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#+TITLE: Homework 6
#+AUTHOR: Elizabeth Hunt
#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX: \setlength\parindent{0pt}
#+OPTIONS: toc:nil
* Question One
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_SRC c
// 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_SRC
And by this method passing in ~tests/roots.t.c~ we know they converged within ~tolerance~ before
10 iterations.
* Question Two
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 $\epsilon$.
* Question Three
See ~test/roots.t.c -> UTEST(root, bisection_with_error_assumption)~
and the software manual entry ~bisect_find_root_with_error_assumption~.
* Question Four
See ~test/roots.t.c -> UTEST(root, fixed_point_newton_method)~
and the software manual entry ~fixed_point_newton_method~.
* Question Five
See ~test/roots.t.c -> UTEST(root, fixed_point_secant_method)~
and the software manual entry ~fixed_point_secant_method~.
* Question Six
See ~test/roots.t.c -> UTEST(root, fixed_point_bisection_secant_method)~
and the software manual entry ~fixed_point_bisection_secant_method~.
* Question Seven
The existance of ~test/roots.t.c~'s compilation into ~dist/lizfcm.test~ via ~make~
shows that the compiled ~lizfcm.a~ contains the root methods mentioned; a user
could link the library and use them, as we do in Question Eight.
* Question Eight
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 ~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 ~ 27.269515; P(27.269515) - 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_SRC c
// 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 = 29.75;
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_SRC
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