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Chapter 2 Basic Features
2.1 Simple Math
The MATLAB Command window is the primary means for interacting with
MATLAB. When the Command window is active, statements are entered which can be
executed immediately. For example, simple mathematical calculations can be performed
and the results displayed. The results of each calculation are stored in the default
MATLAB variable named 'ans' which retains its value until its changed.
Example 2.1.
100/5 + 6^
2ans 2(25+(3exp(1)))*
ans = 160 ans = 56 ans = 112 ans = 66.
Alternatively, the results can be assigned to user defined MATLAB variables.
Example 2.1.
t_initial=4; t_final=12; t_elapsed=t_final-t_initial rate= distance=rate(t_elapsed)*
t_elapsed = 8 rate = 5 distance = 40
Note the semicolon suppresses displaying the result of executing a MATLAB
command.
2.2 The MATLAB Workspace
During a MATLAB session, the issued commands and the variables resulting
from those commands are saved in the MATLAB workspace. The numerical values
asssigned to variables are stored there and easily accessed by entering the variables'
names at the MATLAB prompt.
Example 2.2.
l=3; w=12; A=lw; A l,w,A*
A = 36
l = 3 w = 12 A = 36
The 'who' command results in a list of the current MATLAB workspace variables.
Example 2.2.
who
Your variables are:
A distance rate t_final w ans l t_elapsed t_initial
2.4 Comments, Punctuation, and Aborting Execution
MATLAB ignores all text following a % symbol to allow commenting.
Example 2.4.
int_nominal=0.08; % Loan interest rate (per year) int_monthly=int_nominal/12; % Loan interest rate (per month) P=270000; % Loan Amount n_years=30; % Duration of loan (in years) n_months=n_years12; % Duration of loan (in months) x=(1+int_monthly)^n_months; A=P(int_monthlyx)/(x-1) % Monthly installment to repay loan*
A = 1.9812e+
More than one command can appear on the same line if they are separated by a
comma or a semicolon.
Example 2.4.
l=3; w=4; d=5; V=lwd**
V = 60
r=2, h=5 % Cone radius and height V=(1/3)pir^2h % Volume of Cone*
r = 2 h = 5 V = 20.
To improve readability, long expressions or commands can be entered on multiple
lines using an elipsis (...). The break cannot appear in the middle of a variable name.
Comments cannot be continued, i.e. a new % is required on the next line to continue a
comment.
Example 2.4.
x=0.5; sum=1/(1-x) % Computes the sum of the infinite geometric series % 1 + x + x^2 + x^3 + x^4 + x^5 + ......
sum = 2
2.5 Complex Numbers
Arithmetic operations involving complex numbers are easily handled by
MATLAB. Either 'i' or 'j' can be used to represent the square root of minus one.
Example 2.5.
Re_z=3; Im_z=4; z1=Re_z + jIm_z % z1 is a complex number z2=Re_z - jIm_z % z2 is the complex conjugate of z u=sqrt(z1z2) % Find the magnitude of z1 and its conjugate z v=z1/z2 % Complex number v is the quotient of z1 and z w=z2v % Complex number w is the product of z2 and v (restores z1) y=u(v+w)*
z1 = 3.0000 + 4.0000i z2 = 3.0000 - 4.0000i u = 5 v = -0.2800 + 0.9600i w = 3.0000 + 4.0000i y = 13.6000 +24.8000i
The multiplication (*) is not required when 'i' or 'j' is preceded by a number
instead of a variable expression.
The following example illustrates converting a complex number x + iy in
rectangular form to polar form Re jθ^ where R = x^2^ + y^2 and θ = tan −^1 ( y / x ) using the
MATLAB functions 'abs', and 'angle'.
Example 2.5.
x=3; % Real part of complex number y=4; % Imaginary part of complex number z=x+yi % Complex number in rectangular form R=abs(z) % Computes the magnitude of z theta=angle(z) % Computes the angle(rad) of z theta_deg=180theta/pi; % Converts theta to degrees**
z = 3.0000 + 4.0000i R = 5 theta = 0.
In the opposite direction, i.e. converting from polar form Re jθ^ to rectangular form
x + iy where x = R cos( θ )and y = R sin( ) θ , the MATLAB functions 'real' and 'imag'
are used as shown in Example 2.5.3.
Example 2.5.
