lecture 2 notes

lecture_02/README.md

@@ -0,0 +1,174 @@
+
+# Solution to Form #1
+
+
+```octave
+[1,2,3]*[1;2;3]
+```
+
+    ans =
+
+       14
+
+
+
+
+```octave
+[1,2,3]*[1;2;3]=?
+```
+
+
+```octave
+
+```
+
+# The first source of error is roundoff error
+## Just storing a number in a computer requires rounding
+
+
+```octave
+fprintf('realmax = %1.20e\n',realmax)
+fprintf('realmin = %1.20e\n',realmin)
+fprintf('maximum relative error = %1.20e\n',eps)
+
+```
+
+    realmax = 1.79769313486231570815e+308
+    realmin = 2.22507385850720138309e-308
+    maximum relative error = 2.22044604925031308085e-16
+
+
+
+```octave
+s=1;
+for i=1:1000
+    s=s+eps/10;
+end
+s==1
+```
+
+    ans =  1
+
+
+# Freefall Model (revisited)
+## Octave solution (will run same on Matlab)
+
+Set default values in Octave for linewidth and text size
+
+
+```octave
+%plot --format svg
+```
+
+
+```octave
+set (0, "defaultaxesfontname", "Helvetica")
+set (0, "defaultaxesfontsize", 18)
+set (0, "defaulttextfontname", "Helvetica")
+set (0, "defaulttextfontsize", 18) 
+set (0, "defaultlinelinewidth", 4)
+```
+
+Define time from 0 to 12 seconds with `N` timesteps 
+function defined as `freefall`
+
+
+```octave
+function [v_analytical,v_terminal,t]=freefall(N)
+    t=linspace(0,12,N)';
+    c=0.25; m=60; g=9.81; v_terminal=sqrt(m*g/c);
+
+    v_analytical = v_terminal*tanh(g*t/v_terminal);
+    v_numerical=zeros(length(t),1);
+    delta_time =diff(t);
+    for i=1:length(t)-1
+        v_numerical(i+1)=v_numerical(i)+(g-c/m*v_numerical(i)^2)*delta_time(i);
+    end
+    % Print values near 0,2,4,6,8,10,12 seconds
+    indices = round(linspace(1,length(t),7));
+    fprintf('time (s)|vel analytical (m/s)|vel numerical (m/s)\n')
+    fprintf('-----------------------------------------------\n')
+    M=[t(indices),v_analytical(indices),v_numerical(indices)];
+    fprintf('%7.1f | %18.2f | %15.2f\n',M(:,1:3)');
+    plot(t,v_analytical,'-',t,v_numerical,'o-')
+end
+    
+```
+
+
+```octave
+[v_analytical,v_terminal,t]=freefall(120);
+```
+
+    time (s)|vel analytical (m/s)|vel numerical (m/s)
+    -----------------------------------------------
+        0.0 |               0.00 |            0.00
+        2.0 |              18.76 |           18.82
+        4.0 |              32.64 |           32.80
+        6.1 |              40.79 |           40.97
+        8.0 |              44.80 |           44.94
+       10.0 |              46.84 |           46.93
+       12.0 |              47.77 |           47.82
+
+
+
+![svg](output_13_1.svg)
+
+
+# Types of error
+## Freefall is example of "truncation error"
+### Truncation error results from approximating exact mathematical procedure
+
+We approximated the derivative as $\delta v/\delta t\approx\Delta v/\Delta t$
+
+Can reduce error by decreasing step size -> $\Delta t$=`delta_time`
+
+## Another example of truncation error is a Taylor series (or Maclaurin if centered at a=0)
+
+Taylor series:
+$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\frac{f'''(a)}{3!}(x-a)^{3}+...$
+
+We can approximate the next value in a function by adding Taylor series terms:
+
+|Approximation | formula |
+|---|-------------------------|
+|$0^{th}$-order | $f(x_{i+1})=f(x_{i})+R_{1}$ |
+|$1^{st}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+R_{2}$ |
+|$2^{nd}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\frac{f''(x_{i})}{2!}h^{2}+R_{3}$|
+|$n^{th}$-order | $f(x_{i+1})=f(x_{i})+f'(x_{i})h+\frac{f''(x_{i})}{2!}h^{2}+...\frac{f^{(n)}}{n!}h^{n}+R_{n}$|
+
+Where $R_{n}=\frac{f^{(n+1)}(\xi)}{(n+1)!}h^{n+1}$ is the error associated with truncating the approximation at order $n$.
+
+The $n^{th}$-order approximation estimates that the unknown function, $f(x)$, is equal to an $n^{th}$-order polynomial. 
+
+In the Freefall example, we estimated the function with a $1^{st}$-order approximation, so 
+
+$v(t_{i+1})=v(t_{i})+v'(t_{i})(t_{i+1}-t_{i})+R_{1}$
+
+$v'(t_{i})=\frac{v(t_{i+1})-v(t_{i})}{t_{i+1}-t_{i}}-\frac{R_{1}}{t_{i+1}-t_{i}}$
+
+$\frac{R_{1}}{t_{i+1}-t_{i}}=\frac{v''(\xi)}{2!}(t_{i+1}-t_{i})$
+
+or the truncation error for a first-order Taylor series approximation is
+
+$\frac{R_{1}}{t_{i+1}-t_{i}}=O(\Delta t)$
+
+
+1. digital representation of a number is rarely exact
+
+2. arithmetic (+,-,/,\*) causes roundoff error
+
+
+```octave
+fprintf('%1.20f\n',double(pi))
+fprintf('%1.20f\n',single(pi))
+```
+
+    3.14159265358979311600
+    3.14159274101257324219
+
+
+
+```octave
+
+```