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"R3 Font 21" -1 276 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 0 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 22" -1 277 1 {CSTYLE "" -1 -1 "Luci datypewriter" 0 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 23" -1 278 1 {CSTYLE "" -1 -1 "Lucidatypewr iter" 0 14 105 101 100 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 24" -1 279 1 {CSTYLE "" -1 -1 "Lucidatypewrit er" 0 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 25" -1 280 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 0 14 33 1 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 26" -1 281 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 0 14 0 0 36 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 27" -1 282 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 0 14 250 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 28" -1 283 1 {CSTYLE "" -1 -1 "Lucidatypewriter" 0 14 0 0 244 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 284 1 {CSTYLE "" -1 -1 "" 0 1 32 116 50 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "verbatim" -1 285 1 {CSTYLE "" -1 -1 "" 0 0 0 128 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 286 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {SECT 0 {PARA 284 "" 0 "" {TEXT -1 30 " Tridiagonal matrices c ome up." }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "Heating a Metal Rod" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 8 "Problem." }{TEXT -1 281 " The temperature at the ends of a 4 foot metal rod is being maintained at \+ 100 and 150 degrees respectively. Initially, the rod is uniformly 0 degrees everywhere. Three thermometers spaced at 1 foot intervals ar e consulted at 1 minute intervals and the temperatures recorded." }} {PARA 285 "" 0 "" {TEXT -1 211 " x1 \+ x2 x3\n ===================================== ===========\n 100 0 0 \+ 0 150\n" }}{PARA 0 "" 0 "" {TEXT -1 455 " The person in charge of recording the temperatures notes that the t emp at x2 is 90 percent of the average of the temps at x1, x2, and x3 the previous minute, the temp at x1 is 90 percent of the average of \+ 100 and the temps at x1 and x2 the previous minute, and the temp at \+ x3 is 90 percent of the average of 150 and the temps at x2 and x3 the previous minute. What will the temperature readings be in 100 min utes? Will this thing settle down?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 257 8 "Solution" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We have a tridiagona l system of equations!\n" }}{PARA 285 "" 0 "" {TEXT -1 201 "y1 = .9 ( 100 + x1 + x2)/3 ([100] [1 1 0] [x1]) \ny2 = .9 (x 1 + x2 + x3)/3 = .9/3 ([ 0 ] + [1 1 1] [x2])\ny3 = .9 (x2 + x3 + 150)/3 ([150] [0 1 1] [x3])\n" }}{PARA 0 "" 0 "" {TEXT -1 183 "We want to track the sequence of temperature readi ngs t0, t1, t2, t3, ... where t0 is any initial set of temperatures a nd the next set is obtained from the last set by the equation\n" }} {PARA 285 "" 0 "" {TEXT -1 107 "t(n+1) = .9/3 (etmps + A tn), where etmps = ( 100, 0, 150) and\n A = [[1,1,0],[1,1,1],[0,1,1]]. \n" }}{PARA 0 "" 0 "" {TEXT -1 36 " Load in the linear algebra package ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;with(linalg);" }} {PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }} {PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,GramSchmidtG%,Jor danBlockG%)LUdecompG%)QRdecompG%*WronskianG%'addcolG%'addrowG%$adjG%(a djointG%&angleG%(augmentG%(backsubG%%bandG%&basisG%'bezoutG%,blockmatr ixG%(charmatG%)charpolyG%)choleskyG%$colG%'coldimG%)colspaceG%(colspan G%*companionG%'concatG%%condG%)copyintoG%*crossprodG%%curlG%)definiteG %(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotprodG%*eigenvalsG%,eig