{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 10 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 20 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - \+ Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 1 2 258 1 }{PSTYLE "AC - Note" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Author" -1 258 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 4 4 1 0 1 0 2 2 259 1 } {PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Normal Text " -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 258 51 "Newton Divided Difference Meth od of Interpolation--" }{TEXT -1 9 "Graphical" }{TEXT 259 1 " " } {TEXT 257 0 "" }}{PARA 258 "" 0 "" {TEXT 256 7 "\251 2003 " }{TEXT -1 143 "Nathan Collier, Autar Kaw, Jai Paul , Michael Keteltas, Universit y of South Florida , kaw@eng.usf.edu , http://numericalmethods.eng.usf .edu/mws" }}{PARA 257 "" 0 "" {TEXT -1 199 "NOTE: This worksheet demon strates the use of Maple to illustrate the Newton's Divided Difference Method of interpolation. We limit this worksheet to using first, sec ond, and third order polynomials." }}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 12 "Introduction" }}{PARA 260 "" 0 "" {TEXT 278 43 "The Newton's D ivided Difference Polynomial " }{URLLINK 17 "" 4 "http://numericalmeth ods.eng.usf.edu/mws/gen/05inp/mws_gen_inp_txt_direct.doc" "" }{TEXT 279 68 "method of interpolation (for detailed explanation, you can rea d the " }{URLLINK 17 "textbook notes and examples" 4 "http://numerical methods.eng.usf.edu/mws/gen/05inp/mws_gen_inp_txt_ndd.pdf" "" }{TEXT 281 12 ", and see a " }{URLLINK 17 "Power Point Presentation)" 4 "http ://numericalmethods.eng.usf.edu/mws/gen/05inp/mws_gen_inp_ppt_ndd.ppt " "" }{TEXT 280 27 " is based on the following." }}{PARA 0 "" 0 "" {TEXT -1 67 "The general form of the Newton's divided difference polyn omial for " }{OLE 1 3590 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyy yyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::fyyyyya:nYf::wyyyqy;::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ:::::::: gjh`awjFleVr;V:>r=B:<:=ja^GE=;:::::::::N;?R: yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wAJ=j=>Z:>;F;N;; j?J@>:UJ:n;nYvY::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::=Z<>Z:B:::KgUk:;::::::j:nyyYZDjysy?bm?:;JZC:bKi:UT TAeVYuVYeScEBETVeURcUTYeU;sFWCFl:Lnci:>bli:AR:=r:=b<=b:?FDDk;u`:yayQZ:J:J@M:cc:N[<>:=j>r:ER:N:s:;jD:;B:E:?R:B:E:=b: yyyyI:E:WS:k:E:Qb:B:E:Sb:DJJLJCJMTjAN=yyyxY:ryyYGxyyyy;y:nr;j?< :G;Sj`@Pt\\Pd`QrP@[LHB:qi:;fyB:>l;fB]mtFFcmnvGWMJnC==nHE]:>::::::;C:?B :yay=J:>Z::::::J<>Z:FA=?R:AJ:^:vYxY:B:: ::::f:Z:j::<::::::wqy[:::::::::::::vY xI:;Z:::::::::^ZcTTUUSaEBWTSiEB_tUUURWMEHN^YqIHN^YQJ; N`D>f;Nd=F:;N@>p:]C:K:_;cC:=J:>?s:qAB:>L;ZaTXDpql`q?B>:C:Uk:^:>x;J>JSdZ;>r;F:;B:QKHjw;<:[n:JX=J>JSR:wo:B:WM?JMJ?vYxy:J:^=VY[j=B:;JXE:O V:B:=B:>tnQkSHZ;^^=fE>::NuoKk:B>N:F:nyyyyy]:: yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::fyyyyya:nYf::G:I:K:wAyA:::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::JBpPq=mBil[Aj;J:a:<:=ja^GE=;:::::::::N;?R:yyyyyy :>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyq:[Z:F:i:k:m:o:q:s:u:w:y:;C:Z:::::::F:wyyAbR<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV@cYH_WV>Z:::::::V:>::::::::::^=N:WcjuR:B:>`fi:AR:=r:=b<=r:=B:@JCHRvVZIF:QR:?B\\[[Kb:=::RFF::J:dZP:<::::::C:mS:>Z :J;vCS=[LsfFaMR>`:J:<:::::::>=?R:AJ:^:vYxY:B::::::F;Z:>:: :::::::J?:<::::::wqy[:::::::::::::vYxI:;Z::::::::j;j:D[r;ZhC :]J?NJD;::>ZDNZP\\P<;:jysyA:C:=B:;B:yayQZ:B:;:qT@Vir:;u:;J; \\:B:;xyyQMyyyyYZU;GY::[V:>Z<>ZaTXUeRYEUXQZB:ui;OBBgrJKxWyrJKxW;CjOxW>@C:US:>;N`Dni;V@=Z:>_g`=VYZ:>: [>DZjq:>:C:Uk:^:>x;J>JSd:EB;=Z:F_g`]:jw;Lb:DZJ:Y =B:ua[n\\=?Sy:;;J:C:[q:>;N`DJpMj:B:K;VYZ:>:[>DZjq:>:C:UK:^:>X=J>JS d:my<=Z:>?jw;Lb:DZJ:Y=B:_`_n\\=?Sy:;;J;N`Dj?Ak:>Z:F?jw; Lb:DZJ:Y=B:ua]n>:C:UK:^:>X=J>JSd:_e>=J:L=J:^ZcTTPps=B:nGJSJRpj:J q^=VY;fy>Z:JBG:;J>:=J:>ig@ ;N@fdIF:;jv:_;AF==J:>IJSJ[Xj:>:m=N@nh?F:>IJSjQEj:jv:_;uo:<:kMHjw;<:[N: B:Cb:^D:::y:;;Z:jP>:C:[q:>;N@NbNF:JSjnAk:>:S;N@FnHFZ:JPV;N@>tB>; ;JPtLQtLQBEK:_;Wf;=:SK>JSjkLj:JP:_;qq:JP^]:jw?>B:>LB:Cb::::jIJJ:B:f?