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+% Created 2023-11-11 Sat 13:13
+% 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 6}
+\hypersetup{
+ pdfauthor={Elizabeth Hunt},
+ pdftitle={Homework 6},
+ pdfkeywords={},
+ pdfsubject={},
+ pdfcreator={Emacs 29.1 (Org mode 9.7-pre)}, 
+ pdflang={English}}
+\begin{document}
+
+\maketitle
+\setlength\parindent{0pt}
+\section{Question One}
+\label{sec:org206b859}
+
+For \(g(x) = x + f(x)\) then we know \(g'(x) = 1 + 2x - 5\) and thus \(|g'(x)| \lt 1\) is only true
+on the interval \((1.5, 2.5)\), and for \(g(x) = x - f(x)\) then we know \(g'(x) = 1 - (2x - 5)\)
+and thus \(|g'(x)| < 1\) is only true on the interval \((2.5, 3.5)\).
+
+Because we know the roots of \(f\) are \(2, 3\) (\(f(x) = (x-2)(x-3)\)) then we can only be
+certain that \(g(x) = x + f(x)\) will converge to the root \(2\) if we pick an initial
+guess between \((1.5, 2.5)\), and likewise for \(g(x) = x - f(x)\), \(3\):
+
+\begin{verbatim}
+// tests/roots.t.c
+UTEST(root, fixed_point_iteration_method) {
+  // x^2 - 5x + 6 = (x - 3)(x - 2)
+  double expect_x1 = 3.0;
+  double expect_x2 = 2.0;
+
+  double tolerance = 0.001;
+  uint64_t max_iterations = 10;
+
+  double x_0 = 1.55; // 1.5 < 1.55 < 2.5
+  // g1(x) = x + f(x)
+  double root1 =
+      fixed_point_iteration_method(&f2, &g1, x_0, tolerance, max_iterations);
+  EXPECT_NEAR(root1, expect_x2, tolerance);
+
+  // g2(x) = x - f(x)
+  x_0 = 3.4; // 2.5 < 3.4 < 3.5
+  double root2 =
+      fixed_point_iteration_method(&f2, &g2, x_0, tolerance, max_iterations);
+  EXPECT_NEAR(root2, expect_x1, tolerance);
+}
+\end{verbatim}
+
+And by this method passing in \texttt{tests/roots.t.c} we know they converged within \texttt{tolerance} before
+10 iterations.
+\section{Question Two}
+\label{sec:orga0f5b42}
+
+Yes, we showed that for \(\epsilon = 1\) in Question One, we can converge upon a root in the range \((2.5, 3.5)\), and
+when \(\epsilon = -1\) we can converge upon a root in the range \((1.5, 2.5)\).
+
+See the above unit tests in Question One for each \(\epsilon\).
+\section{Question Three}
+\label{sec:org19aa326}
+
+See \texttt{test/roots.t.c -> UTEST(root, bisection\_with\_error\_assumption)}
+and the software manual entry \texttt{bisect\_find\_root\_with\_error\_assumption}.
+\section{Question Four}
+\label{sec:org722aa6a}
+
+See \texttt{test/roots.t.c -> UTEST(root, fixed\_point\_newton\_method)}
+and the software manual entry \texttt{fixed\_point\_newton\_method}.
+\section{Question Five}
+\label{sec:org587ee52}
+
+See \texttt{test/roots.t.c -> UTEST(root, fixed\_point\_secant\_method)}
+and the software manual entry \texttt{fixed\_point\_secant\_method}.
+\section{Question Six}
+\label{sec:org79bf754}
+
+See \texttt{test/roots.t.c -> UTEST(root, fixed\_point\_bisection\_secant\_method)}
+and the software manual entry \texttt{fixed\_point\_bisection\_secant\_method}.
+\section{Question Seven}
+\label{sec:org4cb47e5}
+
+The existance of \texttt{test/roots.t.c}'s compilation into \texttt{dist/lizfcm.test} via \texttt{make}
+shows that the compiled \texttt{lizfcm.a} contains the root methods mentioned; a user
+could link the library and use them, as we do in Question Eight.
+\section{Question Eight}
+\label{sec:org4a8160d}
+
+The given ODE \(\frac{dP}{dt} = \alpha P - \beta P\) has a trivial solution by separation:
+
+\begin{equation*}
+P(t) = C e^{t(\alpha - \beta)}
+\end{equation*}
+
+And
+
+\begin{equation*}
+P_0 = P(0) = C e^0 = C
+\end{equation*}
+
+So \(P(t) = P_0 e^{t(\alpha - \beta)}\).
