[usth/MATH2.2] Numerical Methods

The future starts now.
author: Nguyễn Gia Phong <vn.mcsinyx@gmail.com> 2019-12-16 21:31:18 +0700
committer: Nguyễn Gia Phong <vn.mcsinyx@gmail.com> 2019-12-16 21:31:18 +0700
commit: e461df7573c2b7b7e26c965d8cf2d8e175d67378 (patch)
tree: 193b8439edac52c1749764395fe5f7787889eda9
parent: c1008fe39217be7f91f0ea23483e747bfbc5743e (diff)
download: cp-e461df7573c2b7b7e26c965d8cf2d8e175d67378.tar.gz
45 files changed, 44595 insertions, 0 deletions
diff --git a/usth/MATH2.2/final/EX1.m b/usth/MATH2.2/final/EX1.m
new file mode 100644
index 0000000..354aa2f
--- /dev/null
+++ b/usth/MATH2.2/final/EX1.m
@@ -0,0 +1,36 @@
+disp ("Question 1:");
+disp ("(a)");
+printf ("11^3 + 12^3 - 7^3 = %d\n", 11^3 + 12^3 - 7^3);
+printf ("15! = %d\n", factorial (15));
+
+disp ("(b)");
+A = [1 2 3
+     4 5 6
+     7 8 9];
+B = eye (3);
+
+disp ("(b.i)");
+disp ("A + B = ");
+disp (A + B);
+
+disp ("(b.ii)");
+disp ("A' = ");
+disp (A');
+
+disp ("(b.iii)");
+disp ("A^-1 = ");
+disp (inv (A));
+
+disp ("(c.i)");
+printf ("x^2 = 19  ->  x = %g\n", sqrt (19));
+disp ("(c.ii)");
+printf ("x^4 = 55  ->  x = %g\n", sqrt (sqrt (19)));
+
+disp ("(d)");
+X = 0 : 30;
+Y = X * 2 + 3;
+plot (X, Y);
+xlabel ("x");
+ylabel ("y = 2x + 3");
+disp ("Press any key to continue...");
+kbhit;
diff --git a/usth/MATH2.2/final/EX2.m b/usth/MATH2.2/final/EX2.m
new file mode 100644
index 0000000..9b8c95e
--- /dev/null
+++ b/usth/MATH2.2/final/EX2.m
@@ -0,0 +1,38 @@
+disp ("Question 2:");
+disp ("(a)");
+pkg load symbolic;
+syms x real;
+solve (sqrt (x) - x + 1 == 0)
+% ans = (sym)
+%   √5   3
+%   ── + ─
+%   2    2
+pkg unload symbolic;
+disp ("To get numerical solutions we can use fzero");
+disp ("With the initial guess of 0, fzero (@(x) sqrt (x) - x + 1, 0) returns");
+fzero (@(x) sqrt (x) - x + 1, 0)
+
+disp ("(b)");
+hold on;
+ezplot (@(x) exp (-x));
+ezplot (@(x) sin (x));
+hold off;
+disp ("Press any key to continue...");
+kbhit;
+
+disp ("(c)");
+s = 0;
+for k = 1 : 1000
+  s += k^3;
+endfor
+printf ("The cubic sum of integers from 1 to 1000 is %d\n", s);
+
+disp ("(d)");
+A = [2 1 4
+     1 2 -5
+     3 -2 4];
+b = [10 1 8]';
+disp ("Using mldivide, [x y z] = ");
+disp (mldivide (A, b)');
+disp ("Using inv, [x y z] = ");
+disp ((inv (A) * b)');
diff --git a/usth/MATH2.2/final/EX3.m b/usth/MATH2.2/final/EX3.m
new file mode 100644
index 0000000..99240f8
--- /dev/null
+++ b/usth/MATH2.2/final/EX3.m
@@ -0,0 +1,16 @@
+disp ("Question 3:");
+disp ("(a)");
+function y = f (x)
+  y = 2 + x.^2 + exp(x.*2 + 1);
+endfunction
+h = 0.005;
+printf ("By forward difference with h = 0.05, f'(1.35) = %g\n",
+        (f (1.35 + h) - f (1.35)) / h);
+
+disp ("(b)");
+disp ("I am unsure if diff is different on Matlab, but on octave,");
+disp ("it's simply taking differences between consecutive elements.");
+x = [1.35, 1.35+h];
+printf ("Using diff with h = 0.05, we get same result, f'(1.35) = %g\n",
+        (diff (f (x)) / h));
+disp ("Using symbolical methods, f'(1.35) = 83.5946 which is quite close.");
diff --git a/usth/MATH2.2/hw/README.md b/usth/MATH2.2/hw/README.md
new file mode 100644
index 0000000..fbb5730
--- /dev/null
+++ b/usth/MATH2.2/hw/README.md
@@ -0,0 +1,22 @@
+# Numerical Methods: Homework 1
+
+My name is Nguyễn Gia Phong, my student ID is BI9-184.
+
+The homework were written in Octave and saved as
+
+* `linear`: interactive script to solve linear equation
+* `quadratic`: interactive script to solve quadratic equation
+* `cubic`: interactive script to solve cubic equation
+* `fibonacci.m`: function `fibonacci (n)` returning a vector containing
+  Fibonacci numbers from F<sub>1</sub> to F<sub>n</sub>
+
+The scripts are named without any file extension following the \*nix traditions.
+The homework can be downloaded as a whole in a ZIP package using the button
+near the top-left corner, or alternatively cloned using `git`:
+
+    git clone https://gist.github.com/McSinyx/1d9ff91d1ecfd60254c188bddb7926d5
+
+If this causes any inconvenience, please let me know.