=J JSjHtj:jR:_;sx;=:];N@N[@F:F@JSJiHj:JRtL>JSJm@j:>: [KHjw;<:[>DZa\\\\wGZ:fYTLEjIJJ:<:f_;>:C:[q:>;N`DJ==k:B:UcEK:_;WW==: UK?JMJ?vYxy:J:^=VY[j=B:;JXE:OV:B:=B:>tnQ;B:Sn:>;;hAKR:kU:SjSMY[AJT`qyXwnak[AJYqOc qq;]Y[AZ:>Z:::fc:FF[MpfF;C:=Lr>HSmnVf:FFcmnvGWms>:;::::::::::::::::::3 :" }{TEXT -1 13 " is given as " }}{PARA 0 "" 0 "" {OLE 1 4614 1 "[xm]B r=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:I:wAy A::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gjdtGxj FleVr;V:>rAZ:j:vCSmlJ::::::::::OJ;@jyyyyyI:;Z:::::::^<>:F:AlqfG[maNFO= ;::::::::_J;@j:j<>:yayA:<::::::=J:nF>:V:Y:G:;:wA?:Jyyyyg:n:v:JyK?j?J@>:UJ:n;v;;JBB:]:_J:V<^R<:TnEj``pkDqqH qqTPtWdZfbk;>ZEZ?GHZhV@cYH_WV;J:<::F:;jqfGWl=F :KJnIJ@>w=J:V:>::::::::::^=N:qdjmR:B:>fuh:AR:=r:=b<=b>AfSHjA@Z EFZLVjhkBN[AFXDJ;a\\AfRHjAZ<>kjjBJ=i\\LVJ:>kkjBJ;[rK=EKKDk ;]a=v;Zj:RDrDDJ;u\\AfR:D;:?FDBXDZL:H:b:K>ZcJ;]>Wlj:gmlJ::::::>^:N:yay=J:B::::::^:f`;J:<:?ja^G>D_mlVH[KRJ: <:::::::>=?R:AJ:^:vYxY:B::::::F;;JgP:;B:F:Y<>:;:::::::::N;;B:yay=J:B:: ::::nYyA<::::::::::::jysy:>:<::::::::DJ;?Zr:Sf:NZ:^=>:EZ:F[=f:V[b<>v[N\\:>Z:>ryyYGwyyyy; WBnv;B:QB:n>^;UTRcETcTX[US>f:^=l;J:@CB:f_;J>>:_c<;V:wRB=Z:F_JN;N@NoTf:JSJhTK;>:MCDK:_;Ge>=J:>_bF;JSJGmj:<:McEK:_;CG<=:KcEK :_;ci:=:M;N@^n;F:>?s:qQJxI;B:>LB:Cb:;Jd:::v=>>:<:Uk:^Z:JrA:K:_KjAJD=l: B:]cEK:_;;c@<:];N@FrOF:>`h>;N@vyEF:>@JSJG\\j:jR:_;qV;=:[KHjwE:B:>Lb:;b ::C:[q:>[:JSd:ECB=Z:f_h^=VY;><:[> DZjq:>:jP@j:^:>X;J>JSdJ:_tWF:JSJm\\k:jP:_;_y> =:U;N@nZJF:V_h>;N@NeGF:V?JSJRdj:jP:_;_e;=:Q;N@^s=F:fah^=VY;;<:[n:>:Cb: ^DPps::jIJJZ:Z:>:US:;J=J:nGJSj<@j:Jq^=VY;> <:[>^Z<::::y:;;Z:jPF:C:[Q:>;N@;Nwuk:B:keEK:_;CIA=:k=N@^kSF:>IJSJuXk:>: k=N@>]OF:>IJSJ;Hk:Jv:_;cR>=J:Nfh>;N@ndEF:>IJSJZ`j:Jv:_;;v;=:?=N@^w>F:N FJSJZDj:IJSJ;GZ:JmtL?JMJ?vYxy:J:^=VY[j=>Z:>:s?tnQ::::WTJWTLB:::::::::::::::::::1:" } {TEXT -1 74 "where n is the order of the polynomial that approximates \+ the function and," }}{PARA 0 "" 0 "" {TEXT -1 15 " " } {OLE 1 4102 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: fyyyyya:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::Jr=js=mBil[Aj;J:M:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:;B:::::: :c:;:=j[vGUMrvC?MoJ::::::::JCNZ;F:f:;jysy;Z::::::j:>:G=;:Aja:MLqnFYM:> ::::::::N<:W:Y:[ Z:FZ:B:::KR;SJJfB^H:;::::j:nyyYZDjysy?bm?Z:>:;`:Z@w:<:::::^=N:qdju;;:;ctgJxlNE>Z:>:AR:=r:=b<=b>AfRHjA@ZEFZ<>kjjBN[AfSDJ;i_LVjrs:Y:>K:?VQryvY>:<:::::vYxy;J:yayQZ:J:JOX:qe:NjDZ:>Z:f:NZ;Z:f:FZ=f:V[b<>j@N<;JM 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Y:^:F:;B:yayQZ:J:JdtZ:Vi:E:Mb:ElrfH=MtFGYMq>>Wlj:gm l>:;:::::JJD_mlVH[KRJ:<:::::::>=?R:AJ:^: vYxY:B::::::v:>u=Z:j:vCJ:Z:f:^[<>b<>ZGN\\:B:;xyyQMyyyyYZmZ:nv;B:QB:n>^;UTRcETcTX[US>FsZ:Vy<>jxM:<:[V: >Z:^ZcTTUUSaEBWTSiEB_tUUURWMEHN^YqIHN^YQJ;N`D>j;neFF: mBn:JSJfQj:JP>;N@nx=FZ:jgtLHjw?^yM:<:[>>Z:B:f ?;JJSJwTj:Jv:_;_F;=:k=N@>c=F:nfh>;N@n?=:?eEs:qQJZ :JBK:^Z<::::y:;;Z:jPF:C:[Q:>;N@;qMXj:>Z:ngh>[:JSjYPj:Jq:_;=q:Jq^=VYZ:J B?:DJ:DZJVdscRYEUXQZLb:DZJ:Y=B:?g;OBOgrJKpIJJ::US:=JZ:fahN;n>N;yayI:;J: ^=VY[j=B:;JXE:tnQ:: ::WTJWTL>Z::::::::::::::::]j:Lj:B:;b:SB:D::1:" }}{PARA 0 "" 0 "" {TEXT -1 24 "where the definition of " }{OLE 1 3590 1 "[xm]Br=WfoRrB:: :wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: NDYmq^H;C:ELq^H_mvJ::::::::gjxovyjFleVr;V:>b =B:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>:::::::: N:;::::::::_Z:vyyuy:>:<::::::A J:n:>:nYN::wAJ=j=B:K:M:OJ:V;^;nYvY:::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::F:DJ::;`:Z@w:;:KCTgJh@SEB:;j;@j:Hj:dj:Dk;q `=v;hjB@JCHRtN\\=@G>::yayY:^:;j:>:yayQZ:J:JIE:cc:Nl;j:F;Hj^?Z:N:s:;jDJ:>Z:f:NZ;Z:f:FZ=f:V[l;fB]mtFFcmnvGWMJ nC==nHE]:>::::::;K;:<::::::C:o[:>:N:YLpJb@>Z::::::::kJ ;@j;>:C:yayA:<::::::EZ:^t;J:<:=B:Y<>:;:::::::::N;vYxI:;Z::::::JywYB::: ::::::::::yay=J:B::::::::F:;J@CB:f_;J>>:_cZ:^hb_=;jw?jx]:JBG:>O@B:f?=J:;JHjw?:sg:tnQ::::WTJWTLB::::::::::::::::::::::::::::: ::::::::::::3:" }{TEXT -1 22 "divided difference is " }{OLE 1 4102 1 " [xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G: jy;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::JBme kFleVr;V:>b?B:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLk NG>::::::::N:V:Y:G:;:wA?:Jyyyyg:n:nY>;F;;J?>:Q:S:UJ:n;v;;JBB:]:_J:V<^Z:B:::[IwCR:yyyxI:::::j:nyyYZDjysy?bm?:;JZC:bKiZ:fccWflwgmwg`_hZfb kgh[_hcwgh?^mn_jB=BKaDBETV:;rA>Z<>kjjBJ;q?;B::::::::vYxy;J<r:=V:::: :::;C:?jysy:>:<::::::C:a^:>:N:YLpJbNHEms>@[;;B:::::::JFNZ;V:;JZ:>:::::::::J?B:yay=J:B::::::nYyA<::::::::::::jysy :>:<::::::::jl;J:@CB:;jP@:K:_c<;V:;v<=B::_;oc;OJ:^_hn_hn?BEK:_;IC;=:qLHjw?^yM:<:[V:B:Cb::::jIJJ:B:f?