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You can browse this item here: sylv.tex
| Confidence | Program | Detection | Match Type | Support
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|---|
| 100%
| dexvert
| LaTeX Document (document/latex)
| magic
| Supported |
| 1%
| dexvert
| Text File (text/txt)
| fallback
| Supported |
| 100%
| file
| LaTeX document text
| default
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| 99%
| file
| LaTeX document, ASCII text, with CR line terminators
| default
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| 100%
| checkBytes
| Printable ASCII
| default
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| 100%
| perlTextCheck
| Likely Text (Perl)
| default
| |
| 100%
| siegfried
| fmt/281 LaTeX (Subdocument)
| default
| |
| 100%
| detectItEasy
| Format: plain text[CR]
| default (weak)
|
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| id metadata |
|---|
| key | value |
|---|
| macFileType | [TEXT] |
| macFileCreator | [ttxt] |
hex view+--------+-------------------------+-------------------------+--------+--------+
|00000000| 5c 73 65 63 74 69 6f 6e | 7b 73 79 6c 76 20 2d 2d |\section|{sylv --|
|00000010| 2d 20 73 6f 6c 75 74 69 | 6f 6e 20 6f 66 20 53 79 |- soluti|on of Sy|
|00000020| 6c 76 65 73 74 65 72 20 | 45 71 75 61 74 69 6f 6e |lvester |Equation|
|00000030| 7d 0d 5c 6c 61 62 65 6c | 7b 66 6e 3a 73 79 6c 76 |}.\label|{fn:sylv|
|00000040| 7d 0d 5c 69 6e 64 65 78 | 7b 73 79 6c 76 28 29 7d |}.\index|{sylv()}|
|00000050| 0d 5c 69 6e 64 65 78 7b | 4c 79 61 70 75 6e 6f 76 |.\index{|Lyapunov|
|00000060| 20 65 71 75 61 74 69 6f | 6e 7d 0d 5c 69 6e 64 65 | equatio|n}.\inde|
|00000070| 78 7b 53 79 6c 76 65 73 | 74 65 72 20 65 71 75 61 |x{Sylves|ter equa|
|00000080| 74 69 6f 6e 7d 0d 5c 62 | 65 67 69 6e 7b 72 61 69 |tion}.\b|egin{rai|
|00000090| 6c 7d 0d 53 79 6c 76 20 | 3a 20 27 73 79 6c 76 27 |l}.Sylv |: 'sylv'|
|000000a0| 20 27 28 27 20 28 20 4d | 61 74 72 69 78 20 28 20 | '(' ( M|atrix ( |
|000000b0| 28 29 20 7c 20 27 2c 27 | 20 20 4d 61 74 72 69 78 |() | ','| Matrix|
|000000c0| 29 20 27 2c 27 20 4d 61 | 74 72 69 78 20 29 20 27 |) ',' Ma|trix ) '|
|000000d0| 29 27 20 3b 0d 5c 65 6e | 64 7b 72 61 69 6c 7d 0d |)' ;.\en|d{rail}.|
|000000e0| 0d 5c 76 65 72 62 2b 73 | 79 6c 76 2b 20 69 73 20 |.\verb+s|ylv+ is |
|000000f0| 61 20 53 79 6c 76 65 73 | 74 65 72 20 65 71 75 61 |a Sylves|ter equa|
|00000100| 74 69 6f 6e 20 73 6f 6c | 76 65 72 2e 20 20 47 69 |tion sol|ver. Gi|
|00000110| 76 65 6e 20 74 68 65 20 | 66 69 72 73 74 20 6d 61 |ven the |first ma|
|00000120| 74 72 69 78 0d 61 72 67 | 75 6d 65 6e 74 20 69 73 |trix.arg|ument is|
|00000130| 20 24 41 24 2c 20 74 68 | 65 20 73 65 63 6f 6e 64 | $A$, th|e second|
|00000140| 20 6d 61 74 72 69 78 20 | 61 72 67 75 6d 65 6e 74 | matrix |argument|
|00000150| 20 69 73 20 24 42 24 2c | 20 61 6e 64 20 74 68 65 | is $B$,| and the|
|00000160| 20 74 68 69 72 64 0d 6d | 61 74 72 69 78 20 61 72 | third.m|atrix ar|
|00000170| 67 75 6d 65 6e 74 20 69 | 73 20 24 43 24 2c 20 5c |gument i|s $C$, \|
|00000180| 76 65 72 62 2b 73 79 6c | 76 2b 20 73 6f 6c 76 65 |verb+syl|v+ solve|
|00000190| 73 20 0d 0d 5c 62 65 67 | 69 6e 7b 64 69 73 70 6c |s ..\beg|in{displ|
|000001a0| 61 79 6d 61 74 68 7d 0d | 41 78 20 2b 20 78 42 20 |aymath}.|Ax + xB |
|000001b0| 3d 20 2d 43 0d 5c 65 6e | 64 7b 64 69 73 70 6c 61 |= -C.\en|d{displa|
|000001c0| 79 6d 61 74 68 7d 0d 0d | 66 6f 72 20 24 78 24 2e |ymath}..|for $x$.|
|000001d0| 20 20 49 66 20 74 68 65 | 72 65 20 61 72 65 20 6f | If the|re are o|
