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7stabiliteitsinterval~:GO7$,$FepF2Fgr-Fgo6#QhoDe~methode~is~een~BDF-methode~(k<
7)~en~heeft~een~oneindig~stabiliteitsgebied.F(F(F(F(F(,
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^blF0F0F0@$32-F_`m6#,&-F_`m6#&F\q6#F^blF0F2F0#F0",+++++"/-Fh]m6$/F[alFgfo-F`gnF
bboFgr>F_elF^\m@%F_el-Fb\o6#%9de~methode~is~nulstabielGC%F`do@%-Fedo6$FgfoFj^m-
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bbele~wortel~met~modulus~1GF(F(F(F(,
I(helpLMMf*F(F(F(F(C;-F]]m6#%cqAlle~methodes~worden~weergegeven~en~gebruikt~als
~2-dimensionale~matrix.~Als~de~methode~de~volgende~is:|+~G-F]]m6$/-FP6$*&&Fh^lF
[gnF0&F^_l6#,&FghlF0F<F0F0/F<;FgrFe^l*&F`dlF0-FP6$*&&FFF[gnF0&F__lFdioF0FfioF0%
!G-F]]m6#%Idan~wordt~de~methode~voorgesteld~door:|+~G-F]]m6#-Fhs6#-F\r6%F^rF`\l
7$7&&Fh^l6#Fgr&Fh^lFfu%"|irG&Fh^lFegl7&&FFF[[p&FFFfuF][p&FFFegl-F]]m6#%`wBespro
ken~procedures:~`AdamsBashforth`,~`AdamsMoulton`,~`MilneSimpson`,~`Nystrom`,~`B
DF`,~`Maximaal`,~`uitschrijven`,~`orde`,~`PLAF`,~`stabvt`,~`grens`,~`stabgebied
`,~`stabinterval`,~`nulstabiel`.|+G-F]]m6#%ctBij~de~constructieprocedures~(`Ada
msBashforth`,~`AdamsMoulton`,~`MilneSimpson`,~`Nystrom`~en~`BDF`)~is~de~enige~p
arameter~een~vergelijking~van~de~vorm:|+G-F]]m6$/%"pGF][p%Q(Geef~de~orde~van~de
~methode),~~~~~~~~~~~~~~~~~~G-F]]m6$/Fe^lF][p%L(Geef~het~aantal~stappen~van~de~
methode),~~G-F]]m6$/FIF][p%M(Geef~de~constructieparameter~(zie~cursus)).G-F]]m6
#%\dnAan~`Maximaal`~geef~je~|dy|dyn~parameter~mee:~een~positief~even~getal~->~h
et~stapgetal.~Als~output~krijg~je~telkens~de~gevraagde~methode~(in~matrixvorm).
~`uitschrijven`~geeft~de~gewone~vorm~van~de~methode~terug.~`orde`~en~`PLAF`~spr
eken~voor~zich,`stabvt`~geeft~de~stabiliteitsveelterm.~`grens`~bepaalt~een~anal
ytische~uitdrukking~voor~de~grens~van~het~stabiliteitsgebied~en~`stabgebied`~ge
eft~een~plot~van~dat~gebied,~waarbij~de~groene~punten~in~het~stabiliteitsgebied
~liggen~en~de~rode~erbuiten.~Als~input~geef~je~de~methode.~Bij~deze~laatste~is~
er~nog~een~optionele~parameter,~nl.~de~afstand~tot~de~randkromme~voor~punten~di
e~moeten~gecontroleerd~worden.~`randkromme`~geeft~een~analytische~uitdrukking~(
in~poolco|azrdinaten)~van~de~randkromme~van~het~stabiliteitsgebied,~`stabinterv