=JL=J:^Z<::::y:;;J<<:UK:^:>X=J>JS>Jsc<=Z:>IK:_;A H;=:k=N@Fi=F:nFK:_;Wk:Jk^=VY;;<:[>;J>:<:Uk:^:>X;J>JS>V[U;=Z:f GKJ:N@FV=:UMHjw?JB:>L;Z<>ZaTXDpql`q?b:B:UR>Or]grJKpIJJ::UK:^:>X=J>J SdJ:gZ:;JHjw?:sg:B:=J;Dlc`qsLqlp@;B:OjJNk;Z:FZ :JbMYCF_;B:::::::::::::::::::::::::::: ::::::::::::::::1:" }{OLE 1 4614 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B >N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::fyyyyya:nYf::G:I:wAyA:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ ::::::::gj`=OZkFleVr;V:>ZAZ:j:vCSmlJ:::::::: ::OJ;@jyyyyyI:;Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:yayA:<:::: ::=J:nF>:V:Y:G:;:wA?:Jyyyyg:n:v :JyK?j?J@j@>:W:YJ:><:a:c:e:gJ:v<>=F=N=V=nYvY::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::F:DJ:bOpZ;J:<:: ::::j:nyyYZDjysy?bm?:;JZC:bKi:UTTAeVYuVYeScEBETVeURcUTYeU;sFWCFl:Lnc:>xqndJ;DRPNZbZuL;TRCDLCHRmjgrZ:NI:_jp>ZKZ@P>Wlj:gml>:;:::::JJ:eR: >Z:J;vCS=[LsfFaMR>@>Z::::::::kJ;@j;>:C:yayA:<::::::M:;G;B:FZ:vCJ::<::::::wqy[:::::::::::::vYxI:;Z::::::::JCHB:?:@[:B:: ::::::vYxy;J:yayQZ:B:;:[v?vo>JiL:=j>>Z=v_:Gc;YJGvyyuy;B:[B:^;UTRcETcTX[US>x:>;Ny<>:[Z::CZ:f_;j]O^=;B:q i:;sy>Z:JBA:;B:Cb:;N`D^w=>sFFZ:JR>;N@UX:uy>=:[;N@nsHF:F`:>;N@f^CF:F@s:qQ:uI;B:>LB:Cb:: ::jIJJ:^Z:jPN:C:[Y:=:N@UU:kF?=:uL@>:_;_t=O:ScEWcEW;lL>JSjmlj:jg>;N@voC F:fEJSjAMJ?JPJQZf:_;se:=J:VEs:qAB:>L=J:^Z>:<:Uk:^:>X?B:K:_K@I jIlj:Jv>;N@>bCF:>IJSfGct>=B:;Jv:_;AR==:k=N@FqEF:nFK:_;]F<=:k=N@>]=F:>I JSJaAj:>:GMHjw?>B:>LJ>:<:UK;^:>X=J>JS^WUY<=:UM>JSfS[W==B:fGJS j\\Ij:jp^=VY;><:[N:b:;b:;B>aTXDpql`q?b:B:Yx;Ob^grJKpIJJ::Uk:^:>X?J>JSd J@GyDF:f_:>;N@Umhdj:<:U;N@;e:ok:jX^=VY;;<:[>DJ:DZJ:Y=B:gI;ObcgrJKpIJJ: ^:f_;N:C:[q:>;N`DfSyd==:UC:OJ:n>N;yayI:;J:^=VY[j=B:;JXE:;B:=J;Dlc`qsLq lp@;B:OjJNk;Z:FZ:JbMYCF_X?B:AZ:>::::WTJWTL>Z::::::::::: :::::::::::::::::::::::::::::::::::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 107 "The examples that follow illustrate the method in a descriptive m anner. For further explanations, click on " }{URLLINK 17 "text notes" 4 "http://numericalmethods.eng.usf.edu/mws/gen/05inp/mws_gen_inp_txt_n dd.doc" "" }{TEXT -1 31 " for the method on the website." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "with(LinearAlgebra):\nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 104 "Warning, the previous binding of the name GramSchmi dt has been removed and it now has an assigned value\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee n redefined and unprotected\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 261 23 "Section I : Input Data." }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The following is the array of x-y \+ data which is used to interpolate. It is obtained from the " } {URLLINK 17 "physical problem" 4 "http://numericalmethods.eng.usf.edu/ mws/gen/05inp/mws_gen_inp_phy_problem.pdf" "" }{TEXT -1 131 " of velo city of rocket (y-values) vs. time (x-values) data. We are asked to find the velocity at an intermediate point of x=16." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "xy:=[[10,227.04],[0,0],[20,517.35],[15,36 2.78],[30,901.67],[22.5,602.97]]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Value of X at which Y is desired" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xdesired:=16:" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 262 33 "Section II : Big scary functions ." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "n:=rowdim(xy):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for i from 1 to n do\nx[i]:=xy[i,1];\ny[i]:=xy[i,2]; \nend do:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 548 "The following function considers the x data and selects \+ those data points which are close to the desired x value based on the \+ least absolute difference between the x values and the desired x value . This function selects the two closest data points that bracket the desired value of x. It first picks the closest data point to the desi red x value. It then checks if this value is less than or greater than the desired value.If it is less, then, it selects the data point whic h is greater than the desired value and also the closest, and vicevers a." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Finds the absolute difference between the X values and the desi red X value." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for i from 1 to n d o\n co[i]:=abs(x[i]-xdesired);\nend do:\n" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 58 "Identifies the X value with the least absolute dif ference." }{MPLTEXT 1 0 95 "\nc:=co[1]:\nfor i from 2 to n do\n if c \+ > co[i] then\n c:=co[i];\n ci:=i;\n end if:\nend do:\n" }} {PARA 0 "" 0 "" {TEXT -1 179 "If the value with the least absolute dif ference is less than the desired value, then it selects the closest da ta point greater than the desired value to bracket the desired value. " }{MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 316 "if x[ci] < xd esired then\n q:=1;\n for i from 1 to n