+
+[![Creative Commons License](https://i.creativecommons.org/l/by-sa/4.0/80x15.png)](http://creativecommons.org/licenses/by-sa/4.0/)
+This work is licensed under a [Creative Commons Attribution-ShareAlike 4.0 International License](http://creativecommons.org/licenses/by-sa/4.0/).
diff --git a/usth/MATH2.2/hw/cubic b/usth/MATH2.2/hw/cubic
new file mode 100755
index 0000000..ca24739
--- /dev/null
+++ b/usth/MATH2.2/hw/cubic
@@ -0,0 +1,25 @@
+#!/usr/bin/env octave
+disp ("Solve quadratic equation a*x^3 + b*x^2 + c*x + d = 0");
+a = input ("a = ");
+while (a == 0)
+  disp ("a must be nonzero");
+  a = input ("a = ");
+end
+b = input ("b = ");
+c = input ("c = ");
+d = input ("d = ");
+
+% Using the trigonometric and hyperbolic solutions on Wikipedia:
+% https://en.wikipedia.org/wiki/Cubic_function
+p = (3*a*c - b^2) / (3*a^2);
+q = (2*b^3 - 9*a*b*c + 27*a^2*d) / (27*a^3);
+if (p == 0)
+  disp ("The given cubic equation has only one distinct solution:");
+  x = cbrt(-q) - b/a/3
+else
+  r = 2 * sqrt(-p/3);
+  disp ("The three solutions of the given quadratic equation are:");
+  for k = 0 : 2
+    x = r * cos((acos(3*q/p/r) - 2*pi*k) / 3) - b/a/3
+  end
+end
diff --git a/usth/MATH2.2/hw/fibonacci.m b/usth/MATH2.2/hw/fibonacci.m
new file mode 100644
index 0000000..340c108
--- /dev/null
+++ b/usth/MATH2.2/hw/fibonacci.m
@@ -0,0 +1,7 @@
+function fib = fibonacci (n)
+  fib = 0 : n;
+  for i = 2 : n
+    fib(i + 1) = fib(i) + fib(i - 1);
+  end
+  fib = fib(2:end);
+end
diff --git a/usth/MATH2.2/hw/homework.pdf b/usth/MATH2.2/hw/homework.pdf
new file mode 100644
index 0000000..9d3f2e9
--- /dev/null
+++ b/usth/MATH2.2/hw/homework.pdf
Binary files differdiff --git a/usth/MATH2.2/hw/linear b/usth/MATH2.2/hw/linear
new file mode 100755
index 0000000..fad7e67
--- /dev/null
+++ b/usth/MATH2.2/hw/linear
@@ -0,0 +1,11 @@
+#!/usr/bin/env octave
+disp ("Solve quadratic equation ax + b = 0");
+a = input ("a = ");
+while (a == 0)
+  disp ("a must be nonzero");
+  a = input ("a = ");
+end
+b = input ("b = ");
+
+disp ("The two solutions of the given linear equation is:");
+x = -b / a
diff --git a/usth/MATH2.2/hw/quadratic b/usth/MATH2.2/hw/quadratic
new file mode 100755
index 0000000..b6ea91a
--- /dev/null
+++ b/usth/MATH2.2/hw/quadratic
@@ -0,0 +1,14 @@
+#!/usr/bin/env octave
+disp ("Solve quadratic equation a*x^2 + b*x + c = 0");
+a = input ("a = ");
+while (a == 0)
+  disp ("a must be nonzero");
+  a = input ("a = ");
+end
+b = input ("b = ");
+c = input ("c = ");
+
+sd = sqrt(b*b - a*c*4);
+disp ("The two solutions of the given quadratic equation are:");
+x0 = (-b - sd) / a / 2
+x1 = (-b + sd) / a / 2
diff --git a/usth/MATH2.2/labwork/1/bisect.m b/usth/MATH2.2/labwork/1/bisect.m
new file mode 100644
index 0000000..1d9e99f
--- /dev/null
+++ b/usth/MATH2.2/labwork/1/bisect.m
@@ -0,0 +1,36 @@
+function [x fx ea iter] = bisect (f, xl, xu, es = 0.0001, maxit = 50)
+  % uses bisection method to find the root of f,
+  % with xl and xu being lower and upper guesses,
+  % es being desired relative error
+  % and maxit being maximum allowable iterations
+  % to return the real root x, fx = f(x),
+  % approximate relative error ea (%)
+  % and number of iterations iter
+  nargin < 3 && error ('bisect requires at least 3 arguments');
+  [fl fu iter] = deal (f (xl), f (xu), 0);
+  if (fl == 0)
+    [x fx ea] = deal (xl, 0, 0);
+    return;
+  elseif (fu == 0)
+    [x fx ea] = deal (xu, 0, 0);
+    return;
+  end
+  fl * fu < 0 || error ('no sign change');
+
+  [x ea] = deal (xl, 100);
+  while (ea > es && iter++ < maxit) % yes, I use Octave only
+    [xold x] = deal (x, (xl + xu) / 2);
+    fx = f (x);
+    if (fx == 0)
+      ea = 0;
+      break;
+    elseif (x)
+      ea = abs ((x - xold) / x) * 100;
+    end
+    if (f (xl) * fx < 0)
+      xu = x;
+    else
+      xl = x;
+    end
+  end
+end
diff --git a/usth/MATH2.2/labwork/1/intro b/usth/MATH2.2/labwork/1/intro
new file mode 100755
index 0000000..555f874
--- /dev/null
+++ b/usth/MATH2.2/labwork/1/intro
@@ -0,0 +1,62 @@
+#!/usr/bin/env octave
+% Exercise I.1.1
+x = 1 : 5
+printf ("x (3) + x (2) = %f\n", x (3) + x (2));
+
+% Exercise I.1.2
+x = x'
+
+% Exercise I.2.1
+A = zeros (3, 3)
+A += eye (3)
+A += [0 0 0; 2 0 0; 3 4 0]
+B = A'
+A + B
+A - B
+A * B  % matmul
+A .* B % element-wise mul
+A(1:2, 1:2)
+A(1:3)
+A(1:3, 1) % col
+A(:, 1) % col
+
+% Exercise I.3
+function avg = mean (v)
+  avg = sum (v) / numel (v);
+end
+mean (x)
+
+% Exercise I.4
+function price = ebill (kwh)
+  l = [50 50 100 100 100 Inf];
+  p = [1.484 1.533 1.786 2.242 2.503 2.587];
+  for i = 1 : 6
+    if (kwh > l(i))
+      kwh -= l(i);
+    else
+      l(i) = kwh;
+      kwh = 0;
+    end
+  end
+  price = sum (l .* p);
+end
+
+% Exercise I.5.1
+a = (@(x) x*x') (1 : 1000)
+b = sum (arrayfun (@(x) (-1)^x / (1 + x*2), [0 : 501]))
+c = sum (arrayfun (@(x) 1 / (1 + x*2)^2 / (3 + x*2)^2, [0 : 420])) - (pi^2 - 8)/16
+
+% Exercise I.5.2
+function p = permutations (n, k)
+  p = factorial (n) / factorial (n - k);
+end
+function c = combinations (n, k)
+  c = permutations (n, k) / factorial (k);
+end
+
+% Exercise I.6
+log (2) / log (1.1)
+
+% Exercise I.7
+x = linspace(-5, 5);
+plot (x, x + 1, 'color', 'green');