envaluesG%-eigenvectorsG%+eigenvectsG%,entermatrixG%&equalG%,exponenti alG%'extendG%,ffgausselimG%*fibonacciG%+forwardsubG%*frobeniusG%*gauss elimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hadamardG%(hermiteG%(h essianG%(hilbertG%+htransposeG%)ihermiteG%*indexfuncG%*innerprodG%)int basisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)jacobianG%'jordanG%'ke rnelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG%'matrixG%&minorG%(mi npolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normalizeG%*nullspaceG%'or thogG%*permanentG%&pivotG%*potentialG%+randmatrixG%+randvectorG%%rankG %(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rrefG%*scalarmulG%-sin gularvalsG%&smithG%&stackG%*submatrixG%*subvectorG%)sumbasisG%(swapcol G%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*transposeG%,vandermondeG%* vecpotentG%(vectdimG%'vectorG%*wronskianG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We define a word to build tridiagonal matrices easily. \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "tridiag := proc(a,b,c,n)\n lo cal B, i, j;\n B := matrix(n,n);\n for i from 1 to n do for j from 1 to n do \n if j = i -1 then B[i,j] := a \n elif j = 1+i th en B[i,j] := c\n elif j = i then B[i,j] :=b else B[i,j] := 0 fi od o d;\n evalm(B);\n end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 " \n\n To set up the particular problem we started with, let " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "n := 3:\nA :=tridiag(1.,1.,1.,n); \netmps := vector([ 100,seq( 0,i=1..n-2),150]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7%$\"\"\"\"\"!F*F,7%F*F*F*7%F,F*F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&etmpsG-%'VECTORG6#7%\"$+\"\"\"! \"$]\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 " \nWe can start with a n initially cold rod, where the thermometers\n read 0 degrees accross \+ the board." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "step := vector([0,0,0 ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%stepG-%'VECTORG6#7%\"\"!F)F) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " step := evalm(.9/3*(et mps+A&*step));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%stepG-%'VECTORG6# 7%$\"+++++I!\")\"\"!$\"+++++XF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "for i from 1 to 15 do step := evalm(.9/3*(etmps+A&*step)) ;\n print(evalm(step));\n \+ od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+++++R!\")$ \"++++]AF)$\"++++]eF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7 %$\"++++X[!\")$\"+++++OF)$\"++++IpF)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'VECTORG6#7%$\"+++]Lb!\")$\"+++]7YF)$\"++++fwF)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'VECTORG6#7%$\"+++!Q/'!\")$\"+++]T`F)$\"+++X\"=)F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+++f:k!\")$\"++]- qeF)$\"++])ob)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+ +Xo&o'!\")$\"+++v_iF)$\"++I2G))F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'VECTORG6#7%$\"+].`\")o!\")$\"+]A&*HlF)$\"++pCC!*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+!yWM-(!\")$\"+])=2t'F)$\"+X(fi;*F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+*3\\i7(!\")$\"+9 q7woF)$\"+!e$4p#*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$ \"+IGr+s!\")$\"+/4W\")pF)$\"+xhcV$*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+>hkas!\")$\"+tfrdqF)$\"+C@](R*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+F'3PH(!\")$\"+j#fH6(F)$\"+IacO%* F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+n.+At!\")$\"+' **pH:(F)$\"+2u&[Y*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7% $\"+36\\Ut!\")$\"+J$[>=(F)$\"+?#[`[*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+K=Ldt!\")$\"+)HOH?