do\n if x[i] > xdesired t hen\n nex[q]:=x[i];\n q:=q+1;\n end if;\n end do;\n\n b :=nex[1]:\n for i from 2 to q-1 do\n if b > nex[i] then\n b:= nex[i];\n end if:\n end do:\n\n for i from 1 to n do\n if x[i] =b then bi:=i end if;\n end do;\nend if:\n" }}{PARA 0 "" 0 "" {TEXT -1 179 "If the value with the least absolute difference is greater tha n the desired value, then it selects the closest data point less than \+ the desired value to bracket the desired value." }{MPLTEXT 1 0 1 "\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 335 "if x[ci] > xdesired then\n q:=1;\n \+ for i from 1 to n do\n if x[i] < xdesired then\n nex[q]:=x[i] ;\n q:=q+1;\n end if;\n end do;\n\n b:=nex[1]:\n for i from 2 to q-1 do\n if b < nex[i] then\n b:=nex[i];\n end if:\n \+ end do:\n\n for i from 1 to n do\n if x[i]=b then bi:=i end if;\n end do;\nend if:\n\nfirsttwo:=:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 316 "If more than two values \+ are desired, the same procedure as above is followed to choose the 2 d atapoints which bracket the desired value. In addition, the following \+ function selects the subsequent values that are closest to the desired value and puts all the values into a matrix, maintaining the original data order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 959 "for i from 1 to n do\n A[i,2]:=i;\n A[i,1]:=c o[i];\nend do:\n\nfor passnum from 1 to n-1 do\n for i from 1 to n-pa ssnum do\n if A[i,1]>A[i+1,1] then\n temp1:=A[i,1];\n tem p2:=A[i,2];\n A[i,1]:=A[i+1,1];\n A[i,2]:=A[i+1,2];\n A [i+1,1]:=temp1;\n A[i+1,2]:=temp2;\n end if:\n end do:\nend d o:\n\nfor i from 1 to n do\n A[i,3]:=i;\nend do:\n\nfor passnum from \+ 1 to n-1 do\n for i from 1 to n-passnum do\n if A[i,2]>A[i+1,2] th en\n temp1:=A[i,1];\n temp2:=A[i,2];\n temp3:=A[i,3];\n A[i,1]:=A[i+1,1];\n A[i,2]:=A[i+1,2];\n A[i,3]:=A[i+1, 3];\n A[i+1,1]:=temp1;\n A[i+1,2]:=temp2;\n A[i+1,3]:=t emp3;\n end if:\n end do:\nend do:\n\nfor i from 1 to n do\n d[i] :=A[i,3];\nend do:\n\nif d[firsttwo[2]]<>2 then\n temp:=d[firsttwo[2] ];\n d[firsttwo[2]]:=1;\n for i from 1 to n do\n if i <> firsttwo [1] and i <> firsttwo[2] and d[i] <= temp then\n d[i]:=d[i]+1;\n end if:\n end do;\n d[firsttwo[1]]:=1;\nend if:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "ranger:=proc(ar,n)\nlocal i,xmin,x max,xrange;\nxmin:=ar[1]:\nxmax:=ar[1]:\nfor i from 1 to n do\nif ar[i ] > xmax then xmax:=ar[i] end if;\nif ar[i] < xmin then xmin:=ar[i] en d if;\nend do;\nxrange:=xmin..xmax;\nend proc: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Plotting the given values of X and Y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(xy,ranger(x,n),style=POINT,color=BLUE,symbol=CIR CLE,symbolsize=30);" }}{PARA 13 "" 1 "" {GLPLOT2D 535 535 535 {PLOTDATA 2 "6(-%'CURVESG6#7(7$$\"#5\"\"!$\"3#***********RqA!#:7$$F*F* F/7$$\"#?F*$\"3A+++++]t^F-7$$\"#:F*$\"3t**********zFOF-7$$\"#IF*$\"3f* *********p;!*F-7$$\"3+++++++]A!#;$\"3F+++++qHgF--%'SYMBOLG6$%'CIRCLEGF <-%+AXESLABELSG6$Q!6\"FL-%&STYLEG6#%&POINTG-%'COLOURG6&%$RGBGF/F/$\"*+ +++\"!\")-%%VIEWG6$;F/F;%(DEFAULTG" 1 5 4 1 30 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 34 "Section III: Linear Interpolation." }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 7 "Given " }{OLE 1 3590 1 "[xm]Br=WfoRrB:::wk;nyyI;G: ;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:E Lq^H_mvJ::::::::gjpW?[kFleVr;V:>Z>Z:j:vCSmlJ ::::::::::OJ;@jyyyyyI:;Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:ya yA:<::::::=J:nF>:V:Y:G:;:wA?:Jy k<:v:j>J?>:QJ:^;f;;JAjA>:wAyA:::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::=Z<>Z:B:::kU:S:;vI>J:::::F:wyyAbR<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZh V@cYH_WV>Z::JtaMrMJ@>x>^;;J:::j;J:::A:;:::::::^=N:qdju;;:[WueJ\\uLEB:; j;@j:Hj:dj:DJ;]\\LVjrs:YR:hj:DJ>==]J?DJ;m\\LVjss:Y::Zcb:?FEkkjBJ;m\\vyyuyA:C:=Z:vYxY;<:;:GW;Vir:eR:N:s:; jZ:f:NZ;J::Gc;YJGfB]mtF FcmnvGWMJnC==nHE=;:::::JJZ:>Z::::::J<>:m\\:>:N:YLpJbNHEms>@[ ;;B:::::::JFNZ;V:;J\\:B:;xyyQMyyyyYjM:GY:^;UTRcETcTX[US>FsZ:Vy<>jx]:JBA:;B:Cb:;;JSdJ:AJ@LJ;>:MCEK:_;Wb:=J:JSJT;N@;Y:]u:=B:JSjv@j:>:[KH jw?JB:>L=:Cb::::jIJJ::C:[q:>;N@;N[Ej:>Z:Fif>;N@>T=:kEEO:G;Ojyyiy I:;J:^]:jw?:sg:B:=b:?bBaTXaEWEUUB:OjJNk;Z:FZ:JbMYCF_H;B:^yyyyyA:<::3:" }{OLE 1 3590 1 "[xm]Br=WfoRr B:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::NDYmq^H;C:ELq^H_mvJ::::::::gjd^I[kFleVr;V :>Z>Z:j:vCSmlJ::::::::::OJ;@jyyyyyI:;Z:::::::^<>:F:AlqfG[maNFO=;:::::: ::_J;@j:j<>:yayA:<::::::=J:nF>:V:Y:G:;:wA?:Jyk<:v:j>J?