al`~geeft~het~stabiliteitsinterval~en~`nulstabiel`~controleert~of~een~methode~n
ulstabiel~is.~Als~input~geef~je~de~methode~als~matrix.|+G-Fb\o6#%H~~~~~AdamsBas
hforth:=proc(R::equation)|+G-Fb\o6#%F~~~~~AdamsMoulton:=proc(R::equation)|+G-Fb
\o6#%F~~~~~MilneSimpson:=proc(R::equation)|+G-Fb\o6#%A~~~~~Nystrom:=proc(R::equ
ation)|+G-Fb\o6#%=~~~~~BDF:=proc(R::equation)|+G-Fb\o6#%@~~~~~Maximaal:=proc(k:
:posint)|+G-Fb\o6#%B~~~~~orde:=proc(methode::matrix)|+G-Fb\o6#%J~~~~~uitschrijv
en:=proc(methode::matrix)|+G-Fb\o6#%B~~~~~PLAF:=proc(methode::matrix)|+G-Fb\o6#
%D~~~~~stabvt:=proc(methode::matrix)|+G-Fb\o6#%C~~~~~grens:=proc(methode::matri
x)|+G-Fb\o6#%hn~~~~~stabgebied:=proc(methode::matrix,GRENSWAARDE::realcons)|+G-
Fb\o6#%J~~~~~stabinterval:=proc(methode::matrix)|+G-Fb\o6#%H~~~~~nulstabiel:=pr
oc(methode::matrix)|+GO%%NULLGF(F(F(F(,
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F0F0FgrC&>F\p-F[`l6$FihpFibn>Fap-Fhel6$<#-Fidl6$/&F\p6$FepF0&-F`ip6$Far*(FNF0Fd
jpF0-F\qF^hmF0Fc[q/FepFihp<#-Fidl6$Fb[qFi[q-F[flFcalO7$-Fidl6$&FjipF`gl/Fep;F0,
&F3F0F2F07#F[\qO7$F`\qF^cnF(F(F(F(,
I+take_step2f*6&FhgpFdglFh_pFhep6%Fe^lF^_lFJF(F(C$>FarF^hp@%1F3Far-Fgo6#%Ouw~st
ap~is~kleiner~dan~het~aantal~beginwaardenGC%>Fgq-Feep6%F8F1FN?(FepF^[mF0F3Fbr>F
gq-Fegp6'FepF8F1FNFgqOFgqF(F(F(F(,
I$numf*6&'%#npGFf^l'F`dlFj^mFdglFh_p6'F^_lFghlFe^lF<FiamF(F(C%>Far-Fi\q6&F3F1FN
F8?(F\qF0F0-Figl6#&FNF]apFbr>&F\pF]jn-Ffjm6%7#-Fidl6$7$*&FJF0F8F0&&Far6#FJF]jn/
FJ;F0F3Fc[n/Febm-%&COLORG6&%$RGBG,$-%%randGF(#F0".++++++"F_`qF_`q-Fe\n6#<#-Fidl
6$F]_q/F\q;F0Fi^qF(F(F(F(,
I$LAFf*6%FdglFh_p'F^^nFf^l6%F^_lFe^lF%F(F(C&@%/FeoF`y>FepF1>Fep,&FdcnF0F`yF0>Fg
qFiil>Far-Fi\q6&FfjpF3F8.F`dlO-F]fm6$f*FfblF(F)F(-FA6%F3/F`dlFgrFiblF(F(6$F%Fep
-Fhs6#,&-F[`l6#7#-F]gm6#-Fh]m6$/Fgbl,&F]gpF0*&FgqF0FjaqF0F0F`fpF0&Far6#FfjpF2F(
F(F(F(,
I$GAFf*F_^q6&F^_lFfjnF<FiamF(F(C&>FarFf^q>Fgq-Fe_p6$FNFgbl?(FepF0F0Fi^qFbr>&F\q
F`gl-Ffjm6%7#-Fidl6$7$Fd_q,&-Fh]m6$/Fgbl,&&FNF\dpF0Fd_qF0&FgqF`glF0&&Far6#Fa`nF
`glF2/FJFd\qFc[nFj_q-Fe\n6#<#-Fidl6$F\dq/FepFj`qF(F(F(F(,
I4helpOP_PROBLEEM_LMMf*F(F(F(F(C6-F]]m6#%^qDezelfde~afspraken~i.v.m.~de~input~v
an~methodes~blijft~gelden~(typ~`helpLMM(~)`~voor~meer~uitleg).|+G-F]]m6#%eqBesp
roken~procedures:~`exact`,~`genereer_beginwaarden`,~`take_step`,~`take_step2`,~
