parent
c1008fe392
commit
e461df7573
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@ -0,0 +1,36 @@
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disp ("Question 1:");
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disp ("(a)");
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printf ("11^3 + 12^3 - 7^3 = %d\n", 11^3 + 12^3 - 7^3);
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printf ("15! = %d\n", factorial (15));
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disp ("(b)");
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A = [1 2 3
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4 5 6
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7 8 9];
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B = eye (3);
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disp ("(b.i)");
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disp ("A + B = ");
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disp (A + B);
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disp ("(b.ii)");
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disp ("A' = ");
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disp (A');
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disp ("(b.iii)");
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disp ("A^-1 = ");
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disp (inv (A));
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disp ("(c.i)");
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printf ("x^2 = 19 -> x = %g\n", sqrt (19));
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disp ("(c.ii)");
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printf ("x^4 = 55 -> x = %g\n", sqrt (sqrt (19)));
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disp ("(d)");
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X = 0 : 30;
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Y = X * 2 + 3;
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plot (X, Y);
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xlabel ("x");
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ylabel ("y = 2x + 3");
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disp ("Press any key to continue...");
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kbhit;
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@ -0,0 +1,38 @@
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disp ("Question 2:");
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disp ("(a)");
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pkg load symbolic;
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syms x real;
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solve (sqrt (x) - x + 1 == 0)
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% ans = (sym)
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% √5 3
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% ── + ─
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% 2 2
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pkg unload symbolic;
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disp ("To get numerical solutions we can use fzero");
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disp ("With the initial guess of 0, fzero (@(x) sqrt (x) - x + 1, 0) returns");
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fzero (@(x) sqrt (x) - x + 1, 0)
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disp ("(b)");
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hold on;
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ezplot (@(x) exp (-x));
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ezplot (@(x) sin (x));
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hold off;
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disp ("Press any key to continue...");
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kbhit;
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disp ("(c)");
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s = 0;
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for k = 1 : 1000
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s += k^3;
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endfor
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printf ("The cubic sum of integers from 1 to 1000 is %d\n", s);
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disp ("(d)");
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A = [2 1 4
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1 2 -5
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3 -2 4];
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b = [10 1 8]';
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disp ("Using mldivide, [x y z] = ");
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disp (mldivide (A, b)');
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disp ("Using inv, [x y z] = ");
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disp ((inv (A) * b)');
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@ -0,0 +1,16 @@
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disp ("Question 3:");
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disp ("(a)");
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function y = f (x)
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y = 2 + x.^2 + exp(x.*2 + 1);
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endfunction
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h = 0.005;
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printf ("By forward difference with h = 0.05, f'(1.35) = %g\n",
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(f (1.35 + h) - f (1.35)) / h);
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disp ("(b)");
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disp ("I am unsure if diff is different on Matlab, but on octave,");
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disp ("it's simply taking differences between consecutive elements.");
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x = [1.35, 1.35+h];
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printf ("Using diff with h = 0.05, we get same result, f'(1.35) = %g\n",
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(diff (f (x)) / h));
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disp ("Using symbolical methods, f'(1.35) = 83.5946 which is quite close.");
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@ -0,0 +1,22 @@
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# Numerical Methods: Homework 1
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My name is Nguyễn Gia Phong, my student ID is BI9-184.
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The homework were written in Octave and saved as
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* `linear`: interactive script to solve linear equation
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* `quadratic`: interactive script to solve quadratic equation
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* `cubic`: interactive script to solve cubic equation
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* `fibonacci.m`: function `fibonacci (n)` returning a vector containing
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Fibonacci numbers from F<sub>1</sub> to F<sub>n</sub>
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The scripts are named without any file extension following the \*nix traditions.
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The homework can be downloaded as a whole in a ZIP package using the button
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near the top-left corner, or alternatively cloned using `git`:
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git clone https://gist.github.com/McSinyx/1d9ff91d1ecfd60254c188bddb7926d5
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If this causes any inconvenience, please let me know.
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[](http://creativecommons.org/licenses/by-sa/4.0/)
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This work is licensed under a [Creative Commons Attribution-ShareAlike 4.0 International License](http://creativecommons.org/licenses/by-sa/4.0/).
