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|00000000| 5c 73 65 63 74 69 6f 6e | 7b 6f 64 65 20 2d 2d 2d |\section|{ode ---|
|00000010| 20 6f 72 64 69 6e 61 72 | 79 20 64 69 66 66 65 72 | ordinar|y differ|
|00000020| 65 6e 74 69 61 6c 20 65 | 71 75 61 74 69 6f 6e 20 |ential e|quation |
|00000030| 73 6f 6c 76 65 72 7d 0d | 5c 6c 61 62 65 6c 7b 66 |solver}.|\label{f|
|00000040| 6e 3a 6f 64 65 7d 0d 5c | 69 6e 64 65 78 7b 6f 64 |n:ode}.\|index{od|
|00000050| 65 28 29 7d 0d 5c 69 6e | 64 65 78 7b 64 69 66 66 |e()}.\in|dex{diff|
|00000060| 65 72 65 6e 74 69 61 6c | 20 65 71 75 61 74 69 6f |erential| equatio|
|00000070| 6e 73 7d 0d 5c 69 6e 64 | 65 78 7b 69 6e 74 65 67 |ns}.\ind|ex{integ|
|00000080| 72 61 74 69 6f 6e 7d 0d | 5c 69 6e 64 65 78 7b 6f |ration}.|\index{o|
|00000090| 72 64 69 6e 61 72 79 20 | 64 69 66 66 65 72 65 6e |rdinary |differen|
|000000a0| 74 69 61 6c 20 65 71 75 | 61 74 69 6f 6e 20 73 6f |tial equ|ation so|
|000000b0| 6c 76 65 72 7d 0d 5c 69 | 6e 64 65 78 7b 73 6f 6c |lver}.\i|ndex{sol|
|000000c0| 75 74 69 6f 6e 21 64 69 | 66 66 65 72 65 6e 74 69 |ution!di|fferenti|
|000000d0| 61 6c 20 65 71 75 61 74 | 69 6f 6e 73 7d 0d 5c 62 |al equat|ions}.\b|
|000000e0| 65 67 69 6e 7b 72 61 69 | 6c 7d 0d 4f 44 45 20 3a |egin{rai|l}.ODE :|
|000000f0| 20 27 6f 64 65 27 20 27 | 28 27 20 46 75 6e 63 74 | 'ode' '|(' Funct|
|00000100| 69 6f 6e 20 27 2c 27 20 | 4d 61 74 72 69 78 20 27 |ion ',' |Matrix '|
|00000110| 2c 27 20 53 63 61 6c 61 | 72 20 27 2c 27 20 53 63 |,' Scala|r ',' Sc|
|00000120| 61 6c 61 72 20 28 20 27 | 2c 27 20 28 53 63 61 6c |alar ( '|,' (Scal|
|00000130| 61 72 20 28 27 2c 27 20 | 28 20 53 63 61 6c 61 72 |ar (',' |( Scalar|
|00000140| 20 28 27 2c 27 20 28 53 | 63 61 6c 61 72 20 3f 29 | (',' (S|calar ?)|
|00000150| 3f 29 3f 29 3f 29 3f 29 | 3f 29 20 27 29 27 20 3b |?)?)?)?)|?) ')' ;|
|00000160| 0d 5c 65 6e 64 7b 72 61 | 69 6c 7d 0d 0d 5c 76 65 |.\end{ra|il}..\ve|
|00000170| 72 62 2b 6f 64 65 2b 20 | 69 73 20 61 20 66 69 72 |rb+ode+ |is a fir|
|00000180| 73 74 20 6f 72 64 65 72 | 20 4f 72 64 69 6e 61 72 |st order| Ordinar|
|00000190| 79 20 44 69 66 66 65 72 | 65 6e 74 69 61 6c 20 45 |y Differ|ential E|
|000001a0| 71 75 61 74 69 6f 6e 20 | 73 6f 6c 76 65 72 2e 0d |quation |solver..|
|000001b0| 49 74 20 69 6e 74 65 67 | 72 61 74 65 73 20 7b 5c |It integ|rates {\|
|000001c0| 65 6d 20 73 65 74 73 7d | 20 6f 66 20 66 69 72 73 |em sets}| of firs|
|000001d0| 74 20 6f 72 64 65 72 20 | 64 69 66 66 65 72 65 6e |t order |differen|
|000001e0| 74 69 61 6c 20 65 71 75 | 61 74 69 6f 6e 73 2e 0d |tial equ|ations..|
|000001f0| 0d 5c 62 65 67 69 6e 7b | 76 65 72 62 61 74 69 6d |.\begin{|verbatim|
|00000200| 7d 0d 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |}.      |        |
