Introduction to Numerical Analysis. Marek Kręglewski
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1 Iroduo o Numerl Alss Mre Kręglews
2 Course oe Wee Wee Wee Wee Wee 5 Wee 6 Wee 7 Wee 8 Wee 9 Wee Wee Wee Wee Wee Wee 5 Soluos o oler equos oe vrle: e seo lgorm. Derel lulus. Te Newo-Rpso meod, e se meod. Iegrl lulus. Numerl egro: rpezodl rule d Smpso s rule. Tlor epso error o umerl meod Numerl dereo: orwrd d wrd-deree ormul. Tree-po ormul o umerl dereo. Il vlue-prolem or derel equos: Euler s meod, e Ruge-Ku meods. Te Rrdso s erpolo. Il vlue-prolem or derel equos: Euler s meod, e Ruge-Ku meods. Poloml erpolo: Newo d Lgrge polomls. Meods or solvg ler ssems: ler ssems o equos, Crmer s rule, Guss elmo. Appromo eor: les-squres ppromo. Ler lger, mr verso d e deerm o mr. Roud-o errors: solue error, relve error, sg dgs.
3 Course oe LABORATORY CLASSES. MS Eel geerl roduo. Applo o e MS Eel solvg umerl prolems MANUALS:. E. Seer, Mems or emss, Oord.. A. Rlso, Iroduo o umerl lss.
4 Soluo o equo oe vrle =() START READ, ε, A = = =½(+A/) NO - < ε YES WRITE STOP Tre o operos
5 Algorm oo START d STOP o sequel lgorm INPUT d OUTPUT operos = SUBSTITUTION operos LOOP? CONDITIONAL opero SUBSTITUTION vrle = epresso Clule e vlue o e epresso d sve uder e me o e vrle
6 Coverge proess: =½(+/) L R Iero proess 6
7 Dverge proess: =6-* E E E E E E+7 -.7E L R
8 Soluo o equo oe vrle Bseo meod Soluo o equo ()=,.e. ser or zero pos o e uo (). Ser or e zero po e rge <,>, w: ) e uo () s ouous ) () ges e sg e rge <,>,.e. ()*()< zero po p p p p
9 Bseo Algorm START READ,, ε ()*()< YES p=(+)/ ()*(p)< YES =p NO NO WRITE: orre rge =p Tre o operos NO - <ε YES WRITE, STOP
10 Soluo o equo oe vrle Newo-Rpso meod Te ser o zero po egs po, : ) e uo () d s rs dervve re ouous ) e rs dervve s dere rom zero zero po Te epso Tlor seres:!
11 Newo-Rpso lgorm START READ, ε = = - ( ) / ( ) NO - < ε YES WRITE STOP Tre o operos
12 Soluo o equo oe vrle Se Meod Te ser or e zero po egs rom pr o pos(, ), : ) e uo () s ouous ) ( ) ( ), we zero po Te rs dervve rom e Newo-Rpso meod ppromed w epresso:
13 Se meod lgorm START READ,, ε q =( ) q =( ) = ; = q =q ; q =( ) NO = q ( - ) /(q -q ) - < ε YES WRITE STOP Tre o operos
14 Derel Clulus Dervve o uo mesure ow rpdl e depede vrle ges w ges o e depede vrle =() Tge le (α) (slope) = - α = - lm lm d d dervve
15 Derel Clulus Fd e dervve o e uo = Le = - d = ( )-( ) = ( ) -( ) = ( +) -( ) = [( ) + +() ]-( ) = = [ +() ] Aer dvdg I e lm s (.e. ) d d lm Te dervve o e uo = s d/d=
16 Derel Clulus Dervves o some elemer uos ( s os): Fuo =() Dervve d/d= () - l() l() / s() os() os() -s() Le () d z() re derele uos o : d d z d dz d z d dz d z d d d d d d dz d d z d d / z d d dz z d d z Compose uo (u()) d d d du du * d
17 Iegrl Clulus prpl s Te dervve F() o () s e uo su df()/d=() Te dee egrl s e sme g s e dervve uo A dee egrl s e lm o sum o erms () d F F
18 Iegrl Clulus - emples A r moves w os velo v()=5 m/. Clule e dse overs ours. s s v( ) 5m/ * m v d 5d 5 5* 5* m A soe s llg w e elero g() = m/s. A e egg s velo s m/s. Clule e dse e soe overs ewee d d seod o e ll. v v s v g( ) d d os d os d 5 5* 5* 8 6m
19 Numerl egro T T d m m m T T T m * * T T T m m m *, m m T T T T Trpezodl rule T m
