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+--------+-------------------------+-------------------------+--------+--------+
|00000000| 5c 73 65 63 74 69 6f 6e | 7b 6c 79 61 70 20 2d 2d |\section|{lyap --|
|00000010| 2d 20 73 6f 6c 75 74 69 | 6f 6e 20 6f 66 20 74 68 |- soluti|on of th|
|00000020| 65 20 6c 79 61 70 75 6e | 6f 76 20 65 71 75 61 74 |e lyapun|ov equat|
|00000030| 69 6f 6e 7d 0d 5c 6c 61 | 62 65 6c 7b 66 6e 3a 6c |ion}.\la|bel{fn:l|
|00000040| 79 61 70 7d 0d 5c 69 6e | 64 65 78 7b 6c 79 61 70 |yap}.\in|dex{lyap|
|00000050| 28 29 7d 0d 5c 69 6e 64 | 65 78 7b 4c 79 61 70 75 |()}.\ind|ex{Lyapu|
|00000060| 6e 6f 76 20 65 71 75 61 | 74 69 6f 6e 7d 0d 5c 69 |nov equa|tion}.\i|
|00000070| 6e 64 65 78 7b 53 79 6c | 76 65 73 74 65 72 20 65 |ndex{Syl|vester e|
|00000080| 71 75 61 74 69 6f 6e 7d | 0d 5c 62 65 67 69 6e 7b |quation}|.\begin{|
|00000090| 72 61 69 6c 7d 0d 4c 79 | 61 70 75 6e 6f 76 20 3a |rail}.Ly|apunov :|
|000000a0| 20 27 6c 79 61 70 27 20 | 27 28 27 20 4d 61 74 72 | 'lyap' |'(' Matr|
|000000b0| 69 78 20 27 2c 27 20 28 | 20 28 29 20 7c 20 4d 61 |ix ',' (| () | Ma|
|000000c0| 74 72 69 78 29 20 20 27 | 2c 27 20 4d 61 74 72 69 |trix) '|,' Matri|
|000000d0| 78 20 20 27 29 27 20 3b | 0d 5c 65 6e 64 7b 72 61 |x ')' ;|.\end{ra|
|000000e0| 69 6c 7d 0d 0d 5c 76 65 | 72 62 2b 6c 79 61 70 2b |il}..\ve|rb+lyap+|
|000000f0| 20 69 73 20 75 73 65 64 | 20 74 6f 20 73 6f 6c 76 | is used| to solv|
|00000100| 65 20 74 68 65 20 67 65 | 6e 65 72 61 6c 20 66 6f |e the ge|neral fo|
|00000110| 72 6d 20 6f 66 20 74 68 | 65 20 53 79 6c 76 65 73 |rm of th|e Sylves|
|00000120| 74 65 72 0d 65 71 75 61 | 74 69 6f 6e 2c 20 61 6e |ter.equa|tion, an|
|00000130| 64 20 69 74 73 20 73 70 | 65 63 69 61 6c 20 66 6f |d its sp|ecial fo|
|00000140| 72 6d 20 69 6e 20 74 68 | 65 20 4c 79 61 70 75 6e |rm in th|e Lyapun|
|00000150| 6f 76 20 65 71 75 61 74 | 69 6f 6e 2e 20 20 49 74 |ov equat|ion. It|
|00000160| 20 69 73 0d 76 65 72 79 | 20 73 69 6d 69 6c 61 72 | is.very| similar|
|00000170| 20 74 6f 20 74 68 65 20 | 5c 76 65 72 62 2b 73 79 | to the |\verb+sy|
|00000180| 6c 76 2b 20 66 75 6e 63 | 74 69 6f 6e 2c 20 65 78 |lv+ func|tion, ex|
|00000190| 63 65 70 74 20 74 68 61 | 74 20 74 68 65 20 61 72 |cept tha|t the ar|
|000001a0| 67 75 6d 65 6e 74 73 0d | 6e 65 65 64 20 6e 6f 74 |guments.|need not|