|00000210| 20 20 64 79 28 69 29 2f | 64 74 20 3d 20 66 28 74 |  dy(i)/|dt = f(t|
|00000220| 2c 79 28 31 29 2c 79 28 | 32 29 2c 2e 2e 2e 2c 79 |,y(1),y(|2),...,y|
|00000230| 28 4e 29 29 0d 20 20 20 | 20 20 20 20 20 20 20 20 |(N)).   |        |
|00000240| 20 20 20 20 20 79 28 69 | 29 20 67 69 76 65 6e 20 |     y(i|) given |
|00000250| 61 74 20 20 74 20 2e 0d | 0d 53 79 6e 74 61 78 3a |at  t ..|.Syntax:|
|00000260| 20 6f 64 65 20 28 20 72 | 68 73 66 2c 20 79 73 74 | ode ( r|hsf, yst|
|00000270| 61 72 74 2c 20 74 73 74 | 61 72 74 2c 20 74 65 6e |art, tst|art, ten|
|00000280| 64 2c 20 64 74 6f 75 74 | 2c 20 72 65 6c 65 72 72 |d, dtout|, relerr|
|00000290| 2c 20 61 62 73 65 72 72 | 20 29 0d 0d 20 20 20 20 |, abserr| )..    |
|000002a0| 20 20 20 20 72 68 73 66 | 20 20 20 20 41 20 66 75 |    rhsf|    A fu|
|000002b0| 6e 63 74 69 6f 6e 20 74 | 68 61 74 20 65 76 61 6c |nction t|hat eval|
|000002c0| 75 61 74 65 73 20 64 79 | 28 69 29 2f 64 74 20 61 |uates dy|(i)/dt a|
|000002d0| 74 20 74 2e 20 54 68 65 | 20 66 75 6e 63 74 69 6f |t t. The| functio|
|000002e0| 6e 0d 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |n.      |        |
|000002f0| 20 20 74 61 6b 65 73 20 | 74 77 6f 20 61 72 67 75 |  takes |two argu|
|00000300| 6d 65 6e 74 73 20 61 6e | 64 20 72 65 74 75 72 6e |ments an|d return|
|00000310| 73 20 64 79 2f 64 74 2e | 20 41 6e 20 65 78 61 6d |s dy/dt.| An exam|
|00000320| 70 6c 65 20 74 68 61 74 | 0d 20 20 20 20 20 20 20 |ple that|.       |
|00000330| 20 20 20 20 20 20 20 20 | 20 67 65 6e 65 72 61 74 |        | generat|
|00000340| 65 73 20 64 79 2f 64 74 | 20 66 6f 72 20 56 61 6e |es dy/dt| for Van|
|00000350| 20 64 65 72 20 50 6f 6c | 27 73 20 65 71 75 61 74 | der Pol|'s equat|
|00000360| 69 6f 6e 20 69 73 20 73 | 68 6f 77 6e 0d 20 20 20 |ion is s|hown.   |
|00000370| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 62 65 6c |        |     bel|
|00000380| 6f 77 2e 20 0d 43 61 6e | 20 62 65 20 75 73 65 72 |ow. .Can| be user|
|00000390| 20 6f 72 20 62 6c 74 69 | 6e 20 66 75 6e 63 74 69 | or blti|n functi|
|000003a0| 6f 6e 2e 0d 0d 20 20 20 | 20 20 20 20 20 79 73 74 |on...   |     yst|
|000003b0| 61 72 74 20 20 54 68 65 | 20 69 6e 69 74 69 61 6c |art  The| initial|
|000003c0| 20 76 61 6c 75 65 73 20 | 6f 66 20 79 2c 20 79 28 | values |of y, y(|
|000003d0| 74 73 74 61 72 74 29 2e | 0d 4d 75 73 74 20 62 65 |tstart).|.Must be|
|000003e0| 20 63 6f 6c 75 6d 6e 20 | 6f 72 20 72 6f 77 20 76 | column |or row v|
|000003f0| 65 63 74 6f 72 0d 20 20 | 20 20 20 20 20 20 74 73 |ector.  |      ts|
|00000400| 74 61 72 74 20 20 54 68 | 65 20 69 6e 69 74 69 61 |tart  Th|e initia|
|00000410| 6c 20 76 61 6c 75 65 20 | 6f 66 20 74 68 65 20 69 |l value |of the i|
