenvalues}
\item Location: \texttt{src/eigen.c}
\item Input: a pointer to an invertible matrix \texttt{m}, a matrix whose rows correspond to initial
eigenvector guesses at each "partition" which is computed from a uniform distribution
between the number of rows this "guess matrix" has and the distance between the least
dominant eigenvalue and the most dominant. Additionally, a \texttt{max\_iterations} and a \texttt{tolerance}
that act as stop conditions.
\item Output: a vector of \texttt{doubles} corresponding to the "nearest" eigenvalue at the midpoint of
each partition, via the given guess of that partition.
\end{itemize}
\begin{verbatim}
Array_double *partition_find_eigenvalues(Matrix_double *m,
Matrix_double *guesses,
double tolerance,
size_t max_iterations) {
assert(guesses->rows >=
2); // we need at least, the most and least dominant eigenvalues
double end = dominant_eigenvalue(m, guesses->data[guesses->rows - 1],
tolerance, max_iterations);
double begin =
least_dominant_eigenvalue(m, guesses->data[0], tolerance, max_iterations);
double delta = (end - begin) / guesses->rows;
Array_double *eigenvalues = InitArrayWithSize(double, guesses->rows, 0.0);
for (size_t i = 0; i < guesses->rows; i++) {
double box_midpoint = ((delta * i) + (delta * (i + 1))) / 2;
double nearest_eigenvalue = shift_inverse_power_eigenvalue(
m, guesses->data[i], box_midpoint, tolerance, max_iterations);
eigenvalues->data[i] = nearest_eigenvalue;
}
return eigenvalues;
}
\end{verbatim}
\subsubsection{\texttt{leslie\_matrix}} \subsubsection{\texttt{leslie\_matrix}}
\label{sec:org88d4547} \label{sec:org0637da1}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{leslie\_matrix} \item Name: \texttt{leslie\_matrix}
@ -1165,7 +1276,7 @@ double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance,
\item Input: two pointers to \texttt{Array\_double}'s representing the ratio of individuals in an age class \item Input: two pointers to \texttt{Array\_double}'s representing the ratio of individuals in an age class
\(x\) getting to the next age class \(x+1\) and the number of offspring that individuals in an age \(x\) getting to the next age class \(x+1\) and the number of offspring that individuals in an age
class create in age class 0. class create in age class 0.
\item Output: the leslie matrix generated with the input vectors. \item Output: the leslie matrix generated from the input vectors.
\end{itemize} \end{itemize}
\begin{verbatim} \begin{verbatim}
@ -1185,12 +1296,12 @@ Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio,
} }
\end{verbatim} \end{verbatim}
\subsection{Appendix / Miscellaneous} \subsection{Appendix / Miscellaneous}
\label{sec:org925aa32} \label{sec:orgddf0893}
\subsubsection{Data Types} \subsubsection{Data Types}
\label{sec:org37335a1} \label{sec:org0ec3831}
\begin{enumerate} \begin{enumerate}
\item \texttt{Line} \item \texttt{Line}
\label{sec:orgaf72b30} \label{sec:org1b45662}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/types.h} \item Location: \texttt{inc/types.h}
@ -1203,7 +1314,7 @@ typedef struct Line {
} Line; } Line;
\end{verbatim} \end{verbatim}
\item The \texttt{Array\_<type>} and \texttt{Matrix\_<type>} \item The \texttt{Array\_<type>} and \texttt{Matrix\_<type>}
\label{sec:org82faf8e} \label{sec:org1a49c97}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/types.h} \item Location: \texttt{inc/types.h}
@ -1234,10 +1345,10 @@ typedef struct {
\end{verbatim} \end{verbatim}
\end{enumerate} \end{enumerate}
\subsubsection{Macros} \subsubsection{Macros}
\label{sec:org1f988ea} \label{sec:orge905d93}
\begin{enumerate} \begin{enumerate}
\item \texttt{c\_max} and \texttt{c\_min} \item \texttt{c\_max} and \texttt{c\_min}
\label{sec:org8b37b18} \label{sec:org0267cca}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h} \item Location: \texttt{inc/macros.h}
@ -1250,7 +1361,7 @@ typedef struct {
#define c_min(x, y) (((x) <= (y)) ? (x) : (y)) #define c_min(x, y) (((x) <= (y)) ? (x) : (y))
