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|00000000| 5c 73 65 63 74 69 6f 6e | 7b 73 63 68 6f 72 64 20 |\section|{schord |
|00000010| 2d 2d 2d 20 4f 72 64 65 | 72 65 64 20 53 63 68 75 |--- Orde|red Schu|
|00000020| 72 20 64 65 63 6f 6d 70 | 6f 73 69 74 69 6f 6e 7d |r decomp|osition}|
|00000030| 0d 5c 6c 61 62 65 6c 7b | 66 6e 3a 73 63 68 6f 72 |.\label{|fn:schor|
|00000040| 64 7d 0d 5c 69 6e 64 65 | 78 7b 73 63 68 6f 72 64 |d}.\inde|x{schord|
|00000050| 28 29 7d 0d 5c 69 6e 64 | 65 78 7b 53 63 68 75 72 |()}.\ind|ex{Schur|
|00000060| 20 64 65 63 6f 6d 70 6f | 73 69 74 69 6f 6e 7d 0d | decompo|sition}.|
|00000070| 5c 62 65 67 69 6e 7b 72 | 61 69 6c 7d 0d 53 63 68 |\begin{r|ail}.Sch|
|00000080| 6f 72 64 20 3a 20 27 73 | 63 68 6f 72 64 27 20 27 |ord : 's|chord' '|
|00000090| 28 27 20 4c 69 73 74 20 | 27 2c 27 20 53 63 61 6c |(' List |',' Scal|
|000000a0| 61 72 20 27 2c 27 20 53 | 63 61 6c 61 72 27 29 27 |ar ',' S|calar')'|
|000000b0| 20 3b 0d 5c 65 6e 64 7b | 72 61 69 6c 7d 0d 0d 5c | ;.\end{|rail}..\|
|000000c0| 76 65 72 62 2b 73 63 68 | 6f 72 64 2b 20 69 73 20 |verb+sch|ord+ is |
|000000d0| 61 20 66 75 6e 63 74 69 | 6f 6e 20 75 73 65 64 20 |a functi|on used |
|000000e0| 66 6f 72 20 70 6f 73 74 | 2d 70 72 6f 63 65 73 73 |for post|-process|
|000000f0| 69 6e 67 20 74 68 65 20 | 72 65 73 75 6c 74 20 6f |ing the |result o|
|00000100| 66 0d 5c 76 65 72 62 2b | 73 63 68 75 72 2b 2e 20 |f.\verb+|schur+. |
|00000110| 20 54 68 65 20 6c 69 73 | 74 20 61 72 67 75 6d 65 | The lis|t argume|
|00000120| 6e 74 20 69 73 20 75 6e | 64 65 72 73 74 6f 6f 64 |nt is un|derstood|
|00000130| 20 74 6f 20 62 65 20 74 | 68 65 20 72 65 74 75 72 | to be t|he retur|
|00000140| 6e 20 76 61 6c 75 65 0d | 66 72 6f 6d 20 5c 76 65 |n value.|from \ve|
|00000150| 72 62 2b 73 63 68 75 72 | 2b 2c 20 61 6e 64 20 6d |rb+schur|+, and m|
|00000160| 75 73 74 20 63 6f 6e 74 | 61 69 6e 20 65 6c 65 6d |ust cont|ain elem|
|00000170| 65 6e 74 73 20 5c 76 65 | 72 62 2b 74 2b 20 61 6e |ents \ve|rb+t+ an|
|00000180| 64 20 5c 76 65 72 62 2b | 7a 2b 2e 0d 54 68 65 20 |d \verb+|z+..The |
|00000190| 72 65 73 75 6c 74 20 6f | 66 20 5c 76 65 72 62 2b |result o|f \verb+|
|000001a0| 73 63 68 6f 72 64 2b 20 | 69 73 20 61 20 6c 69 73 |schord+ |is a lis|
|000001b0| 74 20 63 6f 6e 74 61 69 | 6e 69 6e 67 20 65 6c 65 |t contai|ning ele|
|000001c0| 6d 65 6e 74 73 20 5c 76 | 65 72 62 2b 74 2b 20 61 |ments \v|erb+t+ a|
|000001d0| 6e 64 0d 5c 76 65 72 62 | 2b 7a 2b 2e 20 49 74 20 |nd.\verb|+z+. It |
|000001e0| 69 73 20 70 6f 73 73 69 | 62 6c 65 20 74 6f 20 63 |is possi|ble to c|
|000001f0| 6f 6e 63 61 74 65 6e 61 | 74 65 20 75 73 65 20 6f |oncatena|te use o|