|00000420| 6e 64 65 70 65 6e 64 65 | 6e 74 20 76 61 72 69 61 |ndepende|nt varia|
|00000430| 62 6c 65 2e 0d 43 61 6e | 20 61 6c 73 6f 20 62 65 |ble..Can| also be|
|00000440| 20 61 20 6d 61 74 72 69 | 78 20 2d 20 66 69 72 73 | a matri|x - firs|
|00000450| 74 20 65 6c 65 6d 65 6e | 74 20 77 69 6c 6c 20 62 |t elemen|t will b|
|00000460| 65 20 75 73 65 64 2e 0d | 20 20 20 20 20 20 20 20 |e used..|        |
|00000470| 74 65 6e 64 20 20 20 20 | 54 68 65 20 66 69 6e 61 |tend    |The fina|
|00000480| 6c 20 76 61 6c 75 65 20 | 6f 66 20 74 68 65 20 69 |l value |of the i|
|00000490| 6e 64 65 70 65 6e 64 65 | 6e 74 20 76 61 72 69 61 |ndepende|nt varia|
|000004a0| 62 6c 65 2e 0d 43 61 6e | 20 61 6c 73 6f 20 62 65 |ble..Can| also be|
|000004b0| 20 61 20 6d 61 74 72 69 | 78 20 2d 20 66 69 72 73 | a matri|x - firs|
|000004c0| 74 20 65 6c 65 6d 65 6e | 74 20 77 69 6c 6c 20 62 |t elemen|t will b|
|000004d0| 65 20 75 73 65 64 2e 0d | 43 61 6e 6e 6f 74 20 62 |e used..|Cannot b|
|000004e0| 65 20 73 61 6d 65 20 61 | 73 20 74 73 74 61 72 74 |e same a|s tstart|
|000004f0| 0d 0d 20 20 20 20 20 20 | 20 20 64 74 6f 75 74 20 |..      |  dtout |
|00000500| 20 20 54 68 65 20 6f 75 | 74 70 75 74 20 69 6e 74 |  The ou|tput int|
|00000510| 65 72 76 61 6c 2e 20 54 | 68 65 20 76 65 63 74 6f |erval. T|he vecto|
|00000520| 72 20 79 20 77 69 6c 6c | 20 62 65 20 73 61 76 65 |r y will| be save|
|00000530| 64 20 61 74 0d 20 20 20 | 20 20 20 20 20 20 20 20 |d at.   |        |
|00000540| 20 20 20 20 20 74 73 74 | 61 72 74 2c 20 69 6e 63 |     tst|art, inc|
|00000550| 72 65 6d 65 6e 74 73 20 | 6f 66 20 74 73 74 61 72 |rements |of tstar|
|00000560| 74 20 2b 20 64 74 6f 75 | 74 2c 20 61 6e 64 20 74 |t + dtou|t, and t|
|00000570| 65 6e 64 2e 20 49 66 0d | 20 20 20 20 20 20 20 20 |end. If.|        |
|00000580| 20 20 20 20 20 20 20 20 | 64 74 6f 75 74 20 69 73 |        |dtout is|
|00000590| 20 6e 6f 74 20 73 70 65 | 63 69 66 69 65 64 2c 20 | not spe|cified, |
|000005a0| 74 68 65 6e 20 74 68 65 | 20 64 65 66 61 75 6c 74 |then the| default|
|000005b0| 20 69 73 20 74 6f 20 73 | 74 6f 72 65 0d 20 20 20 | is to s|tore.   |
|000005c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 6f 75 74 |        |     out|
|000005d0| 70 75 74 20 61 74 20 31 | 30 31 20 76 61 6c 75 65 |put at 1|01 value|
|000005e0| 73 20 6f 66 20 74 68 65 | 20 69 6e 64 65 70 65 6e |s of the| indepen|
|000005f0| 64 65 6e 74 20 76 61 72 | 69 61 62 6c 65 2e 0d 0d |dent var|iable...|
|00000600| 20 20 20 20 20 20 20 20 | 72 65 6c 65 72 72 20 20 |        |relerr  |
|00000610| 54 68 65 20 72 65 6c 61 | 74 69 76 65 20 65 72 72 |The rela|tive err|
|00000620| 6f 72 20 74 6f 6c 65 72 | 61 6e 63 65 2e 20 44 65 |or toler|ance. De|
|00000630| 66 61 75 6c 74 20 76 61 | 6c 75 65 20 69 73 20 31 |fault va|lue is 1|
