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Elizabeth Hunt 2023-12-11 19:24:33 -07:00
parent 9b73fc2978
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GPG Key ID: 52B3774857EB24B1
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@ -1,4 +1,4 @@
% Created 2023-11-29 Wed 18:33 % Created 2023-12-11 Mon 19:22
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[11pt]{article} \documentclass[11pt]{article}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -15,13 +15,13 @@
\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
\author{Elizabeth Hunt} \author{Elizabeth Hunt}
\date{\today} \date{\today}
\title{LIZFCM Software Manual (v0.5)} \title{LIZFCM Software Manual (v0.6)}
\hypersetup{ \hypersetup{
pdfauthor={Elizabeth Hunt}, pdfauthor={Elizabeth Hunt},
pdftitle={LIZFCM Software Manual (v0.5)}, pdftitle={LIZFCM Software Manual (v0.6)},
pdfkeywords={}, pdfkeywords={},
pdfsubject={}, pdfsubject={},
pdfcreator={Emacs 29.1 (Org mode 9.7-pre)}, pdfcreator={Emacs 28.2 (Org mode 9.7-pre)},
pdflang={English}} pdflang={English}}
\begin{document} \begin{document}
@ -29,8 +29,9 @@
\tableofcontents \tableofcontents
\setlength\parindent{0pt} \setlength\parindent{0pt}
\section{Design} \section{Design}
\label{sec:orge6394d6} \label{sec:org63deaf6}
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
@ -46,8 +47,9 @@ the C programming language. I have a couple tenets for its design:
\item Routines are separated into "modules" that follow a form of separation of concerns \item Routines are separated into "modules" that follow a form of separation of concerns
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:org19eb4ad} \label{sec:org7291327}
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.
@ -71,12 +73,13 @@ 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:orgcdad892} \label{sec:org1ebe7fa}
\subsection{Simple Routines} \subsection{Simple Routines}
\label{sec:org414c16a} \label{sec:orgff18c6b}
\subsubsection{\texttt{smaceps}} \subsubsection{\texttt{smaceps}}
\label{sec:org1f871c1} \label{sec:org443df5e}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{smaceps} \item Name: \texttt{smaceps}
@ -100,8 +103,9 @@ float smaceps() {
return machine_epsilon; return machine_epsilon;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{dmaceps}} \subsubsection{\texttt{dmaceps}}
\label{sec:orga6517f5} \label{sec:org5121603}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{dmaceps} \item Name: \texttt{dmaceps}
@ -125,10 +129,11 @@ double dmaceps() {
return machine_epsilon; return machine_epsilon;
} }
\end{verbatim} \end{verbatim}
\subsection{Derivative Routines} \subsection{Derivative Routines}
\label{sec:org7037c95} \label{sec:org6fd324c}
\subsubsection{\texttt{central\_derivative\_at}} \subsubsection{\texttt{central\_derivative\_at}}
\label{sec:org5004cc1} \label{sec:orge9f0821}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{central\_derivative\_at} \item Name: \texttt{central\_derivative\_at}
@ -157,8 +162,9 @@ double central_derivative_at(double (*f)(double), double a, double h) {
return (y2 - y1) / (x2 - x1); return (y2 - y1) / (x2 - x1);
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{forward\_derivative\_at}} \subsubsection{\texttt{forward\_derivative\_at}}
\label{sec:orgfe0e436} \label{sec:org8720f28}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{forward\_derivative\_at} \item Name: \texttt{forward\_derivative\_at}
@ -187,8 +193,9 @@ double forward_derivative_at(double (*f)(double), double a, double h) {
return (y2 - y1) / (x2 - x1); return (y2 - y1) / (x2 - x1);
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{backward\_derivative\_at}} \subsubsection{\texttt{backward\_derivative\_at}}
\label{sec:orgf50d7e6} \label{sec:org1589b19}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{backward\_derivative\_at} \item Name: \texttt{backward\_derivative\_at}
@ -217,10 +224,11 @@ double backward_derivative_at(double (*f)(double), double a, double h) {
return (y2 - y1) / (x2 - x1); return (y2 - y1) / (x2 - x1);
} }
\end{verbatim} \end{verbatim}
\subsection{Vector Routines} \subsection{Vector Routines}
\label{sec:org125771d} \label{sec:org493841e}
\subsubsection{Vector Arithmetic: \texttt{add\_v, minus\_v}} \subsubsection{Vector Arithmetic: \texttt{add\_v, minus\_v}}
\label{sec:orgf07f7dd} \label{sec:org3912c29}
\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}
@ -249,8 +257,9 @@ Array_double *minus_v(Array_double *v1, Array_double *v2) {
return sub; return sub;
} }
\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:org3f50ecb} \label{sec:orged74cfb}
\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}
@ -282,8 +291,9 @@ double linf_norm(Array_double *v) {
return max; return max;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{vector\_distance}} \subsubsection{\texttt{vector\_distance}}