|00000200| 66 20 5c 76 65 72 62 2b | 73 63 68 6f 72 64 2b 2e |f \verb+|schord+.|
|00000210| 0d 0d 54 68 65 20 62 61 | 73 69 63 20 69 64 65 61 |..The ba|sic idea|
|00000220| 20 6f 66 20 5c 76 65 72 | 62 2b 73 63 68 6f 72 64 | of \ver|b+schord|
|00000230| 2b 20 69 73 20 74 6f 20 | 72 65 61 72 72 61 6e 67 |+ is to |rearrang|
|00000240| 65 20 74 68 65 20 5c 76 | 65 72 62 2b 74 2b 20 6d |e the \v|erb+t+ m|
|00000250| 61 74 72 69 78 20 74 6f | 0d 6d 6f 76 65 20 74 68 |atrix to|.move th|
|00000260| 65 20 64 69 61 67 6f 6e | 61 6c 20 62 6c 6f 63 6b |e diagon|al block|
|00000270| 20 69 6e 20 74 68 65 20 | 72 6f 77 20 67 69 76 65 | in the |row give|
|00000280| 6e 20 62 79 20 74 68 65 | 20 66 69 72 73 74 20 73 |n by the| first s|
|00000290| 63 61 6c 61 72 20 61 72 | 67 75 6d 65 6e 74 0d 74 |calar ar|gument.t|
|000002a0| 6f 20 74 68 65 20 72 6f | 77 20 67 69 76 65 6e 20 |o the ro|w given |
|000002b0| 62 79 20 74 68 65 20 73 | 65 63 6f 6e 64 20 73 63 |by the s|econd sc|
|000002c0| 61 6c 61 72 20 61 72 67 | 75 6d 65 6e 74 2e 0d 0d |alar arg|ument...|
|000002d0| 5c 70 61 72 61 67 72 61 | 70 68 7b 45 78 61 6d 70 |\paragra|ph{Examp|
|000002e0| 6c 65 7d 0d 5c 69 6e 64 | 65 78 7b 65 78 61 6d 70 |le}.\ind|ex{examp|
|000002f0| 6c 65 20 6f 66 20 73 63 | 68 6f 72 64 28 29 7d 0d |le of sc|hord()}.|
|00000300| 5c 69 6e 64 65 78 7b 73 | 63 68 6f 72 64 28 29 21 |\index{s|chord()!|
|00000310| 65 78 61 6d 70 6c 65 7d | 0d 5c 62 65 67 69 6e 7b |example}|.\begin{|
|00000320| 76 65 72 62 61 74 69 6d | 7d 0d 3e 20 72 6f 6f 74 |verbatim|}.> root|
|00000330| 32 6f 6e 32 20 3d 20 73 | 71 72 74 28 32 29 2f 32 |2on2 = s|qrt(2)/2|
|00000340| 0d 20 72 6f 6f 74 32 6f | 6e 32 20 3d 0d 20 20 20 |. root2o|n2 =.   |
|00000350| 20 30 2e 37 30 37 0d 3e | 20 4d 20 3d 20 5b 20 72 | 0.707.>| M = [ r|
|00000360| 6f 6f 74 32 6f 6e 32 2c | 20 72 6f 6f 74 32 6f 6e |oot2on2,| root2on|
|00000370| 32 2c 30 3b 20 72 6f 6f | 74 32 6f 6e 32 2c 20 72 |2,0; roo|t2on2, r|
|00000380| 6f 6f 74 32 6f 6e 32 2c | 20 30 3b 20 30 2c 30 2c |oot2on2,| 0; 0,0,|
|00000390| 31 5d 0d 20 4d 20 3d 0d | 20 20 20 20 30 2e 37 30 |1]. M =.|    0.70|
|000003a0| 37 20 20 20 20 20 20 30 | 2e 37 30 37 20 20 20 20 |7      0|.707    |
|000003b0| 20 20 20 20 20 20 30 20 | 20 0d 20 20 20 20 30 2e |      0 | .    0.|
|000003c0| 37 30 37 20 20 20 20 20 | 20 30 2e 37 30 37 20 20 |707     | 0.707  |
|000003d0| 20 20 20 20 20 20 20 20 | 30 20 20 0d 20 20 20 20 |        |0  .    |
|000003e0| 20 20 20 20 30 20 20 20 | 20 20 20 20 20 20 20 30 |    0   |       0|
|000003f0| 20 20 20 20 20 20 20 20 | 20 20 31 20 20 0d 3e 20 |        |  1  .> |
|00000400| 52 20 3d 20 73 63 68 75 | 72 28 4d 29 0d 20 52 20 |R = schu|r(M). R |
