General purpose commands Arrays Mathematical operations Programming with Matlab Bi7740: Scientiﬁc computing Matlab remider Vlad Popovici popovici@iba.muni.cz Institute of Biostatistics and Analyses Masaryk University, Brno March 11, 2014 Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Outline 1 General purpose commands 2 Arrays 3 Mathematical operations 4 Programming with Matlab Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Help and paths doc, help, helpdesk, helpwin, lookfor: different ways of obtaining help docopt: (on UNIX) path where the help ﬁles reside addpath, path, pathtool, rmpath: manage search paths for functions lasterr, lastwarn: last error and warning type, what, which: list a ﬁle, list a directory, locate functions and ﬁles Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Creating arrays var = [val1 val2 ...] row vector from values var = [val1; val2; ...] column vector from values; equivalent to var = [val1 val2 ...]' (' is the transposition operator) equally spaced data: ::; is is 1, omit :. Example: 1:2:10, 1:10, 1.5:0.1:2.1. last value might not be in the generated sequence decreasing sequences: as above, with negative : 20:−5:0 equally spacing of an interval: linspace(,,) and values will be in the sequence logarithmically equally space points: logspace Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Creating 2D arrays - matrices var = [a11, a12,...; a21, a22,...; ...] from values you can use vector generators, as before, for the rows of the matrix zeros(m,n), ones(m,n), eye(m): all-zeros, all-ones, or identity matrix with m rows and n columns transpose of X: X' Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Arrays dimension size(a), size(b), size(b,2) length(a): for vectors numel(b): total number of elements Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Addressing the elements of an array by index: a = 1:3:10; a(1) and b = [1:4;5:8]; b(1,2) by index vectors: a(1:2), b(1,:), b(2,2:end): end gives the last index in the corresponding dimension of an array all elements as a row vector: a(:), b(:): column-by-column ordering of matrix elements (column-major) you can use single index for n-dimensional arrays too! Pay attention to ordering of elements: b(1:5) Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Extending/reducing arrays arrays are dynamical structures, you can add a new value beyond the dimension of the array: b(3,3)= 13: Matlab adds 0s to ﬁll-in to the required new dimension this operation is expensive: better pre-allocate the array! concatenation: [a a], [a;a]: the arrays must be consistent general concatenation: cat(...) delete elements: replace them by empty arrays: c = b; c(3,:)= []. You have to keep the shape of the array! Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab (Re-)shaping the array reshape(b, ). The total number of elements must not change! reshape(b, [1,2,2,3]) Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Strings s = 'ab cd ef' s = char('aaa', 'bb', 'c') rows are the strings given Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Basic operators The arrays must have conforming shapes/sizes, depending on the operation. +,−,*,^ classical matrix operators (addition, subtraction, multiplication, power) /: matrix division, related to the inverse (we discussed) .*,.^,./ element-wise operations scalars are multicast: [1 2 3; 4 5 6] * 2 and operation is kept element-wise Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Some basic built-in functions for arrays Usually, the summaries are column-wise - see the help for details min(b), min(b, 4), [m,n] = min(b) max(...) mean(...), median(...), std(...) sum(...), prod(...) dot(...,..): dot(a,a) is the same as a*a' Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Random arrays rand(m,n): random uniformly distributed numbers between 0 and 1, in the form of a matrix (or vector, or scalar, depending on m and n) randperm(n): random permutation of 1:n vector randperm(n,k) random selection of k elements out of 1:n randi(imax, m, n): random uniformly distributed integers between 1 and imax, in the form of a matrix (or vector, or scalar) randn(...): normally distributed random numbers: try hist(randn(1000, 1)) Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Relational operators Check help precedence for precedence of operators! relational operators: <, >, =<, ≥, ==, ∼= the result is either logical 1 (true) or logical 0 (false) and has the shape of the operands (after bringing them to compatible shapes) example: mod(1:10,5)< 3 logical results can be used to address elements of arrays: a = randn(4); a(a < 0.5) what happend? Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Logical operators and functions AND: x & y, and(x, y) OR: x | y, or(x, y) NOT: ∼x, not(x) XOR: xor(x,y) all(x), any(x), find(x) Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Conditional branching 1 if 2 3 [elseif 4 ] 5 .......... 6 [else 7 ] 8 end One-liner: if instruction; end Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab 1 switch 2 case value1 3 4 case {value2, value3,...} 5 6 .......... 7 otherwise 8 9 end Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Loops 1 for iterator = sequence 2 3 end 4 5 while 6 7 end continue skips the rest of the block and jumps to the next iteration (if any) break: breaks out of the loop; if it is outside a loop, it terminates the script/function Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Functions One visible function per .m ﬁle. 1 function [] = () 2 % document your function here 3 4 % process arguments 5 if nargin < ... 6 ... 7 end 8 9 10 11 % prepare the return values 12 if nargout... 13 ... 14 end 15 end % or return: not mandatory, but a good practice Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab usually, the variables created in the functions are local use global to declare that a variable is from the global environment (put globals at the top of the ﬁle) the ﬁle with the function must be in the search path of Matlab Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Anonymous functions 1 = @(arguments) 2 3 cube = @(x) x.^3 must be a single valid Matlab expression you can use also variables from the context in which the anonymous function is deﬁned anonymous functions can be used as parameters for other functions Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Inline functions 1 = inline('mathematical expression ... as a string') 2 3 square = inline('x .^ 2') 4 square(1:5) 5 quad = inline('x .^ 2 + y .^ 2 + dot(x,y)', 'x', 'y') if you do not provide explicit parameters, they are deduced from the expression and ordered alphabetically in the argument list you cannot use i and j as variables the inline functions can be used as arguments for other functions Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Functions as arguments @function_name gives you a function handle, that can be used for calling the function itself. 1 function y = funplot(F, x0, x1) 2 x = linspace(x0, x1, 100); 3 y = F(x); 4 plot(x, y); 5 return 6 7 funplot(@cos, 0, 2*pi) 8 funplot(square, −3, 3) 9 funplot(@(x) x .^ 3, 0, 5) Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Subfunctions In a ﬁle "main_function.m": 1 function ...= main_function(...) 2 end 3 4 function ...= subfunction_1(...) 5 ... 6 end 7 8 function ...= subfunction_2(...) 9 ... 10 end Vlad Bi7740: Scientiﬁc computing General purpose commands Arrays Mathematical operations Programming with Matlab Nested functions 1 function ...= A(...) 2 function ...= B(...) 3 ... 4 end 5 ... 6 function ...= C(...) 7 ... 8 end 9 ... 10 end Vlad Bi7740: Scientiﬁc computing