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1.1.6 Mathematical Functions

Mathematical functions and special reserved constants/variables defined in MATLAB are listed in Table 1.3.

Table 1.3 Functions and variables inside MATLAB.

Function	Remark	Function	Remark
`cos(x)`		`exp(x)`	Exponential function
`sin(x)`		`log(x)\|`	Natural logarithm
`tan(x)`		`log10(x)`	Common logarithm
`acos(x)`	cos⁻¹(x)	`abs(x)`	Absolute value
`asin(x)`	sin⁻¹(x)	`angle(x)`	Phase of a complex number (rad)
`atan(x)`	`‐`π/2 ≤ tan⁻¹(x) ≤ π/2	`sqrt(x)`	Square root
`atan2(y,x)`	`‐`π ≤ tan⁻¹(y/x) ≤ π	`real(x)`	Real part
`cosh(x)`	(e^x + e^−x)/2	`imag(x)`	Imaginary part
`sinh(x)`	(e^x − e^−x)/2	`conj(x)`	Complex conjugate
`tanh(x)`	(e^x − e^−x)/(e^x + e^−x)	`round(x)`	The nearest integer (round‐off)
`acosh(x)`	cosh⁻¹(x)	`fix(x)`	The nearest integer toward 0
`asinh(x)`	sinh⁻¹(x)	`floor(x)`	The greatest integer ≤x
`atanh(x)`	tanh⁻¹(x)	`ceil(x)`	The smallest integer ≥x
`max`	Maximum and its index	`sign(x)`	1 (positive)/0/−1 (negative)
`min`	Minimum and its index	`mod(y,x)`	Remainder of y/x
`sum`	Sum	`rem(y,x)`	Remainder of y/x
`prod`	Product	`eval(f)`	Evaluate an expression
`norm`	Norm	`feval(f,a)`	Function evaluation
`sort`	Sort in the ascending order	`polyval`	Value of a polynomial function
`clock`	Present time	`poly`	Polynomial with given roots
`find`	Index of element(s) satisfying given condition	`roots`	Roots of polynomial
`flops(0)`	Reset the flops count to zero	`tic`	Start a stopwatch timer
`flops`	Cumulative # of floating point operations (no longer available)	`toc`	Read the stopwatch timer (elapsed time from tic)
`date`	Present date	`magic`	Magic square
	Reserved variables with special meaning
`i, j`		`pi`	π
`eps`	Machine epsilon	`Inf, inf`	Largest number (∞)
realmax, realmin	Largest/smallest positive number	`NaN`	Not_a_Number (undetermined)
`end`	The end of for‐loop or if, while, case statement or an array index	`break`	Exit while/for loop
`nargin`	# of input arguments	`nargout`	# of output arguments
`varargin`	Variable input argument list	`varargout`	Variable output argument list

MATLAB also allows us to define our own function and store it in a file named after the function name so that it can be used as if it were a built‐in function. For instance, we can define a scalar‐valued function:

and a vector‐valued function

as follows:

function y=f1(x) y=1./(1+8*x.̂2); function y=f49(x) y(1)= x(1)̂2+4*x(2)̂2 -5; y(2)=2*x(1)̂2-2*x(1)-3*x(2) -2.5;

Once we store these functions as M‐files, each named “f1.m” and “f49.m” after the function names, respectively, we can call and use them as needed inside another M‐file or in the MATLAB Command window.

>f1([0 1]) % several values of a scalar function of a scalar variable ans= 1.0000 0.1111 >f49([0 1]) % a value of a 2D vector function of a vector variable ans= -1.0000 -5.5000 >feval('f1',[0 1]), feval('f49',[0 1]) % equivalently, yields the same ans= 1.0000 0.1111 ans= -1.0000 -5.5000

1 (Q7) With the function f1(x) defined as a scalar function of a scalar variable, we enter a vector as its input argument to obtain a seemingly vector‐valued output. What's going on?
2 (A7) It is just a set of function values [f1(x1) f1(x2) …] obtained at a time for several values [x1 x2 …] of x. In expectation of one‐shot multioperation, it is a good practice to put a dot ( .) just before the arithmetic operators *(multiplication), /(division), and ̂(power) in the function definition so that the element‐by‐element (element‐wise or term‐wise) operation can be done any time.

Note that we can define a simple function not only in an independent M‐file but also inside a script by using the function handle operator @, the ‘ inline()’ command, or just in a form of literal expression that can be evaluated by the command ‘ eval()’.

>f1h=@(x)1./(1+8*x.̂2); % Using function handle >f1i=inline('1./(1+8*x.̂2)','x'); % Usng inline() >f1h([0 1]), feval(f1h,[0 1]), f1i([0 1]), feval(f1i,[0 1]) ans= 1.0000 0.1111 ans= 1.0000 0.1111 >f1='1./(1+8*x.̂2)'; x=[0 1]; eval(f1) ans= 1.0000 0.1111

As far as a polynomial function is concerned, it can simply be defined as its coefficient vector arranged in descending order. It may be called to yield its value for certain value(s) of its independent variable by using the command ‘ polyval()’.

>p=[1 0 –3 2]; % Polynomial function p(x) = x3-3x + 2 >polyval(p,[0 1]) % Polynomial function values at x=0 and 1 ans= 2.0000 0.0000

The multiplication of two polynomials can be performed by taking the convolution of their coefficient vectors representing the polynomials in MATLAB, since

where

This operation can be done by using the MATLAB built‐in command ‘ conv()’ as illustrated beneath:

>a=[1 –1]; b=[1 1 1]; c=conv(a,b) c= 1 0 0 -1 % meaning that (x-1)(x2 + x + 1) = x3 + 0 ⋅ x2 + 0 ⋅ x-1

But, in case you want to multiply a polynomial by only xⁿ, you can simply append n zeros to the right end of the polynomial coefficient vector to extend its dimension.

>a=[1 2 3]; c=[a 0 0] % equivalently, c=conv(a,[1 0 0]) c= 1 2 3 0 0 % meaning that (x2 + 2x + 3)x2 = x4 + 2x3 + 3x2 + 0 ⋅ x + 0

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