|00000410| 3d 0d 20 20 20 74 20 20 | 20 20 20 20 20 20 20 20 |=.   t  |        |
|00000420| 20 20 7a 20 20 20 20 20 | 20 20 20 20 20 20 20 0d |  z     |       .|
|00000430| 3e 20 6e 65 77 3d 20 20 | 73 63 68 6f 72 64 28 52 |> new=  |schord(R|
|00000440| 2c 31 2c 33 29 0d 20 6e | 65 77 20 3d 0d 20 20 20 |,1,3). n|ew =.   |
|00000450| 74 20 20 20 20 20 20 20 | 20 20 20 20 20 7a 20 20 |t       |     z  |
|00000460| 20 20 20 20 20 20 20 20 | 20 20 0d 3e 20 6e 65 77 |        |  .> new|
|00000470| 2e 74 0d 2d 31 2e 31 31 | 65 2d 31 36 20 20 20 20 |.t.-1.11|e-16    |
|00000480| 20 20 20 20 20 20 30 20 | 20 20 20 20 20 20 20 20 |      0 |        |
|00000490| 20 30 20 20 0d 20 20 20 | 20 20 20 20 20 30 20 20 | 0  .   |     0  |
|000004a0| 20 20 20 20 20 20 20 20 | 31 20 20 20 20 20 20 20 |        |1       |
|000004b0| 20 20 20 30 20 20 0d 20 | 20 20 20 20 20 20 20 30 |   0  . |       0|
|000004c0| 20 20 20 20 20 20 20 20 | 20 20 30 20 20 20 20 20 |        |  0     |
|000004d0| 20 20 31 2e 34 31 20 20 | 0d 3e 20 6e 65 77 2e 7a |  1.41  |.> new.z|
|000004e0| 0d 20 20 20 2d 30 2e 37 | 30 37 20 20 20 20 20 20 |.   -0.7|07      |
|000004f0| 20 20 20 20 30 20 20 20 | 20 20 20 30 2e 37 30 37 |    0   |   0.707|
|00000500| 20 20 0d 20 20 20 20 30 | 2e 37 30 37 20 20 20 20 |  .    0|.707    |
|00000510| 20 20 20 20 20 20 30 20 | 20 20 20 20 20 30 2e 37 |      0 |     0.7|
|00000520| 30 37 20 20 0d 20 20 20 | 20 20 20 20 20 30 20 20 |07  .   |     0  |
|00000530| 20 20 20 20 20 20 20 20 | 31 20 20 20 20 20 20 20 |        |1       |
|00000540| 20 20 20 30 20 0d 3e 20 | 6e 65 77 2e 7a 2a 6e 65 |   0 .> |new.z*ne|
|00000550| 77 2e 74 2a 6e 65 77 2e | 7a 27 0d 20 20 20 20 30 |w.t*new.|z'.    0|
|00000560| 2e 37 30 37 20 20 20 20 | 20 20 30 2e 37 30 37 20 |.707    |  0.707 |
|00000570| 20 20 20 20 20 20 20 20 | 20 30 20 20 0d 20 20 20 |        | 0  .   |
|00000580| 20 30 2e 37 30 37 20 20 | 20 20 20 20 30 2e 37 30 | 0.707  |    0.70|
|00000590| 37 20 20 20 20 20 20 20 | 20 20 20 30 20 20 0d 20 |7       |   0  . |
|000005a0| 20 20 20 20 20 20 20 30 | 20 20 20 20 20 20 20 20 |       0|        |
|000005b0| 20 20 30 20 20 20 20 20 | 20 20 20 20 20 31 20 20 |  0     |     1  |
|000005c0| 0d 5c 65 6e 64 7b 76 65 | 72 62 61 74 69 6d 7d 0d |.\end{ve|rbatim}.|
|000005d0| 0d 53 65 65 20 61 6c 73 | 6f 3a 20 50 61 67 65 7e |.See als|o: Page~|
|000005e0| 5c 70 61 67 65 72 65 66 | 7b 66 6e 3a 73 63 68 75 |\pageref|{fn:schu|
|000005f0| 72 7d 20 5c 76 65 72 62 | 2b 73 63 68 75 72 28 29 |r} \verb|+schur()|
|00000600| 2b 2e 0d 0d 5c 6e 65 77 | 70 61 67 65 0d 0d 0d 0d |+...\new|page....|
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