From e461df7573c2b7b7e26c965d8cf2d8e175d67378 Mon Sep 17 00:00:00 2001 From: Nguyễn Gia Phong Date: Mon, 16 Dec 2019 21:31:18 +0700 Subject: [usth/MATH2.2] Numerical Methods The future starts now. --- usth/MATH2.2/labwork/1/bisect.m | 36 + usth/MATH2.2/labwork/1/intro | 62 + usth/MATH2.2/labwork/1/labwork1.pdf | Bin 0 -> 165522 bytes usth/MATH2.2/labwork/1/ratpoison.m | 15 + usth/MATH2.2/labwork/1/ratpoison.py | 8 + usth/MATH2.2/labwork/2/2a.tikz | 515 + usth/MATH2.2/labwork/2/labwork2.pdf | Bin 0 -> 68137 bytes usth/MATH2.2/labwork/2/report.pdf | Bin 0 -> 191189 bytes usth/MATH2.2/labwork/2/report.tex | 88 + usth/MATH2.2/labwork/3/GaussPivot.m | 25 + usth/MATH2.2/labwork/3/LU.m | 10 + usth/MATH2.2/labwork/3/LU_pivot.m | 20 + usth/MATH2.2/labwork/3/gauss.m | 12 + usth/MATH2.2/labwork/3/labwork3.pdf | Bin 0 -> 72465 bytes usth/MATH2.2/labwork/3/subst.m | 17 + usth/MATH2.2/labwork/4/cubic.tikz | 521 + usth/MATH2.2/labwork/4/f10-linear.tikz | 482 + usth/MATH2.2/labwork/4/f10-poly.tikz | 731 + usth/MATH2.2/labwork/4/f10-spline.tikz | 500 + usth/MATH2.2/labwork/4/labwork4.pdf | Bin 0 -> 72397 bytes usth/MATH2.2/labwork/4/linear.tikz | 248 + usth/MATH2.2/labwork/4/nearest.tikz | 521 + usth/MATH2.2/labwork/4/report.pdf | Bin 0 -> 159435 bytes usth/MATH2.2/labwork/4/report.tex | 123 + usth/MATH2.2/labwork/4/spline.tikz | 521 + usth/MATH2.2/labwork/5/labwork5.pdf | Bin 0 -> 32550 bytes usth/MATH2.2/labwork/5/profit.m | 15 + usth/MATH2.2/labwork/5/profit.tikz | 39631 +++++++++++++++++++++++++++++++ usth/MATH2.2/labwork/5/report.pdf | Bin 0 -> 288040 bytes usth/MATH2.2/labwork/5/report.tex | 133 + 30 files changed, 44234 insertions(+) create mode 100644 usth/MATH2.2/labwork/1/bisect.m create mode 100755 usth/MATH2.2/labwork/1/intro create mode 100644 usth/MATH2.2/labwork/1/labwork1.pdf create mode 100644 usth/MATH2.2/labwork/1/ratpoison.m create mode 100644 usth/MATH2.2/labwork/1/ratpoison.py create mode 100644 usth/MATH2.2/labwork/2/2a.tikz create mode 100644 usth/MATH2.2/labwork/2/labwork2.pdf create mode 100644 usth/MATH2.2/labwork/2/report.pdf create mode 100644 usth/MATH2.2/labwork/2/report.tex create mode 100644 usth/MATH2.2/labwork/3/GaussPivot.m create mode 100644 usth/MATH2.2/labwork/3/LU.m create mode 100644 usth/MATH2.2/labwork/3/LU_pivot.m create mode 100644 usth/MATH2.2/labwork/3/gauss.m create mode 100644 usth/MATH2.2/labwork/3/labwork3.pdf create mode 100644 usth/MATH2.2/labwork/3/subst.m create mode 100644 usth/MATH2.2/labwork/4/cubic.tikz create mode 100644 usth/MATH2.2/labwork/4/f10-linear.tikz create mode 100644 usth/MATH2.2/labwork/4/f10-poly.tikz create mode 100644 usth/MATH2.2/labwork/4/f10-spline.tikz create mode 100644 usth/MATH2.2/labwork/4/labwork4.pdf create mode 100644 usth/MATH2.2/labwork/4/linear.tikz create mode 100644 usth/MATH2.2/labwork/4/nearest.tikz create mode 100644 usth/MATH2.2/labwork/4/report.pdf create mode 100644 usth/MATH2.2/labwork/4/report.tex create mode 100644 usth/MATH2.2/labwork/4/spline.tikz create mode 100644 usth/MATH2.2/labwork/5/labwork5.pdf create mode 100644 usth/MATH2.2/labwork/5/profit.m create mode 100644 usth/MATH2.2/labwork/5/profit.tikz create mode 100644 usth/MATH2.2/labwork/5/report.pdf create mode 100644 usth/MATH2.2/labwork/5/report.tex (limited to 'usth/MATH2.2/labwork') 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 Binary files /dev/null and b/usth/MATH2.2/labwork/1/labwork1.pdf differ diff --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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differ diff --git a/usth/MATH2.2/labwork/2/report.pdf b/usth/MATH2.2/labwork/2/report.pdf new file mode 100644 index 0000000..6711c94 Binary files /dev/null and b/usth/MATH2.2/labwork/2/report.pdf differ diff --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 Binary files /dev/null and b/usth/MATH2.2/labwork/3/labwork3.pdf differ diff --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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diff --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} -- cgit 1.4.1