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@ -0,0 +1,25 @@
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#!/usr/bin/env octave
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disp ("Solve quadratic equation a*x^3 + b*x^2 + c*x + d = 0");
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a = input ("a = ");
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while (a == 0)
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disp ("a must be nonzero");
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a = input ("a = ");
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end
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b = input ("b = ");
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c = input ("c = ");
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d = input ("d = ");
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% Using the trigonometric and hyperbolic solutions on Wikipedia:
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% https://en.wikipedia.org/wiki/Cubic_function
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p = (3*a*c - b^2) / (3*a^2);
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q = (2*b^3 - 9*a*b*c + 27*a^2*d) / (27*a^3);
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if (p == 0)
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disp ("The given cubic equation has only one distinct solution:");
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x = cbrt(-q) - b/a/3
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else
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r = 2 * sqrt(-p/3);
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disp ("The three solutions of the given quadratic equation are:");
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for k = 0 : 2
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x = r * cos((acos(3*q/p/r) - 2*pi*k) / 3) - b/a/3
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end
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end
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@ -0,0 +1,7 @@
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function fib = fibonacci (n)
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fib = 0 : n;
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for i = 2 : n
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fib(i + 1) = fib(i) + fib(i - 1);
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end
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fib = fib(2:end);
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end
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Binary file not shown.
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@ -0,0 +1,11 @@
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#!/usr/bin/env octave
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disp ("Solve quadratic equation ax + b = 0");
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a = input ("a = ");
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while (a == 0)
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disp ("a must be nonzero");
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a = input ("a = ");
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end
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b = input ("b = ");
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disp ("The two solutions of the given linear equation is:");
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x = -b / a
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@ -0,0 +1,14 @@
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#!/usr/bin/env octave
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disp ("Solve quadratic equation a*x^2 + b*x + c = 0");
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a = input ("a = ");
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while (a == 0)
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disp ("a must be nonzero");
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a = input ("a = ");
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end
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b = input ("b = ");
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c = input ("c = ");
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sd = sqrt(b*b - a*c*4);
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disp ("The two solutions of the given quadratic equation are:");
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x0 = (-b - sd) / a / 2
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x1 = (-b + sd) / a / 2
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@ -0,0 +1,36 @@
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function [x fx ea iter] = bisect (f, xl, xu, es = 0.0001, maxit = 50)
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% uses bisection method to find the root of f,
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% with xl and xu being lower and upper guesses,
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% es being desired relative error
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% and maxit being maximum allowable iterations
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% to return the real root x, fx = f(x),
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% approximate relative error ea (%)
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% and number of iterations iter
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nargin < 3 && error ('bisect requires at least 3 arguments');
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[fl fu iter] = deal (f (xl), f (xu), 0);
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if (fl == 0)
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[x fx ea] = deal (xl, 0, 0);
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return;
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elseif (fu == 0)
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[x fx ea] = deal (xu, 0, 0);
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return;
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end
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fl * fu < 0 || error ('no sign change');
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[x ea] = deal (xl, 100);
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while (ea > es && iter++ < maxit) % yes, I use Octave only
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[xold x] = deal (x, (xl + xu) / 2);
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fx = f (x);
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if (fx == 0)
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ea = 0;
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break;
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elseif (x)
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ea = abs ((x - xold) / x) * 100;
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end
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if (f (xl) * fx < 0)
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xu = x;
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else
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xl = x;
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end
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end
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end
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@ -0,0 +1,62 @@
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#!/usr/bin/env octave
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% Exercise I.1.1
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x = 1 : 5
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printf ("x (3) + x (2) = %f\n", x (3) + x (2));
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% Exercise I.1.2
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x = x'
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% Exercise I.2.1
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A = zeros (3, 3)
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A += eye (3)
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A += [0 0 0; 2 0 0; 3 4 0]
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B = A'
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A + B
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A - B
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A * B % matmul
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A .* B % element-wise mul
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A(1:2, 1:2)
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A(1:3)
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A(1:3, 1) % col
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A(:, 1) % col
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% Exercise I.3
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function avg = mean (v)
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avg = sum (v) / numel (v);
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end
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mean (x)
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% Exercise I.4
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function price = ebill (kwh)
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l = [50 50 100 100 100 Inf];
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p = [1.484 1.533 1.786 2.242 2.503 2.587];
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for i = 1 : 6
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if (kwh > l(i))
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kwh -= l(i);
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else
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l(i) = kwh;
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kwh = 0;
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end
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end
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price = sum (l .* p);
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end
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% Exercise I.5.1
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a = (@(x) x*x') (1 : 1000)
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b = sum (arrayfun (@(x) (-1)^x / (1 + x*2), [0 : 501]))
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c = sum (arrayfun (@(x) 1 / (1 + x*2)^2 / (3 + x*2)^2, [0 : 420])) - (pi^2 - 8)/16
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% Exercise I.5.2
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function p = permutations (n, k)
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p = factorial (n) / factorial (n - k);
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end
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function c = combinations (n, k)
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c = permutations (n, k) / factorial (k);
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end
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% Exercise I.6
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log (2) / log (1.1)
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% Exercise I.7
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x = linspace(-5, 5);
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plot (x, x + 1, 'color', 'green');
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Binary file not shown.