diff --git a/usth/MATH2.2/labwork/1/labwork1.pdf b/usth/MATH2.2/labwork/1/labwork1.pdf
new file mode 100644
index 0000000..d8ce2d9
--- /dev/null
+++ b/usth/MATH2.2/labwork/1/labwork1.pdf
Binary files differdiff --git a/usth/MATH2.2/labwork/1/ratpoison.m b/usth/MATH2.2/labwork/1/ratpoison.m
new file mode 100644
index 0000000..f2db823
--- /dev/null
+++ b/usth/MATH2.2/labwork/1/ratpoison.m
@@ -0,0 +1,15 @@
+function [x fx ea i] = ratpoison (f, df, x0, es = 0.00000001, imax = 20)
+  nargin < 2 && error ('ratpoison requires at least 2 ingredients');
+  [x fx dfx ea i] = deal (x0, f (x0), df (x0), 1, 0);
+  while (ea > es && i++ < imax)
+    [xold x] = deal (x, x - fx/dfx);
+    [fx dfx] = deal (f (x), df (x));
+    if (fx == 0)
+      ea = 0;
+      break;
+    elseif (x)
+      % just drop the percent BS
+      ea = abs ((x - xold) / x);
+    end
+  end
+end
diff --git a/usth/MATH2.2/labwork/1/ratpoison.py b/usth/MATH2.2/labwork/1/ratpoison.py
new file mode 100644
index 0000000..16aa65d
--- /dev/null
+++ b/usth/MATH2.2/labwork/1/ratpoison.py
@@ -0,0 +1,8 @@
+def ratpoison(f, df, x, es=10**-8, imax=20):
+    ea, i = 1, 0
+    while ea > es and i < imax:
+        i += 1
+        xold, x = x, x - f(x)/df(x)
+        if f(x) == 0: return x, 0, 0, i
+        if x: ea = abs((x - xold) / x)
+    return x, f(x), ea, i
diff --git a/usth/MATH2.2/labwork/2/2a.tikz b/usth/MATH2.2/labwork/2/2a.tikz
new file mode 100644
index 0000000..c27b94f
--- /dev/null
+++ b/usth/MATH2.2/labwork/2/2a.tikz
@@ -0,0 +1,515 @@
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diff --git a/usth/MATH2.2/labwork/2/labwork2.pdf b/usth/MATH2.2/labwork/2/labwork2.pdf
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+\documentclass[a4paper,12pt]{article}
+\usepackage[english,vietnamese]{babel}
+\usepackage{amsmath}
+\usepackage{lmodern}
+\usepackage{hyperref}
+\usepackage{tikz}
+
+\newcommand{\exercise}[1]{\noindent\textbf{#1.}}
+\renewcommand{\thesection}{\Roman{section}}
+\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
+
+\title{Numerical Method: Labwork 2 Report}
+\author{Nguyễn Gia Phong--BI9-184}
+\date{Fall 2019}
+
+\begin{document}
+\maketitle
+\setcounter{section}{2}
+\section{Polynomial}
+\exercise{1.c} At the time of writing, function \verb|fzero|
+in Octave have not support the \verb|Display| option
+just yet\footnote{Bug report: \url{https://savannah.gnu.org/bugs/?56954}}.
+However, the implementation of this option is rather trivial,
+thus I made a quick patch (which is also attached at the bug report).
+Using this, one can easily display all the iterations as followed:
+
+\begin{verbatim}
+octave:1> fzero (@(x) x.^2 - 9, 0, optimset ('display', 'iter'))
+
+Search for an interval around 0 containing a sign change:
+Func-eval 1,  how = initial,  a = 0,  f(a) = -9,  b = 0,  f(b) = -9
+Func-eval 2,  how = search,  a = 0,  f(a) = -9,  b = 0.099,  f(b) = -8.9902
+Func-eval 3,  how = search,  a = 0,  f(a) = -9,  b = 0.1025,  f(b) = -8.98949
+Func-eval 4,  how = search,  a = 0,  f(a) = -9,  b = 0.095,  f(b) = -8.99098
+Func-eval 5,  how = search,  a = 0,  f(a) = -9,  b = 0.11,  f(b) = -8.9879
+Func-eval 6,  how = search,  a = 0,  f(a) = -9,  b = 0.075,  f(b) = -8.99437
+Func-eval 7,  how = search,  a = 0,  f(a) = -9,  b = 0.15,  f(b) = -8.9775
+Func-eval 8,  how = search,  a = 0,  f(a) = -9,  b = 0,  f(b) = -9
+Func-eval 9,  how = search,  a = 0,  f(a) = -9,  b = 0.35,  f(b) = -8.8775
+Func-eval 10,  how = search,  a = 0,  f(a) = -9,  b = -0.4,  f(b) = -8.84
+Func-eval 11,  how = search,  a = 0,  f(a) = -9,  b = 1.1,  f(b) = -7.79
+Func-eval 12,  how = search,  a = 0,  f(a) = -9,  b = -4.9,  f(b) = 15.01
+
+Search for a a zero in the interval [-4.9, 0]:
+Func-eval 13,  how = initial,  x = 0,  f(x) = -9
+Func-eval 14,  how = interpolation,  x = -1.83673,  f(x) = -5.62641  (NaN%)
+Func-eval 15,  how = interpolation,  x = -3.36837,  f(x) = 2.3459  (141.7%)
+Func-eval 16,  how = interpolation,  x = -3.19097,  f(x) = 1.1823  (-49.6%)
+Func-eval 17,  how = interpolation,  x = -2.99725,  f(x) = -0.0164972  (-101.4%)
+Func-eval 18,  how = interpolation,  x = -3.00258,  f(x) = 0.0154927  (193.9%)
+Func-eval 19,  how = interpolation,  x = -3,  f(x) = 3.07975e-07  (-100.0%)
+Func-eval 20,  how = interpolation,  x = -3,  f(x) = -7.10543e-15  (-100.0%)
+Func-eval 21,  how = interpolation,  x = -3,  f(x) = 5.32907e-15  (169.7%)
+
+Algorithm converged
+
+ans = -3.0000
+\end{verbatim}
+
+To answer the question in part b, (since I believe these parts are linked
+to each other), the current implementation of \verb|fzero| search for
+the second bracket over quantitative chages below if \verb|X0| if it is a
+single scalar, thus $[-4.9, 0]$ is gotten and the found solution is negative:
+
+\begin{verbatim}
+[-.01 +.025 -.05 +.10 -.25 +.50 -1 +2.5 -5 +10 -50 +100 -500 +1000]
+\end{verbatim}
+
+\section{Non-linear Systems}
+\exercise{1.a} These statements were used to plot the given functions:
+\begin{verbatim}
+ezplot(@(x1, x2) x1 .^ 2 + x1 .* x2 - 10)
+hold on
+ezplot(@(x1, x2) x2 + 3 .* x1 .* x2 .^ 2 - 57)
+\end{verbatim}
+
+As shown in the graphs (where $x_1^2 + x_1 x_2 = 10$ are the blue lines
+and $x_2 + 3 x_1 x_2 = 57$ are the yellow ones), the solutions of $(x_1, x_2)$
+are quite close to $(2, 3)$ and $(4.5, -2)$.
