Iroduo o Numerl Alss Mre Kręglews
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.
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.
Soluo o equo oe vrle =() START READ, ε, A = = =½(+A/) NO - < ε YES WRITE STOP Tre o operos
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
Coverge proess: =½(+/) 8.5.5.5.5.69756.69756.9.9 6 L R Iero proess 6
Dverge proess: =6-*..59.59.79.79-6.5896-6.5896 -.65 -.65-9.55657-9.55657-875.576-875.576-7.595E+ -7.595E+ -5.7686E+ -5.7686E+ -.768E+7 -.768E+7 -.7E+95 8 6 - - L R -6-8 -
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
Bseo Algorm START READ,, ε ()*()< YES p=(+)/ ()*(p)< YES =p NO NO WRITE: orre rge =p Tre o operos NO - <ε YES WRITE, STOP
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:!
Newo-Rpso lgorm START READ, ε = = - ( ) / ( ) NO - < ε YES WRITE STOP Tre o operos
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:
Se meod lgorm START READ,, ε q =( ) q =( ) = ; = q =q ; q =( ) NO = q ( - ) /(q -q ) - < ε YES WRITE STOP Tre o operos
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
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=
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
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
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
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
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
All egro emple I ( ) d ()= d 68 ()= d 5 5 5 5 5 5 9766,
Numerl egro emple I ( ) d () 6 78 76 T ( ) T ( ) S( ) 78 *78 78 *78 68 695 Clulo resuls T(=) 78 76 T(=) 695 9 S(=) 68 9766,67 I (ure) 68 9766, ()= ()= T( ) T( ) S( ) 76 *6 76 76 *6 76 9766 9 T() T()-I 78 695 Errors o e rpezodl rule error T() T()-I 76 969,6 9,6
Geomer seres /* S S S S S S r r We = ) Te sum s equl o ) s seres epso o e uo
Mlur Seres ) (,,, oss 6 ) "( ) ( d d d d Tus ()! "()! ()!!!! ()! "() ()
Tlor Seres ) (,,, oss 6 ) "( ) ( d d d d Tus ) (! ) "(! ) (!!!! ) (! ) "( ) (
Mlur Seres - emple e, Clule e vlue (6) usg e Mlur seres e " e e * Cll e Tlor seres
Numerl dereo ) ( ) ( lm ) ( ) ( lm Oe-sde ppromo: L R ) ( ) ( ) ( ) ( - + (+) () (-) Te verge o R L (erl dervve): L R ) ( ) ( Deo o dervve
Dereo e error!!!!!!! /:!!! dervve error! : /!! dervve error Oe-sde dervve Cerl dervve _
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/^.869.657...697.5.5.576.67.9.5.556.5.969....57...5.9.67 ()=[(+)-()]/ + (+) () error error/.869.8768 -.565 -.565 Te deresg sep mmzes e error. Te resuls re dere or dere meods..5.5.576.8 -.5.5 -.56....7898 -.5. -.55
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
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
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.
Derel equo s order dn N d N dn/d Nl -5. 5-5 66.57. 5-5 67.879. 5-65.. 6.5 -.5 5.5 5.5.5-56.5 8.85 6.6 5.65-78.5 9.7877 7.7 7.85-9.65.978 8.8.965-9.5 8.56 9.9.955-9.7656.9.97656 -.888 6.7797..888 -..8677.. -.7.7875..7 -.65.59..65 -.58.988 5.5.58 -.559.558 6.6.559 -.769.56 7.7.769 -.85.68 8.8.85 -.97. 9.9.97 -.95.785.95 -.77.5 8 6 5.86. N Nl.5.5
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
Derel equo d order e e - Il odos: e e e e Geerl soluo ludg e l odos: e e os
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:
Numerl soluo I o. For = : m v m s () () v() () -.8997 -.9 -.6799.98865 -.6799 -.98865.9699.985958 -.956 -.98596.55988.89785 -.567 -.89785 5.65985.87 -.68 -.8 6.78598.7778 -.777 -.7778 7.96979.65 -.886 -.65 8.7976.5687 -.97 -.569 9.7897.97 -.995 -.97.89969.895 -.95 -.895.98966.5675 -.875 -.568.57796.985 -.77 -.98.7696 -.58 -.8558.579.85957 -.88 -.987.878 5.9695 -. -.59.6 6.995 -.566 -.9988.566 7.598 -.6975 -.958.6975 8.5695 -.888 -.88.8887 9.879 -.96 -.788.96.67999 -.75 -.6777.75.78896 -.976 -.755.976.87979 -.5979 -.99.5979.69 -.97 -.798.965.597 -.595 -.777.5955 - - - Oe-sde epresso or e dervve 5 5 - - - - v () v() ()
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:
Numerl soluo II o. For = : m v m v v s () () v(+/) () -.655 -.8997.996 -.958 -.99.6799.965877 -.66 -.965877.9699.977 -.58 -.977.55988.86588 -.5559 -.86588 5.65985.7968 -.6597 -.7968 6.78598.7679 -.75 -.7679 7.96979.68 -.886 -.68 8.7976.995 -.897 -.995 9.7897.897 -.976 -.895.89969.5795 -.98977 -.579.98966.959 -.99799 -.955.57796 -. -.99778.6.7696 -.7 -.9858.7.85957 -.685 -.969.685 5.9695 -.898 -.896.8987 6.995 -.597 -.86.5969 7.598 -.6 -.75758.65 8.5695 -.7898 -.658.78976 9.879 -.795 -.555.795.67999 -.86696 -.566.86696.78896 -.96 -.95.96.87979 -.96657 -.9.966569.69 -.997 -.67.9977.597 -.999997.6769.9999975 Cerl dervve.5.5 -.5 5 5 - -.5.5.5 -.5 - -.5 -.5.5.5 - -.5 v () v(+/) ()
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
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
Rrdso s erpolo emple I d 68 Numerl resuls w e rpezod meod: T() 78 695 T T 695 78 T 695 695 68
Rrdso s erpolo emple ()=l() l()=/ ()=[(+)-(-)]/(*) F() /.8.8.5.59..78857...775.5 -.87..6.9555...65.88 -.5.8.8.969....57 -...9.67 Te erl dervve meod error, us p=. = F()-F()
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
Clulo o e poloml vlue Nurl orm o e poloml w Te Horer s seme o e lulo w
Clulo o e poloml vlue Algorm START red, { }, w= =- w=w*+ =- YES NO wre w STOP
Tre o e lulo w ()=+- + = = = =- = Clule e vlue o e poloml =. w *-= *+= *+= - Te vlue = s.