|00000640| 2e 65 2d 36 2e 0d 0d 20 | 20 20 20 20 20 20 20 61 |.e-6... |       a|
|00000650| 62 73 65 72 72 20 20 54 | 68 65 20 61 62 73 6f 6c |bserr  T|he absol|
|00000660| 75 74 65 20 65 72 72 6f | 72 20 74 6f 6c 65 72 61 |ute erro|r tolera|
|00000670| 6e 63 65 2e 20 41 74 20 | 65 61 63 68 20 73 74 65 |nce. At |each ste|
|00000680| 70 2c 20 6f 64 65 0d 20 | 20 20 20 20 20 20 20 20 |p, ode. |        |
|00000690| 20 20 20 20 20 20 20 72 | 65 71 75 69 72 65 73 20 |       r|equires |
|000006a0| 74 68 61 74 3a 0d 0d 20 | 20 20 20 20 20 20 20 20 |that:.. |        |
|000006b0| 20 20 20 20 20 20 20 61 | 62 73 28 6c 6f 63 61 6c |       a|bs(local|
|000006c0| 20 65 72 72 6f 72 29 20 | 3c 3d 20 61 62 73 28 79 | error) |<= abs(y|
|000006d0| 29 2a 72 65 6c 65 72 72 | 20 2b 20 61 62 73 65 72 |)*relerr| + abser|
|000006e0| 72 0d 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |r.      |        |
|000006f0| 20 20 0d 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |  .     |        |
|00000700| 20 20 20 46 6f 72 20 65 | 61 63 68 20 63 6f 6d 70 |   For e|ach comp|
|00000710| 6f 6e 65 6e 74 20 6f 66 | 20 74 68 65 20 6c 6f 63 |onent of| the loc|
|00000720| 61 6c 20 65 72 72 6f 72 | 20 61 6e 64 20 73 6f 6c |al error| and sol|
|00000730| 75 74 69 6f 6e 0d 20 20 | 20 20 20 20 20 20 20 20 |ution.  |        |
|00000740| 20 20 20 20 20 20 76 65 | 63 74 6f 72 73 2e 20 54 |      ve|ctors. T|
|00000750| 68 65 20 64 65 66 61 75 | 6c 74 20 76 61 6c 75 65 |he defau|lt value|
|00000760| 20 69 73 20 31 2e 65 2d | 36 2e 0d 0d 0d 20 20 20 | is 1.e-|6....   |
|00000770| 20 20 20 20 20 54 68 65 | 20 46 6f 72 74 72 61 6e |     The| Fortran|
|00000780| 20 73 6f 75 72 63 65 20 | 63 6f 64 65 20 66 6f 72 | source |code for|
|00000790| 20 6f 64 65 28 29 20 69 | 73 20 63 6f 6d 70 6c 65 | ode() i|s comple|
|000007a0| 74 65 6c 79 20 65 78 70 | 6c 61 69 6e 65 64 20 61 |tely exp|lained a|
|000007b0| 6e 64 0d 20 20 20 20 20 | 20 20 20 64 6f 63 75 6d |nd.     |   docum|
|000007c0| 65 6e 74 65 64 20 69 6e | 20 74 68 65 20 74 65 78 |ented in| the tex|
|000007d0| 74 2c 20 22 43 6f 6d 70 | 75 74 65 72 20 53 6f 6c |t, "Comp|uter Sol|
|000007e0| 75 74 69 6f 6e 20 6f 66 | 20 4f 72 64 69 6e 61 72 |ution of| Ordinar|
|000007f0| 79 0d 20 20 20 20 20 20 | 20 20 44 69 66 66 65 72 |y.      |  Differ|
|00000800| 65 6e 74 69 61 6c 20 45 | 71 75 61 74 69 6f 6e 73 |ential E|quations|
|00000810| 3a 20 54 68 65 20 49 6e | 69 74 69 61 6c 20 56 61 |: The In|itial Va|
|00000820| 6c 75 65 20 50 72 6f 62 | 6c 65 6d 22 20 62 79 20 |lue Prob|lem" by |
|00000830| 0d 20 20 20 20 20 20 20 | 20 4c 2e 20 46 2e 20 53 |.       | L. F. S|
|00000840| 68 61 6d 70 69 6e 65 20 | 61 6e 64 20 20 4d 2e 20 |hampine |and  M. |
