Appendix A

MATLAB Functions

This appendix provides you with an overview of the MATLAB functions. Each function helps you perform a specific task within MATLAB. The tables contain the function name and a short description of the task that the function performs. If you need more information than this appendix gives you, you can obtain additional information by typing help <function_name> in the MATLAB command window and pressing Enter. Of course, you also have the information found in this book, and you can search in Help. See Chapter 3 for some useful information on using the Help resources that MATLAB provides.

Table A-1 Arithmetic Functions

Function	Description
uplus	Unary plus — comparable to + acting on one object
plus	Plus — comparable to + adding two objects
uminus	Unary minus — comparable to – acting on one object
minus	Minus — comparable to – subtracting two objects
mtimes	Matrix multiplication — comparable to *
times	Array multiply — comparable to .*
rdivide	Right-array division — comparable to ./
mrdivide	Solves systems of linear equations xA = B for x — comparable to /
ldivide	Left-array division — comparable to .\
mldivide	Solves systems of linear equations Ax = B for x — comparable to \
power	Array power — comparable to .^
mpower	Matrix power — comparable to ^
diff	Differences and approximate derivatives
prod	Product of array elements
sum	Sum of array elements
ceil	Rounds toward positive infinity
fix	Rounds toward zero
floor	Rounds toward negative infinity
idivide	Integer division with rounding option
mod	Modulus after division
rem	Remainder after division
round	Rounds to nearest integer

Table A-2 Trigonometric Functions

Function	Description
sin	Sine of argument in radians
sind	Sine of argument in degrees
asin	Inverse sine; result in radians
asind	Inverse sine; result in degrees
sinh	Hyperbolic sine of argument in radians
asinh	Inverse hyperbolic sine
cos	Cosine of argument in radians
cosd	Cosine of argument in degrees
acos	Inverse cosine; result in radians
acosd	Inverse cosine; result in degrees
cosh	Hyperbolic cosine
acosh	Inverse hyperbolic cosine
tan	Tangent of argument in radians
tand	Tangent of argument in degrees
atan	Inverse tangent; result in radians
atand	Inverse tangent; result in degrees
atan2	Four-quadrant inverse tangent
atan2d	Four-quadrant inverse tangent; result in degrees
tanh	Hyperbolic tangent
atanh	Inverse hyperbolic tangent
csc	Cosecant of argument in radians
cscd	Cosecant of argument in degrees
acsc	Inverse cosecant; result in radians
acscd	Inverse cosecant; result in degrees
csch	Hyperbolic cosecant
acsch	Inverse hyperbolic cosecant
sec	Secant of argument in radians
secd	Secant of argument in degrees
asec	Inverse secant; result in radians
asecd	Inverse secant; result in degrees
sech	Hyperbolic secant
asech	Inverse hyperbolic secant
cot	Cotangent of argument in radians
cotd	Cotangent of argument in degrees
acot	Inverse cotangent; result in radians
acotd	Inverse cotangent; result in degrees
coth	Hyperbolic cotangent
acoth	Inverse hyperbolic cotangent
hypot	Square root of sum of squares

Table A-3 Exponentials, Logarithms, Powers, and Roots

Function	Description
exp	Exponential
expm1	Computes exp(x)–1 accurately for small values of x
log	Natural logarithm
log10	Common (base 10) logarithm
log1p	Computes log(1+x) accurately for small values of x
log2	Determines base 2 logarithm and dissect floating-point numbers into exponent and mantissa
nthroot	Real nth root of real numbers
pow2	Base 2 power and scale floating-point numbers
reallog	Natural logarithm for nonnegative real arrays
realpow	Array power for real-only output
realsqrt	Square root for nonnegative real arrays
sqrt	Square root

Table A-4 Complex Number Functions

Function	Description
abs	Absolute value and complex magnitude
angle	Phase angle
complex	Construct complex data from real and imaginary components
conj	Complex conjugate
i	Imaginary unit
imag	Imaginary part of complex number
isreal	Checks whether input is a real array
j	Imaginary unit
real	Real part of complex number
sign	Signum() function, which returns 1 if the corresponding element is greater than 0, 0 if the corresponding element is zero, and –1 if the corresponding element is less than 0.