`num`,~`LAF`~en~`GAF`.|+G-F]]m6#%hqDe~procedure~`exact`~berekent~de~exacte~oplo
ssing~van~een~lineair~probleem.~Als~het~probleem~hetvolgende~is:|+G-F]]m6#/-F`g
n6$-F^_lFfblFgbl*&%"AGF0FefqF0-F]]m6#/-F^_l6#&FgblF[[p&F^_lF[[p-F]]m6#%fp~dan~w
ordt~dit~voorgesteld~door~een~lijst~met~drie~argumenten,~nl.~`matrix`,~`x0`~en~
`y0`.|+G-Fb\o6#%S~~~~~probleem['matrix']:=matrix(m,n,[[|ir],|ir,[|ir]]);|+G-Fb\
o6#%9~~~~~probleem['x0']:=0;|+G-Fb\o6#%;~~~~~probleem['y0']:=[|ir];|+G-F]]m6#%[
uOpgelet:~zowel~de~quotes~rond~de~namen~`matrix`,~`x0`~en~`y0`,~als~de~namen~ze
lf~mogen~niet~veranderd~worden.~Om~`exact`~dan~toe~te~passen,~doe~je~hetvolgend
e:|+G-Fb\o6#%9~~~~~exact(probleem,x);|+G-F]]m6#%`ioDeze~`x`~is~de~variabele~waa
rin~men~`y`~zal~uitdrukken.~Belangrijk~om~weten~is~dat~de~volgende~procedures~e
nkel~werken~met~expliciete~methodes.~De~procedure~`genereer_beginwaarden`~berek
ent~het~aantal~(exacte)~beginwaarden~dat~nodig~is~om~de~methode~te~kunnen~toepa
ssen.~Als~input~geef~je~de~methode,~het~probleem~en~de~gewenste~`h`-waarde.~Men
~kan~ook~het~symbool~`h`~ingeven.~De~procedure~`take_step`~berekent~y[x0~+~(n-1
)~h]~a.d.h.v~een~lijst~[y[x0],~...~,y[x0~+~(n-2)~h]].~Als~input~geef~je~dus~de~
`n`,~de~methode,~het~probleem,~de~`h`-waarde~en~voorgenoemde~lijst.~De~procedur
e~`take_step2`~doet~hetzelfde,~maar~begint~van~nul.~Het~genereert~zelf~de~begin
waarden~en~berekent~de~gevraagde~waarden~tot~het~gewenste~punt:~`x0~+~(n-1)~h`.
~Men~moet~dus~enkel~de~`n`,~de~methode,~het~probleem~en~de~`h`-waarde~ingeven.~
`LAF`~berekent~een~reeksontwikkeling~van~de~lokale~afknottingsfout~in~`h`.~Men~
geeft~de~methode~en~het~probleem~mee,~en~optioneel~kan~nog~een~derde~parameter~
gebruikt~worden~om~zelf~de~orde~van~de~ontwikkeling~te~geven.~`num`~geeft~een~p
lot~van~de~numerieke~oplossing~en~`GAF`~een~plot~van~de~globale~afknottingsfout
.~Voor~beide~heeft~men~dezelfde~parameters:~het~aantal~te~berekenen~punten,~de~
`h`-waarde,~de~methode~en~het~probleem.|+G-Fb\o6#%?~~~~~exact:=proc~(probleem,x
)|+G-Fb\o6#%hn~~~~~genereer_beginwaarden:=proc(methode::matrix,probleem,h)|+G-F
b\o6#%hn~~~~~take_step:=proc(n::posint,methode::matrix,probleem,h,y)|+G-Fb\o6#%
gn~~~~~take_step2:=proc(n::posint,methode::matrix,probleem,h)|+G-Fb\o6#%T~~~~~L
AF:=proc(methode::matrix,probleem,t::posint)|+G-Fb\o6#%jn~~~~~num:=proc(n::posi