|000001b0| 20 62 65 20 75 70 70 65 | 72 20 74 72 69 61 6e 67 | be uppe|r triang|
|000001c0| 75 6c 61 72 2e 0d 0d 47 | 69 76 65 6e 20 74 68 65 |ular...G|iven the|
|000001d0| 20 66 69 72 73 74 20 6d | 61 74 72 69 78 20 61 72 | first m|atrix ar|
|000001e0| 67 75 6d 65 6e 74 20 69 | 73 20 24 41 24 2c 20 74 |gument i|s $A$, t|
|000001f0| 68 65 20 73 65 63 6f 6e | 64 20 6d 61 74 72 69 78 |he secon|d matrix|
|00000200| 20 61 72 67 75 6d 65 6e | 74 0d 69 73 20 24 42 24 | argumen|t.is $B$|
|00000210| 2c 20 61 6e 64 20 74 68 | 65 20 74 68 69 72 64 20 |, and th|e third |
|00000220| 6d 61 74 72 69 78 20 61 | 72 67 75 6d 65 6e 74 20 |matrix a|rgument |
|00000230| 69 73 20 24 43 24 2c 20 | 5c 76 65 72 62 2b 6c 79 |is $C$, |\verb+ly|
|00000240| 61 70 2b 20 73 6f 6c 76 | 65 73 20 0d 0d 5c 62 65 |ap+ solv|es ..\be|
|00000250| 67 69 6e 7b 64 69 73 70 | 6c 61 79 6d 61 74 68 7d |gin{disp|laymath}|
|00000260| 0d 41 78 20 2b 20 78 42 | 20 3d 20 2d 43 0d 5c 65 |.Ax + xB| = -C.\e|
|00000270| 6e 64 7b 64 69 73 70 6c | 61 79 6d 61 74 68 7d 0d |nd{displ|aymath}.|
|00000280| 0d 66 6f 72 20 24 78 24 | 2e 20 20 0d 0d 49 66 20 |.for $x$|. ..If |
|00000290| 74 68 65 72 65 20 61 72 | 65 20 6f 6e 6c 79 20 74 |there ar|e only t|
|000002a0| 77 6f 20 61 72 67 75 6d | 65 6e 74 73 2c 20 20 69 |wo argum|ents, i|
|000002b0| 74 20 69 73 20 61 73 73 | 75 6d 65 64 20 24 42 20 |t is ass|umed $B |
|000002c0| 3d 20 41 5e 7b 54 7d 24 | 2c 20 61 6e 64 20 74 68 |= A^{T}$|, and th|
|000002d0| 65 0d 4c 79 61 70 75 6e | 6f 76 20 65 71 75 61 74 |e.Lyapun|ov equat|
|000002e0| 69 6f 6e 20 69 73 20 73 | 6f 6c 76 65 64 2e 20 20 |ion is s|olved. |
|000002f0| 4e 6f 74 65 20 74 68 61 | 74 20 69 6e 20 5c 76 65 |Note tha|t in \ve|
|00000300| 72 62 2b 73 79 6c 76 2b | 2c 20 79 6f 75 20 6c 65 |rb+sylv+|, you le|
|00000310| 61 76 65 20 6f 75 74 0d | 74 68 65 20 63 6f 6d 6d |ave out.|the comm|
|00000320| 61 20 61 6e 64 20 74 68 | 65 20 61 72 67 75 6d 65 |a and th|e argume|
|00000330| 6e 74 20 2d 2d 2d 20 5c | 76 65 72 62 2b 6c 79 61 |nt --- \|verb+lya|
|00000340| 70 2b 20 72 65 71 75 69 | 72 65 73 20 62 6f 74 68 |p+ requi|res both|
|00000350| 20 63 6f 6d 6d 61 73 2e | 0d 0d 24 5c 4c 6f 6e 67 | commas.|..$\Long|
|00000360| 72 69 67 68 74 61 72 72 | 6f 77 24 20 54 68 69 73 |rightarr|ow$ This|
|00000370| 20 69 73 20 6e 6f 74 20 | 61 6e 20 5c 52 4c 61 42 | is not |an \RLaB|
