simponic/lizfcm
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

recompile software manual after hw 7 and hw 7 p6

Browse Source
This commit is contained in:
Elizabeth Hunt 2023-11-27 15:13:34 -07:00
parent 0981ffa00c
commit f0b420e8cd
Signed by: simponic
GPG Key ID: 52B3774857EB24B1
6 changed files with 310 additions and 77 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
doc/software_manual.pdf
View File

Binary file not shown.

231
doc/software_manual.tex
View File

@ -1,4 +1,4 @@
% Created 2023-11-15 Wed 14:43
% Created 2023-11-27 Mon 15:10
% Intended LaTeX compiler: pdflatex
\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
@ -30,7 +30,7 @@
\setlength\parindent{0pt}
\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
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
@ -47,7 +47,7 @@ the C programming language. I have a couple tenets for its design:
in files, and not individual files per function.
\end{itemize}
\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
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
in the standard method.
\section{The LIZFCM API}
\label{sec:org2144095}
\label{sec:orgd18dc24}
\subsection{Simple Routines}
\label{sec:orgc9edf4b}
\label{sec:orgcc14949}
\subsubsection{\texttt{smaceps}}
\label{sec:org449b8ec}
\label{sec:orgd908a7a}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{smaceps}
@ -101,7 +101,7 @@ float smaceps() {
}
\end{verbatim}
\subsubsection{\texttt{dmaceps}}
\label{sec:org9a9ac05}
\label{sec:org53d7f6f}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{dmaceps}
@ -126,9 +126,9 @@ double dmaceps() {
}
\end{verbatim}
\subsection{Derivative Routines}
\label{sec:orgc31ab7b}
\label{sec:orgd7542a0}
\subsubsection{\texttt{central\_derivative\_at}}
\label{sec:org83dc368}
\label{sec:orgf572396}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{central\_derivative\_at}
@ -158,7 +158,7 @@ double central_derivative_at(double (*f)(double), double a, double h) {
}
\end{verbatim}
\subsubsection{\texttt{forward\_derivative\_at}}
\label{sec:orgf1ec748}
\label{sec:org4e1fa4a}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{forward\_derivative\_at}
@ -188,7 +188,7 @@ double forward_derivative_at(double (*f)(double), double a, double h) {
}
\end{verbatim}
\subsubsection{\texttt{backward\_derivative\_at}}
\label{sec:orga2827be}
\label{sec:org50d656f}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{backward\_derivative\_at}
@ -218,9 +218,9 @@ double backward_derivative_at(double (*f)(double), double a, double h) {
}
\end{verbatim}
\subsection{Vector Routines}
\label{sec:org4bc395e}
\label{sec:orgfb4a8e7}
\subsubsection{Vector Arithmetic: \texttt{add\_v, minus\_v}}
\label{sec:orgcc76baa}
\label{sec:org06284bd}
\begin{itemize}
\item Author: Elizabeth Hunt
\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}
\subsubsection{Norms: \texttt{l1\_norm}, \texttt{l2\_norm}, \texttt{linf\_norm}}
\label{sec:org015b19a}
\label{sec:org2849a93}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name(s): \texttt{l1\_norm}, \texttt{l2\_norm}, \texttt{linf\_norm}
@ -283,7 +283,7 @@ double linf_norm(Array_double *v) {
}
\end{verbatim}
\subsubsection{\texttt{vector\_distance}}
\label{sec:org78137a7}
\label{sec:org4274c99}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{vector\_distance}
@ -303,7 +303,7 @@ double vector_distance(Array_double *v1, Array_double *v2,
}
\end{verbatim}
\subsubsection{Distances: \texttt{l1\_distance}, \texttt{l2\_distance}, \texttt{linf\_distance}}
\label{sec:orgd71d562}
\label{sec:orge4d3e3f}
