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

Formattazione del testo

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.=) convergence the fixed pointof20 methodGlobal convergence.If Il1 (¢ ' 1EM <max ✗][ ee✗ a , solution * [ ] * (ho] ¢ *li )e =✗✗ ✗• a ,snethodthe convenga• ! lei '| M' * ( a)• bE✗ ✗ --Local method21 Newtonof "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, <✗ ✗. - of mehhodquadrati Newton22 'sconvergence. simple rootsfor -1-0I(f' *1=0f-] (*[ *cif- e ✗✗ ✗e a , such that7 08 > :e,if I"I *' 8<✗ ✗- =(lei // *' |'/ • +1 * « ✗✗c✗ _✗- smethodNewtonof forLocal23 convergence. mortsmultiple"" [ 1f. ]eC >for soneE ma , '" (' *") )f- =/") 0* ( *( f'such f-( * )that f- O* = ✗=✗ ✗ =✗✗if "7s I*I S0 : <> ✗✗ - methoolthe convengasteepest methodDefinition descent24 . ¢the ofminimum' for" initial gness✗ ') là'la ' )'&(-11 ¢ le O✗ya✗✗ >= - ,thatsuchWheal ya : )appmoximated'"'

'( ("( ¢¢ le))giga ✗ cony→ ya× - of descent methodConvergence steepest25 . unique minimum *inwith]¢ C' [ e ✗c- ana ., ¢O'that I '( ( K l/)such IK 0 ) E> - ×× yy -f-If K le ✗ o{ Kara =7 sufficiente small*I ' 1'°✗ ✗- linearesnethodThe convengolonge forhe andfor (i. sufficiente 0,1sama € )ee . | lei' | '*)le| |-11 *⇐ ✗(✗ ✗ ✗_- ALGEBRALINEARQUESTIONSstheoremGerschgomin70 '. AX of¥ ( si 7 suchindexeigenrralne thatis✗n :u|il [ la istE- ai j i=/PROOF : )( AÌ XÌXÈ witheigenrrchor associated =thatsuchChoose inde i : )Ìelement ofbiggest(jlI Iil = imax ✗✗ ÈÈeigenzertortheconsidera = ,¥I lzjlil 1 j1 «=z , XÈequation AÌ givestheof =throwiThe :-✗

§ Zig= aij [ d✗ = ai 2- e.g.j g-a- .ii j #i71 fachomizationSchur. T triangolareand] unitariaAmatriceK :A UTUT= the of eigenralnesofdiagonaleElements T A→on definite mahmiusof positiveEigenzalues74 . is POSITIVE DEFINITEmatrixsimmetrieA EIGENit VALUESPOSITIVEhasPROOF : diagonalizable )isAassume int( T -A-: -Atsimmetrie A Tt^T -: = =_ xtttnt' A✗ × = ×txt( ntx= È ?( 0ti became) all a>Tx o>= . definite75 positivesimmetrieof2 mamm-. matmices definitepositiveandsimmetrieA )( (A)11HA = max o,PROOF : unitaria ti OUNA >U U= ,,Il È Il 11 UÌUN 11max mai=. ,11 Il IlIl 1 1=✗ =✗, < È= TIE tizi,dimia=of invertiteforSVD82 Gisborne squame. matrice ¥ inrcmtibleèusts matrixuniqueSVD segnare .Singular values Aofeigewoalues→ti .PROOF :inventarleA À A inventarleAtandA andpositive definite )leonine . ..Eigenrralnes Eigenvalnesof AT A of AAT= 0>Set ti Vand that ATA= matrixti diagonale :VTÀAV EA= =U

sone < rp .ÀHA ¥BIIKe Il A- BE what B)with- =peÀ closetis the tomatrix A inI-mobeniusnsmm.PROOF : IlIta HUEVTHB B=- -e eIl VT ÙB Ha= { -Il [ UTBV Il= - re ÈÈ ÈCHE CIÌHA la (A)= +- ci- , ÈSmollett value of this uhmeepression C =À È TU ✓= dq e .. .with tmolenius90 SVD norme. definition89 thebeethoven for ..fÈÈÀIIIl A- =ePROOF : UÈVTÀ Il UEVTIl /Il leA- = _+ È=p IlE - rfÈÈ= de.q . .END OFALGEBRALINEARQUESTIONSeliminazionefsmwamdDefinition91 . )inter( liiLmatrixtriangolare -1-0Lower of bSolution L :=✗bn1) × =, ln ÌÉ2) ¥las lela✗ ✗=a - sl.ae substitutionboulevardDefinition92 . )V. In interni) (materiatriangolareUpper -1-0✗ n ;of bSolution L :=✗la1) ✗ =a tl nn Èoaiuas2) ¥ lela✗ ✗=a - sNL fece factomizatisnbased LUDefinition93 solution on. A invertibileb withA +✗ )(LUA 1=0lii Mii= ,1) salve forLy e= y2)