|00000380| 5c 20 62 75 69 6c 74 2d | 69 6e 20 66 75 6e 63 74 |\ built-|in funct|
|00000390| 69 6f 6e 2e 20 20 54 68 | 69 73 20 66 75 6e 63 74 |ion. Th|is funct|
|000003a0| 69 6f 6e 0d 69 73 20 6e | 6f 72 6d 61 6c 6c 79 20 |ion.is n|ormally |
|000003b0| 6c 6f 61 64 65 64 20 6f | 6e 20 73 74 61 72 74 2d |loaded o|n start-|
|000003c0| 75 70 20 66 72 6f 6d 20 | 74 68 65 20 5c 76 65 72 |up from |the \ver|
|000003d0| 62 2b 6c 79 61 70 2e 72 | 2b 20 66 69 6c 65 20 69 |b+lyap.r|+ file i|
|000003e0| 6e 20 74 68 65 0d 73 74 | 61 6e 64 61 72 64 20 5c |n the.st|andard \|
|000003f0| 76 65 72 62 2b 72 6c 69 | 62 2b 20 64 69 72 65 63 |verb+rli|b+ direc|
|00000400| 74 6f 72 79 2e 20 55 73 | 65 20 6f 66 20 74 68 65 |tory. Us|e of the|
|00000410| 20 5c 76 65 72 62 2b 2d | 72 2b 20 6f 70 74 69 6f | \verb+-|r+ optio|
|00000420| 6e 2c 20 69 6e 63 6f 72 | 72 65 63 74 6c 79 20 0d |n, incor|rectly .|
|00000430| 73 65 74 74 69 6e 67 20 | 74 68 65 20 5c 76 65 72 |setting |the \ver|
|00000440| 62 2b 52 4c 41 42 5f 4c | 49 42 5f 44 49 52 2b 20 |b+RLAB_L|IB_DIR+ |
|00000450| 65 6e 76 69 72 6f 6e 6d | 65 6e 74 61 6c 20 76 61 |environm|ental va|
|00000460| 72 69 61 62 6c 65 2c 20 | 6f 72 20 6d 6f 64 69 66 |riable, |or modif|
|00000470| 79 69 6e 67 0d 5c 76 65 | 72 62 2b 6c 79 61 70 2e |ying.\ve|rb+lyap.|
|00000480| 72 2b 20 6d 61 79 20 6d | 61 6b 65 20 74 68 69 73 |r+ may m|ake this|
|00000490| 20 66 75 6e 63 74 69 6f | 6e 20 75 6e 61 76 61 69 | functio|n unavai|
|000004a0| 6c 61 62 6c 65 2e 0d 0d | 0d 5c 70 61 72 61 67 72 |lable...|.\paragr|
|000004b0| 61 70 68 7b 45 78 61 6d | 70 6c 65 7d 0d 5c 69 6e |aph{Exam|ple}.\in|
|000004c0| 64 65 78 7b 65 78 61 6d | 70 6c 65 20 6f 66 20 6c |dex{exam|ple of l|
|000004d0| 79 61 70 28 29 7d 0d 5c | 69 6e 64 65 78 7b 6c 79 |yap()}.\|index{ly|
|000004e0| 61 70 28 29 21 65 78 61 | 6d 70 6c 65 7d 0d 5c 62 |ap()!exa|mple}.\b|
|000004f0| 65 67 69 6e 7b 76 65 72 | 62 61 74 69 6d 7d 0d 3e |egin{ver|batim}.>|
|00000500| 20 61 20 3d 20 72 61 6e | 64 28 33 2c 33 29 0d 20 | a = ran|d(3,3). |
|00000510| 61 20 3d 0d 20 20 20 20 | 30 2e 33 34 31 20 20 20 |a =. |0.341 |
|00000520| 20 20 20 30 2e 34 35 35 | 20 20 20 20 20 20 30 2e | 0.455| 0.|
|00000530| 31 38 38 20 20 0d 20 20 | 20 20 30 2e 36 32 33 20 |188 . | 0.623 |
|00000540| 20 20 20 20 30 2e 30 31 | 37 38 20 20 20 20 20 20 | 0.01|78 |