+
+\begin{figure}[!h]
+  \centering
+  \scalebox{0.37}{\input{2a.tikz}}
+\end{figure}
+
+I would also like to note that I am personally impressed how gnuplot
+(which is utilised by Octave) is able to export to TikZ graphics with ease.
+\end{document}
diff --git a/usth/MATH2.2/labwork/3/GaussPivot.m b/usth/MATH2.2/labwork/3/GaussPivot.m
new file mode 100644
index 0000000..7764b3a
--- /dev/null
+++ b/usth/MATH2.2/labwork/3/GaussPivot.m
@@ -0,0 +1,25 @@
+function x = GaussPivot (A, b)
+  [m n] = size (A);
+  m ~= n && error ('Matrix A must be square');
+  nb = n + 1;
+  Aug = [A b];
+  % forward elimination
+  for k = 1 : n - 1
+    % partial pivoting
+    [big i] = max (abs (Aug(k:n, k)));
+    ipr = i + k - 1;
+    if ipr ~= k
+      Aug([k ipr], :) = Aug([ipr k], :);
+    end
+    for i = k + 1 : n
+      coeff = Aug(i, k) / Aug(k, k);
+      Aug(i, k:nb) = Aug(i, k:nb) - coeff * Aug(k, k:nb);
+    end
+  end
+  % back substitution
+  x = zeros (n, 1);
+  x(n) = Aug(n, nb) / Aug(n, n);
+  for i = n - 1 : -1 : 1
+    x(i) = (Aug(i, nb) - Aug(i, i+1:n) * x(i+1:n)) / Aug(i, i);
+  end
+end
diff --git a/usth/MATH2.2/labwork/3/LU.m b/usth/MATH2.2/labwork/3/LU.m
new file mode 100644
index 0000000..860486d
--- /dev/null
+++ b/usth/MATH2.2/labwork/3/LU.m
@@ -0,0 +1,10 @@
+function [L U] = LU (A)
+  [n n1] = size (A);
+  [L U] = deal (eye (n), A);
+  for k = 1:n
+    for i = k + 1 : n
+      L(i, k) = U(i, k) / U(k, k);
+      U(i, :) = U(i, :) - L(i, k)*U(k, :);
+    end
+  end
+end
diff --git a/usth/MATH2.2/labwork/3/LU_pivot.m b/usth/MATH2.2/labwork/3/LU_pivot.m
new file mode 100644
index 0000000..c506697
--- /dev/null
+++ b/usth/MATH2.2/labwork/3/LU_pivot.m
@@ -0,0 +1,20 @@
+function [L U P] = LU_pivot (A)
+  [n _] = size (A);
+  [L P U] = deal (eye (n), eye (n), A);
+  for k = 1:n
+    [pivot m] = max (abs (U(k:n, k)));
+    m = m + k - 1;
+    if (m ~= k)
+      U([m k], :) = U([k m], :);  % interchange rows m and k in U
+      P([m k], :) = P([k m], :);  % interchange rows m and k in P
+      if k >= 2;  % very important point
+        % interchange rows m and k in columns 1:k-1 of L
+        L([m k], 1:k-1) = L([k m], 1:k-1);
+      end
+    end
+    for i = k + 1 : n
+      L(i, k) = U(i, k) / U(k, k);
+      U(i, :) -= L(i, k)*U(k, :);
+    end
+  end
+end
diff --git a/usth/MATH2.2/labwork/3/gauss.m b/usth/MATH2.2/labwork/3/gauss.m
new file mode 100644
index 0000000..09128dc
--- /dev/null
+++ b/usth/MATH2.2/labwork/3/gauss.m
@@ -0,0 +1,12 @@
+function x = gauss (A, b)
+  issquare (A) || error ('Matrix A must be square');
+  Aug = [A b];
+  [m n] = size (Aug);
+  % forward elimination
+  for k = 1 : m - 1
+    for l = k + 1 : m
+      Aug(l, k:n) = Aug(l, k:n) - Aug(k, k:n)*Aug(l, k)/Aug(k, k);
+    end
+  end
+  x = ubst (Aug);
+end
diff --git a/usth/MATH2.2/labwork/3/labwork3.pdf b/usth/MATH2.2/labwork/3/labwork3.pdf
new file mode 100644
index 0000000..4e0c7e0
--- /dev/null
+++ b/usth/MATH2.2/labwork/3/labwork3.pdf
Binary files differdiff --git a/usth/MATH2.2/labwork/3/subst.m b/usth/MATH2.2/labwork/3/subst.m
new file mode 100644
index 0000000..5c6e783
--- /dev/null
+++ b/usth/MATH2.2/labwork/3/subst.m
@@ -0,0 +1,17 @@
+function x = subst (aug)
+  [m n] = size (aug);
+  x = zeros (m, 1);
+  if (istril (aug))
+    x(m) = aug(1, n) / aug(1, m);
+    for k = 2 : m
+      x(k) = (aug(k, n) - aug(k, k+1:m)*x(1:k-1)) / aug(k, k);
+    end
+  elseif (istriu (aug))
+    x(m) = aug(m, n) / aug(m, m);
+    for k = m - 1 : -1 : 1
+      x(k) = (aug(k, n) - aug(k, k+1:m)*x(k+1:m)) / aug(k, k);
+    end
+  else
+    error ('aug must be a triangular matrix');
+  end
+end
diff --git a/usth/MATH2.2/labwork/4/cubic.tikz b/usth/MATH2.2/labwork/4/cubic.tikz
new file mode 100644
index 0000000..d169c2d
--- /dev/null
+++ b/usth/MATH2.2/labwork/4/cubic.tikz
@@ -0,0 +1,521 @@
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diff --git a/usth/MATH2.2/labwork/4/report.pdf b/usth/MATH2.2/labwork/4/report.pdf
new file mode 100644
index 0000000..0b8e407
--- /dev/null
+++ b/usth/MATH2.2/labwork/4/report.pdf
Binary files differdiff --git a/usth/MATH2.2/labwork/4/report.tex b/usth/MATH2.2/labwork/4/report.tex
new file mode 100644
index 0000000..41d69ba
--- /dev/null
+++ b/usth/MATH2.2/labwork/4/report.tex
@@ -0,0 +1,123 @@
+\documentclass[a4paper,12pt]{article}
+\usepackage[english,vietnamese]{babel}
+\usepackage{amsmath}
+\usepackage{booktabs}
+\usepackage{enumerate}
+\usepackage{lmodern}
+\usepackage{tikz}
+
+\newcommand{\exercise}[1]{\noindent\textbf{#1.}}
+
+\title{Numerical Methods: Labwork 4 Report}
+\author{Nguyễn Gia Phong--BI9-184}
+\date{\dateenglish\today}
+
+\begin{document}
+\maketitle
+\section{Curve Fitting Problems}
+\exercise{3}  From the given table, we define the two vectors
+\begin{verbatim}
+octave> x = [0.00000 0.78540 1.57080 2.35620 ...