>:Q:S:UJ:n;v;;Jyky;::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::j:b:;B:<::JjMo>D:<:::: ::j:nyyYZDjysy?bm?:;JZC:bKi:UTTAeVYuVYeScEBETVeURcUTYeU;sFWCFl:Lnc:>pMoA=]J?DJ;m\\LVjss:Y::Zcb:?FE:ve?jeDZ:N_D:;B:E:?R:>Z:f:FZ=f:V[^<>v>N::::::;C:?B:yay= J::N:YLpJbNHEms>@[;;B:::::::JFNZ;V:;JZ:j::<::::::wqy[:::::::::::::vYxI:;Z::: :::::jysy;JBJ]AZ:V[:>:G;Sj`@Pt\\Pd`QrP@[LHB:qi:;fyB: >l;J:@CB:JSdZ;>w;F:JSJT>:<:Uk:^ :>x;J>JS>v;n`=FZ:B:]CEK:_;CU:=J:F@s:qQ:[:JBG:^Z<:::W=v=>>:<:UK:^:>X=J> JS>Jsf:=J:JSJb=j:JvlL?JMJ?vyyuy=J:>:s:qQBv::Uk:^:>X;j;::::WTJWTLB:yay=:::1:" } {TEXT -1 55 " fit a linear interpolant through the data. Noting " }{OLE 1 4102 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :fyyyyya:nYf::G:jy;::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::Jrep?@mBil[Aj;J:M:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<:::::: JDJ:j:VBYmp>HYLkNG>::::::::N:; ::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyq<:v:>:M:OJ:V;^;f;;JAjA>: [Z:FZ:B::::::::::F:wyyAbR<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV@cYH_WV>Z:::::::V:>::::::::::^= N:qdju;;:;EteJ\\uLEB:;j;@j:Hj:dj:Dk;aa=v[;r<=b:KFFFA FXHjA::NjcB:<:::::::yayY:^:F:;jysy?B:>:^lAjwD:?f:F:MZ=Fb;B:?JHJ:f:F[:Gc;YJGfB]mtFFcmnvGWMJ nC==nHE=;:::::JJZ:B::::::^:FE@>Z::::::::kJ ;@j;>:C:yayA:<::::::GB:;J\\E:;B:FZ:vCJ:\\:>Z:>ryyYGwyyyy;?GY:l;J:@CB:f_;J>JSdJ:AJEF:;N@n`>F:>?s:qQJxI;B:>LB:Cb::::jIJJ:B:f?=J JSj@@j:>:[KHjw?JB:>L=:Cb::::jIJJ:^Z:jP>:C:[q:>;N@ ;NcMj:B:kM>JSJrHj:<:GM>>:_;wl:jv^=VYZ:JB?:DZaTXDpql`q?B>:jPF:C:[Q:>;N`DJxAj:>Z:fAO:GkAN;yyyxy:>:;JHjw?:sg:B:=b:?bBaTX aEWEUUB:OjJNk;Z::[Hw_jNDN^YQ<>:UK:^:>X=j;B:;:::Ja@Na`^:B:AldfCElysywyy >:::::::oKh>:;::j::Z;kb;SR:kU:S:;::::::j@Wy;F:;JRTm>@JnMW>@:<::::::3:" } {TEXT -1 6 " and " }{OLE 1 4102 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B >N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::fyyyyya:nYf::G:jy;::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:: :::::n:;`:Z@[::Jrep?@mBil[Aj;J:M:<:=ja^GE=;:::::::::N ;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyq<:v: >:M:OJ:V;^;f;;JAjA>:[Z:FZ:B:::krHER:yyyxI:::::j:nyyYZDjysy ?bm?:;JZC:bKi:UTTAeVYuVYeScEBETVeURcUTYeU;sFWCFl:Lnc:AR:=r:= b<=b>AVXHjA@ZEFZ<>kkjBN[AfSDJ;]\\LVjrs:Y::J;a\\:ryvY::::::V:^:F: :N_AjeDZ:NjD:;B:E:?R:B:E:=b:yyyyI:E:WS:k: E:Qb:B:E:Sb:;c:;f;_:Gc;YJGfB]mtFFcmnvGWMJnC==nHE=;:::::JJZ:> Z::::::JD_mlVH[KR<:;B:::::::JF:C:yayA:<::::::GZ:^r Z:>:::::::::J?>Z:vYxI:;Z::::::Jy;<::::::::::::jysy:>:<::::: :::yayA:[B:Z: ^ZcTTUUSaEBWTSiEB_tUUURWMEHN^YqIHN^YQJ;N`D>j;Ns?F:;N@>\\>F:>?s:qQJxI;B:>LB:Cb::::jIJJ:B:f?=JL=J:^Z<::::y:;;Z:jP>:C:[q:>;;JS>JSG;=Z:>IK:_;Ot:=J:nFK:_; wl:jv^=VYZ:JB?:DZaTXDpql`q?B>:jPF:C:[Q:>;N`DJeAj:>Z: fAO:G;OjyyiyI:;J:^=VY[j=B:;JXE:OV:B:JbMYCF_:C:[q:V::B> N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::fyyyyya:nYf::wyyyqy;:::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ: :::::::gjlka[kFleVr;V:>Z>Z:j:vCSmlJ::::::::: :OJ;@jyyyyyI:;Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:yayA:<::::: :=J:nF>:V:Y:G:;:wA?:Jyk<:v:j >>:OJ:V;^;f;;JAjA>:wAyA::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::=Z<>Z:B:::kgOQR:>Z:::::::F:wyyAbR<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV@cYH_WV> Z:::::::V:>::::::::::^=N:qdjI;;:kIieJHLxDJ:VZ;FZ=FZDFZLVj`s:YR:hj:DJ>A =]J?DJ;]\\LVjrc:?VDZ:vYxY;<:;:qx:vdZ:B::::::^:VB;B:;:?ja^G>D_mlV H[KRJ:<:::::::>=?R:AJ:^:vYxY:B::::::n::<::::::::yayA:[B:Z:^ZcTTUUSaEBWTSiEB_tUUUR WMEHN^YqIHN^YQJ;N`D>j;nrJSJu@j:JN^=VY;sy>Z:JB[: ^Z>:<:Uk:^:>x;J>JS>v;^W=Z:F`:^=VY;><:[n:>:Cb::::jIJJ:B:f?;JJSJ;GZ:JmN;n>N;yyyxy:>:;JHjw?:sg:B:=b:?bBaTXa EWEUUB:OjJNk;Z:FZ:JbMYCF_Z:vYxI::::: :::5:" }{TEXT -1 14 " is given by " }{OLE 1 4102 1 "[xm]Br=WfoRrB:::w k;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:jy;:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::JcvGY Mt>^:fBWMtNHm=;:::::::n:;`:Z@[::JR\\QA@mBil[Aj;JZPZ:B :F:YLpfF>:::::::::J?NZ;vyyyyy=J:B:::::::c:;:=j[vGUMrvC?MoJ::::::::JCNZ ;F:f:vYxY:B::::::F:;JmJ:j;vCj^nGGmq>:;::::::::_Z:vyyuy:>:<::::::AJ:n:> :nYN::wyyyqj>J?