|00000550| 30 2e 39 34 36 20 20 0d | 20 20 20 20 30 2e 31 38 |0.946 .| 0.18|
|00000560| 32 20 20 20 20 20 20 30 | 2e 30 37 31 20 20 20 20 |2 0|.071 |
|00000570| 20 20 30 2e 37 39 39 20 | 20 0d 3e 20 62 20 3d 20 | 0.799 | .> b = |
|00000580| 72 61 6e 64 28 33 2c 33 | 29 0d 20 62 20 3d 0d 20 |rand(3,3|). b =. |
|00000590| 20 20 20 30 2e 33 36 39 | 20 20 20 20 20 20 30 2e | 0.369| 0.|
|000005a0| 34 38 31 20 20 20 20 20 | 20 30 2e 39 33 31 20 20 |481 | 0.931 |
|000005b0| 0d 20 20 20 20 30 2e 38 | 39 35 20 20 20 20 20 20 |. 0.8|95 |
|000005c0| 30 2e 37 39 31 20 20 20 | 20 20 20 20 30 2e 39 38 |0.791 | 0.98|
|000005d0| 20 20 0d 20 20 20 20 30 | 2e 38 33 36 20 20 20 20 | . 0|.836 |
|000005e0| 20 20 30 2e 37 33 33 20 | 20 20 20 20 20 30 2e 31 | 0.733 | 0.1|
|000005f0| 39 35 20 20 0d 3e 20 63 | 20 3d 20 72 61 6e 64 28 |95 .> c| = rand(|
|00000600| 33 2c 33 29 0d 20 63 20 | 3d 0d 20 20 20 20 30 2e |3,3). c |=. 0.|
|00000610| 34 35 39 20 20 20 20 20 | 20 20 30 2e 34 36 20 20 |459 | 0.46 |
|00000620| 20 20 20 20 30 2e 37 34 | 39 20 20 0d 20 20 20 30 | 0.74|9 . 0|
|00000630| 2e 30 35 38 39 20 20 20 | 20 20 20 30 2e 32 32 35 |.0589 | 0.225|
|00000640| 20 20 20 20 20 20 30 2e | 33 38 35 20 20 0d 20 20 | 0.|385 . |
|00000650| 20 20 30 2e 32 38 31 20 | 20 20 20 20 20 30 2e 39 | 0.281 | 0.9|
|00000660| 35 37 20 20 20 20 20 20 | 30 2e 36 34 32 20 20 0d |57 |0.642 .|
|00000670| 3e 20 52 65 73 53 79 6c | 76 20 3d 20 6c 79 61 70 |> ResSyl|v = lyap|
|00000680| 28 61 2c 62 2c 63 29 0d | 20 52 65 73 53 79 6c 76 |(a,b,c).| ResSylv|
|00000690| 20 3d 0d 20 20 20 2d 30 | 2e 38 30 34 20 20 20 20 | =. -0|.804 |
|000006a0| 20 20 30 2e 38 36 38 20 | 20 20 20 20 20 2d 31 2e | 0.868 | -1.|
|000006b0| 32 34 20 20 0d 20 20 20 | 20 30 2e 35 38 33 20 20 |24 . | 0.583 |
|000006c0| 20 20 20 30 2e 30 31 31 | 34 20 20 20 20 20 2d 30 | 0.011|4 -0|
|000006d0| 2e 33 37 35 20 20 0d 20 | 20 20 20 20 30 2e 35 35 |.375 . | 0.55|
|000006e0| 20 20 20 20 20 2d 30 2e | 38 32 35 20 20 20 20 2d | -0.|825 -|
|000006f0| 30 2e 30 39 34 32 20 0d | 3e 20 61 2a 52 65 73 53 |0.0942 .|> a*ResS|
|00000700| 79 6c 76 2b 52 65 73 53 | 79 6c 76 2a 62 20 2b 20 |ylv+ResS|ylv*b + |
|00000710| 63 0d 20 34 2e 34 34 65 | 2d 31 36 20 20 20 38 2e |c. 4.44e|-16 8.|
|00000720| 38 38 65 2d 31 36 20 20 | 20 31 2e 34 34 65 2d 31 |88e-16 | 1.44e-1|