nt,h::realcons,methode::matrix,probleem)|+G-Fb\o6#%jn~~~~~GAF:=proc(n::posint,h
::realcons,methode::matrix,probleem)|+GOFd_pF(F(F(F(,
I$rhof*6#'F_oFg^m6#%#PIGF(F(C$>Far-FaanFg`m-Fe]m6#-%'tcoeffGF[]mF(F(F(F(,
I&sigmaf*F\jqF^jqF(F(C$>FarFbjq-Fe]m6#,$-%'lcoeffGF[]mF2F(F(F(F(,
I.zetstapgelijkf*6$'%"PGF\r'%"CGF\r6&%%PI_PG%%PI_CGF\\pFboF(F(C(>FarFg\m>Fgq-Fh
\mF``m>Fep-FaboFban>F\q-Fabo6$FgqF[al@&2F\qFep>Fgq*&)F[al,&FepF0F\qF2F0FgqF02Fe
pF\q>Far*&)F[al,&F\qF0FepF2F0FarF0-Fe]mFc_nF(F(F(F(,
I'simplef*6#'F_jq-Fg^m6$%)anythingGF[al6%F[oF<FeilF(F(C%>Far-F]gm6#-%'factorGF_
p@%-Fedo6$F[^r%"*GC$>FepF0?(FgqF0F0-F`cm6#7#FarFbr@$32Fgr-Fabo6$&FarF\tF[al0-%$
remG6%F\_rF[alF[alFgr>Fep*&FepF0F\_rF0>FepF3-Fe]m6#-Fdfm6%-Faan6%FepF[alF\^rFca
nFdanF(F(F(F(,
I)stabvtPCf*6#%'PCpaarG6-%%M_muGFeil%'beta_kG%"HGF_jq%"WG%#P_G%#C_G%$mu_G%#t_G%
#L_GF(F(C1>F[cl&F36#.Fd[r>F^bl&F36#.Ff[r>Ffdl&F36#.%#muG>F_el&F36#.F^^n>Ffel&F3
6#.F`o>Fep&F^bl6$F^r-FiglFhfo@$330FfelQ$LinF(0FfelQ"nF(0FfelQ%LoutF(-Fgo6#Q^oDe
~laatste~parameter~in~PCpaar~kan~enkel~"n",~"Lin"~of~"Lout"~zijn.F(@$3/FfdlFgrF
gbr-Fgo6#QfqMen~kan~enkel~lokale~extrapolatie~toepassen~als~men~minstens~|dy|dy
nmaal~de~corrector~~~~~~~~~~~~~~~~~~~toepastF(@$F`cr-Fe]m6#-Fh\mFecl>F\q*&FepF0
F`dlF0>Farf*FfblF(F)F(*(FhblF0,&F0F0F3F2F0,&F0F0FhblF2F2F(F(6$Fe`rFfdl>F\p-F`[r
6$F[clF^bl@$30F_elFgr0F_elF0-Fgo6#Q<~'t'~kan~enkel~0~of~1~zijn.F(@%/FfelFhbr@%/
F_elFgr>Fgq,&&F\pF_vF0*&-FarF]jnF0&F\pFfuF0F0>Fgq,&*(FepF0)F[al-Fbw6$,&-FiglFec
lF0F2F0,&FabrF0F2F0F0F`erF0F0*&-F^am6#FberF0,&*&-Fi^l6#FcerF0-Fgjq6#F`erF0F0*&-
Fi^lFefrF0-FgjqFcfrF0F2F0F0C$>Fap*&-FhcnFhfoF0,&-FhcnFeclF0F\grF2F2@&/FfelFfbr@
%F]er>Fgq,&*&FhfnF0F`erF0F0*&,&-Far6#,&F\qF0*&FapF0F\qF0F0F0FapF2F0FcerF0F0>Fgq
,&*(FepF0FgerF0,&FdgrF0*&FapF0FcerF0F2F0F0*&FggrF0F`frF0F0/FfelFjbr@%F]er>Fgq,&
FdgrF0*&,(FberF0FjgrF0FapF2F0FcerF0F0>Fgq,&F]hrF0*&,&FberF0FjgrF0F0F`frF0F0-Fe]
m6#-F_]rF\tF(F(F(F(,
I'ordePCf*F\`r6.%#p_GF\\p%%ORDEG%'ORDE_nG%$lafGFc`rFd`rFe`rFf`rFg`r%#PCG%&probl
GF(F(C.>FapFj`r>F[clF^ar>F^blFbar>FfdlFgar>F_elF[br@$330F_elFfbr0F_elFhbr0F_elF
jbrF[cr@$3/F^blFgrFajr-Fgo6#Q]qMen~kan~enkel~lokale~extrapolatie~toepassen~als~
men~minstens~|dy|dynmaal~de~corrector|+~~~~~~~~~toepastF(>Fgq-FailFecl>Far-Fail