20 Numerl egro S d m S S S m m m m *, / m m m m S S S S / Smpso s rule S m/ m mus e eve
21 All egro emple I ( ) d ()= d 68 ()= d ,
22 Numerl egro emple I ( ) d () T ( ) T ( ) S( ) 78 *78 78 * Clulo resuls T(=) T(=) S(=) ,67 I (ure) , ()= ()= T( ) T( ) S( ) 76 * * T() T()-I Errors o e rpezodl rule error T() T()-I ,6 9,6
23 Geomer seres /* S S S S S S r r We = ) Te sum s equl o ) s seres epso o e uo
24 Mlur Seres ) (,,, oss 6 ) "( ) ( d d d d Tus ()! "()! ()!!!! ()! "() ()
25 Tlor Seres ) (,,, oss 6 ) "( ) ( d d d d Tus ) (! ) "(! ) (!!!! ) (! ) "( ) (
26 Mlur Seres - emple e, Clule e vlue (6) usg e Mlur seres e " e e * Cll e Tlor seres
27 Numerl dereo ) ( ) ( lm ) ( ) ( lm Oe-sde ppromo: L R ) ( ) ( ) ( ) ( - + (+) () (-) Te verge o R L (erl dervve): L R ) ( ) ( Deo o dervve
28 Dereo e error!!!!!!! /:!!! dervve error! : /!! dervve error Oe-sde dervve Cerl dervve _
29 Clulo o dervve Clule e dervve o l() e po = usg e erl dervve meod d oe-sde meod or dere sep leg : ()=l() l()=/ l()=.986 ()=[(+)-(-)]/(*) () () error ^ error/^ ()=[(+)-()]/ + (+) () error error/ Te deresg sep mmzes e error. Te resuls re dere or dere meods
30 Derel equo s order Derel equo or rdove de Suggesed soluo: Ceg e orreess: Susuo: Le sde equl o rg sde, we: dn d N N dn d e N e e e e Cos deermed rom e l odo: Fl l soluo: rdove de os N e N N N N N e
31 Rdove de Derel equo or e rdove de dn d N Al soluo: Hl-le perod : N N N e e N e l N l N l
32 Derel equo e Euler meod Te equo ( s ow uo): Approme epresso or e dervve: Aer rsormo: Smpled oo: d d d d, d, d d d Te ls epresso llows or sep sep lulo o e uo (). Te vlue o e uo e zero sep deermed rom l odo.
33 Derel equo s order dn N d N dn/d Nl N Nl.5.5
34 Derel equo d order Hrmo osllo F p = m - elero F w = - - poso Assumpo: m= = Te le o ores: F p = F w Tus =- " d d Spe soluos: e e e e e e e Geerl soluo: e e Coss deermed rom l odos F
35 Derel equo d order e e - Il odos: e e e e Geerl soluo ludg e l odos: e e os
36 Numerl soluo I d d v d d d dv were : Suesve pplo o pprome epressos or e rs dervve: v v v v v v v v Noo: v v v From e equo resuls:
37 Numerl soluo I o. For = : m v m s () () v() () Oe-sde epresso or e dervve v () v() ()
38 Numerl soluo II d d v d d d dv were : Approme epressos or e erl-dervves: v v v v v v v v Noo: v v v From e equo resuls:
39 Numerl soluo II o. For = : m v m v v s () () v(+/) () Cerl dervve v () v(+/) ()
40 Rrdso s erpolo We lulg e umerl resul w e sep, s possle o esme e resul e lm? F p r O r p F() e resul or e sep = F() poel resul or = p e order o e umerl error Le s lule e umerl resul F or wo dere sep legs (q) F F F F p r O p q q O r q p p p q q /* q p
41 Rrdso s erpolo o. r O q q F F F p s error o ger order d e proess e oued. Te mos reque sep ge q=, d e: r O F F F p p p p p p p p p q q F F q q q F q q F q Susruo o o equos
42 Rrdso s erpolo emple I d 68 Numerl resuls w e rpezod meod: T() T T T
43 Rrdso s erpolo emple ()=l() l()=/ ()=[(+)-(-)]/(*) F() / Te erl dervve meod error, us p=. = F()-F()
44 Te erpolo poloml Te uo () s gve s le,.e. s vlues re ow (+) pos (odes) ( ), ( ), ( ),, ( ). Prolem: d poloml o e - order su s: w( )= ( ) w( )= ( ) w( )= ( ) w () s lled e erpolo poloml. Gols o e erpolo: smple preseo o e uo vlues (oees) eeuo o meml operos usg e poloml deermo o e ermede vlues o e uo
