hw6

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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
+
+