\label{sec:org77ad0f5} \label{sec:org20a5773}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{vector\_distance} \item Name: \texttt{vector\_distance}
@ -302,8 +312,9 @@ double vector_distance(Array_double *v1, Array_double *v2,
return dist; return dist;
} }
\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:org83b7946} \label{sec:orgac16178}
\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}
@ -327,8 +338,9 @@ double linf_distance(Array_double *v1, Array_double *v2) {
return vector_distance(v1, v2, &linf_norm); return vector_distance(v1, v2, &linf_norm);
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{sum\_v}} \subsubsection{\texttt{sum\_v}}
\label{sec:orgb0e87d6} \label{sec:org876aafa}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{sum\_v} \item Name: \texttt{sum\_v}
@ -345,8 +357,9 @@ double sum_v(Array_double *v) {
return sum; return sum;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{scale\_v}} \subsubsection{\texttt{scale\_v}}
\label{sec:org08dbec0} \label{sec:orgf1d236c}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{scale\_v} \item Name: \texttt{scale\_v}
@ -363,8 +376,9 @@ Array_double *scale_v(Array_double *v, double m) {
return copy; return copy;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{free\_vector}} \subsubsection{\texttt{free\_vector}}
\label{sec:org36eb399} \label{sec:org2ca163d}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{free\_vector} \item Name: \texttt{free\_vector}
@ -380,8 +394,9 @@ void free_vector(Array_double *v) {
free(v); free(v);
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{add\_element}} \subsubsection{\texttt{add\_element}}
\label{sec:org987fa04} \label{sec:org7a99233}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{add\_element} \item Name: \texttt{add\_element}
@ -399,8 +414,9 @@ Array_double *add_element(Array_double *v, double x) {
return pushed; return pushed;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{slice\_element}} \subsubsection{\texttt{slice\_element}}
\label{sec:orged034ec} \label{sec:org6c07c99}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{slice\_element} \item Name: \texttt{slice\_element}
@ -417,8 +433,9 @@ Array_double *slice_element(Array_double *v, size_t x) {
return sliced; return sliced;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{copy\_vector}} \subsubsection{\texttt{copy\_vector}}
\label{sec:org3d6d716} \label{sec:org81f7cc1}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{copy\_vector} \item Name: \texttt{copy\_vector}
@ -436,8 +453,9 @@ Array_double *copy_vector(Array_double *v) {
return copy; return copy;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{format\_vector\_into}} \subsubsection{\texttt{format\_vector\_into}}
\label{sec:orgb4ab2db} \label{sec:orgd168171}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{format\_vector\_into} \item Name: \texttt{format\_vector\_into}
@ -465,10 +483,11 @@ void format_vector_into(Array_double *v, char *s) {
strcat(s, "\n"); strcat(s, "\n");
} }
\end{verbatim} \end{verbatim}
\subsection{Matrix Routines} \subsection{Matrix Routines}
\label{sec:orgd984ad2} \label{sec:org5c45c12}
\subsubsection{\texttt{lu\_decomp}} \subsubsection{\texttt{lu\_decomp}}
\label{sec:org184e384} \label{sec:orgf1e0ac3}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{lu\_decomp} \item Name: \texttt{lu\_decomp}
@ -528,7 +547,7 @@ Matrix_double **lu_decomp(Matrix_double *m) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{bsubst}} \subsubsection{\texttt{bsubst}}
\label{sec:org78c1bb9} \label{sec:orgec7e4b5}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{bsubst} \item Name: \texttt{bsubst}
@ -553,7 +572,7 @@ Array_double *bsubst(Matrix_double *u, Array_double *b) {
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fsubst}} \subsubsection{\texttt{fsubst}}
\label{sec:org9d7af79} \label{sec:org72ff2ed}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{fsubst} \item Name: \texttt{fsubst}
@ -579,8 +598,9 @@ Array_double *fsubst(Matrix_double *l, Array_double *b) {
return x; return x;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{solve\_matrix\_lu\_bsubst}} \subsubsection{\texttt{solve\_matrix\_lu\_bsubst}}
\label{sec:org85d9237} \label{sec:orga735557}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -615,8 +635,9 @@ Array_double *solve_matrix_lu_bsubst(Matrix_double *m, Array_double *b) {
return x; return x;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{gaussian\_elimination}} \subsubsection{\texttt{gaussian\_elimination}}
\label{sec:orgd188f79} \label{sec:org71d2519}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -629,30 +650,32 @@ applying reduction to all other rows. The general idea is available at \url{http
\begin{verbatim} \begin{verbatim}