(F)$\"+l*)=+&*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 295 "So we can see that there is some apparent convergence of the t emperature distribution to a stable configuration. We can try to rep resent this graphically some way. For example, a simple movie showing a plot of the temperature curve as it changes over time is easily do ne and can be useful. " }}{PARA 0 "" 0 "" {TEXT -1 136 " All we have \+ to do here is keep a list of plots and display them later. First a li ttle word (unnecessary but it improves readibality). " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 " plotstep := proc(step,n)\nlocal i;\n plot([[0,0],[0,100],\n se q([10*i,step[i]],i = 1 .. n),\n [40,150]])\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 328 "n := 3:\nA :=tridiag(1.,1.,1.,n); \netmps := vector([ 100,seq( 0,i=1..n-2),150]);\nstep := vector([0,0,0 ]);\nmovie := plotstep(step,n):\n step := evalm(.9/3*(etmps+A&*step)); \n movie := movie,plotstep(step,n):\nfor i from 1 to 30 do step := eva lm(.9/3*(etmps+A&*step)) ;\n movie := movie, plotstep(step,n):\n pri nt(evalm(step));\n od :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%' MATRIXG6#7%7%$\"\"\"\"\"!F*F,7%F*F*F*7%F,F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&etmpsG-%'VECTORG6#7%\"$+\"\"\"!\"$]\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%stepG-%'VECTORG6#7%\"\"!F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%stepG-%'VECTORG6#7%$\"+++++I!\")\"\"!$\"+++++XF +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+++++R!\")$\"+++ +]AF)$\"++++]eF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+ +++X[!\")$\"+++++OF)$\"++++IpF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' VECTORG6#7%$\"+++]Lb!\")$\"+++]7YF)$\"++++fwF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+++!Q/'!\")$\"+++]T`F)$\"+++X\"=)F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+++f:k!\")$\"++]-qeF )$\"++])ob)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"++Xo &o'!\")$\"+++v_iF)$\"++I2G))F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'V ECTORG6#7%$\"+].`\")o!\")$\"+]A&*HlF)$\"++pCC!*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+!yWM-(!\")$\"+])=2t'F)$\"+X(fi;*F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+*3\\i7(!\")$\"+9q7w oF)$\"+!e$4p#*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+ IGr+s!\")$\"+/4W\")pF)$\"+xhcV$*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'VECTORG6#7%$\"+>hkas!\")$\"+tfrdqF)$\"+C@](R*F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'VECTORG6#7%$\"+F'3PH(!\")$\"+j#fH6(F)$\"+IacO%*F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+n.+At!\")$\"+'** pH:(F)$\"+2u&[Y*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$ \"+36\\Ut!\")$\"+J$[>=(F)$\"+?#[`[*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+K=Ldt!\")$\"+)HOH?(F)$\"+l*)=+&*F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+R/3ot!\")$\"+Ir8=sF)$\"+yv$4^* F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+r_'eP(!\")$\"+ Xl9HsF)$\"+8Cs=&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$ \"+YN]\"Q(!\")$\"+p-7PsF)$\"+)ogV_*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+Yre&Q(!\")$\"+``*GC(F)$\"+)GW%G&*F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+]Za)Q(!\")$\"+P!yqC(F)$\"+#*=S J&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+Poo!R(!\")$ \"+/u5]sF)$\"+zRaL&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6# 7%$\"+s#QAR(!\")$\"+m9I_sF)$\"+9a4N&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+A>O$R(!\")$\"+X0*QD(F)$\"+k!>i`*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+Td<%R(!\")$\"+f9/bsF)$\"+$ )G.P&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+g^w%R(! \")$\"+C](eD(F)$\"+-BiP&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECT ORG6#7%$\"+a?