Optimization

314 Appendix A

acotd Inverse cotangent, result in degrees

acoth Inverse hyperbolic cotangent

hypot Square root of sum of squares

Exponential

exp Exponential

expm1 Compute exp(x) – 1 accurately

log Natural logarithm

log1p Compute log(1 + x) accurately

log10 Common (base 10) logarithm

log2 Base 2 logarithm and dissect oating point number

pow2 Base 2 power and scale oating point number

sqrt Square root

nthroot Real nth root of real numbers

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

isreal True for real array

Rounding and Remainder

x Round toward zero

oor Round toward minus innity

ceil Round toward plus innity

round Round toward nearest integer

mod Modulus (signed remainder after division)

rem Remainder after division

sign Signum

MATLAB also provides values of useful constants.

pi 3.14159265…

i Imaginary unit

j Same as i

eps Floating-point relative precision

realmin Smallest oating-point number

realmax Largest oating-point number

Inf Innity

NaN Not-a-number

315Appendix A

A.4 Matrix Operations

Examples of zeros, ones, and rand functions are given below.

>> Y = zeros(3,2)

Y =

0 0

0 0

0 0

>> X = ones(2,3)

X =

1 1 1

1 1 1

>> Z = rand(2)

Z =

0.1656 0.2630

0.6020 0.6541

The rand function generates a uniformly generated random number

between 0 and 1.

The matrix A can be saved in the same directory, for a later use, by the

command:

>> save –ascii aa X

Sometimes it is necessary to clear all variables and functions from the

command window. This is done with the command

>> clear all

Now if A is punched in the command prompt it results in an error.

>> A

??? Undened function or variable ‘A’.

To get back the saved value of matrix A, use the load command

>> A=load('aa')

A =

1 2 3

4 -1 -2

5 6 7

To know about a function name, use help from the menu or simply type

help functionnane in the command. For example,

>> help clc

316 Appendix A

CLC Clear command window.

CLC clears the command window and homes the cursor.

If one is not able to recollect the function name, use the lookfor command.

For example, to get the name of absolute function:

>> lookfor absolute

abs Absolute value

genelowvallter Filters genes with low absolute expression levels

imabsdiff Absolute difference of two images

meanabs Mean of absolute elements of a matrix or matrices

sumabs Sum of absolute elements of a matrix or matrices

mae Mean absolute error performance function

sae Sum absolute error performance function

dmae Mean absolute error performance derivative function

circlepick Pick bad triangles using an absolute tolerance

mad Mean/median absolute deviation

The concatenation of the matrices is shown with the following example.

>> A = [1 2 3; 4 -1 -2; 5 6 7]

A =

1 2 3

4 -1 -2

5 6 7

>> B = [8;9;10]

B =

8

9

10

>> Z = [A B]

Z =

1 2 3 8

4 -1 -2 9

5 6 7 10

Suppose we want to delete the second column of the Z matrix. This can be

done by

>> Z(:,2)=[]

Z =

1 3 8

4 -2 9

5 7 10

317Appendix A

The inverse of the square matrix can be computed by

>> inv(Z)

ans =

-0.3517 0.1102 0.1822

0.0212 -0.1271 0.0975

0.1610 0.0339 -0.0593

The eigenvalues of the square matrix are computed by

>> eig(Z)

ans =

17.1878

-2.3534

-5.8344

Some of the array operators are

+ Addition

– Subtraction

.* Element-by-element multiplication

./ Element-by-element division

.\ Element-by-element left division

.^ Element-by-element power

.' Unconjugated array transpose

For example,

>> U = [1 2 3]

U =

1 2 3

>> V = [-1 -2 -3]

V =

-1 -2 -3

>> U.*V

ans =

-1 -4 -9

The display of numbers is controlled by the format command. Typical

commands are

format short

format long

The previous command can be brought back into the command prompt

using the key ↑.

318 Appendix A

A.5 Plotting

If x and y are two vectors then plot(x, y) makes a graph. For example, con-

sider the following example.

>> x = 0:0.01:2*pi;

>> y = cos(x);

>> y1 = sin(x);

>> plot(x,y,'--',x,y1,'r:')

>> xlabel('0 \leq x \leq 2\pi')

>> ylabel('Sine and Cosine functions')

>> legend('cos(x)', 'sin(x)')

>> title('Multiple Plots')

Figure A.2 is displayed on the desktop and can be edited using the gure

menu.

The following example demonstrates how to make a contour plot (Figure

A.3).

>> [X,Y] = meshgrid(-2:.01:2,-2:.01:3);

>> Z = X.^2+Y.^2;

>> v=[1;2;3;4;5;6;7;8;9;10;11;12];

>> [c,h] = contour(X,Y,Z,v); clabel(c,h);

Figure A.2

Multiple plots.