Riassunti digitali (apple pencil) di tutte le domande di teoria dell'esame Numerical Methods for Engineering

Questions Numerical Methods

Definition 1 absolute errore

| I{ ✗= ✗-abs apparsa.

Definition 2 relative emmor

I I✗✗E - apparsa= .mee ✗ court

Definition 3 operation

f. )mlfm.nlalgomithrn f- )ok )=: y . . . . ..È comprate0C arithmetica f-# tooperations= ii O= (unknown 0C )is is0ttsually gin ) used0C =,

Definition 4 floating point numbers

1) am.pe( Q✗ = aria- , , .. .Tessawhore signs number: ofm significant: dig its0 1= ,P basis expo nent pof: e :usnal.ly )(E 0 0c- < >= e emin max,

5 Prosperity of floats

like commutativoihypropertiesarithmeticusual are!walidnecessarieNOT

Definition 6 machine epsilon / accuracy

^ mp -Em = 2Relative floating apprsximatisnF. in pointerrore][✗ e Tmaxr- min ,Ix Il(float 4 E- ✗ mII ✗Proof : Pe'( 1) 0assume amaanae •=✗ - mai. . . . ..,obhained(If )float moundingis la :× ×floatI pe/) "( - 7,49E 7× - × →2 9 61 9→,nII pe -7✗ ¥ pi e-e, Èpe mI Rootkit -pi e-e× - =IN 2zmachine

8 Nate epsilonon in floating point

mepresentablepossiblesmallest number Eon"pe 1 =/ Em=appresa -. byrelativeunarzidable causissmallestEm error→ pointfloating arithmetic

Definition 9 method iterativa

Sequence : )' le'' ^lei "' )le( iterativa-( bele ) -+1 :✗✗ ✗✗ g= , .. ,., functionitematisn:g'' '' ° " iniziale✗✗ → guesses, ,.> .

Definition 10 method iterativa of convergence

f- ( ) 0problem =: ×Convergencelocal : *Iterativa solutiontheconvergente tomehhod is loosely ✗initialtif such I*'" sufficiente simall' " isIthatguess × ✗ ✗-'•' *line ✗ = ✗le → a Convergenceglobal : Intervalin tothe [ ]econvergentiis globallymehhodIterativa a ,[* ' ]finitasolution " liifthe c-✗ aguess ✗ ,' *leilion ✗✗ =fa → a

Definition 11 bisection method

(a)f- f-that b )such]° Of- ([( e <C- a , ÷'"' "' li" li =✗== aa

1) ' "f-) )lei "' f-" (f- la )ofcheck siga ×,, fsandsolution

2) then 0ofIf =areElsa ''"' "If ( )f- 0)f- la "' '"If (f-)f- )( 0< : e✗ < :×alle )+1 allei alle )-11 ( lei= ✗=) ( leile -11( ) (f leile -11( (µ✗= =) )filealle -113)

-111)'• + +✗ = 2Repeat

Definition 12 convergence of bisection method

* that f-such (hobisechionsnehhod *) -0The ✗ -✗convengaif : ÷lei' |*I E✗ ✗_Proof : )"that each Interval ' (in *] *"' 7 0[Given f-e : =✗a ✗, ,* "'point "'container7 in [only ]all e✗ome a , .lilei * | "' ' lei'[ ]/ a-b.E✗ a =✗_ , le2

Definition 13 Newton snehhod

]c' [ le(fLinoleum ) e a: × , ]' ['Initial e° c-guest ✗: a ,Newton ' smethod FÈlei')( lett 70le✗ ✗= _ ' lei )( 'f ✗

Definition 14 equation of nonlinear uof residuale

f-' solution of'• 1×1=0appmoximation theof✗ ' • ' I Methoditerativi( )I f- the Theis Ofresi dual✗

Definition 15 chomdmehhool

][ liiniziale inof f-' 1×1=0'° aguest✗ ,)•'(f-'•' ( a)8 ✗b✗ = -- )fla( )f- b -leilei( (la fa() 0-11 G 7✗ += ✗✗

Definition 16 smart methsd

'' ]'^' [° ec-✗ ✗ a ,, lei )') (f-leig lelei(' ( ( -1 ) ✗✗✗ -= ✗- ))'• a-( ( "' 'f- f✗ ✗-)( le +1 ( lei falei(g 71✗ = ✗ + ✗