\begin{itemize}
\item Author: Elizabeth Hunt
\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}
\subsubsection{\texttt{sum\_v}}
\label{sec:orgb188125}
\label{sec:org94e6241}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{sum\_v}
@ -346,7 +346,7 @@ double sum_v(Array_double *v) {
}
\end{verbatim}
\subsubsection{\texttt{scale\_v}}
\label{sec:org0a828aa}
\label{sec:orgbc8e308}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{scale\_v}
@ -364,7 +364,7 @@ Array_double *scale_v(Array_double *v, double m) {
}
\end{verbatim}
\subsubsection{\texttt{free\_vector}}
\label{sec:orgfff2e8b}
\label{sec:org42b8bd4}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{free\_vector}
@ -381,7 +381,7 @@ void free_vector(Array_double *v) {
}
\end{verbatim}
\subsubsection{\texttt{add\_element}}
\label{sec:orgf002846}
\label{sec:org69937a3}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{add\_element}
@ -400,7 +400,7 @@ Array_double *add_element(Array_double *v, double x) {
}
\end{verbatim}
\subsubsection{\texttt{slice\_element}}
\label{sec:org8ef8f62}
\label{sec:org9958a86}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{slice\_element}
@ -418,7 +418,7 @@ Array_double *slice_element(Array_double *v, size_t x) {
}
\end{verbatim}
\subsubsection{\texttt{copy\_vector}}
\label{sec:org6794d79}
\label{sec:org6f1daca}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{copy\_vector}
@ -437,7 +437,7 @@ Array_double *copy_vector(Array_double *v) {
}
\end{verbatim}
\subsubsection{\texttt{format\_vector\_into}}
\label{sec:orgaaea3a7}
\label{sec:orgb326fc6}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{format\_vector\_into}
@ -466,9 +466,9 @@ void format_vector_into(Array_double *v, char *s) {
}
\end{verbatim}
\subsection{Matrix Routines}
\label{sec:org7b74a8b}
\label{sec:org8bc3f25}
\subsubsection{\texttt{lu\_decomp}}
\label{sec:orgb5ebd91}
\label{sec:org0d25547}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{lu\_decomp}
@ -528,7 +528,7 @@ Matrix_double **lu_decomp(Matrix_double *m) {
}
\end{verbatim}
\subsubsection{\texttt{bsubst}}
\label{sec:org4b2bdc3}
\label{sec:orge607e22}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{bsubst}
@ -553,7 +553,7 @@ Array_double *bsubst(Matrix_double *u, Array_double *b) {
}
\end{verbatim}
\subsubsection{\texttt{fsubst}}
\label{sec:orgf6f799e}
\label{sec:org3da184d}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{fsubst}
@ -580,7 +580,7 @@ Array_double *fsubst(Matrix_double *l, Array_double *b) {
}
\end{verbatim}
\subsubsection{\texttt{solve\_matrix\_lu\_bsubst}}
\label{sec:org789acbf}
\label{sec:orgf2bb0ea}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c}
@ -616,7 +616,7 @@ Array_double *solve_matrix_lu_bsubst(Matrix_double *m, Array_double *b) {
}
\end{verbatim}
\subsubsection{\texttt{gaussian\_elimination}}
\label{sec:orge5cbe95}
\label{sec:org6a7faa2}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c}
@ -671,7 +671,7 @@ Matrix_double *gaussian_elimination(Matrix_double *m) {
}
\end{verbatim}
\subsubsection{\texttt{solve\_matrix\_gaussian}}
\label{sec:org9c2b7c3}
\label{sec:org5379c90}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c}
@ -704,7 +704,7 @@ Array_double *solve_matrix_gaussian(Matrix_double *m, Array_double *b) {
}
\end{verbatim}
\subsubsection{\texttt{m\_dot\_v}}
\label{sec:org4c184b5}
\label{sec:org333640a}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c}
@ -725,7 +725,7 @@ Array_double *m_dot_v(Matrix_double *m, Array_double *v) {
}
\end{verbatim}
\subsubsection{\texttt{put\_identity\_diagonal}}