Matrix_double *gaussian_elimination(Matrix_double *m) { Matrix_double *gaussian_elimination(Matrix_double *m) {
uint64_t h = 0; uint64_t h = 0, k = 0;
uint64_t k = 0;
Matrix_double *m_cp = copy_matrix(m); Matrix_double *m_cp = copy_matrix(m);
while (h < m_cp->rows && k < m_cp->cols) { while (h < m_cp->rows && k < m_cp->cols) {
uint64_t max_row = 0; uint64_t max_row = h;
double total_max = 0.0; double max_val = 0.0;
for (uint64_t row = h; row < m_cp->rows; row++) { for (uint64_t row = h; row < m_cp->rows; row++) {
double this_max = c_max(fabs(m_cp->data[row]->data[k]), total_max); double val = fabs(m_cp->data[row]->data[k]);
if (c_max(this_max, total_max) == this_max) { if (val > max_val) {
max_val = val;
max_row = row; max_row = row;
} }
} }
if (max_row == 0) { if (max_val == 0.0) {
k++; k++;
continue; continue;
} }
if (max_row != h) {
Array_double *swp = m_cp->data[max_row]; Array_double *swp = m_cp->data[max_row];
m_cp->data[max_row] = m_cp->data[h]; m_cp->data[max_row] = m_cp->data[h];
m_cp->data[h] = swp; m_cp->data[h] = swp;
}
for (uint64_t row = h + 1; row < m_cp->rows; row++) { for (uint64_t row = h + 1; row < m_cp->rows; row++) {
double factor = m_cp->data[row]->data[k] / m_cp->data[h]->data[k]; double factor = m_cp->data[row]->data[k] / m_cp->data[h]->data[k];
@ -670,8 +693,9 @@ Matrix_double *gaussian_elimination(Matrix_double *m) {
return m_cp; return m_cp;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{solve\_matrix\_gaussian}} \subsubsection{\texttt{solve\_matrix\_gaussian}}
\label{sec:org2f14966} \label{sec:org230915f}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -703,8 +727,10 @@ Array_double *solve_matrix_gaussian(Matrix_double *m, Array_double *b) {
return solution; return solution;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{m\_dot\_v}} \subsubsection{\texttt{m\_dot\_v}}
\label{sec:orgc3739fc} \label{sec:org83c8351}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -724,8 +750,9 @@ Array_double *m_dot_v(Matrix_double *m, Array_double *v) {
return product; return product;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{put\_identity\_diagonal}} \subsubsection{\texttt{put\_identity\_diagonal}}
\label{sec:org6e2dda0} \label{sec:orge3fcb3e}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -742,8 +769,9 @@ Matrix_double *put_identity_diagonal(Matrix_double *m) {
return copy; return copy;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{slice\_column}} \subsubsection{\texttt{slice\_column}}
\label{sec:org6ec00b6} \label{sec:org95e39ba}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -765,8 +793,9 @@ Matrix_double *slice_column(Matrix_double *m, size_t x) {
return sliced; return sliced;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{add\_column}} \subsubsection{\texttt{add\_column}}
\label{sec:org46ea124} \label{sec:org9a2ad93}
\begin{itemize} \begin{itemize}
\item Author: Elizabet Hunt \item Author: Elizabet Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -788,8 +817,9 @@ Matrix_double *add_column(Matrix_double *m, Array_double *v) {
return pushed; return pushed;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{copy\_matrix}} \subsubsection{\texttt{copy\_matrix}}
\label{sec:org0fb04f5} \label{sec:org63765c0}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -807,8 +837,9 @@ Matrix_double *copy_matrix(Matrix_double *m) {
return copy; return copy;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{free\_matrix}} \subsubsection{\texttt{free\_matrix}}
\label{sec:org1e7bf4e} \label{sec:orgc337967}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c} \item Location: \texttt{src/matrix.c}
@ -825,8 +856,9 @@ void free_matrix(Matrix_double *m) {
free(m); free(m);
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{format\_matrix\_into}} \subsubsection{\texttt{format\_matrix\_into}}
\label{sec:org7d927f5} \label{sec:org6b188b4}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{format\_matrix\_into} \item Name: \texttt{format\_matrix\_into}
@ -843,7 +875,7 @@ void format_matrix_into(Matrix_double *m, char *s) {
strcpy(s, "empty"); strcpy(s, "empty");
for (size_t y = 0; y < m->rows; ++y) { for (size_t y = 0; y < m->rows; ++y) {
char row_s[256]; char row_s[5192];
strcpy(row_s, ""); strcpy(row_s, "");
format_vector_into(m->data[y], row_s); format_vector_into(m->data[y], row_s);
@ -853,9 +885,9 @@ void format_matrix_into(Matrix_double *m, char *s) {
} }
\end{verbatim} \end{verbatim}
\subsection{Root Finding Methods} \subsection{Root Finding Methods}
\label{sec:org2d9e027} \label{sec:org352ccdf}
\subsubsection{\texttt{find\_ivt\_range}} \subsubsection{\texttt{find\_ivt\_range}}
\label{sec:org13aac0a} \label{sec:orgb9a0d16}
\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 +919,7 @@ 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:org37c0d43} \label{sec:org25382b3}