+>            3.14159 3.92699 4.71239 5.49779 6.28319];
+octave> fx = [0.00000 0.70711 1.00000 0.70711 ...
+>             0.00000 -0.70711 -1.00000 -0.70711 0.00000];
+\end{verbatim}
+
+\begin{enumerate}[(a)]
+  \item \verb|f(3.00000)| and \verb|f(4.50000)| can be interpolated by
+\begin{verbatim}
+octave> points = [3.00000 4.50000];
+octave> linear = interp1 (x, fx, points)
+linear =
+   0.12748  -0.92080
+\end{verbatim}
+    To further illustrate this, we can then plot these point along
+    with the linearly interpolated line:
+    \verb|plot (points, linear, "o", x, fx)|
+    \begin{figure}[!h]
+      \centering
+      \scalebox{0.36}{\input{linear.tikz}}
+    \end{figure}
+
+  \item For convenience purposes, we define a thin wrapper around \verb|interp1|
+\begin{verbatim}
+octave> interpolate = @(X, method) interp1 (
+> x, fx, X, method, "extrap");
+\end{verbatim}
+    Anonymous function had to be used because named functions somehow do not
+    support closure.  Now we can use \verb|interpolate (points, method)|
+    to approximate \verb|f(3.00000)| and \verb|f(4.50000)|
+    and obtain the table below
+    \begin{center}
+      \begin{tabular}{c r r r}
+        \toprule
+        method & nearest & cubic & spline \\
+        \midrule
+        f(3.00000) & 0 & 0.13528 & 0.14073 \\
+        f(4.50000) & -1 & -0.96943 & -0.97745\\
+        \bottomrule
+      \end{tabular}
+    \end{center}
+
+    Next, we use some plots to better visualize these interpolation methods.
+\begin{verbatim}
+octave> interplot = @(mark, line, method) plot (
+> mark, interpolate (mark, method), "o",
+> line, interpolate (line, method));
+octave> B = linspace (x(1), x(end));
+octave> interplot (points, B, "nearest")
+\end{verbatim}
+\scalebox{0.61}{\input{nearest.tikz}}
+\begin{verbatim}
+octave> interplot (points, B, "cubic")
+\end{verbatim}
+\scalebox{0.61}{\input{cubic.tikz}}
+\begin{verbatim}
+octave> interplot (points, B, "spline")
+\end{verbatim}
+\scalebox{0.61}{\input{spline.tikz}}
+
+    One can easily notice while \verb|nearest| simply chooses the nearest
+    neighbor, \verb|cubic| and \verb|spline| both try to \textit{smoothen}
+    the curve.  This leads to the fact that \verb|nearest|'s approximations
+    strays from \verb|linear|'s in the opposite dirrection when compared to
+    the other two's.  It also explains why \verb|cubic|'s and \verb|spline|'s
+    results are quite close to each other.
+
+  \item Since we are already extrapolating (by providing the \verb|extrap|
+    argument to \verb|interp1|), interpolating for \verb|f(10)| is rather
+    straightforward:
+\begin{verbatim}
+octave> interpolate (10, "spline")
+ans =  1.4499
+octave> C = linspace (0, 10);
+octave> interplot (10, C, "spline")
+\end{verbatim}
+\scalebox{0.39}{\input{f10-spline.tikz}}
+\begin{verbatim}
+octave> interpolate (10, "linear")
+ans =  3.3463
+octave> interplot (10, C, "linear")
+\end{verbatim}
+\scalebox{0.39}{\input{f10-linear.tikz}}
+
+    From the existing data, we can make a guess that \verb|f|
+    is a cubic function and regression fits quite well:
+\begin{verbatim}
+octave> p = polyfit (x, fx, 3)
+p =
+   0.084488  -0.796282   1.681694  -0.043870
+octave> polyval (p, 10)
+ans =  21.633
+octave> plot (x, fx, "o", C, polyval (p, C))
+\end{verbatim}
+\scalebox{0.62}{\input{f10-poly.tikz}}
+
+    In all these cases, due to the missing data, the value of \verb|f| at 10
+    tends to go \textit{wild}, i.e. far away from the given data in \verb|fx|.
+    If anything, the interpolated/extrapolated ones looks more harmonic,
+    while regression simply fit the curve into the function of the given form.
+    It is not obvious that either technique is better is this case,
+    since the amount of given data is too small.