Fcal@'51FgqFar32FarFgq2,&FgqF0FarF2F^bl>F\qFgq3Fa[s/FearFc[s>F\qFgq>F\q,&FarF0F
^blF0@'/F_elFhbr>FepF\q3/F_elFjbr/FgqFar>FepF_rC,>&FdblF]ap-&%&linalGF[r6#7#7#F
0>&FdblF\dpFgr>&FdblF_dpFi\s>&FfelF[arFap>&FfelF_arF[cl>&FfelFcarF^bl>&FfelFhar
Ffdl>&FfelF\brF_el>F\p&-%&LAFPCG6$FdblFfelFfu>Fep-%(ldegreeG6$-Fbap6$F\pFg^mF`d
l-Fe]mF`glF(F(F(F(,
I-stabgebiedPCf*6$F]`rFh^m69Fc\mF\_mFd\mF]_mF^_mF__mF`_mFa_mF<FJFe^lF%FghlFb_mF
c_mFd_mFe_mFf_mFg_mF'Fh_mFi_mFg`rF(F(C(>8:F[br@$330Fj^sFfbr0Fj^sFhbr0Fj^sFjbr-F
go6#QZDe~laatste~parameter~kan~enkel~"n",~"Lin"~of~"Lout"~zijn.F(@$3/Fe`rFgrF__
s-Fgo6#Q\sMen~kan~enkel~lokale~extrapolatie~toepassen~als~men~minstens~|dy|dynm
aal~de~corrector~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~toepastF(>Far-Fj_rF_p@
%F\`m>FgqF^`m>Fgq$Fc`mF2@%/-FaboF[]mF0Fb_nC.>FepFj\mFh`m?(FfdlF0F0-F`cm6#7#FepF
br>&F^blFhco-F^am6#-Fh]m6$Fbam&FepFhco>Fdjm-Fiam6&<#-Fidl6$7%-F]bm6#Fj`s-F_bmFh
asF`bm/Ffdl;F0Ff`sFc[n/%'symbolG%&crossGFdbm>FfelF0>F\pFg[o>FaemF[cm>FapF[cm?(F
(F0F0F(3/FfelF02FccmFgqC%>FdblF0?(FfdlFgrF0F\pFbr?(F_elF0F0Ff`sFbrC*>F\q-Fh]m6$
/Fgbl-F_dm6#*(FfdlF0Fc]mF0F\pF2&F^bl6#F_el>F[em-F_dm6#,&-F]bmF]jnF0*&F`^mF0-F_b
mF]jnF0F0>F^em,&F[emF0FgqF2>89,&F[emF0FgqF0>Fcdm-Fcem6%/-F_dm6#-Fh]m6$FghmFarFg
rF[alFiem>Fafm-F]fm6$F_`m7#Fcdm>Fjbm&-Fdfm6#FafmFffm@%2FjbmF0C$>Faem7%-F]gm6#Fa
emF^imFdim>FfelF`hmC(>Fap7%-F]gmFcalF^imFdim>Fdbl,&FdblF0F^rF0>Fcdm-Fcem6%/-F_d
m6#-Fh]m6$/F`dlF_dsFarFgrF[alFiem>FafmFjds>FjbmF^es@%FbesC$>Faem7%Ffes-F_dm6#7$
-F]bm6#F_ds-F_bmFbgs-F_dm6#7$Fags,$FcgsF2>FfelF`hmC$>Fap7%F\fsF^gsFdgs>FdblF^fs
>Fgq,$FgqF_jm@$0-F`cmFgesFgr>F[fm-Ffjm6%7#-Fidl6$&Faem6#F]cl/F]clF_\nFc[nFe[n>F
[cl-Ffjm6%7#-Fidl6$&FapFihs/F]cl;F0,&FdblF0F2F0Fc[nFa\n@%/FahsFgr-Fe\n6#<$F[clF
djm-Fe\n6#<%F[clF[fmFdjmF(F(F(F(,
I-randkrommePCf*F\`r6&F<Fc\mFd\mFg`rF(F(C)>F\qF[br@$330F\qFfbr0F\qFhbr0F\qFjbrF
a_s@$3Ff_sFfjs-Fgo6#QarMen~kan~enkel~lokale~extrapolatie~toepassen~als~men~mins
tens~|dy|dynmaal~de~corrector~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~toepastF(>FgqF[`s>Fe
p-Fhel6$FgqF`dlF\]m-Fe]m6#-Fidl6$-Fh]m6$Fj]mFdfp/FarF[bsF(F(F(F(,
I/stabintervalPCf*F\`r6+FbjnFcjnFdjnFJFejnFfjnFgjnFg`rFe`rF(F(C/>FfdlFbar>F^blF
[br@$330F^blFfbr0F^blFhbr0F^blFjbrF[cr@$3F`crFb\tFjjs>FarF[`s>F[clFb]o>FepFf]o>
F\qFgr>F\pFbr>FgqFgr?(F(F0F0F(2F[clFepC'?(F(F0F0F(F\pC%>F\qF_r>Fap-Fhel6$/-Fh]m
6$/F`dl,&FgqF2*&F\qF0FepF0F2FarFgrF[al@$2F0&-Fdfm6#-F]fm6$F_`m7#FapFffm>F\pF^\m
>Fgq,&FgqF0*&F`zF0FepF0F0>F\qFgr>F\pFbr>Fep,$FepF[`oF]`oO7$,$FgqF2FgrF(F(F(F(,
I'helpPCf*F(F(F(F(C6Fgeq-F]]m6#%[qBesproken~procedures:~`ordePC`,~`stabvtPC`,~`
stabgebiedPC`~,`randkrommePC`~en~`stabintervalPC`.|+G-F]]m6#%iuDe~procedure~`or
dePC`~berekent~de~orde~van~het~PC-paar.~Als~input~geef~je~een~lijst~met~5~argum
enten,~nl.~`P`,~`C`,~`mu`,~`t`~en~`L`.~Men~kan~dit~op~de~volgende~manier~doen:|
+G-Fb\o6#%H~~~~~PCpaar['P']:=AdamsBashforth(p=2);|+G-Fb\o6#%F~~~~~PCpaar['C']:=
AdamsMoulton(k=4);|+G-Fb\o6#%7~~~~~PCpaar['mu']:=3;|+G-Fb\o6#%6~~~~~PCpaar['t']
:=0;|+G-Fb\o6#%:~~~~~PCpaar['L']:="Lin";|+G-F]]m6#%duLet~op:~de~namen~`P`,~`C`,
~`mu`,~`t`~en~`L`~mogen~niet~veranderd~worden~en~de~enkele~quotes~errond~zijn~o
ok~belangrijk!~Om~de~procedure~te~gebruiken~kan~je~dan~ingeven:|+G-Fb\o6#%5~~~~
~ordePC(PCpaar);G-F]]m6#%IVoor~deze~L~zijn~er~drie~mogelijkheden:|+G-F]]m6#%co"
n":~geen~lokale~extrapolatie,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~G-F]]m6#
%jn"Lin":~in~elke~loop~gebeurt~lokale~extrapolatie,~~~~~~~~~~~~~~~G-F]]m6#%en"L
out":~enkel~lokale~extrapolatie~op~het~einde,~na~de~lus.G-F]]m6#%b^lVoor~de~pro
cedures~`stabvtPC`,~`stabgebiedPC`,~`randkrommePC`~en~`stabintervalPC`~geldt~de
zelfde~afspraak.~Als~input~bij~`stabgebiedPC`~is~er~nog~een~optionele~parameter
~`GRENSWAARDE`~analoog~met~`stabgebied`~(zie~helpLMM(~)~).~`stabgebiedPC`~is~ee
n~uitgebreide~procedure~en~kan~dus~enige~tijd~in~beslag~nemen.|+G-Fb\o6#%;~~~~~
ordePC:=proc(PCpaar)|+G-Fb\o6#%=~~~~~stabvtPC:=proc(PCpaar)|+G-Fb\o6#%W~~~~~sta
bgebiedPC:=proc(PCpaar,GRENSWAARDE::realcons)|+G-Fb\o6#%A~~~~~randkrommePC:=pro
c(PCpaar)|+G-Fb\o6#%C~~~~~stabintervalPC:=proc(PCpaar)|+GF(F(F(F(,
I,take_stepPCf*6&FhgpFhepFh_pF]`r67FfjnF__lFb`rF^_lFe^lF<FJ%$f_jG%-take_stepLMM
G%$f_tG%#k_G%$y_jGF%Fgfq%#x1G%#y1GFc`rFd`rFe`rFf`rFg`rF(F(C8>FdjmF\ip>F[fm&F1F\
dp>Fafm&F1F_dp>Fjbm&FNF[ar>Faem&FNF_ar>Fcdm&FNFcar>F[em&FNFhar>F^em&FNF\br@$330
F^emFhbr0F^emFfbr0F^emFjbr-Fgo6#Q^oDe~laatste~parameter~in~PCpaar~mag~enkel~"n"
,~"Lin"~of~"Lout"~zijn.F(@$3/FcdmFgrFhdt-Fgo6#QdqMen~kan~enkel~lokale~extrapola
tie~toepassen~als~men~minstens~|dy|dynmaal~de~corrector~~~~~~~~~~~~~~~~toepast.
F(>Ffel,&-FiglFacmF0F2F0>F\p,&-FiglFgesF0F2F0@&0&Fjbm6$F^r,&FfelF0F0F0Fgr-Fgo6#
QNDe~predictor~moet~een~expliciete~methode~zijnF(/&Faem6$F^rF`^oFgr-Fgo6#QNDe~c
orrector~moet~een~impliciete~methode~zijnF(>Fgqf*F`dnF(F)F(FbhpF(F(Fdhp>Fep*&-F
hcnFgesF0,&-FhcnFacmF0F\gtF2F2>F\q-Feep6%FjbmF1F8?(FapF0F0FfelFbr>&F_elFcal-Fgq
6#Fbal@$2FfelF\pC$>Far-Fe_p6$F1Fgbl?(F[clF^ftF0F\pFbrC$>F\q7$-Fidl6$Fbal/Fap;F0
,&F[clF0F2F0-Fffp6#-Fh]m6$<#/Fgbl,&F[fmF0*&FehtF0F8F0F0Far>&F_elFecl-Fgq6#&F\qF
ecl>Ffdlf*6*'Fghl%(integerGFhepFgfqFcctFdctFdglF^_lF`ct6%FeilF<%*k_methodeGF(F(
C&>Fep,&-Figl6#9)F0F2F0>Far-F[`l6$Fhhp;F0-Figl6#F1?(FgqF0F0FepFbr>Far-F`ip6$Far
-F`ip6&*&&F_jt6$F0FgqF0&9*6#,*F3F0FepF2FgqF0F2F0F0*(F8F0&F_jt6$F^rFgqF0&9+Fa[uF
0F2F0O7$-Fidl6$&F`[uF\t/FgqFd\qF^cnF(F(F(@$/FearFgrC$?(F[clF^ftF0F3Fbr>F\q-Fegp
6'F[clFjbmF1F8F\qOF\q?(F[cl,&-Fbw6$FfelF\pF0F0F0F0F3FbrC'>&FdblF[[p&-Ffdl6*F[cl
F8FdjmF[fmFafmFjbmF\qF_elFecl?(FapF0F0FcdmFbrC'>&F^blFgfl-Fgq6#&FdblFgfl>&FdblF
calFfhp?(F]clF0F0F\pFbr>Fh]u-F`ip6$Fh]u-F`ip6&*&&Faem6$F0F]clF0&F\q6#,*F[clF0F\
pF2F]clF0F2F0F0*(F8F0&Faem6$F^rF]clF0&F_elFc^uF0F2F0>Fh]u-F`ip6$Fh]u*(FcftF0F8F
0Fc]uF0@$/F^emFfbr>Fh]u-F`ip6&*&,&F0F0FepF0F0Fh]uF0*&FepF0F\]uF0F0F2@$/F^emFjbr
>&Fdbl6#Fcdm-F`ip6&*&Fc_uF0Fh_uF0Fd_uF0F2>F\q7$Faht-Fffp6#Fh_u@'/F[emF0>F_it&F^
bl6#,&FcdmF0F2F0/F[emFgr>F_itF`it-Fgo6#%Ut~(vierde~parameter~in~PCpaar)~kan~enk
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^ar>Fj^sFbar>8;Fgar>8<F[br@$30FgguFgr0FgguF0-Fgo6#Qent~(voorlaatste~parameter~i
n~PCpaar)~kan~enkel~0~of~1~zijn.F(@%F\`m>FarF8>Far,&-Fbw6$-FailFbim-FailFbgsF0F
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F(6$FgfqFep>F\qFghu>F\pFhhu-Fb\o6$%5predictor:~(orde~%d)GF\q-F]]mFbim-Fb\o6$%5c
orrector:~(orde~%d)GF\p-F]]mFbgs@%0Fj^sFgr@)/FiguFhbr@%/FgguF0-Fb\o6$%=uitgevoe
rde~mode:~P(EC)^%d|+|+GFj^s-Fb\o6$%>uitgevoerde~mode:~P(EC)^%dE|+|+GFj^s/FiguFf
br@%Feju-Fb\o6$%>uitgevoerde~mode:~P(ECL)^%d|+|+GFj^s-Fb\o6$%?uitgevoerde~mode:
~P(ECL)^%dE|+|+GFj^s/FiguFjbr@%Feju-Fb\o6$%>uitgevoerde~mode:~P(EC)^%dL|+|+GFj^
s-Fb\o6$%?uitgevoerde~mode:~P(EC)^%dLE|+|+GFj^sFa_s@'5FcjuF\[v-Fb\o6#%6uitgevoe
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gepast~op~het~probleem~y'=y:|+G>Fap-Fbw6$-FiglFbim-FiglFbgs>Fdbl%$expG-Fb\o6#%1
na~de~predictor:G-F]]m6#&-F[aq6%F^emF[iuFarFfu@%0FiguFfbr@$FajuC$?(F_elF0F0Fj^s
FbrC*-Fb\o6$%.iteratie~%d:|+GF_el-Fb\o6#%.~~fout~na~C:|+G>&8=F[arF^em>&Fe^vF_ar
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I3helpOP_PROBLEEM_PCf*F(F(F(F(C*-F]]m6#%[zDezelfde~afspraken~i.v.m.~de~input~va
n~methodes~blijft~gelden~(typ~`helpLMM(~)`~voor~meer~uitleg),~voor~de~input~van
~problemen~(typ~`helpOP_PROBLEEM_LMM(~)`~voor~meer~uitleg)~en~voor~de~input~van
~PC-paren~(typ~`helpPC(~)`voor~meer~uitleg).|+G-F]]m6#%joBesproken~procedures:~
`take_stepPC`,~`LAFPC`,~`numPC`,~`GAFPC`~en~`opvolging`.|+G-F]]m6#%cjmDe~proced
ure~`take_stepPC`~berekent~alle~numerieke~waarden~tot~een~gegeven~punt~(analoog
~met~`take_step2`~uit~`LMM.m`).~Als~input~geef~je~het~aantal~te~berekenen~punte
n,~de~`h`-waarde,~het~probleem~en~het~PC-paar.~Aan~`LAFPC`~(lokale~afknottingsf
out)~geef~je~het~probleem,~het~PC-paar~en~(optioneel)~een~getal~tot~waar~je~de~
LAF~wil~ontwikkelen.~`numPC`~geeft~een~plot~van~de~numerieke~oplossing~en~`GAFP
C`~van~de~globale~afknottingsfout.~Telkens~geef~je~het~aantal~punten,~de~`h`-wa
arde,~het~probleem~en~het~PC-paar.~Opvolging~is~een~procedure~die~per~stap~dat~
er~uitgevoerd~wordt,~de~lokale~afknottingsfout~op~het~probleem~y'~=~y~berekent.
~Deze~procedure~heeft~enkel~het~PCpaar~nodig,~en~optioneel~een~getal~tot~waar~d
e~ontwikkeling~van~de~LAF~moet~gebeuren.G-Fb\o6#%V~~~~~take_stepPC:=proc(n::pos
int,h_,probleem,PCpaar)|+G-Fb\o6#%M~~~~~LAFPC:=proc(probleem,PCpaar,m::posint)|
+G-Fb\o6#%Z~~~~~numPC:=proc(np::posint,h::realcons,probleem,PCpaar)|+G-Fb\o6#%Z
~~~~~GAFPC:=proc(np::posint,h::realcons,probleem,PCpaar)|+G-Fb\o6#%I~~~~~opvolg
ing:=proc(PCpaar,m::integer)|+GF(F(F(F(,
I&schurf*6$'%%polyGFg^m'FgblF]bs6&%0constructmatrixG%'recursG%$dimG%)polynoomGF
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