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Some Fundamental definitions of Error Analysis:

Absolute and Relative Errors:

Absolute Error: Suppose that tx and ax denote the true and approximate values of a

datum then the error incurred on approximating tx by ax is given by

at xxerrore

and the absolute error ae i.e. magnitude of the error is given by

ata xxe

Relative Error: Relative Error or normalized error re in representing a true

datum tx by an approximate value ax is defined by

t

a

t

at

rx

x

x

xx

valuetrue

errorabsolutee

1

and

.100% rr ee

Example:

Given the number is approximated using n = 5 decimal digits (ie. 3.1415).

Determine the relative error due to chopping and express it as a per cent.

05-9293481e9.265358971415.3 errore

05-2150871e2.9492553605)/-9293481e9.26535897(/)1415.3( re

or 150871%294925536200.003-2150871e2.94925536(%) re

Norms for Vectors and Matrices

Norms for vectors:

Let T

nxxxx ),,,( 21 be a vector.

On a vector space V , a norm is a function from V to be the set of nonnegative

real that obeys those three postulates.

0x

if 0x , Vx

2

xx if R , Vx

yxyx if Vyx , (the ‘triangle inequality’)

We can think of x as the length or magnitude of the vector. There are some

important examples:

1l -norm

n

i

ixx1

1

2l -norm

21

1

2

2)(

n

i

ixx

pl -norm 1p

pn

i

p

ipxx 1

1

)(

l -norm

ini

xx

1max

Example1. Find the 1-, 2- and infinity norm for )3,2,1(Tu .

Solution: We have

6321|||| 1 u

14321|||| 222

2 u

3|||| u

Example2. Let nT Ru )1,....,1,1( be the vector of all 1. Find the 1-, 2- and infinity

norm for u .

Solution: We have

n 1...1|||| 1u

n 222

2 1...11||||u

1|||| u

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Example 3. Let nT n Ru ),....,2,1( , find the 1- and infinity norm foru .

Solution: We have

2

)1(|||| 1

nnu

n||||u

Matrix Norm:

The norm of a square matrix A is a non-negative real number denoted A . There are

several different ways of defining a matrix norm, but they all share the following

properties:

)( 1|I| matrix identity the for

)0 0 ( 0|A| AA iffwith

) ( scalars all forAA

BABA

BAAB

Given a particular vector norm , and matrix A, the norm of A is defined as follows:

}1,:sup{ uRuAuA n

There are some important examples:

}1,,sup{111 uRuAuA n

}1,,sup{222 uRuAuA n

}1,,sup{

uRuAuA n

)(2

AAA T Where )( AAT is the largest eigenvalue of AAT .

n

j

ijni

aA1

1max

n

i

ijnj

aA1

11max

The Frobenius norm of a matrix A , denoted F

A , is a matrix norm similar

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to the Euclidean norm that is defined to be the square root of the sum of the

absolute squares of the elements of )(: ijaA That is,

)(1 1

2

AAtraA Tm

i

n

j

ijF

( Tr means the trace of A.)

Definition: The trace of a square matrix is the sum of the entries on the main

diagonal.

n

i

iiatr1

)(A

Example 1: Compute the 1,-2-, ∞-, and Frobenious norms of the matrix

12

11A .

Solution: We have that

.3max{2,3}

A

.3max{3,2}1

A

2-norm of 2/1

2 )(|||| AAAA t

e.g.

12

11A Then

23

35

12

11

11

21AAT

Its characteristic equation is:

09)-)(2-(5 023

35det

This gives 0172 2

4572/1

The eigenvalues are: 851.61 and 1459.02

Therefore, .618.2851.6),max( 212 A

Frobenious norm: .6458.271211 2222 F

A

Example 2: Determine 1,-2-, ∞-, and Frobenious norms of the matrix

23

21A .

Solution: We have that

5

.5max{3,5}

A

.4max{4,4}1

A

2-norm of 2/1

2 )(|||| AAAA t

e.g.

23

21A Then

88

810

23

21

22

31AAT

Its characteristic equation is:

064)-)(8-(10 088

810det

This gives 016182 2

260182/1

Therefore, .1306.4),max( 212 A

Frobenious norm: .2426.418)2(32)1( 2222 F

A

Example 3: Determine 1,-2-, ∞-, and Frobenious norms of the matrix

211

121

011

A

Solution: We have that

.4

A

.41A

541

462

123

211

121

011

210

121

111

AAt

The eigenvalues are:

77 ,77 ,0 321

Therefore, 106.377)(2

AAA t

Frobenious norm: .7417.3F

A

Exercise 1:

Determine spectral radius, 1,-2-, ∞-, and Frobenious norms of the following matrix:

6

987

654

321

A

205

132

011

B

121

112

211

C .

Solution:

Exercise 2:

Let A be an mxn matrix:

nmRA

0001

0001

0

0001

0001

Determine 1,-2-, ∞-, and Frobenious norms of the matrix A.

Exercise 3:

Let A be an mxn matrix:

7

nmRA

0000

0000

0

0000

1111

Determine 1,-2-, ∞-, and Frobenious norms of the matrix A.

Matrix Eigenvalue:

Let A be an nxn matrix. An eigenvector is A is a nonzero vector xxA such that

for some scalar . A scalar is called an eigenvalue of A if there is a

nontrivial solution x of xxA . Such an x is called an eigenvector

corresponding to .

If xAx is an eigenvalue equation (and we assume that x is not a zero vector),

then

0I)-det(A 0I)x-(A xAx

This leads to a characteristic polynomial in : )det( IApA

is an eigenvalue of A only if 0Ap .

Spectral radius of an operator A is

||max)()(

AA

= ||max

1i

ni

( )(A is the set of all its eigenvalues of A ).

Example:

Let

52

12A be the matrix and we want to compute its eigenvalues. Its

characteristic equation is:

02)-)(5-(2 052

12det

This gives 01272 0)4)(3(

Therefore, A has two eigenvalues: 3 and 4.

Then the spectral radius of a A is .4)( A

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Inner Product

Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner

product on V is a function that associates a real number <u, v> with each pair of

vectors u and v and satisfies the following axioms.

〉,〈〉,〈 uvvu

〉,〈〉,〈〉,〈 wuvuwvu

〉,〈〉,〈 vuvu cc

0〉,〈 vv and 0〉,〈 vv if and only if 0v

Note:

. space for vectorproduct inner general,

, )for product inner Euclidean (productdot

V

nR

vu

vu

Associated Norm

〉,〈||||〉,〈|||| 2 uuuuuu

Let u and v be vectors in an inner product space V.

(1) Cauchy-Schwarz inequality:

vuvu ,

(2) Triangle inequality:

vuvu

The distance between two vectors u, v ∈ V is defined as:

vuvu ),(d

(3) Pythagorean theorem:

u and v are orthogonal if and only if

222 |||||||||||| vuvu

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Example: Let V = C[a, b] be the vector space of all continuous functions f : [a, b] →

R. For f, g ∈ C[a, b], define inner product:

b

a

dxxgxfgf )()(,

We now define the function versions of the same three norms we have just studied.

For functions f ∈ C[a, b] we define

b

a

dxxf2

2)(f

b

a

dxxf )(f 1

)(f max xfbxa

Exercise 4:

Let xxf )( and .2)( 2 xxxg (f, g ∈ C[-1, 1],)

1. Compute ., gf

Solution: We have

2. Compute norm .f

Solution: We have

3. Compute norm .g

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Solution: We have

4. Compute d(f, g).

Solution: We have

Exercise 5:

Let xxf )( and .2/)13()( 2 xxg (f, g ∈ C[-1, 1],)

Show that f and g are orthogonal.

Solution: We have to show that < f, g >= 0. We have

So, f ⊥ g.

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Condition Number

The condition number of a square matrix is the maximum possible error

magnification factor for solving bAx , over all right-hand sides.

Surprisingly, there is a compact formula for the condition number of a square matrix.

.)()( 1 AAAcondA

Example:

9615

654

431

A

2593.08889.16667.3

3704.05556.26667.4

0741.01111.03333.01A

20A

6667.81 A

3340.1736667.8*201 AA .

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Notes on using MATLAB

Formats for printing numbers.

format short 3.1416

format short e 3.1416e+00

format long 3.14159265358979

format long e 3.141592653589793e+00

Vectors and scalars are special cases. Matrices can be created as follows, A = [1, 1, 1,

1; 1, 2, 3, 4]. This creates a 2×4 matrix A whose first row is (1,1,1,1) and whose

second row is (1,2,3,4).

The dimensions of a matrix A can be found by typing size A.

To create a vector, type x=[1,2,3,4]. The commas are optional, x=[1 2 3 4] gives the

same result.

Thus, x=[0 .2 .4 .6 .8 1] can be created by typing x=0:.2:1.

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Built-in functions.

pi 3.1415

zeros(3,3) 3×3 matrix of zeros

eye(5) 5×5 identity matrix

ones(10) vector of length 10 with all entries =1

abs(x) absolute value

sqrt(x) square root, e.g. i=sqrt(-1)

real(z), imag(z) real, imaginary parts of a complex

number

conj(z) complex conjugate

atan2(y,x) polar angle of the complex number x + iy

sin(x), cos(x) trig functions

sinh(x), cosh(x) hyperbolic functions

exp(x) exponential function

log(x) natural logarithm

Example of a loop.

for i = 1:4 x(i) = i; end

Example of a conditional.

if a==0; x = a+1; elseif a < 0; x = a-1; else; x = a+1; end

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Plotting.

plot(x,y) linear plot, uses defaults limits, x and y are vectors

grid draw grid lines on graphics screen

title(’text’) prints a title for the plot

xlabel(’text’) prints a label for the x-axis

ylabel(’text’) prints a label for the y-axis

axis([0, 1, -2, 2]) overides default limits for plotting

hold on superimpose all subsequent plots

hold off turns off a previous hold on

clg clear graphics screen

mesh 3-d plot; type help mesh for details

contour contour plot; type help contour for details

subplot several plots in a window; type help subplot for details

Example: To plot a Gaussian function, type the following lines:

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Matrix functions.

x = A\b gives the solution of Ax=b

[v,d] = eig(A) eigenvalues in d, eigenvectors in v

inv(A) inverse of a square matrix

rank(A) matrix rank

*, + matrix product and sum

.*, .+ element by element product and sum

’ transpose, e.g. A’

ˆ power, e.g. A ˆ 2

. ˆ element by element power, e.g. A.ˆ 2

m-files. An m-file is a file that contains a sequence of MATLAB commands. Some

m-files are built into MATLAB. A user can create a new m-file using an editor. For

example, an m-file called fourier.m could be created containing the lines:

% % Plot a trigonometric function. %

x = 0:.01:1;

y=sin(2*pi*x);

plot(x,y)

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In this case, typing fourier would produce a plot of a sine curve. (Note: % in an

m-file denotes a comment line.)

In order to pass arguments to and from an m-file, the word “function” must be on the

first line. For example:

function [x,y] = fourier(n,xmax)

% % Plot a trigonometric function. %

x=0:.01:xmax;

y=sin(n*pi*x);

plot(x,y)

Typing [x,y] = fourier(2,7); plots a sine curve. After execution, the vectors x and y are

available for further calculations.

Useful commands:

type fft lists the contents of the m-file fft.m

save A stores a matrix in a file called A.mat

save saves all variables in a file called matlab.mat

load temp retrieves all the variables from file temp.mat

print prints the current graphics window