\end{verbatim} \end{verbatim}
\item \texttt{InitArray} \item \texttt{InitArray}
\label{sec:org04ec2d7} \label{sec:orgdb147d1}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h} \item Location: \texttt{inc/macros.h}
@ -1270,7 +1381,7 @@ typedef struct {
}) })
\end{verbatim} \end{verbatim}
\item \texttt{InitArrayWithSize} \item \texttt{InitArrayWithSize}
\label{sec:org4aff8f6} \label{sec:org0b33fa5}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h} \item Location: \texttt{inc/macros.h}
@ -1290,7 +1401,7 @@ typedef struct {
}) })
\end{verbatim} \end{verbatim}
\item \texttt{InitMatrixWithSize} \item \texttt{InitMatrixWithSize}
\label{sec:org3457577} \label{sec:orgcc5a2bb}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h} \item Location: \texttt{inc/macros.h}

17
homeworks/hw-7.org
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@ -1,4 +1,4 @@
#+TITLE: Homework 6 #+TITLE: Homework 7
#+AUTHOR: Elizabeth Hunt #+AUTHOR: Elizabeth Hunt
#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} #+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX: \setlength\parindent{0pt} #+LATEX: \setlength\parindent{0pt}
@ -58,4 +58,19 @@ See also the entry ~Eigen-Adjacent -> partition_find_eigenvalues~ in the LIZFCM
documentation. documentation.
* Question Six * Question Six
Consider we have the results of two methods developed in this homework: ~least_dominant_eigenvalue~, and ~dominant_eigenvalue~
into ~lambda_0~, ~lambda_n~, respectively. Also assume that we have the method implemented as we've introduced,
~shift_inverse_power_eigenvalue~.
Then, we begin at the midpoint of ~lambda_0~ and ~lambda_n~, and compute the
~new_lambda = shift_inverse_power_eigenvalue~
with a shift at the midpoint, and some given initial guess.
1. If the result is equal (or within some tolerance) to ~lambda_n~ then the closest eigenvalue to the midpoint
is still the dominant eigenvalue, and thus the next most dominant will be on the left. Set ~lambda_n~
to the midpoint and reiterate.
2. If the result is greater or equal to ~lambda_0~ we know an eigenvalue of greater or equal magnitude
exists on the right. So, we set ~lambda_0~ to this eigenvalue associated with the midpoint, and
re-iterate.
3. Continue re-iterating until we hit some given maximum number of iterations. Finally we will return
~new_lambda~.

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@ -0,0 +1,107 @@
% Created 2023-11-27 Mon 15:13
% Intended LaTeX compiler: pdflatex
\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{longtable}
\usepackage{wrapfig}
\usepackage{rotating}
\usepackage[normalem]{ulem}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{capt-of}
\usepackage{hyperref}
\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
\author{Elizabeth Hunt}
\date{\today}
\title{Homework 7}
\hypersetup{
pdfauthor={Elizabeth Hunt},
pdftitle={Homework 7},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 29.1 (Org mode 9.7-pre)},
pdflang={English}}
\begin{document}
\maketitle
\setlength\parindent{0pt}
\section{Question One}
\label{sec:org8ef0ee6}
See \texttt{UTEST(eigen, dominant\_eigenvalue)} in \texttt{test/eigen.t.c} and the entry
\texttt{Eigen-Adjacent -> dominant\_eigenvalue} in the LIZFCM API documentation.
\section{Question Two}
\label{sec:orgbdba5c1}
See \texttt{UTEST(eigen, leslie\_matrix\_dominant\_eigenvalue)} in \texttt{test/eigen.t.c}
and the entry \texttt{Eigen-Adjacent -> leslie\_matrix} in the LIZFCM API
documentation.
\section{Question Three}
\label{sec:org19b04f4}
See \texttt{UTEST(eigen, least\_dominant\_eigenvalue)} in \texttt{test/eigen.t.c} which
finds the least dominant eigenvalue on the matrix:
\begin{bmatrix}
2 & 2 & 4 \\
1 & 4 & 7 \\
0 & 2 & 6
\end{bmatrix}
which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\) and should thus produce \(5 - \sqrt{17}\).
See also the entry \texttt{Eigen-Adjacent -> least\_dominant\_eigenvalue} in the LIZFCM API
documentation.