+\end{enumerate}
+\end{document}
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diff --git a/usth/MATH2.2/labwork/5/labwork5.pdf b/usth/MATH2.2/labwork/5/labwork5.pdf
new file mode 100644
index 0000000..52512a5
--- /dev/null
+++ b/usth/MATH2.2/labwork/5/labwork5.pdf
Binary files differdiff --git a/usth/MATH2.2/labwork/5/profit.m b/usth/MATH2.2/labwork/5/profit.m
new file mode 100644
index 0000000..b196205
--- /dev/null
+++ b/usth/MATH2.2/labwork/5/profit.m
@@ -0,0 +1,15 @@
+function z = profit (x, y)
+  A = [7 11; 10 8];
+  b = [77; 80];
+  c = [150; 175];
+  [m n] = size (x);
+  z = -inf (m, n);
+  for s = 1 : m
+    for t = 1 : n
+      r = [x(s, t); y(s, t)];
+      if A * r <= b
+        z(s, t) = dot (c, r);
+      endif
+    endfor
+  endfor
+endfunction
diff --git a/usth/MATH2.2/labwork/5/profit.tikz b/usth/MATH2.2/labwork/5/profit.tikz
new file mode 100644
index 0000000..e91ea31
--- /dev/null
+++ b/usth/MATH2.2/labwork/5/profit.tikz
@@ -0,0 +1,39631 @@
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diff --git a/usth/MATH2.2/labwork/5/report.pdf b/usth/MATH2.2/labwork/5/report.pdf
new file mode 100644
index 0000000..8a690f2
--- /dev/null
+++ b/usth/MATH2.2/labwork/5/report.pdf
Binary files differdiff --git a/usth/MATH2.2/labwork/5/report.tex b/usth/MATH2.2/labwork/5/report.tex
new file mode 100644
index 0000000..16daad0
--- /dev/null
+++ b/usth/MATH2.2/labwork/5/report.tex
@@ -0,0 +1,133 @@
+\documentclass[a4paper,12pt]{article}
+\usepackage[english,vietnamese]{babel}
+\usepackage{amsmath}
+\usepackage{booktabs}
+\usepackage{enumerate}
+\usepackage{lmodern}
+\usepackage{siunitx}
+\usepackage{tikz}
+
+\newcommand{\exercise}[1]{\noindent\textbf{#1.}}
+
+\title{Numerical Methods: Linear Programming}
+\author{Nguyễn Gia Phong--BI9-184}
+\date{\dateenglish\today}
+
+\begin{document}
+\maketitle
+
+Given the production contraints and profits of two grades of heating gas
+in the table below.
+\begin{center}
+  \begin{tabular}{l c c c}
+    \toprule
+    & \multicolumn{2}{c}{Product}\\
+    Resource & Regular & Premium & Availability\\
+    \midrule
+    Raw gas (\si{\cubic\metre\per\tonne}) & 7 & 11 & 77\\
+    Production time (\si{\hour\per\tonne}) & 10 & 8 & 80\\
+    Storage (\si{\tonne}) & 9 & 6\\
+    \midrule
+    Profit (\si{\per\tonne}) & 150 & 175\\
+    \bottomrule
+  \end{tabular}
+\end{center}
+
+\begin{enumerate}[(a)]
+  \item Let two nonnegative numbers $x_1$ and $x_2$ respectively be
+    the quantities in tonnes of regular and premium gas to be produced.
+    The constraints can then be expressed as
+    \[\begin{cases}
+      7x_1 + 11x^2 &\le 77\\
+      10x_1 + 8x_2 &\le 80\\
+      x_1 &\le 9\\
+      x_2 &\le 6
+    \end{cases}
+    \iff \begin{bmatrix}
+      7 & 11\\
+      10 & 8\\
+      1 & 0\\
+      0 & 1
+    \end{bmatrix}\begin{bmatrix}
+      x_1\\ x_2
+    \end{bmatrix}
+    \le \begin{bmatrix}
+      77\\ 80\\ 9\\ 6
+    \end{bmatrix}\]
+
+    The total profit is the linear function to be maximized:
+    \[\Pi(x_1, x_2) = 150x_1 + 175x_2 = \begin{bmatrix}
+      150\\ 175
+    \end{bmatrix}\cdot\begin{bmatrix}
+      x_1\\ x_2
+    \end{bmatrix}\]
+
+    Let $\mathbf x = \begin{bmatrix}
+      x_1\\ x_2
+    \end{bmatrix}$, $A = \begin{bmatrix}
+      7 & 11\\
+      10 & 8\\
+      1 & 0\\
+      0 & 1
+    \end{bmatrix}$, $\mathbf b = \begin{bmatrix}
+      77\\ 80\\ 9\\ 6
+    \end{bmatrix}$ and $\mathbf c = \begin{bmatrix}
+      150\\ 175
+    \end{bmatrix}$, the canonical form of the problem is
+    \[\max\left\{\mathbf c^\mathrm T
+                 \mid A\mathbf x\le b\land\mathbf x\ge 0\right\}\]
+  \item Due to the absense of \verb|linprog| in Octave, we instead use GNU GLPK:
+\begin{verbatim}
+octave> x = glpk (c, A, b, [], [], "UUUU", "CC", -1)
+x =
+   4.8889
+   3.8889
+\end{verbatim}
+    Contraint type \verb|UUUU| is used because all contraints are inequalities
+    with an upper bound and \verb|CC| indicates continous values of $\mathbf x$.
+    With the sense of -1, GLPK looks for the maximization\footnote{I believe
+    \emph{minimization} was a typo in the assignment, since it is trivial that
+    $\Pi$ has the minimum value of 0 at $\mathbf x = \mathbf 0$.} of
+    $\Pi(4.8889, 3.8889) = 1413.9$.
+
+    The two blank arguments are for the lower and upper bounds of $\mathbf x$,
+    default to zero and infinite respectively.  Alternatively we can use
+    the following to obtain the same result
+\begin{verbatim}
+glpk (c, [7 11; 10 8], [77; 80], [], [9 6], "UU", "CC", -1)
+\end{verbatim}
+
+  \item Within the constrains, the profit can be calculated using the following
+    function, which takes two meshes of $x$ and $y$ as arguments
+\begin{verbatim}
+function z = profit (x, y)
+  A = [7 11; 10 8];
+  b = [77; 80];
+  c = [150; 175];
+  [m n] = size (x);
+  z = -inf (m, n);
+  for s = 1 : m
+    for t = 1 : n
+      r = [x(s, t); y(s, t)];
+      if A * r <= b
+        z(s, t) = dot (c, r);
+      endif
+    endfor
+  endfor
+endfunction
+\end{verbatim}
+
+    Using this, the solution space is then plotted using \verb|ezsurf|,
+    which color each grid by their relative values (the smallest is dark purple
+    and the largest is bright yellow):
+\begin{verbatim}
+ezsurf (@(x1, x2) constraints (x1, x2), [0 9 0 6], 58)
+\end{verbatim}
+\pagebreak
+
+    Since the plot is just a part of a plane, we can rotate it for a better view
+    without losing any information about its shape.