>&R(!\")$\"+W(ykD(F)$\"+*>\\!Q&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+S7]&R(!\")$\"++g\"pD(F)$\"+#Qe$Q&*F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+s^s&R(!\")$\"+'o KsD(F)$\"+9BeQ&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$ \"+et)eR(!\")$\"+^?YdsF)$\"++XuQ&*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'VECTORG6#7%$\"+B[+'R(!\")$\"+t\"GwD(F)$\"+l>')Q&*F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "We can look at all of these plots togethe r, and get a good sense of the convergence." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plots[display]([movie]);" }}{PARA 13 "" 1 "" {INLPLOT "6B-%'CURVESG6$7(7$\"\"!F(7$F($\"$+\"F(7$$\"#5F(F(7$$\"#?F(F( 7$$\"#IF(F(7$$\"#SF($\"$]\"F(-%'COLOURG6&%$RGBG$F.!\"\"F(F(-F$6$7(F'F) 7$F-F3F/7$F3$\"#XF(F5F:-F$6$7(F'F)7$F-$\"#RF(7$F0$\"1++++++]A!#97$F3$ \"1++++++]eFPF5F:-F$6$7(F'F)7$F-$\"1++++++X[FP7$F0$\"#OF(7$F3$\"1+++++ +IpFPF5F:-F$6$7(F'F)7$F-$\"1+++++]LbFP7$F0$\"1+++++]7YFP7$F3$\"1++++++ fwFPF5F:-F$6$7(F'F)7$F-$\"1+++++!Q/'FP7$F0$\"1+++++]T`FP7$F3$\"1+++++X \"=)FPF5F:-F$6$7(F'F)7$F-$\"1+++++f:kFP7$F0$\"1++++]-qeFP7$F3$\"1++++] )ob)FPF5F:-F$6$7(F'F)7$F-$\"1,+++Xo&o'FP7$F0$\"1+++++v_iFP7$F3$\"1,+++ I2G))FPF5F:-F$6$7(F'F)7$F-$\"1+++].`\")oFP7$F0$\"1******\\A&*HlFP7$F3$ \"1++++pCC!*FPF5F:-F$6$7(F'F)7$F-$\"1+++!yWM-(FP7$F0$\"1+++])=2t'FP7$F 3$\"1,++X(fi;*FPF5F:-F$6$7(F'F)7$F-$\"1+++*3\\i7(FP7$F0$\"1*****R,Fh(o FP7$F3$\"1,++!e$4p#*FPF5F:-F$6$7(F'F)7$F-$\"1,++IGr+sFP7$F0$\"1+++/4W \")pFP7$F3$\"1+++xhcV$*FPF5F:-F$6$7(F'F)7$F-$\"1+++>hkasFP7$F0$\"1,++t frdqFP7$F3$\"1+++C@](R*FPF5F:-F$6$7(F'F)7$F-$\"1*****pi3PH(FP7$F0$\"1+ ++j#fH6(FP7$F3$\"1+++IacO%*FPF5F:-F$6$7(F'F)7$F-$\"1+++n.+AtFP7$F0$\"1 +++'**pH:(FP7$F3$\"1+++2u&[Y*FPF5F:-F$6$7(F'F)7$F-$\"1+++36\\UtFP7$F0$ 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F:-F$6$7(F'F)7$F-$\"1+++s^s&R(FP7$F0$\"1+++'oKsD(FP7$F3$\"1+++9BeQ&*FP F5F:-F$6$7(F'F)7$F-$\"1+++et)eR(FP7$F0$\"1+++^?YdsFP7$F3$\"1++++XuQ&*F PF5F:-F$6$7(F'F)7$F-$\"1+++B[+'R(FP7$F0$\"1,++t\"GwD(FP7$F3$\"1+++l>') Q&*FPF5F:" 2 397 397 397 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 17747 21586 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "\nOr we can look at them in sequence, to get another feeling the \+ convergence of the approximation toward a limiting set of temp reading s.\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plots[display]([movie],ins equence=true);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%(ANIMATEG6B7#-%'CURVE SG6$7(7$\"\"!F,7$F,$\"$+\"F,7$$\"#5F,F,7$$\"#?F,F,7$$\"#IF,F,7$$\"#SF, $\"$]\"F,-%'COLOURG6&%$RGBG$F2!\"\"F,F,7#-F(6$7(F+F-7$F1F7F37$F7$\"#XF ,F9F>7#-F(6$7(F+F-7$F1$\"#RF,7$F4$\"1++++++]A!#97$F7$\"1++++++]eFVF9F> 7#-F(6$7(F+F-7$F1$\"1++++++X[FV7$F4$\"#OF,7$F7$\"1++++++IpFVF9F>7#-F(6 $7(F+F-7$F1$\"1+++++]LbFV7$F4$\"1+++++]7YFV7$F7$\"1++++++fwFVF9F>7#-F( 6$7(F+F-7$F1$\"1+++++!Q/'FV7$F4$\"1+++++]T`FV7$F7$\"1+++++X\"=)FVF9F>7 #-F(6$7(F+F-7$F1$\"1+++++f:kFV7$F4$\"1++++]-qeFV7$F7$\"1++++])ob)FVF9F >7#-F(6$7(F+F-7$F1$\"1,+++Xo&o'FV7$F4$\"1+++++v_iFV7$F7$\"1,+++I2G))FV F9F>7#-F(6$7(F+F-7$F1$\"1+++].`\")oFV7$F4$\"1******\\A&*HlFV7$F7$\"1++ ++pCC!*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++!yWM-(FV7$F4$\"1+++])=2t'FV7$F7$ \"1,++X(fi;*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++*3\\i7(FV7$F4$\"1*****R,Fh(o FV7$F7$\"1,++!e$4p#*FVF9F>7#-F(6$7(F+F-7$F1$\"1,++IGr+sFV7$F4$\"1+++/4 W\")pFV7$F7$\"1+++xhcV$*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++>hkasFV7$F4$\"1, ++tfrdqFV7$F7$\"1+++C@](R*FVF9F>7#-F(6$7(F+F-7$F1$\"1*****pi3PH(FV7$F4 $\"1+++j#fH6(FV7$F7$\"1+++IacO%*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++n.