|00000730| 35 20 20 0d 20 35 2e 35 | 35 65 2d 31 37 20 20 2d |5 . 5.5|5e-17 -|
|00000740| 31 2e 31 31 65 2d 31 36 | 20 20 20 36 2e 36 36 65 |1.11e-16| 6.66e|
|00000750| 2d 31 36 20 20 0d 20 32 | 2e 32 32 65 2d 31 36 20 |-16 . 2|.22e-16 |
|00000760| 20 20 20 20 20 20 20 20 | 20 30 20 20 20 32 2e 32 | | 0 2.2|
|00000770| 32 65 2d 31 36 20 20 0d | 3e 20 52 65 73 4c 79 61 |2e-16 .|> ResLya|
|00000780| 70 20 3d 20 6c 79 61 70 | 28 61 2c 2c 63 29 0d 20 |p = lyap|(a,,c). |
|00000790| 52 65 73 4c 79 61 70 20 | 3d 0d 20 20 20 20 2d 33 |ResLyap |=. -3|
|000007a0| 2e 37 35 20 20 20 20 20 | 20 20 37 2e 31 31 20 20 |.75 | 7.11 |
|000007b0| 20 20 20 2d 30 2e 39 31 | 36 20 20 0d 20 20 20 20 | -0.91|6 . |
|000007c0| 2d 33 2e 30 37 20 20 20 | 20 20 20 20 32 2e 31 35 |-3.07 | 2.15|
|000007d0| 20 20 20 20 20 20 20 31 | 2e 32 32 20 20 0d 20 20 | 1|.22 . |
|000007e0| 20 20 20 32 2e 32 39 20 | 20 20 20 20 20 20 2d 34 | 2.29 | -4|
|000007f0| 2e 32 20 20 20 20 20 2d | 30 2e 34 32 37 20 20 0d |.2 -|0.427 .|
|00000800| 3e 20 61 2a 52 65 73 4c | 79 61 70 2b 52 65 73 4c |> a*ResL|yap+ResL|
|00000810| 79 61 70 2a 61 27 20 2b | 20 63 0d 20 20 20 20 20 |yap*a' +| c. |
|00000820| 20 20 20 30 20 20 20 38 | 2e 38 38 65 2d 31 36 20 | 0 8|.88e-16 |
|00000830| 20 20 32 2e 35 35 65 2d | 31 35 20 20 0d 20 31 2e | 2.55e-|15 . 1.|
|00000840| 39 34 65 2d 31 35 20 20 | 2d 32 2e 33 39 65 2d 31 |94e-15 |-2.39e-1|
|00000850| 35 20 20 20 31 2e 31 31 | 65 2d 31 36 20 20 0d 2d |5 1.11|e-16 .-|
|00000860| 31 2e 31 31 65 2d 31 35 | 20 20 2d 31 2e 31 31 65 |1.11e-15| -1.11e|
|00000870| 2d 31 36 20 20 2d 31 2e | 33 33 65 2d 31 35 20 20 |-16 -1.|33e-15 |
|00000880| 0d 5c 65 6e 64 7b 76 65 | 72 62 61 74 69 6d 7d 0d |.\end{ve|rbatim}.|
|00000890| 0d 53 65 65 20 61 6c 73 | 6f 3a 20 50 61 67 65 7e |.See als|o: Page~|
|000008a0| 5c 70 61 67 65 72 65 66 | 7b 66 6e 3a 73 63 68 75 |\pageref|{fn:schu|
|000008b0| 72 7d 20 5c 76 65 72 62 | 2b 73 63 68 75 72 28 29 |r} \verb|+schur()|
|000008c0| 2b 2c 0d 50 61 67 65 7e | 5c 70 61 67 65 72 65 66 |+,.Page~|\pageref|
|000008d0| 7b 66 6e 3a 73 79 6c 76 | 7d 20 5c 76 65 72 62 2b |{fn:sylv|} \verb+|
|000008e0| 73 79 6c 76 28 29 2b 2e | 0d 0d 5c 6e 65 77 70 61 |sylv()+.|..\newpa|
|000008f0| 67 65 0d 0d 0d 0d | |ge.... | |
+--------+-------------------------+-------------------------+--------+--------+