Lecture One: Fundamental of Computational Economics

Lecture One: Fundamental ofComputational Economics

Li, Xihao∗

Department of Economics and Social Sciences (DiSES)

Universita Politecnica delle Marche

October, 2013

∗Department of Economics and Social Sciences (DiSES), Universita Politecnica delle Marche, Piazzale

Martelli 8, 60121 Ancona, Italy. Email: xihao.li@gmail.com.

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Introduction: I

Goal of Lecture One: to have general idea about how touse computational tools to work out problems.

Tools:General tools: Matlab, GNU Octave

Special purpose tools: e.g., statistical package: Eview,Sdata, R

We use GNU Octave for demonstration!Main textbook: (”Scientific Computing with MATLABand Octave”, Quarteroni Alfio, Saleri Fausto, GervasioPaola 3rd ed., 2010)

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Introduction: I

Goal of Lecture One: to have general idea about how touse computational tools to work out problems.

Tools:General tools: Matlab, GNU Octave

Special purpose tools: e.g., statistical package: Eview,Sdata, R

We use GNU Octave for demonstration!

Main textbook: (”Scientific Computing with MATLABand Octave”, Quarteroni Alfio, Saleri Fausto, GervasioPaola 3rd ed., 2010)

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Introduction: I

Goal of Lecture One: to have general idea about how touse computational tools to work out problems.

Tools:General tools: Matlab, GNU Octave

Special purpose tools: e.g., statistical package: Eview,Sdata, R

We use GNU Octave for demonstration!Main textbook: (”Scientific Computing with MATLABand Octave”, Quarteroni Alfio, Saleri Fausto, GervasioPaola 3rd ed., 2010)

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

ABC of GNU Octave

GNU Octave (Octave in short) is free, under GNU GeneralPublic License.

Learning by doing!

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: I

Clean up the working environment:

> clear all;

> clc;

display format in Octave:

> a = 1/7;

> a

format long:

> format long

> a

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: II

epsilon, the roundoff error:

> eps

finite scope of float number:

> realmin

> realmax

Infinity representation:

> Inf

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: III

Numercial computation: computer is NOT the same asMath!

Example:

> a = 1; b=1; while a+b ∼ = a; b=b/2; end> b

Another example:

> x = 1.e-15;> ((1+x)-1)/x

NaN:

> 0/0

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: IV

Complex number:

> z = complex(1,1)

Consider abs(z) ( cos( angle(z) ) + i * sin( angle(z) ) ):

modulus of complex number:

> abs(z)

angle of complex number:

> angle(z)

Compare with 45 degree of angle, pi/4:

> angle z = pi/4

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: V

graphical presentation:

> compass(z)

real and imaginary part:

> real(z)

> imag(z)

conjuecture of z:

> conj(z)

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: VI

Matrices:

> A = [ 1,2; 3, 4 ]

> A complex = [ complex(1,1), 2; 3, 4 ]

zero matrix:

> B = zeros(3,3)

identity matrix:

> id = eye(2, 2)

Note the dimension of matrix, try A * B

determinant of matrix:

> det(A)Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: VII

inverse of matrix:

> inv(A)

try complex matrix:

> det(A complex)

> inv(A complex)

singular matrix:

> A sing = [1,0; 3, 0]

> det(A sing)

> inv(A sing)

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: VIII

diagonal matrix:

> v = [1:2:5]

> dv = diag(v)

transpose of matrix:

> A’

Hermitian of complex matrix:

> A complex’

transpose of diagonal matrix:

> dv’

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: IX

vectors:

> v = zeros(10,1)

> v’

> size(v)

> size(v’)

unit vector:

> vone = ones(10,1)

random number vector:

> vrand = rand(100,1);

> plot(vrand)Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: X

dot product:

> vrand = rand(10,1)

> dot(vrand, vone)

dot product with unit vector is equivalent to sum:

> sum(vrand)

Euclidean norm of vector:

> norm(vrand)

> norm(vone)

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: XI

element-by-element operation .∗, .\, .∧:

> vrand .∗ ones(10,1)

> vrand .∗ ones(10,1) - vrand

Question: does it work with: vrand * ones(10,1)?

Question: vrand .∗ ones(10,1) - vrand is the same asvrand .∗ ones(10,1) .− vrand ?

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: XII

simple way to write a function:

> func = ’x.∧ 2 + 2 * x + 1’;

standard way to write a function:

> function y = fun (x); y = x.∧2 + 2 * x + 1 + 10; endfunction

plot functions:

> lim = [-7,5];> figure;> subplot(2,1,1); fplot(func, lim); legend(”x.∧2 + 2 * x + 1”);> hold on;> subplot(2,1,2); fplot(’fun’, lim, ’-r’); legend(”x.∧2 + 2 * x +1 + 10”);> hold off;

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: XII

another way to plot functions:

> xx = [-7:0.1:5];> v func = xx.∧2 + 2 * xx + 1;> v fun = xx.∧2 + 2 * xx + 1 + 10;

> figure;> subplot(2,1,1); plot(xx, v func, ’c’, ’Linewidth’, 5);legend(”x.∧2 + 2 * x + 1”);> hold on;> subplot(2,1,2); plot(xx, v fun, ’-r’, ’Linewidth’, 5);legend(”x.∧2 + 2 * x + 1 + 10”);> hold off;

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Action: XIII

numerical solution of f(x) = 0:

> function y = f (x)> y = x.∧2 - 2 * x - 3;> endfunction

find f(x) = 0:

> fzero(’f’, -2)

> fzero(’f’, 4)

> fzero(’f’, [-2,0])

Question: What if fzero(’f’, [-2,4])? Why?

Li, Xihao Lecture Notes

Lecture One: Fundamental of Computational Economics Introduction First Tour Exercise

Exercise

Try the aforementioned questions.

Read Chapter 1 of textbook for introduction.

Li, Xihao Lecture Notes

Install GNU Octave in Windows I

The complete instruction is here:http://wiki.octave.org/Octave_for_Windows

To install Octave-3.6.4-mingw + octaveforge pkgs:

download:1 binary file ”Octave3.6.4 gcc4.6.2 20130329.7z”2 package file ” Octave3.6.4 gcc4.6.2 pkgs 20130331.7z”

extract these two files to the path: ” C:/Octave/”

copy file ”octave3.6.4 gcc4.6.2.lnk” and”octave3.6.4 gcc4.6.2 docs.lnk” to any convenient lo-cation and edit its properties

Li, Xihao Lecture Notes

Install GNU Octave in Windows IIlaunch Octave console and and execute the follow-ing commands:> pkg rebuild -auto> pkg rebuild -noauto ad> pkg rebuild -noauto nan> pkg rebuild -noauto gsl> pkg rebuild -auto javarestart Octave.

(optional) install ”Notepad++” as editor:download Notepad++ from http://notepad-plus-plus.org/ and install it on your system

edit ” C:/Octave//share /octave /site /m /startup /octaverc”:

uncomment the line:EDITOR(’C:/Program Files/Notepad++/notepad++.exe’);

Li, Xihao Lecture Notes

Install GNU Octave in Windows IIIadd the following lines:> edit (”editor”, sprintf (”%s %%s”, EDITOR ()))> edit mode async

Li, Xihao Lecture Notes