Table A-5 Discrete Math Functions

Function	Description
factor	Prime factors
factorial	Factorial function
gcd	Greatest common divisor
isprime	Array elements that are prime numbers
lcm	Least common multiple
nchoosek	Binomial coefficient or all combinations
perms	All possible permutations
primes	Generates list of prime numbers
rat, rats	Rational fraction approximation

Table A-6 Polynomial Functions

Function	Description
poly	Polynomial with specified roots
polyder	Polynomial derivative
polyeig	Polynomial eigenvalue problem
polyfit	Polynomial curve fitting
polyint	Integrates the polynomial analytically
polyval	Polynomial evaluation
roots	Polynomial roots

Table A-7 Special Functions

Function	Description
erf	Error function
erfc	Complementary error function
erfcinv	Inverse complementary error function
erfcx	Scaled complementary error function
erfinv	Inverse error function

Table A-8 Cartesian, Polar, and Spherical Functions

Function	Description
cart2pol	Transforms Cartesian coordinates to polar or cylindrical coordinates
cart2sph	Transforms Cartesian coordinates to spherical coordinates
pol2cart	Transforms polar or cylindrical coordinates to Cartesian coordinates
sph2cart	Transforms spherical coordinates to Cartesian coordinates

Table A-9 Constants and Test Matrix Functions

Function	Description
eps	Floating-point relative accuracy
Inf	Infinity
pi	Ratio of circle’s circumference to its diameter
NaN	Not-a-Number
isfinite	Array elements that are finite
isinf	Array elements that are infinite
isnan	Array elements that are NaN
gallery	Test matrices
magic	Magic square

Table A-10 Matrix Operation Functions

Function	Description
cross	Vectors cross product
dot	Vectors dot product
kron	Kronecker tensor product
transpose	Transposes

Table A-11 Linear Equation Functions

Function	Description
inv	Matrix inverse
linsolve	Solves linear system of equations

Table A-12 Eigenvalue Functions

Function	Description
eig	Eigenvalues and eigenvectors
eigs	Largest eigenvalues and eigenvectors of matrix
sqrtm	Matrix square root

Table A-13 Matrix Analysis Functions

Function	Description
det	Matrix determinant
norm	Vector and matrix norms
rank	Rank of matrix
rref	Reduced row echelon form
trace	Sum of diagonal elements

Table A-14 Matrix Functions

Function	Description
expm	Matrix exponential
logm	Matrix logarithm
sqrtm	Matrix square root
arrayfun	Applies function to each element of array

Table A-15 Statistical Functions

Function	Description
corrcoef	Correlation coefficients
cov	Covariance matrix
max	Largest elements in array
mean	Average or mean value of array
median	Median value of array
min	Smallest elements in array
mode	Most frequent values in array
std	Standard deviation
var	Variance

Table A-16 Random Number Generator

Function	Description
rng	Controls random number generation
rand	Uniformly distributed pseudo-random numbers
randn	Normally distributed pseudo-random numbers
randi	Uniformly distributed pseudo-random integers

Table A-17 1-D Interpolation

Function	Description
interp1	1-D data interpolation (table lookup)
Spline	Cubic spline data interpolation

Table A-18 Gridded Data Interpolation

Function	Description
interp2	2-D data interpolation (table lookup)
interp3	3-D data interpolation (table lookup)
interpn	nD data interpolation (table lookup)
griddedInterpolant	Interpolant for gridded data
ndgrid	Rectangular grid in nD space
meshgrid	Rectangular grid in 2D and 3D space

Table A-19 Scattered Data Interpolation

Function	Description
griddata	Interpolates scattered data

Table A-20 Optimization Functions

Function	Description
fminbnd	Finds minimum of single-variable function on fixed intervals
fminsearch	Finds minimum of unconstrained multivariable function using derivative-free method
fzero	Finds root of continuous function of one variable

Table A-21 Ordinary Differential Equation Functions

Function	Description
ode23	Solve nonstiff differential equations; low-order method

Table A-22 Sparse Matrix Manipulation Functions

Function	Description
spy	Visualizes a sparsity pattern
find	Finds indices and values of nonzero elements

Table A-23 Elementary Polygons

Function	Description
polyarea	Area of polygon
inpolygon	Points inside polygonal region
rectint	Rectangle intersection area

Appendix B

MATLAB’s Plotting Routines

This appendix lists the plotting routines in MATLAB with a brief description and example. To save space, some entries show a single encompassing example for multiple commands. An example may also use an ellipsis (...) to show the continuation of a pattern. To assist you, we have included several functions to generate test matrices, for example, rand(), magic(), peaks(), cylinder(), ellipsoid(), and sphere().