2.6 Mathematical Functions
MATLAB supports a variety of built- in mathematical functions. A list of
mathematical functions grouped into several categories, along with other MATLAB
information, appears in a Help Window by clicking on 'Help' from the menubar of the
MATLAB Command window. Alternatively, the command 'helpwin' can be typed on
the command line at the MATLAB prompt. The categories of mathematical functions
are:
matlab/elmat - Elementary matrices and matrix manipulation matlab/elfun - Elementary math functions matlab/specfun - Specialized math functions matlab/datafun - Data analysis and Fourier transforms matlab/polyfun - Interpolation and polynomials matlab/funfun - Function functions and ode solvers matlab/sparfun - Sparse matrices
A complete listing of all the functions in a specific category is obtained by double
clicking the category. For example, Elementary math functions includes all of
the following:
Trigonometric. sin - Sine. sinh - Hyperbolic sine. asin - Inverse sine. asinh - Inverse hyperbolic sine. cos - Cosine. cosh - Hyperbolic cosine. acos - Inverse cosine. acosh - Inverse hyperbolic cosine. tan - Tangent. tanh - Hyperbolic tangent. atan - Inverse tangent. atan2 - Four quadrant inverse tangent. atanh - Inverse hyperbolic tangent. sec - Secant. sech - Hyperbolic secant. asec - Inverse secant. asech - Inverse hyperbolic secant. csc - Cosecant. csch - Hyperbolic cosecant. acsc - Inverse cosecant. acsch - Inverse hyperbolic cosecant. cot - Cotangent. coth - Hyperbolic cotangent. acot - Inverse cotangent.
acoth - Inverse hyperbolic cotangent.
Exponential. exp - Exponential. log - Natural logarithm. log10 - Common (base 10) logarithm. log2 - Base 2 logarithm.
pow2 - Base 2 power and scale floating point number. sqrt - Square root. nextpow2 - Next higher power of 2.
Complex. abs - Absolute value. angle - Phase angle. complex - Construct complex data from real and imaginary parts. conj - Complex conjugate. imag - Complex imaginary part. real - Complex real part. unwrap - Unwrap phase angle. isreal - True for real array. cplxpair - Sort numbers into complex conjugate pairs.
Rounding and remainder. fix - Round towards zero. floor - Round towards minus infinity. ceil - Round towards plus infinity. round - Round towards nearest integer. mod - Modulus (signed remainder after division). rem - Remainder after division. sign - Signum.
Also, typing helpwin('elfun')in the Command window will access the
MATLAB Help Window of Elementary math functions directly. Double clicking a
specific function brings up detailed information about that function. For example, the
'rem' function description is
REM Remainder after division. REM(x,y) is x - y.*fix(x./y) if y ~= 0. By convention, REM(x,0) is NaN. The input x and y must be real arrays of the same size, or real scalars.
REM(x,y) has the same sign as x while MOD(x,y) has the same sign as y. REM(x,y) and MOD(x,y) are equal if x and y have the same sign, but differ by y if x and y have different signs.
REM is a built-in function. An M-file for REM would be the same as MOD with "floor" replaced by "fix".
Some of the elementary mathematical functions are illustrated in the following
examples.
Example 2.6.
% A sample of the Trigonometric Functions in MATLAB theta1= pi/6 % Angle (in rad) theta1_deg=180theta1/pi % Angle (in deg) x=sin(theta1) % Sine function theta2=atan(1) % Inverse tangent function (in rad)*
Example 2.6.
% A sample of Rounding and Remainder Functions in MATLAB a=9; b=5; c=-5; x1=fix(a/b) % Round toward zero x2=floor(a/c) % Round toward minus infinity x3=ceil(a/c) % Round toward plus infinity x4=rem(a,b) % Remainder after division x5=sign(b-a) % Signum function
x1 = 1 x2 = - x3 = - x4 = 4 x5 = -
Many of MATLAB's built- in functions return several outputs. In this case, the
left hand side is a row vector with output variable names, separated by commas and
enclosed in square brackets. The cartesian transformation function 'cart2sph'
produces 3 outputs from a set of 3 inputs as described below.
CART2SPH Transform Cartesian to spherical coordinates. [TH,PHI,R] = CART2SPH(X,Y,Z) transforms corresponding elements of data stored in Cartesian coordinates X,Y,Z to spherical coordinates (azimuth TH, elevation PHI, and radius R). The arrays X,Y, and Z must be the same size (or any of them can be scalar). TH and PHI are returned in radians.
TH is the counterclockwise angle in the xy plane measured from the positive x axis. PHI is the elevation angle from the xy plane.
See also CART2POL, SPH2CART, POL2CART.
Two different 2-D coordinate transformation functions are illustrated below.
Example 2.6.
% A sample of Coordinate Transformation Functions in MATLAB x1=3,y1=4, [th1,r1]=cart2pol(x1,y1) % Cartesian to polar cart2pol(x1,y1) % Will return only the first output in 'ans' th2=th1+pi r2=r1; [x2,y2]=pol2cart(th2,r2) % Polar to Cartesian
x1 = 3 y1 = 4 th1 = 0. r1 = 5 ans = 0. th2 = 4. x2 = -3. y2 = -4.