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@ -0,0 +1,15 @@
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function [x fx ea i] = ratpoison (f, df, x0, es = 0.00000001, imax = 20)
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nargin < 2 && error ('ratpoison requires at least 2 ingredients');
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[x fx dfx ea i] = deal (x0, f (x0), df (x0), 1, 0);
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while (ea > es && i++ < imax)
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[xold x] = deal (x, x - fx/dfx);
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[fx dfx] = deal (f (x), df (x));
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if (fx == 0)
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ea = 0;
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break;
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elseif (x)
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% just drop the percent BS
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ea = abs ((x - xold) / x);
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end
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end
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end
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|
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@ -0,0 +1,8 @@
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def ratpoison(f, df, x, es=10**-8, imax=20):
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ea, i = 1, 0
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while ea > es and i < imax:
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i += 1
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xold, x = x, x - f(x)/df(x)
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if f(x) == 0: return x, 0, 0, i
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if x: ea = abs((x - xold) / x)
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return x, f(x), ea, i
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|
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@ -0,0 +1,515 @@
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|||
% Title: gl2ps_renderer figure
|
||||
% Creator: GL2PS 1.4.0, (C) 1999-2017 C. Geuzaine
|
||||
% For: Octave
|
||||
% CreationDate: Fri Sep 27 13:13:59 2019
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||||
\begin{pgfpicture}
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||||
\color[rgb]{1.000000,1.000000,1.000000}
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||||
\pgfpathrectanglecorners{\pgfpoint{0pt}{0pt}}{\pgfpoint{576pt}{432pt}}
|
||||
\pgfusepath{fill}
|
||||
\begin{pgfscope}
|
||||
\pgfpathrectangle{\pgfpoint{0pt}{0pt}}{\pgfpoint{576pt}{432pt}}
|
||||
\pgfusepath{fill}
|
||||
\pgfpathrectangle{\pgfpoint{0pt}{0pt}}{\pgfpoint{576pt}{432pt}}
|
||||
\pgfusepath{clip}
|
||||
\pgfpathmoveto{\pgfpoint{74.880005pt}{399.599976pt}}
|
||||
\pgflineto{\pgfpoint{521.279968pt}{47.520004pt}}
|
||||
\pgflineto{\pgfpoint{74.880005pt}{47.520004pt}}
|
||||
\pgfpathclose
|
||||
\pgfusepath{fill,stroke}
|
||||
\pgfpathmoveto{\pgfpoint{74.880005pt}{399.599976pt}}
|
||||
\pgflineto{\pgfpoint{521.279968pt}{399.599976pt}}
|
||||
\pgflineto{\pgfpoint{521.279968pt}{47.520004pt}}
|
||||
\pgfpathclose
|
||||
\pgfusepath{fill,stroke}
|
||||
\color[rgb]{0.850000,0.850000,0.850000}
|
||||
\pgfsetlinewidth{0.500000pt}
|
||||
\pgfsetdash{{16pt}{0pt}}{0pt}
|
||||
\pgfpathmoveto{\pgfpoint{88.846191pt}{399.599976pt}}
|
||||
\pgflineto{\pgfpoint{88.846191pt}{47.520004pt}}
|
||||
\pgfusepath{stroke}
|
||||
\pgfpathmoveto{\pgfpoint{159.892960pt}{399.599976pt}}
|
||||
\pgflineto{\pgfpoint{159.892960pt}{47.520004pt}}
|
||||
\pgfusepath{stroke}
|
||||
\pgfpathmoveto{\pgfpoint{230.939713pt}{399.599976pt}}
|
||||
\pgflineto{\pgfpoint{230.939713pt}{47.520004pt}}
|
||||
\pgfusepath{stroke}
|
||||
\pgfpathmoveto{\pgfpoint{301.986481pt}{399.599976pt}}
|
||||
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|
||||
\pgfusepath{stroke}
|
||||
\pgfpathmoveto{\pgfpoint{373.033234pt}{399.599976pt}}
|
||||
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|
||||
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|
||||
\pgfpathmoveto{\pgfpoint{444.079987pt}{399.599976pt}}
|
||||
\pgflineto{\pgfpoint{444.079987pt}{47.520004pt}}
|
||||
\pgfusepath{stroke}
|
||||
\pgfpathmoveto{\pgfpoint{515.126709pt}{399.599976pt}}
|
||||
\pgflineto{\pgfpoint{515.126709pt}{47.520004pt}}
|
||||
\pgfusepath{stroke}