\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 +950,7 @@ 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:orga63019e} \label{sec:org4b9cb72}
\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}
@ -945,8 +977,9 @@ double bisect_find_root_with_error_assumption(double (*f)(double), double a,
return root; return root;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fixed\_point\_iteration\_method}} \subsubsection{\texttt{fixed\_point\_iteration\_method}}
\label{sec:org9325db0} \label{sec:org4cee2bd}
\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}
@ -976,8 +1009,9 @@ double fixed_point_iteration_method(double (*f)(double), double (*g)(double),
max_iterations - 1); max_iterations - 1);
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fixed\_point\_newton\_method}} \subsubsection{\texttt{fixed\_point\_newton\_method}}
\label{sec:orgb571ad9} \label{sec:org93e3999}
\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}
@ -1004,8 +1038,9 @@ double fixed_point_newton_method(double (*f)(double), double (*fprime)(double),
max_iterations - 1); max_iterations - 1);
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{fixed\_point\_secant\_method}} \subsubsection{\texttt{fixed\_point\_secant\_method}}
\label{sec:org1b33cb8} \label{sec:orgf3f0711}
\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 +1067,7 @@ 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:org9dad67f} \label{sec:orgeaef048}
\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}
@ -1082,10 +1117,11 @@ double fixed_point_secant_bisection_method(double (*f)(double), double x_0,
return root; return root;
} }
\end{verbatim} \end{verbatim}
\subsection{Linear Routines} \subsection{Linear Routines}
\label{sec:org89d7d4f} \label{sec:orge3b6d97}
\subsubsection{\texttt{least\_squares\_lin\_reg}} \subsubsection{\texttt{least\_squares\_lin\_reg}}
\label{sec:orgda34435} \label{sec:orgcc90c4a}
\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}
@ -1114,10 +1150,11 @@ Line *least_squares_lin_reg(Array_double *x, Array_double *y) {
return line; return line;
} }
\end{verbatim} \end{verbatim}
\subsection{Eigen-Adjacent} \subsection{Eigen-Adjacent}
\label{sec:orgc7e86aa} \label{sec:orga3c637f}
\subsubsection{\texttt{dominant\_eigenvalue}} \subsubsection{\texttt{dominant\_eigenvalue}}
\label{sec:org511efa7} \label{sec:org0306c8a}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{dominant\_eigenvalue} \item Name: \texttt{dominant\_eigenvalue}
@ -1161,7 +1198,7 @@ double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance,
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{shift\_inverse\_power\_eigenvalue}} \subsubsection{\texttt{shift\_inverse\_power\_eigenvalue}}
\label{sec:org2f9dea2} \label{sec:orgc29637a}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{least\_dominant\_eigenvalue} \item Name: \texttt{least\_dominant\_eigenvalue}
@ -1208,8 +1245,9 @@ double shift_inverse_power_eigenvalue(Matrix_double *m, Array_double *v,
return lambda; return lambda;
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{least\_dominant\_eigenvalue}} \subsubsection{\texttt{least\_dominant\_eigenvalue}}
\label{sec:org847cb9b} \label{sec:org5df73a2}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{least\_dominant\_eigenvalue} \item Name: \texttt{least\_dominant\_eigenvalue}
@ -1227,7 +1265,7 @@ double least_dominant_eigenvalue(Matrix_double *m, Array_double *v,
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{partition\_find\_eigenvalues}} \subsubsection{\texttt{partition\_find\_eigenvalues}}
\label{sec:orgb6cc77c} \label{sec:org3dde7af}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{partition\_find\_eigenvalues} \item Name: \texttt{partition\_find\_eigenvalues}
@ -1268,7 +1306,7 @@ Array_double *partition_find_eigenvalues(Matrix_double *m,
} }
\end{verbatim} \end{verbatim}
\subsubsection{\texttt{leslie\_matrix}} \subsubsection{\texttt{leslie\_matrix}}
\label{sec:org2bb3502} \label{sec:orgca10ed3}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Name: \texttt{leslie\_matrix} \item Name: \texttt{leslie\_matrix}
@ -1295,13 +1333,128 @@ Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio,
return leslie; return leslie;
} }
\end{verbatim} \end{verbatim}
\subsection{Jacobi / Gauss-Siedel}
\label{sec:org91c563c}
\subsubsection{\texttt{jacobi\_solve}}
\label{sec:org2cd6098}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{jacobi\_solve}
\item Location: \texttt{src/matrix.c}
\item Input: a pointer to a diagonally dominant square matrix \(m\), a vector representing
the value \(b\) in \(mx = b\), a double representing the maximum distance between
the solutions produced by iteration \(i\) and \(i+1\) (by L2 norm a.k.a cartesian
distance), and a \texttt{max\_iterations} which we force stop.