+
+    \scalebox{0.69}{\input{profit.tikz}}
+\end{enumerate}
+\end{document}
diff --git a/usth/MATH2.2/project/contour.png b/usth/MATH2.2/project/contour.png
new file mode 100644
index 0000000..dfd4dc1
--- /dev/null
+++ b/usth/MATH2.2/project/contour.png
Binary files differdiff --git a/usth/MATH2.2/project/heatrans.m b/usth/MATH2.2/project/heatrans.m
new file mode 100644
index 0000000..d5d2e04
--- /dev/null
+++ b/usth/MATH2.2/project/heatrans.m
@@ -0,0 +1,15 @@
+function T = heatrans (cp, lambda, rho, Tg, Td, T0, L, dx, dt, N)
+  alpha = lambda / rho / cp;
+  beta = alpha * dt / dx^2;
+  M = round (L / dx);
+  side = repelem (beta, M);
+  A = diag (side, -1) + diag (repelem (1 - 2*beta, M + 1)) + diag (side, 1);
+  A(1, :) = A(end, :) = 0;
+  A(1, 1) = A(end, end) = 1;
+
+  T = repelem (T0, M + 1);
+  [T(1) T(end)] = deal (Tg, Td);
+  for k = 2 : N
+    T(:, k) = A * T(:, k - 1);
+  end
+end
diff --git a/usth/MATH2.2/project/mesh.png b/usth/MATH2.2/project/mesh.png
new file mode 100644
index 0000000..75de2d1
--- /dev/null
+++ b/usth/MATH2.2/project/mesh.png
Binary files differdiff --git a/usth/MATH2.2/project/report.pdf b/usth/MATH2.2/project/report.pdf
new file mode 100644
index 0000000..88b7496
--- /dev/null
+++ b/usth/MATH2.2/project/report.pdf
Binary files differdiff --git a/usth/MATH2.2/project/report.tex b/usth/MATH2.2/project/report.tex
new file mode 100644
index 0000000..e1c36bf
--- /dev/null
+++ b/usth/MATH2.2/project/report.tex
@@ -0,0 +1,155 @@
+\documentclass[a4paper,12pt]{article}
+\usepackage[english,vietnamese]{babel}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{graphicx}
+\usepackage{hyperref}
+\usepackage{lmodern}
+\usepackage{mathtools}
+\usepackage{siunitx}
+
+\newcommand{\exercise}[1]{\noindent\textbf{#1.}}
+\newcommand{\sifrac}[2]{\displaystyle\SI[per-mode=fraction]{#1}{#2}}
+
+\title{Numerical Methods: Heat Transfer}
+\author{Nguyễn Gia Phong--BI9-184}
+\date{\dateenglish\today}
+
+\begin{document}
+\maketitle
+
+Given a bar of length $L = \SI{0.4}{\meter}$ consisting of homogeneous
+and isotropic material with the initial temperature of $T_0 = \SI{0}{\celsius}$.
+Suppose it is perfectly insulated with the exception of the ends with
+the temperature of $T_g = \SI{100}{\celsius}$ and $T_d = \SI{50}{\celsius}$.
+The thermal properties of material will be taken constant.
+\begin{itemize}
+  \item Specific heat capacity:
+    $c_p = \sifrac{900}{\joule\per\kilogram\per\celsius}$
+  \item Thermal conductivity:
+    $\lambda = \sifrac{237}{\watt\per\meter\per\celsius}$
+  \item Density:
+    $\rho = \sifrac{2700}{\kilogram\per\cubic\meter}$
+  \item Thermal diffusivity:
+    $\alpha = \dfrac{\lambda}{\rho c_p}
+            = \sifrac{9.753e-5}{\square\meter\per\second}$
+\end{itemize}
+
+The heat transfer in this bar can be described by the following
+partial differential equation
+\[\frac{\partial T}{\partial t}
+  = \alpha\frac{\partial^2 T}{\partial x^2}\tag{$*$}\]
+where the temperature $T$ is a function of postition $x$ and time $t$.
+
+To solve the problem numerically, we devide space and time into equal intervals
+of norms $\Delta x$ and $\Delta t$ respectively and let $M = L/\Delta x$.
+Consequently, the spartial coordinate is defined as $x_i = (i - 1)\Delta x$
+with $i \in [1\,..\,M + 1]$ and the temporal one is $t_n = n\Delta t$
+with $n \in \mathbb N^*$. With these definitions\footnote{I believe $i = 0$
+and $n = 0$ in the assignment papers are typos since then the domain of $x_i$
+would exceed $L$ and $t_0$ would be negative.}, we denote $T_i^n = T(x_i, t_n)$.
+Using numerical methods, we may start approximating the solutions of ($*$).