Definition 17 Newton method modified

' )] iniziale' [ (° e0 guessE > e✗ a, , methodModified Newton :'"lei )f-8 ' ( ×E✗ = - 5'f- •lei ('( fE)+ ✗✗ -( )le ' lei-11 leiS '✗ fa= ✗ + 70✗

Definition 18 fixed point method

¢ ' ['] ]° e] [:[ ee c-✗→ aaa , , , functioniterativa¢¢ )Emelin ( × :: =×methodpointFixed :'tetti fa 71¢ lei( ' )= ✗✗

19 Local fired of method point convergence fixed

[point andin ]* etoIf has ✗ aa ,Ilif (' 1*I ¢ <✗ method.esfired pointthethatsuch7s convergenti0> ' I Sif 'I *° <✗✗ -Proof :' ^ ][q ( esince c- a. * I ¢I le87 'for 4)thatS MI0 such 1> < <✗× -'If 'I I*° 8< :✗ ✗- Lagrange*I ¢' ' /' ' ) )(*I "( °I^ MI |'¢ g*E <✗=✗ ✗ - Pointmid×✗- ✗- 0remTHE8'•I I' and*Iterativa <✗ ✗-, ( ) 'MI|leiI '•' leI ^¢ '" )* |¢ |*(- - *E= ✗✗✗ ✗_ ✗- ✗-€ < -. " Ma' 'I °M S* |E <✗ ✗-maM 1 ÷< O.=)

20 convergence the fixed point of method Global convergence

If Il1 (¢ ' 1EM <max ✗][ ee✗ a , solution * [ ] * (ho] ¢ *li )e =✗✗ ✗• a ,snethodthe convenga• ! lei '| M' * ( a)• bE✗ ✗ --

21 Local method Newton of convergence

s. simple motsfor ) Of (*C' I *[] ][ liIf f- eand =E ✗:✗ e aa , , )'f- ( 0* =/✗snethod isNewtonthe convergenti'thatsuch7 S 0 a> ' I Sif 'I *° <✗✗ -Proof : pointfired withNewton smethod method' → È¢ )( = × -× ('f- )×' ( )) IN " (f- f-¢ ×× = >)( )('f × ][(f- e)linee and0(f-andf- *' ( c-) 0* a.=# ✗✗ |H' kit7s S1 Ifor Ix: *= < <✗-Local convergence pointfixedof method :if 'I ° ' ÈI*concierge, <✗ ✗. -

22 Newton of mehhod quadrati convergence

simple rootsfor -1-0I(f' *1=0f-] (*[ *cif- e ✗✗ ✗e a , such that7 08 > :e,if I"I *' 8<✗ ✗- =(lei // *' |'/ • +1 * « ✗✗c✗ _✗-

23 smethod Newton of for Local convergence

mortsmultiple"" [ 1f. ]eC >for soneE ma , '" (' *") )f- =/") 0* ( *( f'such f-( * )that f- O* = ✗=✗ ✗ =✗✗if "7s I*I S0 : <> ✗✗ - methoolthe convenga

Definition 24 steepest descent method

¢the ofminimum' for" initial gness✗ ') là'la ' )'&(-11 ¢ le O✗ya✗✗ >= - ,thatsuchWheal ya : )appmoximated'"' '( ("( ¢¢ le))giga ✗ cony→ ya× -

25 Convergence of descent method steepest

unique minimum *inwith]¢ C' [ e ✗c- ana ., ¢O'that I '( ( K l/)such IK 0 ) E> - ×× yy -f-If K le <y convergence, sheepest descenttheofLinear

26 Convergence

method unique minimum *inwith]¢ C' [ e ✗c- ana .,¢ )* 0(" >✗ o{ Kara =7 sufficiente small*I ' 1'°✗ ✗- linearesnethodThe convengolonge forhe andfor (i. sufficiente 0,1sama € )ee . | lei' | '*)le| |-11 *⇐ ✗(✗ ✗ ✗_-

Algebra Linear Questions

70 Gerschgomin theorem

stheoremGerschgomin70 '. AX of¥ ( si 7 suchindexeigenrralne thatis✗n :u:|il [ la istE- ai j i=/Proof : )( AÌ XÌXÈ w