Transcript Chapter 3 Review

Chapter 3 Review: Built In
MATLAB Functions
Introduction to MATLAB 7
Engineering 161
Predefined MATLAB functions
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MATLAB is rich in predefined or built in
functions that you can use in your
expressions and calculations;
x = [16, 25, 9, 4];
y = sqrt ( x );
yields the vector y with values [4, 5, 3, 2]
sqrt is a built in MATLAB function that operates on
scalars and vectors and matrices.
Predefined MATLAB functions II
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Use the “help” command to learn how to use
a particular function >> help sqrt
Use “MATLAB Help” feature to see a list of all
MATLAB functions that are available and read
the documentation about each of them.
MATLAB functions are varied and powerful
and can be nested, sqrt(sin(abs(x)))
where x can be a matrix, vector or scalar.
Using MATLAB functions
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MATLAB functions come “built in” to MATLAB,
sqrt, sin, abs, max, exp, etc.
MATLAB functions come in specialized
“toolboxes” for special fields of work; the
financial toolbox, statistical toolbox, symbolic
math toolbox are examples.
And, you will be able to write your own
functions and use them just like built in ones.
Some Elementary Math and
Rounding Functions
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abs(x) absolute value
sqrt(x) square root
round(x) rounds to nearest
fix(x) truncates toward 0
floor(x) rounds down
ceil(x) rounds up
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sign(x) sign of x
rem(x,y) remainder of x/y
exp(x) e raised to x power
log(x) natural log of x
log10(x) log to the base 10
log2(x) log to the base 2
Some Trigonometric Functions
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sin(x)
cos(x)
tan(x)
asin(y)
atan(y)
computes the sine of x, x in radians,
computes the cosine of x, x in radians
computes the tangent of x, x in radians
computes the inverse sine, -1 < x < +1
computes the inverse tangent, i.e.,
computes the angle whose tangent is y
Notice that MATLAB wants angles to be expressed in radians, not
degrees. To convert use the relationship
1 degree = pi/180 radians
angle_radians = angle_degrees*(pi/180)
angle_degrees = angle_radians*(180/pi)
(sind(x) permits x to be in degrees)
Other Useful Elementary
Analysis Functions
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See our text for more detailed explanations; see
Appendix A for a list of some of the MATLAB
functions.
max(x), min(x), mean(x), median(x), std(x), sum(x),
prod(x), cumsum(x), cumprod(x), sort(x), size(x),
length(x) are just a few.
Note that x may be a vector (row or column) or a
matrix. For matrices, functions typically work on the
columns of the matrix to yield their results. (Always
best to use help feature to check and make sure.)
Functions with Multiple
Outputs
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Some functions return multiple results,
for example, consider the max function;
[a,b] = max(x) where x is a row vector
of arbitrary length, returns two values,
a will be the maximum value while b
will be the index in the row vector of
the maximum value
Functions with Multiple
Outputs
%
x = [2,4,6.5,-3, 7.9, 4.1];
[a,b] = max(x)
a=
7.9
b=
5
Functions used in Discrete
Mathematics
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factor(x)- finds the factors of x
gcd(x,y)- finds the greatest common denominator of
x and y
lcm(x,y)- finds the least common multiple of x and y
rats(x)- represents x as a fraction
factorial(x)- for x=4 implies factorial(x) = 24
primes(x)- finds all the prime numbers < x
isprime(x)- checks to see if x is a prime number,
function returns 1 if yes; 0 if no.
Random Number Generators I
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MATLAB contains two random number generators, rand(n,m)
for uniformly distributed random numbers and randn(n,m) for
normal or Gaussian distributed random numbers.
Random number sequences can be scaled for different mean
and standard deviation requirements.
Sequences of random numbers are often used in engineering
problem solving.
x = 10*rand(1,100) – 5; creates a row vector x with 100
random numbers uniformly distributed between -5 and 5.
Enter the above command followed by the command plot (x)
to see what happens.
Random Number Generators II
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rand(n)- nxn matrix uniformly distributed (0 to 1)
rand(m,n)- mxn matrix uniformly distributed (0 to 1)
randn(n)- nxn matrix Gaussian distributed
randn(mxn)- mxn matrix Gaussian distributed
mean (x)- computes mean value of a vector x
median(x)- finds the median of the elements of x
std(x)- computes the standard deviation of the
elements of x
var(x)- computes the variance of the elements of x
More on Matrices
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We’ve seen ways to define matrices;
A = [1,3,5] A = [1 3 5] are equivalent
t = -1:0.5:2 yields t = [-1, -0.5, 0, .5, 1, 1.5, 2]
B = [1,2,5;3,4,6] yields the 2X3 matrix of
two rows and three columns
x = linspace(-5,5,10) creates the vector x of
10 values equally spaced between -5 and 5
More on Matrices II
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Also consider
>> a = [3,5]
>> b = [1,a] creates the row vector
where b = [1,3,5]
>> b(2) is the scalar with value 3
>> a' is the column vector, 2 rows and
one column (we say “a transpose”)
More on the colon operator
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For M=[1,2,3,4,5;2,3,4,5,6;3,4,5,6,7]
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x = M(:,1) is the column vector, the 1st column of M
x = M(:,3) is a column vector, the 3rd column of M
y = M(2,:) is a row vector, the second row of M
w = M(2:3,:) is a 2 rows and 5 columns matrix
made up of the last two rows of M
w = M(2:3,4:5) is a 2 rows and 2 columns matrix
made up the elements, [5,6;6,7]
And don’t forget, z = M(2,4), then z = 5
Special Values
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MATLAB includes a number of predefined constants,
special values and special matrices.
Some examples include, pi, i and j {sqrt (-1)}, Inf,
NaN, clock, date, eps, and ans.
MATLAB allows the following statement, pi = 3, but
be careful, pi will take in the value of 3 not 3.14159..
until it is reset to the value of pi. Clearing the
Workspace reestablishes the value of pi.
Same is true for built in function names, avoid it.
Other Useful Functions
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abs(z)- computes the absolute value of a complex number
angle(z)- computes the angle in radians of a complex number
used in its polar representation
complex(x,y)- generates a complex number, you can also write
z = 5 + i*7
real(z)- extracts the real part of a complex number
imag(z)- extracts the imaginary component of a complex
number
isreal(z)- determines if z is real or not
conj(z)- generates the complex conjugate of z
Other Functions
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See the Table on pages 112 - 113 for a list of basic
functions described in Chp. 3. This is a very small
subset of MATLAB built in functions. Don’t forget to
use the Help feature to learn more about these
functions and to find functions that will help you
solve problems.
Also see Table 3.16 on page 110 for special MATLAB
functions and symbols.
Example; Problem 3.3
% Joe Mixsell
% Problem 3.3
% Calculate the number of rabbits after 10 years with a
% starting number of 100 and breeding at a rate of 90 percent/yr
R = 0.9;
% Breeding rate
P0 = 100;
% Initial population
t = 0:1:10;
% 10 years in increments of 1 year
P = P0*exp(R*t);
% Calculate population for years 0 to 10
disp (‘ Years Population’)
% Display column labels
[t’,P’]
% Output table of values
plot ( t, P)
xlabel (‘Years’), ylabel (‘ Annual Population’)
title (‘ Rabbit Population Calculation’)
Chapter 3 Assignments
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Assignments in Chapter 3 include
problems 3.5, 3.10, 3.22, 3.23.