>:Q:S:UJ:n;v;;JBB:]:_J:V<^Z:B:::yay=:: ::::=Jyyy;d:yayQZHQ:>:;`:Z@w<Wd Zfbk;>ZEZ?GHZhV@cYH_WV>Z::jysy:::J:V:Zy=::::::::^=N:qdjeM:>:>^qo:AR:=r:=b<=b>AfSHjA@ZEFZ<>kkjBN[AFXDJ;a\\AfRHjA:<=]:GvDD k;;J>A=J;[rK=EKKDk;]a=v;Zj:RD:>iH>:qe:N\\ =j:F;Hj>A:?JHJ:f:F[::::::;C:?jysy:>:;B:::::: ^:FL@>Z::::::::kJ;@j;>:C:yayA:<::::::I:ct:Z:f:^[<>b<>^GN\\:B:;xyyQMyyyyYjm Z:nv;B:QB:n>^;UTRcETcTX[US>FsZ:Vy<>jxM:<:[V:>Z:^ZcTTUUSaEBWTSiEB_tU UURWMEHN^YqIHN^YQJJSdJ:AJplj:B:MK>JSJu\\j:JN>;N@nrJSJpY j:>:]C:K:_;UU;=:[;N@^w:F:F@s:qQ:[:JBG:;J>:CB:f?;J;N@nq>F:NFJSJCDj:>:k=N@Nj:FZ:Jm^=VYZ:JB ?:DJ:DZJVdscRYEUXQZB:_H;Ob>grJKpIJJ::Uk:^:>X;J>JSd:?b<=Z:f_:>;N@^b@F:V ?K:_;wv:=J:fAO:G;Ojysy=J:>:s:qQBv::US:;J::::WTJWTL:fyyyY;Z::[RmQR:B:::[:B::nYZ::[:B::><:3:" }{TEXT -1 8 ", where " } {OLE 1 4102 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: fyyyyya:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::JR\\QA@mBil[Aj;J:M:<:=ja^GE=;:::::::::N;?R:yyyyyy:>:<:::::: JDJ:j:VBYmp>HYLkNG>::::::::N:; 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JJ:^Z:jP>:C:[q:>;N@;NKMj:B:kM>JSJZHj:<:GM>JSJQF:NFs:qAB:>L;Z:s:qQB v:tnQ::::WT JWTL:B>N:F:nyyyyy]::yyyyyy: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: fyyyyya:nYf::G:jy;:::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::JrduH@mBil[Aj;J:QZ:B:F:YLpfF>:::::::::J?NZ;vyyyyy=J:B:::::: :c:;:=j[vGUMrvC?MoJ::::::::JCNZ;F:f:vYxY:B::::::F:;JmJ:j;vCj^nGGmq>:;: :::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyqj>J?>:Q:S:UJ:n;v;;JBB :]:_J:V<^:R<:TnEB:UTTAeVYuVYeScEBETVeURcUTYeU;sFWCFl:Lnc;:;vkiJLhlEJ:VZ;FZ=FZDFZ LVj\\s:YR:hj:DJ>A=]J?DJ=q]=n;;R:_rZGM;DRNN\\=@IAr=::DrNNZ@P\\\\N\\=pF> ?::[[KZDj\\FHemj^HMmqnG;KaFFJufF;J::::::>^:N:yay= J:B::::::^:`:J:<:::::::>=?R:AJ:^:vYxY:B::::::v:;JJ I:<:=ja:;B:;:::::::::N;;B:yay=J:B::::::nYyA<::::::::::::jysy:>:<:::::: ::j\\:B: ;xyFqyyyyA^E>:my:V[:B:G;Sj`@Pt\\Pd`QrP@[Q>>:oi:;JBB:JZ:JBA:;B:Cb:JSJTQj:jRJSfs;FvCF:>@JSJRPj:jR:_KZEJ^ 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>AfSBD::@=]:?^:F:\\A:D=J=E;\\A:@=>Z:N[@@;DB ;:k=ElrfH=MtFGYMq>>Wlj:gmlJ::::::>^:N:yay=J:>Z::::::JZ:J;vCS=[L sfFaMR>@>Z::::::::kJ;@j;>:C:yayA:<::::::OB:;B:;T;>Z:j::<::::::::=j?jRR>dZ@@@[:>Z :N[;N<:JSR>d:::>@@[:B:xIyA:::::::V:^:;j:>:yayQZ:B:;:[v?vZAJyM:=j>>Z=vi :E:Mb::Gc;YJGv yyuy;B:[B:^;UTRcETcTX[US>XK:oi:;J>J:J<>Z:f_;j< B:cb:Ib:?s:EJ:V\\;;JwKB>Z::C:Uk:f:^<;C;?r:jC>^>N ]O^=;B:qi:;sy>Z:JBA:ZDfw=f`LF:;JR:_;;w==J:F`@>;N@oX:Ux >=:[;N@^]IF:F@JSfWKG<=:];N@v\\@F:;Z@:djgAJZdj:<:];N@Vl?F:b;ZD>v>FV=:TJ Hjw?jxM:<:[>gNF:F_@>;N@>kLF:;JNTJ>J SJMyj:jN:_;SY==:K;N@qU:Oh<=:M;N@>\\CF:>?JSJQTj:jN:_;OD;=:KKHjw;<:[n:>: Cb:^D:::y:;;Z:jPN:C:[q;JS^o?>tFF:>i@>;N@n^DF:>IJSfk=^kKF:>IJSJ@xj:J v:_Kw^^MFZ:Jv:_;?X>=B:nf@>;N@>kHF:>IJSJMpj:Jm:_k@_\\BF:>IJSJoPj:Jv:_kw ^oCF:>IJSJ_\\j:Jm:_;GI;=:k=N@^j=F:nFJS>j>n?=:?e;s:qAB:>L;Z<>ZaTXDpq l`q?B>:jPV:C:[Q;>;N`D^OOw<=:Uc;K:_k@ofJF:f?JSNIGE>=:U; N@Ie:Ow<=:U;N@UNqUj:jP:_kwnw@F:f?JS>J[X:=J:fa@N;;JMJ?vYxy:J:^=VY[j=:sg:>Z:F:?bBaTXaEWEUUB:OB:=K?A::[Hw_jNDN^YQ<>:US:?J<<:[q;V: " 0 "" {MPLTEXT 1 0 14 "datapoints:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "p:=1 :\nfor i from 1 to n do\nif d[i] <= datapoints then\nxdata[p]:=x[i];\n ydata[p]:=y[i];\np:=p+1;\nend if\nend do:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "entries(xdata);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %7#\"#57#\"#?7#\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "entr ies(ydata);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7#$\"&/F#!\"#7#$\"&N<&F &7#$\"&yi$F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 66 "Calculating coefficients of Newton's Divided difference polynomial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b0:=ydata[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#b0G$\"&/F#!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b1:=(yd ata[2]-ydata[1])/(xdata[2]-xdata[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G$\"+++5.H!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 " b2:=((ydata[3]-ydata[2])/(xdata[3]-xdata[2])-(ydata[2]-ydata[1])/(xdat a[2]-xdata[1]))/(xdata[3]-xdata[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#b2G$\"++++mP!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Newton's divided difference formula for quadrat ic interpolation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "fquad(z):=b0+b1*(z-xdata[1])+b2*(z-xdata[ 1])*(z-xdata[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&fquadG6#%\"zG ,($\"*++qK'!\"(!\"\"*&$\"+++5.H!\")\"\"\"F'F1F1*($\"++++mP!#5F1,&F'F1 \"#5F,F1,&F'F1\"#?F,F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Subs tituting the value of the desired X value for z in the above equation, the corresponding Y value is found." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eval(fquad(z),z=xdesired);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++g(=#R!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "fnew:=%:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "The absolute percentage relative approximate e rror between the first order and second order interpolation is" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "epsilon:=abs((fnew-fprev)/fn ew)*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"+T*=5%Q!