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Questions

Numerical

Methods

absolute

Definition

1 errore

. | I

{ ✗

= ✗

-

abs apparsa .

Definition relative

2 emmor

. I I

✗

✗

E - apparsa

= .

mee ✗ court

Definition operation

3. f. )

mlfm.nl

algomithrn f- )

ok )

=

: y . . . . .

.

È comprate

0C arithmetica f-

# to

operations

= i

i O

= (

unknown 0C )

is is

0

ttsually gin ) used

0C =

, floating point

4. Definition number

s

1) am.pe

( Q

✗ = aria

- , , .

. .

Tessa

whore SI GN

s NUMBER

: OF

m SIGNIFICANT

: DIG ITS

0 1

= ,

P BASIS EXPO NENT P

OF

: e :

usnal.ly )

(

E 0 0

c- < >

= e e

min max

,

5 Prosperity of floats

. like commutativo

ihy

properties

arithmetic

usual are

!

walid

necessarie

NOT epsilon

Definition machine /

6 accuracy

. ^ m

p -

Em = 2

Relative floating apprsximatisn

F. in point

errore

]

[

✗ e Tmax

r

- min ,

Ix Il

(

float 4 E

- ✗ m

I

I ✗

PRO OF : Pe

'

( 1) 0

assume ama

anae •

=

✗ - mai

. . . . .

.

,

obhained

(

If )

float mounding

is la :

× ×

float

I pe

/

) "

( - 7,49

E 7

× - × →

2 9 61 9

→

,

n

I

I pe -

7

✗ ¥ pi e

-

e

, È

pe m

I Rootkit -

pi e

-

e

× - =

IN 2

z

machine

8 Nate epsilon

on

. in floating point

mepresentable

possible

smallest number Eon

"

pe 1 =/ Em

=

appresa -

. by

relative

unarzidable causis

smallest

Em error

→ point

floating arithmetic

methssl

Definition iterativa

9 .

Sequence : )

' le

'

' ^

lei "

' )

le

( iterativa

-

( be

le ) -

+1 :

✗

✗ ✗

✗ g

= , .

. ,

.

, function

itematisn

:

g

'

' '

' ° " iniziale

✗

✗ → guesses

, ,

.

> . method

iterativa

of

Definition

10 convergence

. f- ( ) 0

problem =

: ×

CONVERGENCE

LOCAL : *

Iterativa solution

the

convergente to

mehhod is loosely ✗

initial

t

if such I

*

'" sufficiente simall

' " is

I

that

guess × ✗ ✗

-

'•' *

line ✗ = ✗

le → a CONVERGENCE

GLOBAL : Interval

in to

the [ ]

e

convergenti

is globally

mehhod

Iterativa a ,

[

* ' ]

finita

solution " li

if

the c-

✗ a

guess ✗ ,

' *

lei

lion ✗

✗ =

fa → a bisectionmethod

Definition

11 . (a)

f- f-

that b )

such

]

° O

f- (

[

( e <

C- a , ÷

'"

' "

' li

" li =

✗

=

= a

a

1) ' "

f-

) )

lei "

' f-

" (

f- la )

of

check siga ×

,

, fsand

solution

2) then 0

of

If =

are

Elsa '

'"

' "

If ( )

f- 0

)

f- la "

' '

"

If (

f-

)

f- )

( 0

< : e

✗ < :

×

alle )

+1 allei alle )

-11 ( lei

= ✗

=

) ( lei

le -11

( ) (

f lei

le -11

( (

µ

✗

= =

) )

file

alle -11

3) -11

1)

'• + +

✗ = 2

Repeat bisectionmethod

Definition of

12 convergence

. * that f-

such (

ho

bisechionsnehhod *

) -0

The ✗ -

✗

convenga

if : ÷

lei

' |

*

I E

✗ ✗

_

PROOF : )

"

that each Interval ' (

in *

] *

"' 7 0

[

Given f-

e : =

✗

a ✗

, ,

* "

'

point "

'

container

7 in [

only ]

all e

✗

ome a , .

li

lei * | "

' ' lei

'

[ ]

/ a

-

b.

E

✗ a =

✗

_ , le

2

Newton

Definition snehhod

13 . ]

c' [ le

(

f

Linoleum ) e a

: × , ]

' [

'

Initial e

° c-

guest ✗

: a ,

Newton ' smethod FÈ

lei

'

)

( lett 70

le

✗ ✗

= _ ' lei )

( '

f ✗ equation

Definition of nonlinearu

of

residuale

14 . f-

' solution of

'• 1×1=0

appmoximation the

of

✗ ' • ' I METHOD

ITERATIVI

( )

I f- the THE

is OF

RESI DUAL

✗ chomdmehhool

Definition

15 . ]

[ li

iniziale in

of f-

' 1×1=0

'

° a

guest

✗ ,

)

•

'

(

f-

'

•

' ( a)

8 ✗

b

✗ = -

- )

fla

( )

f- b -

lei

lei

( (

la fa

(

) 0

-11 G 7

✗ +

= ✗

✗ smart

Definition methsd

16 . '

' ]

'

^

' [

° e

c-

✗ ✗ a ,

, lei )

'

) (

f-

lei

g le

lei

(

' ( ( -1 ) ✗

✗

✗ -

= ✗

- )

)

'

• a-

( ( "

' '

f- f

✗ ✗

-

)

( le +1 ( lei fa

lei

(

g 71

✗ = ✗ + ✗ Newton method

modified

17 Definition

. ' )

] iniziale

' [ (

° e

0 guess

E > e

✗ a

, , method

Modified Newton :

'"

lei )

f-

8 ' ( ×

E

✗ = - 5

'

f- •

lei (

'

( f

E)

+ ✗

✗ -

( )

le ' lei

-11 lei

S '

✗ fa

= ✗ + 70

✗ point

fired method

18 Definition

.