Table B-1 Basic Plotting Routines

Routine	Description	Example
ezplot	Plots the expression enclosed in quotes	ezplot('exp(-0.4x)sin(x)')
fplot	Same as ezplot, but requires limits	fplot('exp(-0.4x)sin(x)',[0,2*pi,-0.3,0.6])
plot	Plots data passed in by vectors	x=[0:2pi/100:2pi]; y=exp(-0.4x).sin(x); plot(x,y); figure(2) comet(x,y); figure(3) ribbon(x,y);figure(4) y2=100exp(-0.4x).*cos(x); plotyy(x,y,x,y2)
comet	Just like plot, but comet animates the trajectory. It helps to have a larger vector to slow down the comet trace a tad
ribbon	Like plot, but it displays the data as 3D ribbons
plotyy	Plots data where y values may differ greatly — it makes two y-axes
loglog	Plots data on a log scale on both x- and y-axes — y proportional to a power of x is straight on this plot	x=[0:2pi/100:2pi]; y=10*x.^pi; loglog(x,y)
semilogx	X-axis data is on a log scale and the y-axis is on a linear scale — y linearly related to log(x) is straight on this plot	x=[0:2pi/100:2pi]; y=10*log(x)+pi; semilogx(x,y);
semilogy	Y-axis data is on a log scale and x-axis linear — y proportional to exponential of x is straight on this plot	x=[0:2pi/100:2pi]; y=10exp(pix); semilogy(x,y);

Table B-2 Beyond Basics

Routine	Description	Example
area	Acts just like plot except for the fact that it fills in the area for you	x=[0:2pi/100:2pi]; y=exp(-0.4x).sin(x); area(x,y)
pie	Creates a standard pie chart	x=[2,4,6,8]; pie(x); figure(2) pie3(x)
pie3	Adds some 3D pizazz to pie
ezpolar	Creates a polar plot where the distance from origin vs. angle is plotted — argument is function expression	ezpolar('cos(2*x)^2');
polar	Creates a polar plot like ezpolar but does so by accepting vector arguments — x value corresponds to angle and y value to distance from origin	x=[0:2pi/100:2pi]; y=(exp(-0.1x).sin(x)).^2; polar(x,y)
compass	Like polar, but compass shows data as vectors from origin	compass(rand(1,3)-0.5,... rand(1,3)-0.5)
bar	Creates a standard bar chart that can handle both grouping and stacking	x=[8,7,6;13,21,15;32,27,32]; bar(x); figure(2) y=sum(x,2); bar(y); figure(3) bar(x,'stacked'); figure(4) barh(x); figure(5) bar3(x); figure(6) bar3h(x)
barh	Just like bar except plot is horizontal
bar3	Adds a little 3D pizazz to a standard bar chart
bar3h	Creates a 3D horizontal bar chart
fill	Fills polygons (vector inputs define vertices) with the specified color	y=sin([0:2pi/5:2pi]) x=cos([0:2pi/5:2pi]) fill(x,y,'g')

Table B-3 Statistical Plotting Routines

Routine	Description	Example
scatter	Plots (x,y) data points.	x=[0:2pi/100:2pi]; y=10x+pi+10randn(1,101); scatter(x,y); figure(2) stem(x,y); figure(3) errorbar(x,y,10*ones(1,101),ones(1,101)) figure(4); hist(y); figure(5) histogram(y) histc(y,[-40:20:80]) histcounts(y,5)
stem	Like scatter, but adds a line from the x axis to a data point.
errorbar	Like scatter but adds error bars.
hist or histogram	Creates a histogram — a bar chart showing the frequency of occurrence of data vs. value. These two routines handle the bins differently.
histc	Related to hist, but rather than making a plot, it makes a vector of counts.	histc(y,[-40:20:80]) example with histc category
histcounts	Like histc except it creates n bins.	histcounts(y,5) example with histcounts category
stairs	Like scatter, but makes stairsteps when y values change.	x=[0:2pi/10:2pi]; y=10x+pi+10randn(1,11); stairs(x,y)
rose	A cross between polar and histogram; it displays frequency vs. angle.	rose(randn(1,100),5)
pareto	A bar chart arranged with the highest bars first.	histc(randn(1,100),[-4:1:4]) pareto(ans)
spy	A scatter plot of zeros in a matrix.	mymat=rand(5); mymat=(mymat>0.5).*mymat; spy(mymat)
plotmatrix	A scatter plot of all permutations of columns of x and y.	plotmatrix(magic(3),magic(3))