\item Output: the converged-upon solution \(x\) to \(mx = b\)
\end{itemize}
\begin{verbatim}
Array_double *jacobi_solve(Matrix_double *m, Array_double *b,
double l2_convergence_tolerance,
size_t max_iterations) {
assert(m->rows == m->cols);
assert(b->size == m->cols);
size_t iter = max_iterations;
Array_double *x_k = InitArrayWithSize(double, b->size, 0.0);
Array_double *x_k_1 =
InitArrayWithSize(double, b->size, rand_from(0.1, 10.0));
while ((--iter) > 0 && l2_distance(x_k_1, x_k) > l2_convergence_tolerance) {
for (size_t i = 0; i < m->rows; i++) {
double delta = 0.0;
for (size_t j = 0; j < m->cols; j++) {
if (i == j)
continue;
delta += m->data[i]->data[j] * x_k->data[j];
}
x_k_1->data[i] = (b->data[i] - delta) / m->data[i]->data[i];
}
Array_double *tmp = x_k;
x_k = x_k_1;
x_k_1 = tmp;
}
free_vector(x_k);
return x_k_1;
}
\end{verbatim}
\subsubsection{\texttt{gauss\_siedel\_solve}}
\label{sec:org6633923}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{gauss\_siedel\_solve}
\item Location: \texttt{src/matrix.c}
\item Input: a pointer to a \href{https://en.wikipedia.org/wiki/Gauss\%E2\%80\%93Seidel\_method}{diagonally dominant or symmetric and positive definite}
square matrix \(m\), a vector representing
the value \(b\) in \(mx = b\), a double representing the maximum distance between
the solutions produced by iteration \(i\) and \(i+1\) (by L2 norm a.k.a cartesian
distance), and a \texttt{max\_iterations} which we force stop.
\item Output: the converged-upon solution \(x\) to \(mx = b\)
\item Description: we use almost the exact same method as \texttt{jacobi\_solve} but modify
only one array in accordance to the Gauss-Siedel method, but which is necessarily
copied before due to the convergence check.
\end{itemize}
\begin{verbatim}
Array_double *gauss_siedel_solve(Matrix_double *m, Array_double *b,
double l2_convergence_tolerance,
size_t max_iterations) {
assert(m->rows == m->cols);
assert(b->size == m->cols);
size_t iter = max_iterations;
Array_double *x_k = InitArrayWithSize(double, b->size, 0.0);
Array_double *x_k_1 =
InitArrayWithSize(double, b->size, rand_from(0.1, 10.0));
while ((--iter) > 0) {
for (size_t i = 0; i < x_k->size; i++)
x_k->data[i] = x_k_1->data[i];
for (size_t i = 0; i < m->rows; i++) {
double delta = 0.0;
for (size_t j = 0; j < m->cols; j++) {
if (i == j)
continue;
delta += m->data[i]->data[j] * x_k_1->data[j];
}
x_k_1->data[i] = (b->data[i] - delta) / m->data[i]->data[i];
}
if (l2_distance(x_k_1, x_k) <= l2_convergence_tolerance)
break;
}
free_vector(x_k);
return x_k_1;
}
\end{verbatim}
\subsection{Appendix / Miscellaneous} \subsection{Appendix / Miscellaneous}
\label{sec:org7ecff9b} \label{sec:orga72494e}
\subsubsection{Random}
\label{sec:org4940c39}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{rand\_from}
\item Location: \texttt{src/rand.c}
\item Input: a pair of doubles, min and max to generate a random number min
\(\le\) x \(\le\) max
\item Output: a random double in the constraints shown
\end{itemize}
\begin{verbatim}
double rand_from(double min, double max) {
return min + (rand() / (RAND_MAX / (max - min)));
}
\end{verbatim}
\subsubsection{Data Types} \subsubsection{Data Types}
\label{sec:orgd12d26b} \label{sec:org8d3f6e1}
\begin{enumerate} \begin{enumerate}
\item \texttt{Line} \item \texttt{Line}
\label{sec:org5c75544} \label{sec:orgc0df901}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/types.h} \item Location: \texttt{inc/types.h}
@ -1314,7 +1467,7 @@ 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:orgc595e92} \label{sec:org435e816}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/types.h} \item Location: \texttt{inc/types.h}
@ -1344,11 +1497,12 @@ typedef struct {
} Matrix_int } Matrix_int
\end{verbatim} \end{verbatim}
\end{enumerate} \end{enumerate}
\subsubsection{Macros} \subsubsection{Macros}
\label{sec:orgdcaa9de} \label{sec:orga2161be}
\begin{enumerate} \begin{enumerate}
\item \texttt{c\_max} and \texttt{c\_min} \item \texttt{c\_max} and \texttt{c\_min}
\label{sec:org8f831d6} \label{sec:org16ca9c3}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h} \item Location: \texttt{inc/macros.h}
@ -1360,8 +1514,9 @@ typedef struct {
#define c_max(x, y) (((x) >= (y)) ? (x) : (y)) #define c_max(x, y) (((x) >= (y)) ? (x) : (y))
#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:org61bb69c} \label{sec:orgcaff993}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h} \item Location: \texttt{inc/macros.h}
@ -1380,8 +1535,9 @@ typedef struct {
arr; \ arr; \
}) })
\end{verbatim} \end{verbatim}