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|000001f0| 73 2c 20 20 69 74 20 69 | 73 20 61 73 73 75 6d 65 |s, it i|s assume|
|00000200| 64 0d 24 42 20 3d 20 41 | 5e 7b 54 7d 24 2c 20 61 |d.$B = A|^{T}$, a|
|00000210| 6e 64 20 74 68 65 20 4c | 79 61 70 75 6e 6f 76 20 |nd the L|yapunov |
|00000220| 65 71 75 61 74 69 6f 6e | 20 69 73 20 73 6f 6c 76 |equation| is solv|
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|00000240| 61 72 67 75 6d 65 6e 74 | 73 20 6d 75 73 74 20 62 |argument|s must b|
|00000250| 6f 74 68 20 62 65 20 75 | 70 70 65 72 20 74 72 69 |oth be u|pper tri|
|00000260| 61 6e 67 75 6c 61 72 2e | 20 20 49 66 20 74 68 65 |angular.| If the|
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|00000280| 76 65 72 62 2b 6c 79 61 | 70 2b 20 66 75 6e 63 74 |verb+lya|p+ funct|
|00000290| 69 6f 6e 20 69 6d 70 6c | 65 6d 65 6e 74 73 20 61 |ion impl|ements a|
|000002a0| 20 77 72 61 70 70 65 72 | 20 6f 76 65 72 20 74 68 | wrapper| over th|
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|000002c0| 20 70 65 72 66 6f 72 6d | 73 20 73 75 69 74 61 62 | perform|s suitab|
|000002d0| 6c 65 20 53 63 68 75 72 | 20 64 65 63 6f 6d 70 6f |le Schur| decompo|
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|00000330| 65 67 69 6e 7b 76 65 72 | 62 61 74 69 6d 7d 0d 3e |egin{ver|batim}.>|
|00000340| 20 61 20 3d 20 74 72 69 | 75 28 72 61 6e 64 28 34 | a = tri|u(rand(4|
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|00000410| 62 20 3d 20 74 72 69 75 | 28 72 61 6e 64 28 34 2c |b = triu|(rand(4,|
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|000004f0| 0d 20 63 20 3d 0d 20 20 | 20 20 30 2e 38 34 35 20 |. c =. | 0.845 |
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|000005a0| 20 20 30 2e 36 32 33 20 | 20 0d 3e 20 78 20 3d 20 | 0.623 | .> x = |
|000005b0| 73 79 6c 76 28 61 2c 62 | 2c 63 29 0d 20 78 20 3d |sylv(a,b|,c). x =|
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|00000670| 38 33 20 20 0d 3e 20 61 | 2a 78 2b 78 2a 62 20 2b |83 .> a|*x+x*b +|
|00000680| 20 63 0d 2d 31 2e 31 31 | 65 2d 31 36 20 20 2d 35 | c.-1.11|e-16 -5|
|00000690| 2e 35 35 65 2d 31 37 20 | 20 20 20 20 20 20 20 20 |.55e-17 | |
|000006a0| 20 30 20 20 2d 35 2e 35 | 35 65 2d 31 37 20 20 0d | 0 -5.5|5e-17 .|
|000006b0| 20 20 20 20 20 20 20 20 | 30 20 20 20 20 20 20 20 | |0 |
|000006c0| 20 20 20 30 20 20 20 35 | 2e 35 35 65 2d 31 37 20 | 0 5|.55e-17 |
|000006d0| 20 20 20 20 20 20 20 20 | 20 30 20 20 0d 20 20 20 | | 0 . |
|000006e0| 20 20 20 20 20 30 20 20 | 20 20 20 20 20 20 20 20 | 0 | |
|000006f0| 30 20 20 20 20 20 20 20 | 20 20 20 30 20 20 20 20 |0 | 0 |
|00000700| 20 20 20 20 20 20 30 20 | 20 0d 20 20 20 20 20 20 | 0 | . |
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|00000720| 20 20 20 20 20 20 20 20 | 30 20 20 20 20 20 20 20 | |0 |
|00000730| 20 20 20 30 20 0d 5c 65 | 6e 64 7b 76 65 72 62 61 | 0 .\e|nd{verba|
|00000740| 74 69 6d 7d 0d 0d 53 65 | 65 20 61 6c 73 6f 3a 20 |tim}..Se|e also: |
|00000750| 50 61 67 65 7e 5c 70 61 | 67 65 72 65 66 7b 66 6e |Page~\pa|geref{fn|
|00000760| 3a 6c 79 61 70 7d 20 5c | 76 65 72 62 2b 6c 79 61 |:lyap} \|verb+lya|
|00000770| 70 28 29 2b 2c 0d 50 61 | 67 65 7e 5c 70 61 67 65 |p()+,.Pa|ge~\page|
|00000780| 72 65 66 7b 66 6e 3a 73 | 63 68 75 72 7d 20 5c 76 |ref{fn:s|chur} \v|
|00000790| 65 72 62 2b 73 63 68 75 | 72 28 29 2b 2e 0d 0d 5c |erb+schu|r()+...\|
|000007a0| 6e 65 77 70 61 67 65 0d | 0d 0d 0d |newpage.|... |
+--------+-------------------------+-------------------------+--------+--------+