+
+\begin{enumerate}
+  \item The left-hand side of ($*$) can be approximated as
+    \[\frac{\partial T}{\partial t} = \frac{T_i^{n+1} - T_i^n}{\Delta t}\]
+  \item Similarly, the right-hand side is expressed in the following form
+    \[\alpha\frac{\partial^2 T}{\partial x^2}
+      = \alpha\frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{\Delta x^2}\]
+  \item ($*$) is therefore reformulated as
+    \[\frac{T_i^{n+1} - T_i^n}{\Delta t}
+      = \alpha\frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{\Delta x^2}\]
+  \item From the formular above
+    and let $\beta = \alpha\dfrac{\Delta t}{\Delta x^2}$,
+    we get
+    \[T_i^{n+1} = T_i^n + \beta\left(T_{i+1}^n - 2T_i^n + T_{i-1}^n\right)\]
+  \item Boundary conditions:
+    \begin{itemize}
+      \item $\forall n \in \mathbb N^*,\; T_1^n = T(0, t_n) = T_g$
+      \item $\forall n \in \mathbb N^*,\; T_{M+1}^n = T(L, t_n) = T_d$
+      \item $\forall i \in [2\,..\,M],\; T_{i}^1 = T(x_i, 0) = T_0$
+    \end{itemize}
+  \item From (4) and (5), the temperature at point $x_i$ of the bar
+    at time $t_n$ is recursively defined as
+    \[T_i^n = \begin{dcases}
+        T_i^{n-1} + \beta\left(T_{i+1}^{n-1} - 2T_i^{n-1} + T_{i-1}^{n-1}\right)
+        &\text{ if }1 < i \le M \land n > 1\\
+        T_g &\text{ if }i = 1\\
+        T_d &\text{ if }i = M + 1\\
+        T_0 &\text{ otherwise}
+      \end{dcases}\]
+    Since the temperature only depends on the values in the past, values within
+    $(i, n) \in [1\,..\,M+1]\times[1\,..\,N]$ with any $N$ of choice could
+    be computed via dynamic programming:
+    \begin{enumerate}
+      \item Create a 2-dimensional dynamic array $T$ with one-based index
+        and size $(M+1)\times 1$
+      \item Initialize $T$ with $T_1^1 = T_g$, $T_{M+1}^1 = T_d$
+        and $T_i^1 = T_0\ \forall i \in [2\,..\,M]$,
+        where $T_i^n$ is element of row $i$ and column $n$
+      \item For $n = 2$ to $N$
+        \begin{itemize}
+          \item Let $T_1^k = T_g$
+          \item For $i = 2$ to $M$, let $T_i^k = T_i^{k-1}
+            + \beta\left(T_{i+1}^{k-1} - 2T_i^{k-1} + T_{i-1}^{k-1}\right)$
+          \item Let $T_{M+1}^k = T_d$
+        \end{itemize}
+      \item Return $T_i^n$
+    \end{enumerate}
+    Each iteration in (c) can be written in matrix notation as
+    $T^k = AT^{k+1}$, where $T_n$ is column $n$ and $A$ is
+    a matrix of size $(M+1)\times(M+1)$
+    \[A =\begin{bmatrix}
+        1 & 0 & 0 & \cdots & 0 & 0 & 0\\
+        \beta & 1-2\beta & \beta & \cdots & 0 & 0 & 0\\
+        0 & \beta & 1-2\beta & \cdots & 0 & 0 & 0\\
+        \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots\\
+        0 & 0 & 0 & \cdots & 1-2\beta & \beta & 0\\
+        0 & 0 & 0 & \cdots & \beta & 1-2\beta & \beta\\
+        0 & 0 & 0 & \cdots & 0 & 0 & 1
+      \end{bmatrix}\]
+  \item Steps (a) to (c) is then implemented in Octave as
+\begin{verbatim}
+function T = heatrans (cp, lambda, rho, Tg, Td, T0, L,
+                       dx, dt, N)
+  alpha = lambda / rho / cp;
+  beta = alpha * dt / dx^2;
+  M = round (L / dx);
+  side = repelem (beta, M);
+  A = (diag (repelem (1 - 2*beta, M + 1))
+       + diag (side, -1) + diag (side, 1));
+  A(1, :) = A(end, :) = 0;
+  A(1, 1) = A(end, end) = 1;
+
+  T = repelem (T0, M + 1);
+  [T(1) T(end)] = deal (Tg, Td);
+  for k = 2 : N
+    T(:, k) = A * T(:, k - 1);
+  end
+end
+\end{verbatim}
+
+    Choosing $\Delta x = \SI{0.01}{\meter}$, $\Delta t = \SI{0.5}{\second}$
+    and $N = 841$, we define
+\begin{verbatim}
+T = heatrans (900, 237, 2700, 100, 50, 0, 0.4,
+              0.01, 0.5, 841);
+\end{verbatim}
+    then the temperature at point $x_i$ at time $t_n$ is \verb|T(i, n)|.
+    \pagebreak
+
+    To visualize the heat transfer process, we use \verb|mesh| to plot
+    a 3D graph:
+
+    \includegraphics[width=0.841\textwidth]{mesh.png}
+
+    The temperature can be shown more intuitively using \verb|contourf|:
+
+    \includegraphics[width=0.841\textwidth]{contour.png}
+
+    The \verb|script| to reproduce these results along with \verb|heatrans.m|
+    bundled with this report and this document itself are all licensed under a
+    \href{http://creativecommons.org/licenses/by-sa/4.0/}{Creative Commons
+      Attribution-ShareAlike 4.0 International License}.
+\end{enumerate}
+\end{document}
\ No newline at end of file
diff --git a/usth/MATH2.2/project/script b/usth/MATH2.2/project/script
new file mode 100755
index 0000000..99b61f7
--- /dev/null
+++ b/usth/MATH2.2/project/script
@@ -0,0 +1,22 @@
+#!/usr/bin/env octave
+[cp lambda rho] = deal (900, 237, 2700);
+[Tg Td T0] = deal (100, 50, 0);
+[L dx dt N] = deal (0.4, 0.01, 0.5, 841);
+T = heatrans (cp, lambda, rho, Tg, Td, T0, L, dx, dt, N);
+[X Y Z] = deal (0:dx:L, 0:dt:420, T');
+
+figure (1);   # mesh
+mesh (X, Y, Z);
+title ("Temperature of the bar during the first 420 seconds");
+xlabel ("x (m)");
+ylabel ("t (s)");
+zlabel ("T (°C)");
+
+figure (2);   # contour
+contourf (X, Y, Z, 0:7:100, "showtext", "on");
+title ("Temperature of the bar during the first 420 seconds");
+xlabel ("x (m)");
+ylabel ("t (s)");
+
+disp ("Press any key to coninue...");
+kbhit;
author	Nguyễn Gia Phong <vn.mcsinyx@gmail.com>	2019-12-16 21:31:18 +0700
committer	Nguyễn Gia Phong <vn.mcsinyx@gmail.com>	2019-12-16 21:31:18 +0700
commit	e461df7573c2b7b7e26c965d8cf2d8e175d67378 (patch)
tree	193b8439edac52c1749764395fe5f7787889eda9
parent	c1008fe39217be7f91f0ea23483e747bfbc5743e (diff)
download	cp-e461df7573c2b7b7e26c965d8cf2d8e175d67378.tar.gz