+AtFV7 $F4$\"1+++'**pH:(FV7$F7$\"1+++2u&[Y*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++36\\ UtFV7$F4$\"1+++J$[>=(FV7$F7$\"1+++?#[`[*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++ K=LdtFV7$F4$\"1+++)HOH?(FV7$F7$\"1+++l*)=+&*FVF9F>7#-F(6$7(F+F-7$F1$\" 1,++R/3otFV7$F4$\"1+++Ir8=sFV7$F7$\"1*****zdP4^*FVF9F>7#-F(6$7(F+F-7$F 1$\"1,++r_'eP(FV7$F4$\"1+++Xl9HsFV7$F7$\"1+++8Cs=&*FVF9F>7#-F(6$7(F+F- 7$F1$\"1+++YN]\"Q(FV7$F4$\"1+++p-7PsFV7$F7$\"1+++)ogV_*FVF9F>7#-F(6$7( F+F-7$F1$\"1+++Yre&Q(FV7$F4$\"1+++``*GC(FV7$F7$\"1+++)GW%G&*FVF9F>7#-F (6$7(F+F-7$F1$\"1+++]Za)Q(FV7$F4$\"1+++P!yqC(FV7$F7$\"1+++#*=SJ&*FVF9F >7#-F(6$7(F+F-7$F1$\"1+++Poo!R(FV7$F4$\"1,++/u5]sFV7$F7$\"1+++zRaL&*FV F9F>7#-F(6$7(F+F-7$F1$\"1+++s#QAR(FV7$F4$\"1+++m9I_sFV7$F7$\"1+++9a4N& *FVF9F>7#-F(6$7(F+F-7$F1$\"1+++A>O$R(FV7$F4$\"1+++X0*QD(FV7$F7$\"1**** *R1>i`*FVF9F>7#-F(6$7(F+F-7$F1$\"1*****4uvTR(FV7$F4$\"1+++f9/bsFV7$F7$ \"1,++$)G.P&*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++g^w%R(FV7$F4$\"1*****R-veD( FV7$F7$\"1+++-BiP&*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++a?>&R(FV7$F4$\"1+++W( ykD(FV7$F7$\"1+++*>\\!Q&*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++S7]&R(FV7$F4$\" 1++++g\"pD(FV7$F7$\"1+++#Qe$Q&*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++s^s&R(FV7 $F4$\"1+++'oKsD(FV7$F7$\"1+++9BeQ&*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++et)eR (FV7$F4$\"1+++^?YdsFV7$F7$\"1++++XuQ&*FVF9F>7#-F(6$7(F+F-7$F1$\"1+++B[ +'R(FV7$F4$\"1,++t\"GwD(FV7$F7$\"1+++l>')Q&*FVF9F>" 2 397 397 397 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 100 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 186 "\nSo, on the basis of our calculation an d picture drawing, we could say that the three thermometers will rea d about 74, 72.5, and 95.4 degrees from left to right after 30 minutes or so." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 10 "Problem 1" }{TEXT -1 298 ". Does this end temperature distrib ution depend on the initial distribution? Experiment, by changing t he first step from [0,0,0] to some other things like [10,20,10] and ru nning the cells again. Better still, copy the input cells down to here and make the changes here. Discuss the results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 9 "Problem 2" }{TEXT -1 271 ". Suppose there is no heat loss and the 90 percent becomes 100 per cent. What happens? Think about it and try to predict what will hap pen. Then copy down the appropriate input cells and make appropriate \+ changes, and experiment. Discuss. Was your guess supported?\n" }} {PARA 0 "" 0 "" {TEXT 261 9 "Problem 3" }{TEXT -1 144 ". Suppose th e rod is 6 feet long, and there are 5 thermometers. With end temps o f 100 and 150, what is the minimum stable temp of the rod?\n" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "Back \+ to the original problem" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 " Using Eigenvalues and eigenvectors here can help us analyse the original pr oblem. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "First compute them. We could do this directly, but for the mo ment we can do it the fast way, using eigenvects from the linalg packa ge." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " eigs := eigenvects(A); " } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eigsG6%7%$\"+++++5!\"*\"\"\"<#-%'V ECTORG6#7%$!+3y1rq!#5$!#9!#6$\"+7y1rqF27%$\"+kN@9CF)F*<#-F-6#7%$\"+)** *****\\F2$\"+9y1rqF2$\"+-+++]F27%$!+Ec8UTF2F*<#-F-6#7%$\"+++++]F2$!+8y 1rqF2$\"+,+++]F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "So there are \+ three eigenvalues: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eigs[1][1] , eigs[2][1],eigs[3][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+++++5! \"*$\"+kN@9CF%$!+Ec8UT!