45 Clulo o e poloml vlue Nurl orm o e poloml w Te Horer s seme o e lulo w
46 Clulo o e poloml vlue Algorm START red, { }, w= =- w=w*+ =- YES NO wre w STOP
47 Tre o e lulo w ()=+- + = = = =- = Clule e vlue o e poloml =. w *-= *+= *+= - Te vlue = s.
48 Newo orm o e poloml Le,,,, - re gve umers, were e vlues o poloml re gve (e d). Aulr polooml p (=,,,,) re deed p () = p () = - p () = (- )(- ) p ()= (- )(- ) (- - ) Te poloml w () s gve s w p How o deerme e oees?
49 Deermo o e oees () [ l, l+ ] [ l, l+, l+ ] ( ) ( ) ( ) ( ),,,,,,,,,,,,,,
50 Emple )) ) ( )( ( )( ( w w p p p p () [, ] [,.., ] [,, ] = = 8 = 58 =
51 Ler erpolo ler uo: w ()= + ( ) = = + (/ ) ( ) = = + (/ ) Clule, - = =( - )/( - ) - = =( )/( - ) w ()= [( )/( - )] + [( - )/( - )] w ()= [( + )/( - )] + [( - )/( - )] w ()= + [( - )/( - )] (- ) I s Newo poloml w () = p () + p (), were p () = = p () = - = ( - )/( - )
52 Te Ruge ee We erpolg w poloml o g order, eg. o e - order or e uo e rge[-,] or eqds odes ( ) 5 = - + *, =,,,, () w() E E
53 Te Ruge ee Compre e epressos or e uo (dr le)d poloml (gre le):
54 Clulo ur Error soures: Errors o pu d Roudg errors Cug errors Smplo o model Rdom errors Asolue d relve erros: Approme vlue E vlue Asolue error Relve error r
55 Roudg d ug roudg ug,97,,9 -,97 -, -,9 Roudg o deml dgs Error o e umer ½ - Emple:, ½ - =,,5 Roudg o umers edg w dg 5?,5,,5, Error reduo we lulg sum
56 Errors o luled ques Addo d Suro,,,, m,,,,,8 m,,,,,7,75,5 W s e error o sum? W s e error o deree? m,,,,,9 m,,,,,8,89,5
57 Errors o luled ques Addo d Suro Smlrl: Te solue error o sum or deree s equl o e sum o solue errors o ompoes.
58 Reduo o sg dgs %,,,,,,576,576,576,576 r Asolue error Relve error
59 Errors o luled ques Mulplo d dvso r r r r r r r r r r r r r r Smlrl: r r r r Te relve error o produ or dvso s equl o e sum o relve errors o e ors.
60 Use o dere rules or e error rser Clule e roos o e lger qudr equo w e ur o 5 sg dgs ,98 8 7,98,8 Ol sg dgs 8 7,98 55,98 5 sg dgs r,5,8 r,5 55,98 9 6
61 Use o dere rules or e error rser Te Vee s relos ,98,5,786,5,786 55,98 55,98 7,98 8 7, r r
62 Mml errors o luled ques r,,,,,, Te uo s gve All vrles re gve w errors. W s e error o luled qu?
63 Mml error emple 6%,6,,, r,,,
64 Sdrd errors o omple epressos,,,,,,,,, s s s s s s s s s A gve uo Te s re sdrd errors o prmeers. W s e sdrd error o omple vlue?