\item \texttt{InitArrayWithSize} \item \texttt{InitArrayWithSize}
\label{sec:org6bc07d2} \label{sec:orga925ddb}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h} \item Location: \texttt{inc/macros.h}
@ -1400,8 +1556,9 @@ typedef struct {
arr; \ arr; \
}) })
\end{verbatim} \end{verbatim}
\item \texttt{InitMatrixWithSize} \item \texttt{InitMatrixWithSize}
\label{sec:orgb5de1b7} \label{sec:orgf90d7c8}
\begin{itemize} \begin{itemize}
\item Author: Elizabeth Hunt \item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h} \item Location: \texttt{inc/macros.h}

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#+TITLE: Homework 7 #+TITLE: Homework 8
#+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}

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% Created 2023-12-09 Sat 21:43 % Created 2023-12-09 Sat 22:06
% Intended LaTeX compiler: pdflatex % Intended LaTeX compiler: pdflatex
\documentclass[11pt]{article} \documentclass[11pt]{article}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
@ -15,10 +15,10 @@
\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
\author{Elizabeth Hunt} \author{Elizabeth Hunt}
\date{\today} \date{\today}
\title{Homework 7} \title{Homework 8}
\hypersetup{ \hypersetup{
pdfauthor={Elizabeth Hunt}, pdfauthor={Elizabeth Hunt},
pdftitle={Homework 7}, pdftitle={Homework 8},
pdfkeywords={}, pdfkeywords={},
pdfsubject={}, pdfsubject={},
pdfcreator={Emacs 28.2 (Org mode 9.7-pre)}, pdfcreator={Emacs 28.2 (Org mode 9.7-pre)},
@ -29,11 +29,11 @@
\setlength\parindent{0pt} \setlength\parindent{0pt}
\section{Question One} \section{Question One}
\label{sec:orgb6d5cda} \label{sec:org800c743}
See \texttt{UTEST(jacobi, solve\_jacobi)} in \texttt{test/jacobi.t.c} and the entry See \texttt{UTEST(jacobi, solve\_jacobi)} in \texttt{test/jacobi.t.c} and the entry
\texttt{Jacobi / Gauss-Siedel -> solve\_jacobi} in the LIZFCM API documentation. \texttt{Jacobi / Gauss-Siedel -> solve\_jacobi} in the LIZFCM API documentation.
\section{Question Two} \section{Question Two}
\label{sec:org9786314} \label{sec:org6121bef}
We cannot just perform the Jacobi algorithm on a Leslie matrix since We cannot just perform the Jacobi algorithm on a Leslie matrix since
it is obviously not diagonally dominant - which is a requirement. It is it is obviously not diagonally dominant - which is a requirement. It is
certainly not always the case, but, if a Leslie matrix \(L\) is invertible, we can certainly not always the case, but, if a Leslie matrix \(L\) is invertible, we can
@ -53,13 +53,13 @@ but with the con that \(k\) is given by some function on both the convergence cr
nonzero entries in the matrix - which might end up worse in some cases than the LU decomp approach. nonzero entries in the matrix - which might end up worse in some cases than the LU decomp approach.
\section{Question Three} \section{Question Three}
\label{sec:org0ea87d0} \label{sec:org11282e6}
See \texttt{UTEST(jacobi, gauss\_siedel\_solve)} in \texttt{test/jacobi.t.c} which runs the same See \texttt{UTEST(jacobi, gauss\_siedel\_solve)} in \texttt{test/jacobi.t.c} which runs the same
unit test as \texttt{UTEST(jacobi, solve\_jacobi)} but using the unit test as \texttt{UTEST(jacobi, solve\_jacobi)} but using the
\texttt{Jacobi / Gauss-Siedel -> gauss\_siedel\_solve} method as documented in the LIZFCM API reference. \texttt{Jacobi / Gauss-Siedel -> gauss\_siedel\_solve} method as documented in the LIZFCM API reference.
\section{Question Four, Five} \section{Question Four, Five}
\label{sec:org8eea2ae} \label{sec:org22b52a9}
We produce the following operation counts (by hackily adding the operation count as the last element We produce the following operation counts (by hackily adding the operation count as the last element
to the solution vector) and errors - the sum of each vector elements' absolute value away from 1.0 to the solution vector) and errors - the sum of each vector elements' absolute value away from 1.0
using the proceeding patch and unit test. using the proceeding patch and unit test.

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#+TITLE: Homework 9
#+AUTHOR: Elizabeth Hunt
#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX: \setlength\parindent{0pt}
#+OPTIONS: toc:nil
* Question One
With a ~matrix_dimension~ set to 700, I consistently see about a 3x improvement in performance on my
10-thread machine. The serial implementation gives an average ~0.189s~ total runtime, while the below
parallel implementation runs in about ~0.066s~ after the cpu cache has filled on the first run.