\label{sec:org45882fa}
\label{sec:orgdf99950}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c}
@ -743,7 +743,7 @@ Matrix_double *put_identity_diagonal(Matrix_double *m) {
}
\end{verbatim}
\subsubsection{\texttt{slice\_column}}
\label{sec:org520c709}
\label{sec:org90e8eb5}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c}
@ -766,7 +766,7 @@ Matrix_double *slice_column(Matrix_double *m, size_t x) {
}
\end{verbatim}
\subsubsection{\texttt{add\_column}}
\label{sec:org84191b6}
\label{sec:org7f9f9a5}
\begin{itemize}
\item Author: Elizabet Hunt
\item Location: \texttt{src/matrix.c}
@ -789,7 +789,7 @@ Matrix_double *add_column(Matrix_double *m, Array_double *v) {
}
\end{verbatim}
\subsubsection{\texttt{copy\_matrix}}
\label{sec:orgb84b548}
\label{sec:orgcd0ab1c}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c}
@ -808,7 +808,7 @@ Matrix_double *copy_matrix(Matrix_double *m) {
}
\end{verbatim}
\subsubsection{\texttt{free\_matrix}}
\label{sec:org0de0d86}
\label{sec:orgfece6dc}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{src/matrix.c}
@ -826,7 +826,7 @@ void free_matrix(Matrix_double *m) {
}
\end{verbatim}
\subsubsection{\texttt{format\_matrix\_into}}
\label{sec:orgf8ba876}
\label{sec:org3d91cb0}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{format\_matrix\_into}
@ -853,9 +853,9 @@ void format_matrix_into(Matrix_double *m, char *s) {
}
\end{verbatim}
\subsection{Root Finding Methods}
\label{sec:org8f80c14}
\label{sec:orgc546ba0}
\subsubsection{\texttt{find\_ivt\_range}}
\label{sec:orgf7fc734}
\label{sec:orgf0457e8}
\begin{itemize}
\item Author: Elizabeth Hunt
\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}
\subsubsection{\texttt{bisect\_find\_root}}
\label{sec:orgcf0f46b}
\label{sec:orge2c8a6a}
\begin{itemize}
\item Author: Elizabeth Hunt
\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}
\subsubsection{\texttt{bisect\_find\_root\_with\_error\_assumption}}
\label{sec:org64e4346}
\label{sec:org0f219ab}
\begin{itemize}
\item Author: Elizabeth Hunt
\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}
\subsubsection{\texttt{fixed\_point\_iteration\_method}}
\label{sec:orge6f6ba8}
\label{sec:orgd9fe8e6}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{fixed\_point\_iteration\_method}
@ -977,7 +977,7 @@ double fixed_point_iteration_method(double (*f)(double), double (*g)(double),
}
\end{verbatim}
\subsubsection{\texttt{fixed\_point\_newton\_method}}
\label{sec:org9c22d8f}
\label{sec:org4cf044d}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{fixed\_point\_newton\_method}
@ -1005,7 +1005,7 @@ double fixed_point_newton_method(double (*f)(double), double (*fprime)(double),
}
\end{verbatim}
\subsubsection{\texttt{fixed\_point\_secant\_method}}
\label{sec:org446d473}
\label{sec:orgc5cfd8b}
\begin{itemize}
\item Author: Elizabeth Hunt
\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}
\subsubsection{\texttt{fixed\_point\_secant\_bisection\_method}}
\label{sec:orgade170f}
\label{sec:orgf3fe7ad}
\begin{itemize}
\item Author: Elizabeth Hunt
\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}
\subsection{Linear Routines}
\label{sec:orgc389980}
\label{sec:org8b9d4db}
\subsubsection{\texttt{least\_squares\_lin\_reg}}
\label{sec:org850d9f6}
\label{sec:orgca5faa9}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Name: \texttt{least\_squares\_lin\_reg}
@ -1115,9 +1115,9 @@ Line *least_squares_lin_reg(Array_double *x, Array_double *y) {
}
\end{verbatim}
\subsection{Eigen-Adjacent}
\label{sec:org6bea1aa}
\label{sec:orgb053100}