Table B-4 3D Graphics

Routine	Description	Example
ezcontour	Makes a contour plot like a topographic map	ezcontour('cos(x)*cos(y)')
ezcontourf	Same as ezcontour except it fills in the area between contours	ezcontourf('cos(x)*cos(y)')
ezmesh	Creates a 3D perspective plot with an open wireframe mesh	ezmesh('cos(x)*cos(y)')
ezsurf	Creates a 3D perspective plot with a filled surface	ezsurf('cos(x)*cos(y)')
ezmeshc	Combines a contour plot with mesh	ezmeshc('cos(x)*cos(y)')
ezsurfc	Combines a contour plot with a 3D perspective filled surface plot	ezsurfc('cos(x)*cos(y)')
ezplot3	Plots a curve in 3D space; capable of being animated	ezplot3('sin(x)','cos(x)','sin(3x/2)',... [-2pi,2*pi],'animate')
plot3	Like ezplot3, except it takes vector arguments	a=[-2pi:4pi/100:2pi]; x=sin(a); y=cos(a); z=sin(3a/2); plot3(x,y,z); figure(2) scatter3(x,y,z)
scatter3	Like plot3, but shows individual points
stem3	Stem plot of 3D data	stem3(rand(5))
contour	Creates a contour plot with matrix arguments	x=[-2pi:4pi/100:2pi]; y=[-2pi:4pi/100:2pi];z=cos(x)'*cos(y); contour(x,y,z);title('contour');figure(2) contourf(x,y,z);title('contourf');figure(3) contour3(x,y,z);title('contour3');figure(4) surf(x,y,z);title('surf');figure(5) surface(x,y,z);title('surface');figure(6) mesh(x,y,z);title('mesh');figure(7) waterfall(x,y,z);title('waterfall');figure(8) surfc(x,y,z);title('surfc');figure(9) meshc(x,y,z);title('meshc');figure(10) meshz(x,y,z);title('meshz');figure(11) surfl(x,y,z);title('surfl');figure(12) pcolor(z);title('surfl')
contourf	Same as contour, except contourf fills in the contours
contour3	Same as contour, except contour3 provides a 3D perspective
surf or surface	Creates a filled surface
mesh	Creates a wireframe mesh surface
waterfall	Like mesh, but all column lines are omitted
surfc	Same as surf, but with contour plot added
meshc	Wireframe mesh with contour
meshz	Wireframe mesh with a curtain around the plot
surfl	Same as surf, except surfl simulates light and shadow
pcolor	Shows values of the matrix as colors
surfnorm	Creates surface plot with normal vectors	[x,y,z]=peaks; % Test function surfnorm(x,y,z)
fill3	Like fill except in 3D; note that the fill area may not be coplanar	fill3([0,1,1,0],[0,0,1,0],[0,1,0,1],'g')

Table B-5 Vector Fields

Routine	Description	Example
feather	Like compass except it steps once for each element in x and y	feather(rand(1,3)-0.5,... rand(1,3)-0.5)
quiver	Works like feather except quiver plots vectors in an x-y plane	[x,y]=meshgrid([-5:5],[-5:5]); u=ones(11)+(4./(sqrt(x.^2+y.^2)).cos(atan2(y,x))); v=(4./(sqrt(x.^2+y.^2)).sin(atan2(y,x))); v(6,6)=0; u(6,6)=0; quiver(x,y,u,v); hold on streamline(x,y,u,v,[-5,-5,-5],[-1,.01,1]); hold off, figure(2)
streamline	Plots line from the vector field; the example plots 2D streamline on the same plot as quiver
quiver3	Like quiver for 3D; the example adds a uniform velocity field to a 1/r² velocity field	[x,y,z]=meshgrid([-5:2:3],[-3:2:3],[-3:2:3]); r=sqrt(x.^2+y.^2+z.^2); u=ones(4,5,4)+(10./r.^2).cos(atan2(y,x)).sin(acos(z./r)); v=(10./r.^2).sin(atan2(y,x)).sin(acos(z./r)); w=10.*z./r; quiver3(x,y,z,u,v,w);hold on; streamribbon(x,y,z,u,v,w,-5,0,.1);figure(2) coneplot(x,y,z,u,v,w,x,y,z);figure(3) quiver3(x,y,z,u,v,w);hold on; streamtube(x,y,z,u,v,w,-5,0,.1);
streamribbon	Like streamline, but shows ribbons
coneplot	Like quiver3, but coneplot shows velocity as cones
streamtube	Like streamline, but plots tubes (cylindrical 3D flow lines)

MATLAB® For Dummies®

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MATLAB Functions

MATLAB’s Plotting Routines