#+BEGIN_SRC c
#include <math.h>
#include <omp.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#define matrix_dimension 700
int n = matrix_dimension;
float sum;
int main() {
float A[n][n];
float x0[n];
float b[n];
float x1[n];
float res[n];
srand((unsigned int)(time(NULL)));
// not worth parallellization - rand() is not thread-safe
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
A[i][j] = ((float)rand() / (float)(RAND_MAX) * 5.0);
}
x0[i] = ((float)rand() / (float)(RAND_MAX) * 5.0);
}
#pragma omp parallel for private(sum)
for (int i = 0; i < n; i++) {
sum = 0.0;
for (int j = 0; j < n; j++) {
sum += fabs(A[i][j]);
}
A[i][i] += sum;
}
#pragma omp parallel for private(sum)
for (int i = 0; i < n; i++) {
sum = 0.0;
for (int j = 0; j < n; j++) {
sum += A[i][j];
}
b[i] = sum;
}
float tol = 0.0001;
float error = 10.0 * tol;
int maxiter = 100;
int iter = 0;
while (error > tol && iter < maxiter) {
#pragma omp parallel for
for (int i = 0; i < n; i++) {
float temp_sum = b[i];
for (int j = 0; j < n; j++) {
temp_sum -= A[i][j] * x0[j];
}
res[i] = temp_sum;
x1[i] = x0[i] + res[i] / A[i][i];
}
sum = 0.0;
#pragma omp parallel for reduction(+ : sum)
for (int i = 0; i < n; i++) {
float val = x1[i] - x0[i];
sum += val * val;
}
error = sqrt(sum);
#pragma omp parallel for
for (int i = 0; i < n; i++) {
x0[i] = x1[i];
}
iter++;
}
for (int i = 0; i < n; i++)
printf("x[%d] = %6f \t res[%d] = %6f\n", i, x1[i], i, res[i]);
return 0;
}
#+END_SRC
* Question Two
I only see lowerings in performance (likely due to overhead) on my machine using OpenMP until
~matrix_dimension~ becomes quite large, about ~300~ in testing. At ~matrix_dimension=1000~, I see another
about 3x improvement in total runtime (including initialization & I/O which was untouched, so, even further
improvements could be made) on my 10-thread machine; from around ~0.174~ seconds to ~.052~.
#+BEGIN_SRC c
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#ifdef _OPENMP
#include <omp.h>
#else
#define omp_get_num_threads() 0
#define omp_set_num_threads(int) 0
#define omp_get_thread_num() 0
#endif
#define matrix_dimension 1000
int n = matrix_dimension;
float ynrm;
int main() {
float A[n][n];
float v0[n];
float v1[n];
float y[n];
//
// create a matrix
//
// not worth parallellization - rand() is not thread-safe
srand((unsigned int)(time(NULL)));
float a = 5.0;
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
A[i][j] = ((float)rand() / (float)(RAND_MAX)*a);
}
v0[i] = ((float)rand() / (float)(RAND_MAX)*a);
}
//
// modify the diagonal entries for diagonal dominance
// --------------------------------------------------
//
for (int i = 0; i < n; i++) {
float sum = 0.0;
for (int j = 0; j < n; j++) {
sum = sum + fabs(A[i][j]);
}
A[i][i] = A[i][i] + sum;
}
//
// generate a vector of ones
// -------------------------
//
for (int j = 0; j < n; j++) {
v0[j] = 1.0;
}
//
// power iteration test
// --------------------
//
float tol = 0.0000001;
float error = 10.0 * tol;
float lam1, lam0;
int maxiter = 100;
int iter = 0;
while (error > tol && iter < maxiter) {
#pragma omp parallel for
for (int i = 0; i < n; i++) {
y[i] = 0;
for (int j = 0; j < n; j++) {
y[i] = y[i] + A[i][j] * v0[j];
}
}
ynrm = 0.0;
#pragma omp parallel for reduction(+ : ynrm)
for (int i = 0; i < n; i++) {
ynrm += y[i] * y[i];
}
ynrm = sqrt(ynrm);
#pragma omp parallel for
for (int i = 0; i < n; i++) {
v1[i] = y[i] / ynrm;
}
#pragma omp parallel for
for (int i = 0; i < n; i++) {
y[i] = 0.0;
for (int j = 0; j < n; j++) {
y[i] += A[i][j] * v1[j];
}
}
lam1 = 0.0;
#pragma omp parallel for reduction(+ : lam1)
for (int i = 0; i < n; i++) {
lam1 += v1[i] * y[i];
}
error = fabs(lam1 - lam0);
lam0 = lam1;
#pragma omp parallel for
for (int i = 0; i < n; i++) {
v0[i] = v1[i];
}
iter++;
}
printf("in %d iterations, eigenvalue = %f\n", iter, lam1);
}
#+END_SRC
* Question Three
[[https://static.simponic.xyz/lizfcm.pdf]]

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% Created 2023-12-11 Mon 19:24
% 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 9}
\hypersetup{
pdfauthor={Elizabeth Hunt},
pdftitle={Homework 9},
pdfkeywords={},
pdfsubject={},
pdfcreator={Emacs 28.2 (Org mode 9.7-pre)},
pdflang={English}}
\begin{document}
\maketitle
\setlength\parindent{0pt}
\section{Question One}
\label{sec:org69bed2d}
With a \texttt{matrix\_dimension} set to 700, I consistently see about a 3x improvement in performance on my
10-thread machine. The serial implementation gives an average \texttt{0.189s} total runtime, while the below
parallel implementation runs in about \texttt{0.066s} after the cpu cache has filled on the first run.