+
+We're trying to find \(t\) such that \(P(t) = P_\infty\), thus we're finding roots of \(P(t) - P_\infty\).
+
+The following code (in \texttt{homeworks/hw\_6\_p\_8.c}) produces this output:
+
+\begin{verbatim}
+$ gcc -I../inc/ -Wall hw_6_p_8.c ../lib/lizfcm.a -lm -o hw_6_p_8 && ./hw_6_p_8
+
+a ~ 27.303411; P(27.303411) - P_infty = -0.000000
+b ~ 40.957816; P(40.957816) - P_infty = -0.000000
+c ~ 40.588827; P(40.588827) - P_infty = -0.000000
+d ~ 483.611967; P(483.611967) - P_infty = -0.000000
+e ~ 4.894274; P(4.894274) - P_infty = -0.000000
+
+\end{verbatim}
+
+\begin{verbatim}
+// compile & test w/
+//  \--> gcc -I../inc/ -Wall hw_6_p_8.c ../lib/lizfcm.a -lm -o hw_6_p_8
+//  \--> ./hw_6_p_8
+
+#include "lizfcm.h"
+#include <math.h>
+#include <stdio.h>
+
+double a(double t) {
+  double alpha = 0.1;
+  double beta = 0.001;
+  double p_0 = 2;
+  double p_infty = 29.85;
+
+  return p_0 * exp(t * (alpha - beta)) - p_infty;
+}
+
+double b(double t) {
+  double alpha = 0.1;
+  double beta = 0.001;
+  double p_0 = 2;
+  double p_infty = 115.35;
+
+  return p_0 * exp(t * (alpha - beta)) - p_infty;
+}
+
+double c(double t) {
+  double alpha = 0.1;
+  double beta = 0.0001;
+  double p_0 = 2;
+  double p_infty = 115.35;
+
+  return p_0 * exp(t * (alpha - beta)) - p_infty;
+}
+
+double d(double t) {
+  double alpha = 0.01;
+  double beta = 0.001;
+  double p_0 = 2;
+  double p_infty = 155.346;
+
+  return p_0 * exp(t * (alpha - beta)) - p_infty;
+}
+
+double e(double t) {
+  double alpha = 0.1;
+  double beta = 0.01;
+  double p_0 = 100;
+  double p_infty = 155.346;
+
+  return p_0 * exp(t * (alpha - beta)) - p_infty;
+}
+
+int main() {
+  uint64_t max_iterations = 1000;
+  double tolerance = 0.0000001;
+
+  Array_double *ivt_range = find_ivt_range(&a, -5.0, 3.0, 1000);
+  double approx_a = fixed_point_secant_bisection_method(
+      &a, ivt_range->data[0], ivt_range->data[1], tolerance, max_iterations);
+
+  free_vector(ivt_range);
+  ivt_range = find_ivt_range(&b, -5.0, 3.0, 1000);
+  double approx_b = fixed_point_secant_bisection_method(
+      &b, ivt_range->data[0], ivt_range->data[1], tolerance, max_iterations);
+
+  free_vector(ivt_range);
+  ivt_range = find_ivt_range(&c, -5.0, 3.0, 1000);
+  double approx_c = fixed_point_secant_bisection_method(
+      &c, ivt_range->data[0], ivt_range->data[1], tolerance, max_iterations);
+
+  free_vector(ivt_range);
+  ivt_range = find_ivt_range(&d, -5.0, 3.0, 1000);
+  double approx_d = fixed_point_secant_bisection_method(
+      &d, ivt_range->data[0], ivt_range->data[1], tolerance, max_iterations);
+
+  free_vector(ivt_range);
+  ivt_range = find_ivt_range(&e, -5.0, 3.0, 1000);
+  double approx_e = fixed_point_secant_bisection_method(
+      &e, ivt_range->data[0], ivt_range->data[1], tolerance, max_iterations);
+
+  printf("a ~ %f; P(%f) = %f\n", approx_a, approx_a, a(approx_a));
+  printf("b ~ %f; P(%f) = %f\n", approx_b, approx_b, b(approx_b));
+  printf("c ~ %f; P(%f) = %f\n", approx_c, approx_c, c(approx_c));
+  printf("d ~ %f; P(%f) = %f\n", approx_d, approx_d, d(approx_d));
+  printf("e ~ %f; P(%f) = %f\n", approx_e, approx_e, e(approx_e));
+
+  return 0;
+}
+\end{verbatim}
+\end{document}