|00000850| 4b 2e 20 47 6f 72 64 6f | 6e 2e 0d 0d 20 20 20 20 |K. Gordo|n...    |
|00000860| 20 20 20 20 45 78 61 6d | 70 6c 65 3a 0d 20 20 20 |    Exam|ple:.   |
|00000870| 20 20 20 20 20 0d 20 20 | 20 20 20 20 20 20 76 64 |     .  |      vd|
|00000880| 70 6f 6c 20 3d 20 66 75 | 6e 63 74 69 6f 6e 20 28 |pol = fu|nction (|
|00000890| 20 74 20 2c 20 78 20 29 | 20 0d 20 20 20 20 20 20 | t , x )| .      |
|000008a0| 20 20 7b 0d 20 20 20 20 | 20 20 20 20 20 20 6c 6f |  {.    |      lo|
|000008b0| 63 61 6c 20 28 78 70 29 | 0d 20 20 20 20 20 20 20 |cal (xp)|.       |
|000008c0| 20 20 20 78 70 20 3d 20 | 7a 65 72 6f 73 28 32 2c |   xp = |zeros(2,|
|000008d0| 31 29 3b 0d 20 20 20 20 | 20 20 20 20 20 20 78 70 |1);.    |      xp|
|000008e0| 5b 31 5d 20 3d 20 78 5b | 31 5d 20 2a 20 28 31 20 |[1] = x[|1] * (1 |
|000008f0| 2d 20 78 5b 32 5d 5e 32 | 29 20 2d 20 78 5b 32 5d |- x[2]^2|) - x[2]|
|00000900| 3b 0d 20 20 20 20 20 20 | 20 20 20 20 78 70 5b 32 |;.      |    xp[2|
|00000910| 5d 20 3d 20 78 5b 31 5d | 3b 0d 20 20 20 20 20 20 |] = x[1]|;.      |
|00000920| 20 20 20 20 72 65 74 75 | 72 6e 20 78 70 3b 0d 20 |    retu|rn xp;. |
|00000930| 20 20 20 20 20 20 20 7d | 3b 0d 20 20 20 20 20 20 |       }|;.      |
|00000940| 20 20 0d 20 20 20 20 20 | 20 20 20 78 30 20 3d 20 |  .     |   x0 = |
|00000950| 5b 30 3b 20 30 2e 32 35 | 5d 3b 0d 20 20 20 20 20 |[0; 0.25|];.     |
|00000960| 20 20 20 0d 20 20 20 20 | 20 20 20 20 78 62 61 73 |   .    |    xbas|
|00000970| 65 20 3d 20 6f 64 65 28 | 20 76 64 70 6f 6c 2c 20 |e = ode(| vdpol, |
|00000980| 78 30 2c 20 30 2c 20 32 | 30 2c 20 30 2e 30 35 2c |x0, 0, 2|0, 0.05,|
|00000990| 20 31 65 2d 39 2c 20 31 | 65 2d 39 29 3b 0d 5c 65 | 1e-9, 1|e-9);.\e|
|000009a0| 6e 64 7b 76 65 72 62 61 | 74 69 6d 7d 0d 0d 5c 70 |nd{verba|tim}..\p|
|000009b0| 61 72 61 67 72 61 70 68 | 7b 45 78 61 6d 70 6c 65 |aragraph|{Example|
|000009c0| 7d 0d 5c 69 6e 64 65 78 | 7b 65 78 61 6d 70 6c 65 |}.\index|{example|
|000009d0| 20 6f 66 20 6f 64 65 28 | 29 7d 0d 5c 69 6e 64 65 | of ode(|)}.\inde|
|000009e0| 78 7b 6f 64 65 28 29 21 | 65 78 61 6d 70 6c 65 7d |x{ode()!|example}|
|000009f0| 0d 5c 62 65 67 69 6e 7b | 76 65 72 62 61 74 69 6d |.\begin{|verbatim|
|00000a00| 7d 0d 44 75 6d 6d 79 20 | 45 78 61 6d 70 6c 65 0d |}.Dummy |Example.|
|00000a10| 5c 65 6e 64 7b 76 65 72 | 62 61 74 69 6d 7d 0d 0d |\end{ver|batim}..|
|00000a20| 53 65 65 20 61 6c 73 6f | 3a 20 50 61 67 65 7e 5c |See also|: Page~\|
|00000a30| 70 61 67 65 72 65 66 7b | 66 6e 3a 7d 20 5c 76 65 |pageref{|fn:} \ve|
|00000a40| 72 62 2b 28 29 2b 2e 0d | 0d 5c 6e 65 77 70 61 67 |rb+()+..|.\newpag|
|00000a50| 65 0d 0d 0d             |                         |e...    |        |
+--------+-------------------------+-------------------------+--------+--------+