\begin{verbatim}
#include <math.h>
#include <omp.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#define matrix_dimension 700
int n = matrix_dimension;
float sum;
int main() {
float A[n][n];
float x0[n];
float b[n];
float x1[n];
float res[n];
srand((unsigned int)(time(NULL)));
// not worth parallellization - rand() is not thread-safe
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
A[i][j] = ((float)rand() / (float)(RAND_MAX) * 5.0);
}
x0[i] = ((float)rand() / (float)(RAND_MAX) * 5.0);
}
#pragma omp parallel for private(sum)
for (int i = 0; i < n; i++) {
sum = 0.0;
for (int j = 0; j < n; j++) {
sum += fabs(A[i][j]);
}
A[i][i] += sum;
}
#pragma omp parallel for private(sum)
for (int i = 0; i < n; i++) {
sum = 0.0;
for (int j = 0; j < n; j++) {
sum += A[i][j];
}
b[i] = sum;
}
float tol = 0.0001;
float error = 10.0 * tol;
int maxiter = 100;
int iter = 0;
while (error > tol && iter < maxiter) {
#pragma omp parallel for
for (int i = 0; i < n; i++) {
float temp_sum = b[i];
for (int j = 0; j < n; j++) {
temp_sum -= A[i][j] * x0[j];
}
res[i] = temp_sum;
x1[i] = x0[i] + res[i] / A[i][i];
}
sum = 0.0;
#pragma omp parallel for reduction(+ : sum)
for (int i = 0; i < n; i++) {
float val = x1[i] - x0[i];
sum += val * val;
}
error = sqrt(sum);
#pragma omp parallel for
for (int i = 0; i < n; i++) {
x0[i] = x1[i];
}
iter++;
}
for (int i = 0; i < n; i++)
printf("x[%d] = %6f \t res[%d] = %6f\n", i, x1[i], i, res[i]);
return 0;
}
\end{verbatim}
\section{Question Two}
\label{sec:orgbeace25}
I only see lowerings in performance (likely due to overhead) on my machine using OpenMP until
\texttt{matrix\_dimension} becomes quite large, about \texttt{300} in testing. At \texttt{matrix\_dimension=1000}, I see another
about 3x improvement in total runtime (including initialization \& I/O which was untouched, so, even further
improvements could be made) on my 10-thread machine; from around \texttt{0.174} seconds to \texttt{.052}.
\begin{verbatim}
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#ifdef _OPENMP
#include <omp.h>
#else
#define omp_get_num_threads() 0
#define omp_set_num_threads(int) 0
#define omp_get_thread_num() 0
#endif
#define matrix_dimension 1000
int n = matrix_dimension;
float ynrm;
int main() {
float A[n][n];
float v0[n];
float v1[n];
float y[n];
//
// create a matrix
//
// not worth parallellization - rand() is not thread-safe
srand((unsigned int)(time(NULL)));
float a = 5.0;
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
A[i][j] = ((float)rand() / (float)(RAND_MAX)*a);
}
v0[i] = ((float)rand() / (float)(RAND_MAX)*a);
}
//
// modify the diagonal entries for diagonal dominance
// --------------------------------------------------
//
for (int i = 0; i < n; i++) {
float sum = 0.0;
for (int j = 0; j < n; j++) {
sum = sum + fabs(A[i][j]);
}
A[i][i] = A[i][i] + sum;
}
//
// generate a vector of ones
// -------------------------
//
for (int j = 0; j < n; j++) {
v0[j] = 1.0;
}
//
// power iteration test
// --------------------
//
float tol = 0.0000001;
float error = 10.0 * tol;
float lam1, lam0;
int maxiter = 100;
int iter = 0;
while (error > tol && iter < maxiter) {
#pragma omp parallel for
for (int i = 0; i < n; i++) {
y[i] = 0;
for (int j = 0; j < n; j++) {
y[i] = y[i] + A[i][j] * v0[j];
}
}
ynrm = 0.0;
#pragma omp parallel for reduction(+ : ynrm)
for (int i = 0; i < n; i++) {
ynrm += y[i] * y[i];
}
ynrm = sqrt(ynrm);
#pragma omp parallel for
for (int i = 0; i < n; i++) {
v1[i] = y[i] / ynrm;
}
#pragma omp parallel for
for (int i = 0; i < n; i++) {
y[i] = 0.0;
for (int j = 0; j < n; j++) {
y[i] += A[i][j] * v1[j];
}
}
lam1 = 0.0;
#pragma omp parallel for reduction(+ : lam1)
for (int i = 0; i < n; i++) {
lam1 += v1[i] * y[i];
}
error = fabs(lam1 - lam0);
lam0 = lam1;
#pragma omp parallel for
for (int i = 0; i < n; i++) {
v0[i] = v1[i];
}
iter++;
}
printf("in %d iterations, eigenvalue = %f\n", iter, lam1);
}
\end{verbatim}
\section{Question Three}
\label{sec:org33439e0}
\url{https://static.simponic.xyz/lizfcm.pdf}
\end{document}