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?
Deermo o e oees () [ l, l+ ] [ l, l+, l+ ] ( ) ( ) ( ) ( ),,,,,,,,,,,,,,
Emple 7 58 5 8 )) ) ( )( ( )( ( 7 5 5 w w p p p p () [, ] [,.., ] [,, ] 5 66 8 7 96 5 58 9 78 7 8 = = 8 = 58 =
Ler erpolo ler uo: w ()= + ( ) = = + (/ ) ( ) = = + (/ ) Clule, - = =( - )/( - ) - = =( )/( - ) w ()= [( )/( - )] + [( - )/( - )] w ()= [( + )/( - )] + [( - )/( - )] w ()= + [( - )/( - )] (- ) I s Newo poloml w () = p () + p (), were p () = = p () = - = ( - )/( - )
Te Ruge ee We erpolg w poloml o g order, eg. o e - order or e uo e rge[-,] or eqds odes ( ) 5 = - + *, =,,,, () w() -.86.86 -.8.588.8.588 -.6..588.68. -...5.759.79855. -..5.5.5.976.68665.5.5.5.8E-5 -.6767-6.6..5 -.5 -.5-5 -.5-7.575-7.675.5.. -.5.5 5 6.5 9.75.9 8.86..6. -.5.5 -.5E-5 -.5-9.75-56.5-8.8-67.96..8.588 -.588.759 -.98 -.6767 7.575.9 8.85 9.779.97.588.86 -.8.68 -.7986.68665 6.6-7.675-8.86-67.96 -.9 -.9.86
Te Ruge ee Compre e epressos or e uo (dr le)d poloml (gre le):.5.5.5 -.5 - -.5.5.5 -.5
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
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
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
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.
Reduo o sg dgs %,,,,,,576,576,576,576 r Asolue error Relve error
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.
Use o dere rules or e error rser Clule e roos o e lger qudr equo w e ur o 5 sg dgs. 8 8 78 78 7,98 8 7,98,8 Ol sg dgs 8 7,98 55,98 5 sg dgs r,5,8 r,5 55,98 9 6
Use o dere rules or e error rser Te Vee s relos 6 5 6 9 55,98,5,786,5,786 55,98 55,98 7,98 8 7,98 78 78 8 8 r r
Mml errors o luled ques r,,,,,, Te uo s gve All vrles re gve w errors. W s e error o luled qu?
Mml error emple 6%,6,,, r,,,
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?
Emple,, s s s s s s s,,, s s s
Ler regresso 5 5 (, ) 5 (, ) 6 8 Ler regresso: =*+ Gol: Deermo o opmum vlues o d. 66
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 Ler regresso Soluo o e equos ssem w respe o, :
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
Ler regresso emple -5 - -5 - -5 - -5-5 6 7 8 9 7
[m] [g] * * -*- (-*-)^ -sr -sr - - -..6-8 - 9 -.8.6-5 - 5-7 - 9 - -.8.6-9 -8 8 -..6-8 Sum: 5-65 -68..6 = -.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(,)= -.999 7
More ou regresso - qudrs -5 - -5 - -5 - -5-5 6 7 8 9 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
r ov vr Ler regresso oee, S vr S S r=- -<r< r= <r< r= 7
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
Emple o e mr represeo 9,5 5, 8 8 5 8 8 6 8 * de 8 6 8 6 8 6 5 8 8 8 8 8 8 8 8 8 J J J J J J J J T T T T
Vre o e vrle 8 J 6 5 8 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,6 8 8 58 8 8 J T J,9* 8 8 58 8 8 8 Ler regresso oee r ov(, ) 58,6*8,, 9 s s
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
Deovoluo o omple d Epermel d Te d sould e epressed s sum o Guss urves 5 5 5 5 5 5 P e - eg - poso - wd 78
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
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
Algorm 8 P P P P P P P P J Y Correos o e vlues o prmeers { } J T Y J T J
Te les squres meod 5 5 5 5 5 Kro Psmo rozłożoe słdowe 5 5 5 5 8
Mres
Solvg se o ler equos A A A A A Emple + + = + + = + + = 8 A A 8 A =- = =
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
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
Emple os s s os Q φ=-5
Emple (, )=(,) (, )=(,-) += -=- -5 (, )= (, )=,,
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.
Egevlues, de, de I I A B A B
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?
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
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 8 8 8 8 8 8 8,,, 8 s s 8 8