\subsubsection{\texttt{dominant\_eigenvalue}}
\label{sec:org0e70920}
\label{sec:orgc345d49}
\begin{itemize}
\item Author: Elizabeth Hunt
\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) {
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);
double new_lambda =
@ -1156,8 +1160,115 @@ double dominant_eigenvalue(Matrix_double *m, Array_double *v, double tolerance,
return lambda;
}
\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}}
\label{sec:org88d4547}
\label{sec:org0637da1}
\begin{itemize}
\item Author: Elizabeth Hunt
\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
\(x\) getting to the next age class \(x+1\) and the number of offspring that individuals in an age
class create in age class 0.
\item Output: the leslie matrix generated with the input vectors.
\item Output: the leslie matrix generated from the input vectors.
\end{itemize}
\begin{verbatim}
@ -1185,12 +1296,12 @@ Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio,
}
\end{verbatim}
\subsection{Appendix / Miscellaneous}
\label{sec:org925aa32}
\label{sec:orgddf0893}
\subsubsection{Data Types}
\label{sec:org37335a1}
\label{sec:org0ec3831}
\begin{enumerate}
\item \texttt{Line}
\label{sec:orgaf72b30}
\label{sec:org1b45662}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{inc/types.h}
@ -1203,7 +1314,7 @@ typedef struct Line {
} Line;
\end{verbatim}
\item The \texttt{Array\_<type>} and \texttt{Matrix\_<type>}
\label{sec:org82faf8e}
\label{sec:org1a49c97}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{inc/types.h}
@ -1234,10 +1345,10 @@ typedef struct {
\end{verbatim}
\end{enumerate}
\subsubsection{Macros}
\label{sec:org1f988ea}
\label{sec:orge905d93}
\begin{enumerate}
\item \texttt{c\_max} and \texttt{c\_min}
\label{sec:org8b37b18}
\label{sec:org0267cca}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h}
@ -1250,7 +1361,7 @@ typedef struct {
#define c_min(x, y) (((x) <= (y)) ? (x) : (y))
\end{verbatim}
\item \texttt{InitArray}
\label{sec:org04ec2d7}
\label{sec:orgdb147d1}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h}
@ -1270,7 +1381,7 @@ typedef struct {
})
\end{verbatim}
\item \texttt{InitArrayWithSize}
\label{sec:org4aff8f6}
\label{sec:org0b33fa5}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h}
@ -1290,7 +1401,7 @@ typedef struct {
})
\end{verbatim}
\item \texttt{InitMatrixWithSize}
\label{sec:org3457577}
\label{sec:orgcc5a2bb}
\begin{itemize}
\item Author: Elizabeth Hunt
\item Location: \texttt{inc/macros.h}

17
homeworks/hw-7.org
View File

@ -1,4 +1,4 @@
#+TITLE: Homework 6
#+TITLE: Homework 7
#+AUTHOR: Elizabeth Hunt
#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX: \setlength\parindent{0pt}
@ -58,4 +58,19 @@ See also the entry ~Eigen-Adjacent -> partition_find_eigenvalues~ in the LIZFCM
documentation.
* 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~.

BIN
homeworks/hw-7.pdf Normal file
View File

Binary file not shown.

107
homeworks/hw-7.tex Normal file
View File

@ -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
View File

@ -4,19 +4,9 @@
#include <stdio.h>
#include <string.h>
Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio,
Array_double *age_class_offspring) {
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;
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);
}
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;
}
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);
Matrix_double *leslie_matrix(Array_double *age_class_surivor_ratio,
Array_double *age_class_offspring) {
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;
}