\section{Question Four}
\label{sec:orgc58d42d}
See \texttt{UTEST(eigen, shifted\_eigenvalue)} in \texttt{test/eigen.t.c} which
finds the least dominant eigenvalue on the matrix:
\begin{bmatrix}
2 & 2 & 4 \\
1 & 4 & 7 \\
0 & 2 & 6
\end{bmatrix}
which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\) and should thus produce \(2.0\).
With the initial guess: \([0.5, 1.0, 0.75]\).
See also the entry \texttt{Eigen-Adjacent -> shift\_inverse\_power\_eigenvalue} in the LIZFCM API
documentation.
\section{Question Five}
\label{sec:orga369221}
See \texttt{UTEST(eigen, partition\_find\_eigenvalues)} in \texttt{test/eigen.t.c} which
finds the eigenvalues in a partition of 10 on the matrix:
\begin{bmatrix}
2 & 2 & 4 \\
1 & 4 & 7 \\
0 & 2 & 6
\end{bmatrix}
which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\), and should produce all three from
the partitions when given the guesses \([0.5, 1.0, 0.75]\) from the questions above.
See also the entry \texttt{Eigen-Adjacent -> partition\_find\_eigenvalues} in the LIZFCM API
documentation.
\section{Question Six}
\label{sec:orgadc3078}
Consider we have the results of two methods developed in this homework: \texttt{least\_dominant\_eigenvalue}, and \texttt{dominant\_eigenvalue}
into \texttt{lambda\_0}, \texttt{lambda\_n}, respectively. Also assume that we have the method implemented as we've introduced,
\texttt{shift\_inverse\_power\_eigenvalue}.
Then, we begin at the midpoint of \texttt{lambda\_0} and \texttt{lambda\_n}, and compute the
\texttt{new\_lambda = shift\_inverse\_power\_eigenvalue}
with a shift at the midpoint, and some given initial guess.
\begin{enumerate}
\item If the result is equal (or within some tolerance) to \texttt{lambda\_n} then the closest eigenvalue to the midpoint
is still the dominant eigenvalue, and thus the next most dominant will be on the left. Set \texttt{lambda\_n}
to the midpoint and reiterate.
\item If the result is greater or equal to \texttt{lambda\_0} we know an eigenvalue of greater or equal magnitude
exists on the right. So, we set \texttt{lambda\_0} to this eigenvalue associated with the midpoint, and
re-iterate.
\item Continue re-iterating until we hit some given maximum number of iterations. Finally we will return
\texttt{new\_lambda}.
\end{enumerate}
\end{document}

32
src/eigen.c
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@ -4,19 +4,9 @@
#include <stdio.h> #include <stdio.h>
#include <string.h> #include <string.h>
Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio, double least_dominant_eigenvalue(Matrix_double *m, Array_double *v,
Array_double *age_class_offspring) { double tolerance, size_t max_iterations) {
assert(age_class_surivor_ratio->size + 1 == age_class_offspring->size); return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations);
Matrix_double *leslie = InitMatrixWithSize(double, age_class_offspring->size,
age_class_offspring->size, 0.0);
free_vector(leslie->data[0]);
leslie->data[0] = copy_vector(age_class_offspring);
for (size_t i = 0; i < age_class_surivor_ratio->size; i++)
leslie->data[i + 1]->data[i] = age_class_surivor_ratio->data[i];
return leslie;
} }
double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance, double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance,
@ -110,7 +100,17 @@ Array_double *partition_find_eigenvalues(Matrix_double *m,
return eigenvalues; return eigenvalues;
} }
double least_dominant_eigenvalue(Matrix_double *m, Array_double *v, Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio,
double tolerance, size_t max_iterations) { Array_double *age_class_offspring) {
return shift_inverse_power_eigenvalue(m, v, 0.0, tolerance, max_iterations); assert(age_class_surivor_ratio->size + 1 == age_class_offspring->size);
Matrix_double *leslie = InitMatrixWithSize(double, age_class_offspring->size,
age_class_offspring->size, 0.0);
free_vector(leslie->data[0]);
leslie->data[0] = copy_vector(age_class_offspring);
for (size_t i = 0; i < age_class_surivor_ratio->size; i++)
leslie->data[i + 1]->data[i] = age_class_surivor_ratio->data[i];
return leslie;
} }