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The number of significant digit s at least correct in the solution is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sigdig:=floor(2-log10(epsilon/0.5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigdigG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fprev:=fnew:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 " Plotting the quadratic interpolant and the value of Y for the desired \+ X" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "plot([[t,eval(fquad(z),z=t),t=ranger(xdata,datapoints)],xy,[[xdes ired,eval(fquad(z),z=xdesired)]]],z=ranger(x,n),style=[LINE,POINT,POIN T],color=[RED,BLUE,BLUE],symbol=[CROSS,CIRCLE],symbolsize=[40,30],thic kness=3,title=\"Quadratic interpolation\");" }}{PARA 13 "" 1 "" {GLPLOT2D 535 535 535 {PLOTDATA 2 "6)-%'CURVESG6&7S7$$\"#5\"\"!$\"3#** *********RqA!#:7$$\"3vmm;arz@5!#;$\"3'GBh'Q%\\cK#F-7$$\"3GL$e9ui2/\"F1 $\"3YCaEDG,uBF-7$$\"3omm\"z_\"4i5F1$\"3eNO/thsGCF-7$$\"3ummT&phN3\"F1$ \"3WaTTa#[T[#F-7$$\"3CL$e*=)H\\5\"F1$\"3Fl_6a;lRDF-7$$\"3jm;z/3uC6F1$ \"3;\"3y.k<9f#F-7$$\"3)***\\7LRDX6F1$\"3eflPM)H`k#F-7$$\"3km;zR'ok;\"F 1$\"33$GR`G>9q#F-7$$\"3%***\\i5`h(=\"F1$\"3YbXz:imdFF-7$$\"3PLL$3En$47 F1$\"3v!4^O^ue\"GF-7$$\"3ummT!RE&G7F1$\"39AV&4gRu'GF-7$$\"3)*****\\K]4 ]7F1$\"3fS@j(\\?e#HF-7$$\"3)*****\\PAvr7F1$\"3YM$fjm$z%)HF-7$$\"3-++]n Hi#H\"F1$\"3v8i#z]f>/$F-7$$\"3tm;z*ev:J\"F1$\"3G_M75e:%4$F-7$$\"3SLL$3 47TL\"F1$\"3!eey.Rul:$F-7$$\"3QLLLjM?`8F1$\"3[\"e^=T](4KF-7$$\"3)***\\ 7o7Tv8F1$\"3E'R)Hq@&>F$F-7$$\"3OLLLQ*o]R\"F1$\"3pf#3'*>@tK$F-7$$\"32+] 7=lj;9F1$\"3KNA@YYS)Q$F-7$$\"3&***\\PaRF1$\"3))G0^zED/\\F-7$$\"3hmmmw(Gp$>F1$\"3E#>0eTV\"o\\F-7$$ \"3')**\\7oK0e>F1$\"3()))H$=%**eO]F-7$$\"3I+](=5s#y>F1$\"3%zxlf!oT-^F- 7$$\"#?F*$\"3A+++++]t^F--%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F`[l-%&ST YLEG6#%%LINEG-%'SYMBOLG6$%&CROSSG\"#S-F$6&7(F'7$F`[lF`[lFdz7$$\"#:F*$ \"3t**********zFOF-7$$\"#IF*$\"3f**********p;!*F-7$$\"3+++++++]AF1$\"3 F+++++qHgF--Fjz6&F\\[lF`[lF`[lF][l-Fb[l6#%&POINTG-Ff[l6$%'CIRCLEGFe\\l -F$6&7#7$$\"#;F*$\"3v********f(=#RF-F]]lF_]lFe[l-%*THICKNESSG6#\"\"$-% +AXESLABELSG6$Q\"z6\"Q!Fe^l-%&TITLEG6#Q8Quadratic~interpolationFe^l-%% VIEWG6$;F`[lFd\\l%(DEFAULTG" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 265 55 "Section V: Cubic Interpolation (Third or der polynomial)" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "The same method as above is used for 4 data points and the correspond ing coefficients, are calcualted and evaluated in the formula as show n below to find the corresponding Y value for the desired X value." }} {PARA 0 "" 0 "" {TEXT -1 83 "Hence, the four closest data points to th e desired value are chosen in this method." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "datapoints:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "p:=1:\nfor i from 1 to n do\nif d[i] <= datapoints t hen\nxdata[p]:=x[i];\nydata[p]:=y[i];\np:=p+1;\nend if\nend do:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "entries(xdata);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&7#\"#57#\"#?7#\"#:7#$\"$D#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "entries(ydata);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&7#$\"&/F#!\"#7#$\"&N<&F&7#$\"&yi$F&7#$\"&(HgF&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Ca lculating coefficients of Newton's Divided difference polynomial" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "b0:=ydata[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G$\"&/F#! \"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b1:=(ydata[2]-ydata[ 1])/(xdata[2]-xdata[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b1G$\"+ ++5.H!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "b2:=((ydata[3 ]-ydata[2])/(xdata[3]-xdata[2])-(ydata[2]-ydata[1])/(xdata[2]-xdata[1] ))/(xdata[3]-xdata[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G$\"++ ++mP!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "b3:=(((ydata[4] -ydata[3])/(xdata[4]-xdata[3])-(ydata[3]-ydata[2])/(xdata[3]-xdata[2]) )/(xdata[4]-xdata[2])-b2)/(xdata[4]-xdata[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b3G$\"+glmMa!#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Newton's divided difference formul a for cubic interpolation is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "fcubic(z):=b0+b1*(z-xdata[1 ])+b2*(z-xdata[1])*(z-xdata[2])+b3*(z-xdata[1])*(z-xdata[2])*(z-xdata[ 3]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>-%'fcubicG6#%\"zG,*$\"*++qK'!