diff options
Diffstat (limited to 'usth/MATH2.2')
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. + +[](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 @@ +% Title: gl2ps_renderer figure +% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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files differdiff --git a/usth/MATH2.2/labwork/2/report.pdf b/usth/MATH2.2/labwork/2/report.pdf new file mode 100644 index 0000000..6711c94 --- /dev/null +++ b/usth/MATH2.2/labwork/2/report.pdf Binary files differdiff --git a/usth/MATH2.2/labwork/2/report.tex b/usth/MATH2.2/labwork/2/report.tex new file mode 100644 index 0000000..c4481fa --- /dev/null +++ b/usth/MATH2.2/labwork/2/report.tex @@ -0,0 +1,88 @@ +\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 @@ +% Title: gl2ps_renderer figure +% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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--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} diff --git a/usth/MATH2.2/labwork/4/spline.tikz b/usth/MATH2.2/labwork/4/spline.tikz new file mode 100644 index 0000000..9c7c9b3 --- /dev/null +++ b/usth/MATH2.2/labwork/4/spline.tikz @@ -0,0 +1,521 @@ +% Title: gl2ps_renderer figure +% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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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 @@ +% Title: gl2ps_renderer figure +% Creator: GL2PS 1.4.0, (C) 1999-2017 C. 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+\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; |