65 Emple,, s s s s s s s,,, s s s
66 Ler regresso 5 5 (, ) 5 (, ) 6 8 Ler regresso: =*+ Gol: Deermo o opmum vlues o d. 66
67 Ler regresso Bs ssumpos: ) Rdom dsruo o roud e srg le ) Te vro σ depede o Les squres meod:, Deermo o m Φ(,) w respe o d :,, 67
68 68 Ler regresso Soluo o e equos ssem w respe o, :
69 Ler regresso Esmo o vre or : s Esmo o vre or prmeers d : s s Ler orrelo oee r r ov vr s, S vr S S s Te vlue o r sps rom - o +. r> des posve orrelo, r< egve orrelo ewee d. r= des e l o ler orrelo ewee d. 69
70 Ler regresso emple
71 [m] [g] * * -*- (-*-)^ -sr -sr Sum: = -.6 g/m = g s^=.5 s=.7 g s^=. s=.55 g/m s^=. s=.66 g sr= 5 ov(,)= -6.8 sr= - vr()= 8. vr()= 69.6 r(,)=
72 More ou regresso - qudrs IV III I II μ Qudrs: μ I -μ < -μ < (-μ )(-μ )> II -μ > -μ < (-μ )(-μ )< III -μ > -μ > (-μ )(-μ )> IV -μ < -μ > (-μ )(-μ )< ov(, ) ov, 7
73 r ov vr Ler regresso oee, S vr S S r=- -<r< r= <r< r= 7
74 Eess ssem o ler equos J J ε ε ε J T T We ser or soluo, were e vlue o T s mml. J J J J J J J J J ε ε J J J J J J J J J J J T T T T T T T T T T T T T T T T T T T T T T Te opl vlues o prmeers w mmze e sum o squres
75 Emple o e mr represeo 9,5 5, * de J J J J J J J J T T T T
76 Vre o e vrle 8 J Vre s, 5,,9 9,5,,7 ε T ε,,9,,7 ε s,,9,,7, 9 Vres d ovre 5,8 s ov(, ) ov(, ) s s, J T J,9* Ler regresso oee r ov(, ) 58,6*8,, 9 s s
77 Jo Te model uo e ler regresso = * +. Jo s mr o dervves over prmeers, ll pos o d =,,, J We g e d o e poloml o e d order = + * + *, e e Jo es orm o: J
78 Deovoluo o omple d Epermel d Te d sould e epressed s sum o Guss urves P e - eg - poso - wd 78
79 Te les squres meod { }, =:M, M ed prmeers Te error uo (sum over pos): Φ{ } = j [ j (ep) - j ({ }] Prolem To mmze Φ roug modo o { } usg e srg vlues o prmeers { } 79
80 Te error uo d Jo 8 e P e P e P N P P e P Deomposo over N ds Elemes o e Jo
81 Algorm 8 P P P P P P P P J Y Correos o e vlues o prmeers { } J T Y J T J
82 Te les squres meod Kro Psmo rozłożoe słdowe
83 Mres
84 Solvg se o ler equos A A A A A Emple + + = + + = + + = 8 A A 8 A =- = =
85 Mres d geomerl rsormos verso P mrror releo P roo os s s os P φ Trsormo mres re orogol Q Q Q Q Q T Q Q Q Q Q T os s s os os s s os os s s os Q Q Q Q Q T
86 Smlr rsormo o mres Mppg A, rsorms : A I veors d re rsormed o veors roug mppg Q, w s mppg o veor o veor? Q I d, Q A AQ Q I e mr Q s o sgulr, us Q Q Q Q AQ AQ B Te mres A B re wo mres rsormed roug smlr rsormo B Q AQ
87 Emple os s s os Q φ=-5
88 Emple (, )=(,) (, )=(,-) += -=- -5 (, )= (, )=,,
89 Crers equo λ slr, A() I() K() K = A λi rers mr o mr A dek = K(λ) = de(a - λi)=a - λi= rers equo K(λ) = λ + - λ λ λ + = Te roos o e poloml K(λ): λ, λ,, λ -, λ re lled egevlues o mr A. I B = Q - AQ, e rers equo or B K = B λi = Q - AQ - Q - IQ = Q - (A - I)Q, d e deerm dek =B - λi= Q - A - λi Q = A - λi= Two mres reled o e oer roug smlr rsormo ve e sme se o egevlues.
90 Egevlues, de, de I I A B A B
91 Dgol mr d d d d d d d d d d d d d d d d,,,, I D I D D C we rsorm gve qudr mr A o dgol mr D roug smlr rsormo?
92 Dgolzo s os s os s os s os s os s os s os s os s os s os s os s os s os s os s os s os os s s os os s s os os s s os os s s os AQ Q A Q Now we se e odgol elemes o zero: 8 os s s os Aer e rsormo:, s os s os
93 Egevlues d egeveors C - AC s smlr rsormo w dgolzes e mr A. Colums o e mr C o egeveors. I e mr C s orogol, C - =C T, d C - AC = C T AC. os C s 8 s 8 os 8 8 os s 8 8 s os 8 8 Mulplo o e mr A o o sdes e egeveor produes respeve egevlue: os 8 os s os s 8 os s os os os s s os s os s 8 I geerl: 8 8 T A ,,, 8 s s 8 8
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