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "The eigenvectors belonging to the first eigenvalue are all scalar multipl es of the 1 eigenvector that is given. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " eigs[1][3][1];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$!+3y1rq!#5$!#9!#6$\"+7y1rqF)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "\nIn general, each eigenvalue com es with an integer specifying the dimension of the space of eigenvecto rs belonging to that eigenvalue, and a set of basis vectors for that e igenspace. " }}{PARA 0 "" 0 "" {TEXT -1 219 "Going back to the origin al analysis, we want to track the sequence of temperature readings t0 , t1, t2, t3, ... where t0 is any initial set of temperatures and the next set is obtained from the last set by the equation" }}{PARA 0 "" 0 "" {TEXT -1 68 "t(n+1) = .9/3 (etmps + A tn), where etmps = ( 1 00, 0, 150) and" }}{PARA 0 "" 0 "" {TEXT -1 38 " A = [[1,1,0],[ 1,1,1],[0,1,1]].\n" }}{PARA 0 "" 0 "" {TEXT -1 102 "For the moment, l et r = .9/3, and simply compute the first few terms of the sequence t 0, t1, t2, ..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 285 "" 0 "" {TEXT -1 178 "t1 = r(etmps + A t0)\nt2 = r(etmps + A t1) = r(etmps + A r(emtps + A t0))\n =r etmps + r^2 A etmps + r^2 A^2 t0.\nt3 = \+ r etmps + r^2 A etmps + r^3 A^2 etmps + r^3 A^3 t0." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "r :='r':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " f := t0 -> r*( etmps + A*t0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6#%#t0G6\"6$%)operatorG%&arrowGF(*&%\"rG\"\"\",&%&etmpsGF .*&%\"AGF.9$F.F.F.F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "We can see that the nth set tn of temp readings is a partial sum of a geomet ric series of vectors plus a final term. For example, the set t6 look s like :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "expand( (f@@6)(t0));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&%\"rG\"\"\"%&etmpsGF&F&*(%\"AGF&F %\"\"#F'F&F&*(F)F*F%\"\"$F'F&F&*(F)F,F%\"\"%F'F&F&*(F)F.F%\"\"&F'F&F&* (F)F0F%\"\"'F'F&F&*(F)F2F%F2%#t0GF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "\nSo, write etmps as a linear combination of the eige nvectors and sum the geometric series of scalar multiples that pops \+ out for each of the eigenvectors. In each case, " }}{PARA 286 "" 0 "" {TEXT -1 55 "\n(1 + r*lamda + (r*lamda)^2 + ... ) = r *1/(1-r*lamd a)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "eqn := evalm(a*eigs[1][3][1]+ b*eigs[2][3][1]+c*eigs[3][3][1]-etmps);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eqnG-%'VECTORG6#7%,*%\"aG$!+3y1rq!#5%\"bG$\"+)*******\\F-%\"c G$\"+++++]F-!$+\"\"\"\",(F*$!#9!#6F.$\"+9y1rqF-F1$!+8y1rqF-,*F*$\"+7y1 rqF-F.$\"+-+++]F-F1$\"+,+++]F-!$]\"F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sol := solve(\{eqn[1],eqn[2],eqn[3]\},\{a,b,c\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG<%/%\"bG$\"++++]7!\"(/%\"cGF(/% \"aG$\"+-R`NN!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 " Now by summ ing three geometric series, we can compute the limiting set of tempera ture readings.\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "assign (sol): \+ r := .3: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 " evalm(a*r*1 /(1-r*eigs[1][1])*eigs[1][3][1]+b*r*1/(1-r*eigs[2][1])*eigs[2][3][1]+c *r*1/(1-r*eigs[3][1])*eigs[3][3][1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%$\"+wLJ'R(!\")$\"+QX1esF)$\"+B0 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 258 9 "Problem 4" }{TEXT -1 79 " : Use eigenvalue/eigenvector analysis to investigate Problems 1,2, \+ and/or 3." }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Back" 1 "syll322.mw s" "" }}}}}{MARK "0 29 0 2" 55 }{VIEWOPTS 1 1 0 1 1 1803 }