\"(!\"\"*&$\"+++5.H!\")\"\"\"F'F1F1*($ \"++++mP!#5F1,&F'F1\"#5F,F1,&F'F1\"#?F,F1F1**$\"+glmMa!#7F1F6F1F8F1,&F 'F1\"#:F,F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Substituting th e value of the desired X value for z in the above equation, the corres ponding Y value is found." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eval(fcubic(z),z=xdesired);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +!or0#R!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "fnew:=%:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "The absolute percentage relative \+ approximate error between the second order and third order interpolati on is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "epsilon:=abs((fnew -fprev)/fnew)*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"+g <'oK$!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The number of signifi cant digits at least correct in the solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sigdig:=floor(2-log10(epsilon/0.5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sigdigG\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Plotting the cubic interpolant and the value of Y for the desired X" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 244 "plot([[t,eval(fcubic(z),z=t),t=ranger(xdata,datapoin ts)],xy,[[xdesired,eval(fcubic(z),z=xdesired)]]],z=ranger(x,n),style=[ LINE,POINT,POINT],color=[RED,BLUE,BLUE],symbol=[CROSS,CIRCLE],symbolsi ze=[40,30],thickness=3,title=\"Cubic interpolation\");" }}{PARA 13 "" 1 "" {GLPLOT2D 539 539 539 {PLOTDATA 2 "6)-%'CURVESG6&7S7$$\"#5\"\"!$ \"3#***********RqA!#:7$$\"3ML$3FWYs-\"!#;$\"3s5Bcr')>SBF-7$$\"3u;H#oU` 40\"F1$\"3YVrl)p\"H,CF-7$$\"3OLe*)4Whx5F1$\"3Cd8X>[SqCF-7$$\"3UL3F>@X/ 6F1$\"3b#fx+#zTSDF-7$$\"3t;zptA;J6F1$\"3Zkxm_Ya5EF-7$$\"3U$e*)f+Ef:\"F 1$\"3=oawoV'fn#F-7$$\"3-]iS;uc\"=\"F1$\"3znL3$*p6WFF-7$$\"3M$eR(*z&337 F1$\"3;@Ah*p\\]\"GF-7$$\"3)*\\7GQ\">XB\"F1$\"3+0UoCa@')GF-7$$\"3wm;/w! 4W5B\"p)Gw$F-7$$ \"3wm\"H#oKDt:F1$\"3.NY-4&3:%QF-7$$\"3s;zW<5&yf\"F1$\"3C.;;:')>9RF-7$$ \"3RL3-8IQC;F1$\"3mzmV&QPJ*RF-7$$\"3e$eR(*HU>l\"F1$\"3#e*=e+VsvSF-7$$ \"3!*\\P%)RF$fn\"F1$\"3Wy'o4f7\"[TF-7$$\"3>LeRsI%=q\"F1$\"3%=:**G2=oA% F-7$$\"38++D\")4hGM\"R* Q%F-7$$\"37]il(fN,y\"F1$\"3EC!p1M/!oWF-7$$\"3'**\\(o/&o#3=F1$\"3'*\\2% o6Efb%F-7$$\"3nmm\"HF1$\"3+ PUs_W9&)[F-7$$\"3y;H#=v\"*o$>F1$\"3G$e)p*))>m'\\F-7$$\"3I]7yv(*=j>F1$ \"3OUto*G\"R_]F-7$$\"3Ym;/,4!*))>F1$\"3)o.Ir/`o8&F-7$$\"3!*\\iSm!=e,#F 1$\"3qzv$Q=BfA&F-7$$\"3]LL$e\\U#F1$\"3=gK2i$G^%eF-7$$\"3>]PMF,%GA#F1$ \"3t2o(flaP$fF-7$$\"3+++++++]AF1$\"3^***\\(****pHgF--%'COLOURG6&%$RGBG $\"*++++\"!\")$F*F*F`[l-%&STYLEG6#%%LINEG-%'SYMBOLG6$%&CROSSG\"#S-F$6& 7(F'7$F`[lF`[l7$$\"#?F*$\"3A+++++]t^F-7$$\"#:F*$\"3t**********zFOF-7$$ \"#IF*$\"3f**********p;!*F-7$Fez$\"3F+++++qHgF--Fjz6&F\\[lF`[lF`[lF][l -Fb[l6#%&POINTG-Ff[l6$%'CIRCLEGFj\\l-F$6&7#7$$\"#;F*$\"3!*******z;d?RF -F`]lFb]lFe[l-%*THICKNESSG6#\"\"$-%+AXESLABELSG6$Q\"z6\"Q!Fh^l-%&TITLE G6#Q4Cubic~interpolationFh^l-%%VIEWG6$;F`[lFi\\l%(DEFAULTG" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 282 0 "" }{TEXT 283 23 "Section V I: Conclusion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 311 "Maple helped us to apply our knowledge of numerical meth ods of interpolation to find the value of y at a particular value of x using first, second, and third order Newton's Divided Difference Poly nomial method of interpolation. Using Maple functions and plotting rou tines made it easy to illustrate this method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT 285 10 "References" }}{PARA 0 " " 0 "" {TEXT -1 4 "[1] " }{TEXT 287 171 "Nathan Collier, Autar Kaw, Ja i Paul , Michael Keteltas, Holistic Numerical Methods Institute, See h ttp://numericalmethods.eng.usf.edu/mws/gen/05inp/mws_gen_inp_sim_ndd.m ws" }}{PARA 0 "" 0 "" {TEXT 288 73 "http://numericalmethods.eng.usf.ed u/mws/gen/05inp/mws_gen_inp_txt_ndd.pdf" }{TEXT -1 0 "" }}{PARA 259 " " 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 284 10 "Disclaimer" } {TEXT 286 1 ":" }{TEXT -1 248 " While every effort has been made to va lidate the solutions in this worksheet, University of South Florida an d the contributors are not responsible for any errors contained and ar e not liable for any damages resulting from the use of this material. " }}}{MARK "14 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }