Introduction to GNU Octave

Octave is an interactive programming language specifically suited for vectorizable numerical calculations. It provides a high level interface to many standard libraries of numerical mathematics, e.g. LAPACK or BLAS. The syntax of Octave resembles that of Matlab . An Octave program usually runs unmodified on Matlab . Matlab , being commercial software, has a larger function set, and so the reverse does not always work, especially when the program makes use of specialized add-on toolboxes for Matlab .

• help lists the names of all built-in functions and internal variables.
• help name further explains the variable or function ``name''.

octave:1> help eig

• All commands can be typed in at the prompt or read from a script.
• Scripts are plain text files with file suffix .m . They are imported by calling the file name without the suffix and behave as if their content was typed in line by line.
• ; separates several commands within a line. A command terminated by ; does not display its result on-screen.
• , separates two commands without suppressing on-screen output.
• . at the end of the line denotes that an expression continues into the next line.
• Comments are preceded by % .
• Octave is case sensitive.

• varname = expression assigns the result of expression to varname .
• Octave has all the usual mathematical functions + , - , * , / , ^ , sin , cos , exp , acos , abs , etc.
• The operators of elementary logic are:

• whos shows all user-defined variables and functions (see Section 3.1).
• clear name clears name from memory; if no name is given, all variables and functions will be cleared.
• type name displays information about the variable or function name on the screen.

octave:1> x12 = 1/8, long_name = 'A String' x12 = 0.12500 long_name = A String octave:2> sqrt(-1)-i ans = 0 octave:3> x = sqrt(2); sin(x)/x ans = 0.69846

one = 1; two = 2; three = one + two;

octave:1> doless octave:2> whos *** local user variables: prot type rows cols name ==== ==== ==== ==== ==== wd real scalar 1 1 three wd real scalar 1 1 one wd real scalar 1 1 two

Matrices and vectors are the most important building blocks for programming in Octave .

2.1 Vectors

• Row vector: : v = [ 1 2 3 ]
• Column vector: : v = [ 1; 2; 3 ]
• Automatic generation of vectors with constant increment: Start[:Increment]:End

octave:1> x = 3:6 x = 3 4 5 6 octave:2> y = 0:.15:.7 y = 0.00000 0.15000 0.30000 0.45000 0.60000 octave:3> z = pi:-pi/4:0 z = 3.14159 2.35619 1.57080 0.78540 0.00000

A matrix is generated as follows.

octave:1> A = [ 1 2; 3 4] A = 1 2 3 4

octave:2> b = [5; 6]; octave:3> M = [A b] M = 1 2 5 3 4 6

• eye(m,n) produces a matrix with ones on the main diagonal and zeros elsewhere. When , the identity matrix is generated.
• zeros(m,n) generates the zero matrix of dimension .
• ones(m,n) generates an matrix where all entries are .
• rand(m,n) generates a random matrix whose entries are uniformly distributed in the interval .

• + , - , and * denote matrix addition, subtraction, and multiplication.
• A' transposes and conjugates A .
• A.' transposes A .

octave:1> A = [1 2; 3 4]; B = 2*ones(2,2); octave:2> A+B, A-B, A*B ans = 3 4 5 6 ans = -1 0 1 2 ans = 6 6 14 14

2.4 Element-wise operations

While * refers to the usual matrix multiplication, .* denotes element-wise multiplication. Similarly, ./ and .^ denote the element-wise division and power operators.

octave:1> A = [1 2; 3 4]; A.^2 % Element-wise power ans = 1 4 9 16 octave:2> A^2 % Proper matrix power: A^2 = A*A ans = 7 10 15 22

• v(k) is the -th element of the row or column vector .
• A(k,l) is the matrix element .
• v(m:n) is the ``slice'' of the vector from its th to its -th entry.
• A(k,:) is the th row of matrix .
• A(:,l) is the th column of matrix .

• length(v) returns the number of elements of the vector .
• [Rows,Columns] = size(A) returns the number of rows and columns of the matrix .

• reshape(A,m,n) transforms A into an -matrix.
• diag(A) creates a column vector containing the diagonal elements of the matrix A .
• diag(v) generates a matrix with the elements from the vector v on the diagonal.
• diag(v,k) generates a matrix with the elements from the vector v on the k -th diagonal.
• A(k,:) = rv assigns to the k -th row of A the row vector rv .
• A(k,:) = [] deletes the k -th row of A .
• A(:,j) = cv assigns to the j -th column of A the column vector cv .
• A(:,j) = [] deletes the j -th column of A .

octave:1> A = [1 2 3; 4 5 6]; v = [7; 8]; octave:2> A(2,3) = v(2) A = 1 2 3 4 5 8 octave:3> A(:,2) = v A = 1 7 3 4 8 8 octave:4> A(1,1:2) = v' A = 7 8 3 4 8 8

• A\b solves the equation .

• B = inv(A) computes the inverse of .
• [L,U,P] = lu(A) computes the LU-decomposition .
• [Q,R] = qr(A) computes the QR-decomposition .
• R = chol(A) computes the Cholesky decomposition of .
• S = svd(A) computes the singular values of .
• H = hess(A) brings into Hessenberg form.
• E = eig(A) computes the eigenvalues of .
• [V,D] = eig(A) computes a diagonal matrix , containing the eigenvalues of , and a matrix containing the corresponding eigenvectors such that .
• norm(X,p) calculates the -norm of vector . If is a matrix, can only take the values 1 , 2 or inf . The default is .
• cond(A) computes the condition number of with respect to the -norm.

• [i,j,v] = find(A) finds the indices of all nonzero elements of A . The resulting vectors satisfy .
• any(v) returns if the vector v contains nonzero elements.
• any(A) applies any to each of the columns of the matrix A .

3.1 Functions

Traditionally, functions are also stored in plain text files with suffix .m . In contrast to scripts, functions can be called with arguments, and all variables used within the function are local --they do not influence variables defined previously.

A function f , saved in the file named f.m .

function y = f (x) y = cos(x/2)+x; end

In Octave , several functions can be defined in a single script file. Matlab on the other hand, strictly enforces one function per .m file, where the name of the function must match the name of the file. If compatibility with Matlab is important, this restriction should also be applied to programs written in Octave .

A function dolittle , which is saved in the file named dolittle.m .

function [out1,out2] = dolittle (x) out1 = x^2; out2 = out1*x; end

octave:1> [x1,x2]=dolittle(2) x1 = 4 x2 = 8 octave:2> whos *** currently compiled functions: prot type rows cols name ==== ==== ==== ==== ==== wd user function - - dolittle *** local user variables: prot type rows cols name ==== ==== ==== ==== ==== wd real scalar 1 1 x1 wd real scalar 1 1 x2

Obviously, the variables out1 and out2 were local to dolittle . Previously defined variables out1 or out2 would not have been affected by calling dolittle .

global name declares name as a global variable.

A function foo in the file named foo.m :

global N % makes N a global variable; may be set in main file function out = foo(arg1,arg2) global N % makes local N refer to the global N end

The syntax of for - and while -loops is immediate from the following examples:

for n = 1:10 [x(n),y(n)]=dolittle(n); end while t
For -loop backward:
for n = 10:-1:1 .
Conditional branching works as follows.
if x==0 error('x is 0!'); else y = 1/x; end switch pnorm case 1 sum(abs(v)) case inf max(abs(v)) otherwise sqrt(v'*v) end

  
eval(string) evaluates string as Octave code.
  
feval(funcname,arg1, ) is equivalent to calling funcname with arguments arg1, .

Approximate an integral by the midpoint rule:
We define two functions, gauss.m and mpr.m , as follows:
function y = gauss(x) y = exp(-x.^2/2); end function S = mpr(fun,a,b,N) h = (b-a)/N; S = h*sum(feval(fun,[a+h/2:h:b])); end
Now the function gauss can be integrated by calling:
octave:1> mpr('gauss',0,5,500)
Loops and function calls, especially through feval , have a very high computational overhead. Therefore, if possible, vectorize all operations.
We are programming the midpoint rule from the previous section with a for -loop (file name is mpr_long.m ):
function S = mpr_long(fun,a,b,N) h = (b-a)/N; S = 0; for k = 0:(N-1), S = S + feval(fun,a+h*(k+1/2)); end S = h*S; end
We verify that mpr and mpr_long yield the same answer, and compare the evaluation times.
octave:1> t = cputime; > Int1=mpr('gauss',0,5,500); t1=cputime-t; octave:2> t = cputime; > Int2=mpr_long('gauss',0,5,500); t2=cputime-t; octave:3> Int1-Int2, t2/t1 ans = 0 ans = 45.250

  
save data var1 [var2 ] saves the values of variables var1 etc. into the file data .
  
load data reads the file data , restoring the values of the previously saved variables.
  
fprintf(string[,var1, ]) resembles C syntax for formatting output, see man fprintf under Unix.
  
format [long|short] enlarges or reduces the number of decimals displayed. Calling format without argument restores the default.
  
pause Suspends evaluation until a key is pressed.

octave:1> for k = .1:.2:.5, > fprintf('1/%g = %10.2e\n',k,1/k); end 1/0.1 = 1.00e+01 1/0.3 = 3.33e+00 1/0.5 = 2.00e+00

  
plot(x,y[,fmt]) plots a line through the points . With the string fmt you can select line style and color; see help plot for details.
  
semilogx(x,y[,fmt]) like plot with a logarithmic scale for the axis.
  
semilogy(x,y[,fmt]) like plot with a logarithmic scale for the axis.
  
loglog(x,y[,fmt]) like plot with a logarithmic scale for both axes.


   Generate a vector with the -coordinates of the points to be plotted.

x = x_min:step_size:x_max;
y = f(x);
plot(x,y)
plot(x,y) grid
octave:1> x = -10:.1:10; octave:2> y = sin(x).*exp(-abs(x)); octave:3> plot(x,y) octave:4> grid

  
[xx,yy] = meshgrid(x,y) generates the grid data for a 3D plot--two matrices xx whose rows are copies of x , and yy whose columns are copies of y .
  
mesh(x,y,z) plots a surface in 3D.

octave:1> x = -2:0.1:2; octave:2> [xx,yy] = meshgrid(x,x); octave:3> z = sin(xx.^2-yy.^2); octave:4> mesh(x,x,z);

  
title(string) writes string as title for the graphics.
  
xlabel(string) labels the -axis with string .
  
ylabel(string) labels the -axis with string .
  
zlabel(string) labels the -axis with string .
  
axis(v) set axes limits for the plot. v is a vector of form v = (xmin, xmax, ymin, ymax[, zmin zmax]) .
  
hold [on|off] controls whether the next graphics output should or not clear the previous graphics.
  
clg clears the graphics window.

Take a matrix and a vector with
Solve the system of equations . Calculate the LU and QR decompositions, and the eigenvalues and eigenvectors of . Compute the Cholesky decomposition of , and verify that .
A = reshape(1:4,2,2).'; b = [36; 88]; A\b [L,U,P] = lu(A) [Q,R] = qr(A) [V,D] = eig(A) A2 = A.'*A; R = chol(A2) cond(A)^2 - cond(A2)
Compute the matrix-vector product of a random matrix with a random vector in two different ways. First, use the built-in matrix multiplication * . Next, use for -loops. Compare the results and computing times.
A = rand(100); b = rand(100,1); t = cputime; v = A*b; t1 = cputime-t; w = zeros(100,1); t = cputime; for n = 1:100, for m = 1:100 w(n) = w(n)+A(n,m)*b(m); end end t2 = cputime-t; norm(v-w), t2/t1
Running this script yields the following output.
ans = 0 ans = 577.00
Calculate all the roots of the polynomial
Hint: Use the command compan .
Plot these roots as points in the complex plane and draw a unit circle for comparison. (Hint: hold, real, imag ).
bdf6 = [147/60 -6 15/2 -20/3 15/4 -6/5 1/6]; R = eig(compan(bdf6)); plot(R,'+'); hold on plot(exp(pi*i*[0:.01:2])); if any(find(abs(R)>1)) fprintf('BDF6 is unstable\n'); else fprintf('BDF6 is stable\n'); end
Plot the graph of the function
x = -3:0.1:3; [xx,yy] = meshgrid(x,x); z = exp(-xx.^2-yy.^2); figure, mesh(x,x,z); title('exp(-x^2-y^2)');
For each Hilbert matrix where compute the solution to the linear system , ones(n,1) . Calculate the error and the condition number of the matrix and plot both in semi-logarithmic coordinates. (Hint: hilb, invhilb .)
err = zeros(15,1); co = zeros(15,1); for k = 1:15 H = hilb(k); b = ones(k,1); err(k) = norm(H\b-invhilb(k)*b); co(k) = cond(H); end semilogy(1:15,err,'r',1:15,co,'x');
Calculate the least square fit of a straight line to the points , given as two vectors and . Plot the points and the line.
function coeff = least_square (x,y) n = length(x); A = [x ones(n,1)]; coeff = A\y; plot(x,y,'x'); hold on interv = [min(x) max(x)]; plot(interv,coeff(1)*interv+coeff(2)); end
Write a program to integrate an arbitrary function in one variable on an interval numerically using the trapezoidal rule with :
For a function of your choice, check, by generating a doubly logarithmic error plot, that the trapezoidal rule is of order .
function S = trapez(fun,a,b,N) h = (b-a)/N; % fy = feval(fun,[a:h:b]); better: fy = feval(fun,linspace(a,b,N+1)); fy(1) = fy(1)/2; fy(N+1) = fy(N+1)/2; S = h*sum(fy); end function y = f(x) y = exp(x); end for k=1:15; err(k) = abs(exp(1)-1-trapez('f',0,1,2^k)); end loglog(1./2.^[1:15],err); hold on; title('Trapezoidal rule, f(x) = exp(x)'); xlabel('Increment'); ylabel('Error'); loglog(1./2.^[1:15],err,'x');