|
||||
\color[rgb]{0.150000,0.150000,0.150000}
|
||||
\pgfsetdash{}{0pt}
|
||||
\pgfpathmoveto{\pgfpoint{88.846191pt}{51.985016pt}}
|
||||
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|
||||
\pgfusepath{stroke}
|
||||
\pgfpathmoveto{\pgfpoint{88.846191pt}{395.134949pt}}
|
||||
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|
||||
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|
||||
\pgfpathmoveto{\pgfpoint{159.892960pt}{51.985016pt}}
|
||||
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|
||||
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|
||||
\pgfpathmoveto{\pgfpoint{159.892960pt}{395.134949pt}}
|
||||
\pgflineto{\pgfpoint{159.892960pt}{399.599976pt}}
|
||||
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|
||||
\pgfpathmoveto{\pgfpoint{230.939713pt}{51.985016pt}}
|
||||
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||||
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|
||||
\pgfpathmoveto{\pgfpoint{230.939713pt}{395.134949pt}}
|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
\pgfpathmoveto{\pgfpoint{301.986481pt}{395.134949pt}}
|
||||
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||||
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|
||||
\pgfpathmoveto{\pgfpoint{373.033234pt}{51.985016pt}}
|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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||||
{
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||||
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||||
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||||
{
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||||
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|
||||
Binary file not shown.
Binary file not shown.
|
|
@ -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}
|
||||
|
|
@ -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
|
||||
|
|
@ -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
|
||||
|
|
@ -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
|
||||
|
|
@ -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
|
||||
Binary file not shown.
|
|
@ -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
|
||||
|
|
@ -0,0 +1,521 @@
|
|||
% Title: gl2ps_renderer figure
|
||||
% Creator: GL2PS 1.4.0, (C) 1999-2017 C. Geuzaine
|
||||
% For: Octave
|
||||
% CreationDate: Fri Oct 25 16:20:21 2019
|
||||
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|
||||
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|
||||
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|
||||
\pgfusepath{fill}
|
||||
\begin{pgfscope}
|
||||
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|
||||
\pgfusepath{fill}
|
||||
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|
||||
\pgfusepath{clip}
|
||||
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|
||||
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|
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|
||||
\pgfpathclose
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||||
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|
||||
\pgfpathclose
|
||||
\pgfusepath{fill,stroke}
|
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|
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|
||||
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@ -0,0 +1,482 @@
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@ -0,0 +1,500 @@
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Binary file not shown.
|
|
@ -0,0 +1,123 @@
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\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}
|
||||
|
|
@ -0,0 +1,521 @@
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||||
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|
||||
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|
||||
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|
||||
\end{pgfscope}
|
||||
\end{pgfpicture}
|
||||
Binary file not shown.
|
|
@ -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
|
||||
File diff suppressed because it is too large
Load Diff
Binary file not shown.
|
|
@ -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}
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 59 KiB |
|
|
@ -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
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 91 KiB |
Binary file not shown.
|
|
@ -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}
|
||||
|
|
@ -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;
|
||||
Loading…
Reference in New Issue