¢ ' [

'

] ]

° e

] [

:[ e

e c-

✗

→ a

a

a , , , function

iterativa

¢

¢ )

Emelin ( × :

: =

×

method

point

Fixed :

'

tetti fa 71

¢ lei

( ' )

= ✗

✗ Local fired

19 of method

point

convergence

. fixed [

point and

in ]

* e

to

If has ✗ a

a ,

Il

if (

' 1

*

I ¢ <

✗ method.es

fired point

the

that

such

7s convergenti

0

> ' I S

if '

I *

° <

✗

✗ -

PROOF :

' ^ ]

[

q ( e

since c- a. * I ¢

I le

8

7 '

for 4)

that

S M

I

0 such 1

> < <

✗

× -

'

If '

I I

*

° 8

< :

✗ ✗

- LAGRANGE

*

I ¢

' ' /

' ' ) )

(

*

I "

( °

I

^ MI |

'

¢ g

*

E <

✗

=

✗ ✗ - POINT

MID

×

✗

- ✗

- 0rem

THE

8

'•

I I

' and

*

Iterativa <

✗ ✗

-

, ( ) '

MI

|

lei

I '•

' le

I ^

¢ '

" )

* |

¢ |

*

(

- - *

E

= ✗

✗

✗ ✗

_ ✗

- ✗

-

€ < -

. " Ma

' '

I °

M S

* |

E <

✗ ✗

-

ma

M 1 ÷

< O

.

=) convergence the fixed point

of

20 method

Global convergence

.

If Il

1 (

¢ ' 1

EM <

max ✗

]

[ e

e

✗ a , solution * [ ] * (

ho

] ¢ *

li )

e =

✗

✗ ✗

• a ,

snethod

the convenga

• ! lei '

| M

' * ( a)

• b

E

✗ ✗ -

-

Local method

21 Newton

of "

convergence s

. simple mots

for ) O

f (

*

C' I *

[

] ]

[ li

If f- e

and =

E ✗

:

✗ e a

a , , )

'

f- ( 0

* =/

✗

snethod is

Newton

the convergenti

'

that

such

7 S 0 a

> ' I S

if '

I *

° <

✗

✗ -

PROOF : point

fired with

Newton smethod method

' → È

¢ )

( = × -

× (

'

f- )

×

' ( )

) IN " (

f- f-

¢ ×

× = >

)

( )

(

'

f × ]

[

(

f- e

)

linee and

0

(

f-

and

f- *

' ( c-

) 0

* a.

=

# ✗

✗ |

H' kit

7s S

1 I

for Ix

: *

= < <

✗

-

LOCAL CONVERGENCE POINT

FIXED

OF METHOD :

if '

I ° ' È

I

*

concierge

, <

✗ ✗

. - of mehhod

quadrati Newton

22 's

convergence

. simple roots

for -1-0

I

(

f' *

1=0

f-

] (

*

[ *

ci

f- e ✗

✗ ✗

e a , such that

7 0

8 > :

e

,

if I

"

I *

' 8

<

✗ ✗

- =

(

lei /

/ *

' |

'

/ • +1 * « ✗

✗

c

✗ _

✗

- smethod

Newton

of for

Local

23 convergence

. morts

multiple

"

" [ 1

f. ]

e

C >

for sone

E m

a , '" (

' *

"

) )

f- =/

"

) 0

* ( *

( f

'

such f-

( * )

that f- O

* = ✗

=

✗ ✗ =

✗

✗

if "

7s I

*

I S

0 : <

> ✗

✗ - methool

the convenga

steepest method

Definition descent

24 . ¢

the of

minimum

' for

" initial gness

✗ '

) là

'

la ' )

'

&

(

-11 ¢ le O

✗

ya

✗

✗ >

= - ,

that

such

Wheal ya : )

appmoximated

'

"

' '

( (

"

( ¢

¢ le

)

)

giga ✗ con

y

→ ya

× - of descent method

Convergence steepest

25 . unique minimum *

in

with

]

¢ C' [ e ✗

c- an

a .

, ¢

O'

that I '

( ( K l

/

)

such I

K 0 ) E

> - ×

× y

y -

f-

If K le <

y convergence

, sheepest descent

the

of

Linear

26 Convergence

. method unique minimum *

in

with

]

¢ C' [ e ✗

c- an

a .

,

¢ )

* 0

(

" >

✗ o

{ Kara =

7 sufficiente small

*

I ' 1

'

°

✗ ✗

- lineare

snethod

The convengo

longe for

he and

for (

i. sufficiente 0,1

sama € )

e

e . | lei

' | '

*

)

le

| |

-11 *

⇐ ✗

(

✗ ✗ ✗

_

- ALGEBRA

LINEAR

QUESTIONS

stheorem

Gerschgomin

70 '

. A

X of

¥ ( si 7 such

index

eigenrralne that

is

✗

n :

u

:|

il [ la ist

E

- ai j i

=/

PROOF : )

( AÌ XÌ

X

È w