Functions in Matlab

Functions in Matlab Builtin and user defined functions

Builtin Functions • Exponential • exp(x) • sqrt(x) • Logarithmic • log(x) natural logarithm ln • log10(x) • log2(x)

Builtin continued • numeric • ceil(x) round to nearest integer towards + • fix(x) round to nearest integer towards 0 • floor(x) round to nearest integer towards -  • round(x) round to nearest integer • sign(x) +1, 0 or -1 • rem(x,y) Finds remainder of x/y • complex • abs(x) absolute value • angle(x) in complex plane • conj(x) • imag(x) • real(x)

Builtin continued • Trigonometric and their inverse • cos(x) acos(x) • sin(x) asin(x) • tan(x) atan(x) • cot(x) acot(x) • csc(x) acsc(x) • sec(x) asec(x) • atan2(x,y)

Builtin Functions continued • Hyperbolic and their inverse • cosh(x) acosh(x) • coth(x) acoth(x) • csch(x) acsch(x) • sech(x) asech(x) • sinh(x) asinh(x) • tanh(x) atanh(x)

Remarks • All trigonometric functions require the use of radians and not degrees • All buitlin functions can also be used with complex numbers. • To understand the meaning with complex numbers need to go back to basic definitions: • exp(i x) = cos(x) + i sin(x) • sin(x) =(exp(i x) - exp(-i x) )/(2 i) • sinh(x) = (exp(x) – exp(-x) )/2 • Basic example • exp( a + i b) = exp(a) exp(i b) = exp(a) (cos(x) + i sin(x))

Remarks continued • Since a trigonometric function is periodic each value has many inverse values • Matlab will return only one value, the so called 'principle value' • For engineering applications it is important to check that this is the correct value. • Plotting a complex valued function z =f(x+iy) is not possible. It requires a four dimensional space (x,y,u,v) since z = u+iv

Builtin Functions for vectors • max(x) • returns largest value in vector x • [a,b] = max(x) • returns largest value in a and index where found in b • max(x,y) • x and y arrays of same size, returns vector of same length with lager value from corresponding positions in x and y • same type of functions are available for min

Builtin functions for vectors • sort(x) • mean(x) • median(x) • sum(x) • prod(x) • cumsum(x) • returns vector of same size with cummulative sum of x i.e. x=[4,2,3] returns [4,5,9] • cumprod(x) • returns cummulative products in vector of same size

Builtin functions applied to matrices • Matrices (arrays) are stored in column major form • When builtin functions for vectors are applied to a matrix function operates on columns and returns a row vector

User defined functions • Similar to script files except that • temporary variables remain local • function use standardized input and output parameters • Script files good for quick and dirty jobs • Functions can be used over and over again with no side effects • Similar to subroutine, function or procedure in other programming languages like Fortran, PL/I, Pascal, C, C++

Example • f(x,y) = 3*x +6*y^2 • Questions to be answered first: Are x and y allowed to be arrays? • Since Matlab treats everything as an array it is best to allow all parameters to be an array • If x and y are arrays they have to be of the same size and f(x,y) is returned as an array of the same size • In above example x or y could be scalar (i.e. a constant) and function still makes sense

Matlab code placed in file fun.m • function z = fun(x,y) • u=3*x; • z = u + 6*y.^2 ; • Note operator .^ to allow for an array • Here the operator * does not require the array operator .* since it is a multiplication with a constant • + alays operates on the elements of an array thus .+ is not defined • u is a local variable in will not be available after the function is executed • u is used in the example for demonstration only. It would be better to write z = 3*x+6*y.^2;

Remarks • Check that function name fun does not yet exist via: exist fun • The input parameters x, y and the output parameter z are local • fun(3,4) does not create a variable z instead it returns the value in ans • q = fun(x0,y0) gives anwser to q, assuming x0 and y0 are defined • fun([2:5],[7:10]) works since arrays are of the same length

Parameters for functions • Input parameters: 0 or more • parameter list enclosed by ( ) • if there are no parameters can omit () • function show_date • today = date • Example with three input parameters • function volume = box(height, width, depth) • volume = height.*width.*depth;

Parameters for user defined functions • Output parameter 0 or more • parameter list enclosed by [ ] • 0 or 1 output parameter, can omit [ ] • 2 or more parameters separated by commas • first output parameter ist set to ans, if function is called without output parameters • Examples: • function [areaC, cirumC] = circle(radius); • function areaSquare = square(sides) ;

Remarks • Why pass constants like g Earth's gravitation to a function? • g varies with units and forces user to select the proper unit • a better choice is to let the user select at the very beginning, which units are to be used • in this case need to state in function • global g • and in main progra • global g • g = 9.81 • Input parameters are local • even if they are change in the function the original value remains unchanged • if the value in the calling program needs to be changed have to return value as an output parameter

Function handle • Useful as parameter to other functions • Can be considered as an alternate name for a function – but with more capabilities • Example: • sine_handle = @sin • sine_handle(x) same values as sin(x) for all x • x = [0 : 0.01 : 2*pi] ; • y = sin( x ); • plot(x,y) • plot( x, sin(x) ) • plot( [0 : 0.01 : 2*pi] , sin( [0 : 0.01 : 2*pi] )) ;

Function handle continued • In last example everything is on one line • It requires writing the interval twice • It would be more convient to write • gen_plot( function_handle, interval ) • The first parameter has to be a function handle and not just the name of a function • gen_plot( sin, [0 : 0.01 : 2*pi ] ) does not make sense to Matlab, but the following does • gen_plot( sine_handle, [0 : 0.01 : 2*pi] )

Using a function handle • function [] = gen_plot( func_handle, interval ) ; • plot( interval, func_handle(interval) ) ; • When plotting lots of functions it may be useful to have gen_plot available • The example shows how to pass functions as parameters. • Another use is anonymous functions • Assume the user needs to work temporaily with the function x3+3*x – 1 • Instead of writing the function • function y = mypoly(x) ; • y = x.^3+3*x-1 • and storing it as mypoly.m in subdirectory work we can use an anonymous function with the function handle mypoly • mypoly = @(x) x.^3+3*x-1

Using anonymous functions • With a function handle an anonymous function can be used like any other • gen_plot( mypoly, [-10 : 0.01 : 10] ) • fzero( mypoly, 0.0 ) • roots( [1, 0, 3, -1] ) • Without the function handle the anonymous function can be inserted directly as the parameter • gen_plot( @(x) x.^3+3*x-1, [-10 : 0.01 : 10] )

More examples • f1 = @(x) x + 2* exp(-x) -3 • fzero( f1, 0 ) • fzero( f1, 1 ) • Assume f1 had been defined as a function and kept in f1.m then • fzero( f1, 0 ) would be in error • Besides function handle Matlab provides the the following alternate method • fzero( 'f1', 0 ) • fzero( 'sin', 0 ) • fzero( 'x.^3', 0 ) need to use default variable name

Functions in Matlab