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|00000000| 5c 73 65 63 74 69 6f 6e | 7b 71 72 20 2d 2d 2d 20 |\section|{qr --- |
|00000010| 51 52 20 64 65 63 6f 6d | 70 6f 73 69 74 69 6f 6e |QR decom|position|
|00000020| 7d 0d 5c 6c 61 62 65 6c | 7b 66 6e 3a 71 72 7d 0d |}.\label|{fn:qr}.|
|00000030| 5c 69 6e 64 65 78 7b 71 | 72 28 29 7d 0d 5c 69 6e |\index{q|r()}.\in|
|00000040| 64 65 78 7b 51 52 20 64 | 65 63 6f 6d 70 6f 73 69 |dex{QR d|ecomposi|
|00000050| 74 69 6f 6e 7d 0d 5c 69 | 6e 64 65 78 7b 6d 61 74 |tion}.\i|ndex{mat|
|00000060| 72 69 78 21 51 52 20 64 | 65 63 6f 6d 70 6f 73 74 |rix!QR d|ecompost|
|00000070| 69 6f 6e 7d 0d 5c 62 65 | 67 69 6e 7b 72 61 69 6c |ion}.\be|gin{rail|
|00000080| 7d 0d 51 52 20 3a 20 27 | 71 72 27 20 27 28 27 20 |}.QR : '|qr' '(' |
|00000090| 4d 61 74 72 69 78 20 28 | 20 27 2c 27 20 27 22 70 |Matrix (| ',' '"p|
|000000a0| 22 27 29 3f 20 27 29 27 | 20 3b 0d 5c 65 6e 64 7b |"')? ')'| ;.\end{|
|000000b0| 72 61 69 6c 7d 0d 0d 5c | 76 65 72 62 2b 71 72 2b |rail}..\|verb+qr+|
|000000c0| 20 63 6f 6d 70 75 74 65 | 73 20 74 68 65 20 51 52 | compute|s the QR|
|000000d0| 20 64 65 63 6f 6d 70 6f | 73 69 74 69 6f 6e 20 6f | decompo|sition o|
|000000e0| 66 20 74 68 65 20 61 72 | 67 75 6d 65 6e 74 20 6d |f the ar|gument m|
|000000f0| 61 74 72 69 78 2c 20 0d | 72 65 74 75 72 6e 69 6e |atrix, .|returnin|
|00000100| 67 20 61 20 6c 69 73 74 | 20 77 69 74 68 20 74 77 |g a list| with tw|
|00000110| 6f 20 65 6c 65 6d 65 6e | 74 73 20 5c 76 65 72 62 |o elemen|ts \verb|
|00000120| 2b 71 2b 20 61 6e 64 20 | 5c 76 65 72 62 2b 72 2b |+q+ and |\verb+r+|
|00000130| 2e 20 54 68 65 73 65 20 | 61 72 65 0d 74 68 65 20 |. These |are.the |
|00000140| 24 51 24 20 61 6e 64 20 | 24 52 24 20 6d 61 74 72 |$Q$ and |$R$ matr|
|00000150| 69 63 65 73 2c 20 73 75 | 63 68 20 74 68 61 74 2c |ices, su|ch that,|
|00000160| 20 66 6f 72 20 61 6e 20 | 61 72 67 75 6d 65 6e 74 | for an |argument|
|00000170| 20 6d 61 74 72 69 78 20 | 6f 66 20 24 41 24 3a 0d | matrix |of $A$:.|
|00000180| 0d 5c 5b 20 41 20 3d 20 | 51 20 52 20 5c 5d 0d 0d |.\[ A = |Q R \]..|
|00000190| 49 66 20 74 68 65 20 6f | 70 74 69 6f 6e 61 6c 20 |If the o|ptional |
|000001a0| 61 72 67 75 6d 65 6e 74 | 20 5c 76 65 72 62 2b 22 |argument| \verb+"|
|000001b0| 70 22 2b 20 69 73 20 75 | 73 65 64 2c 20 63 6f 6c |p"+ is u|sed, col|
|000001c0| 75 6d 6e 20 70 69 76 6f | 74 69 6e 67 20 69 73 20 |umn pivo|ting is |
|000001d0| 75 73 65 64 2c 0d 77 69 | 74 68 20 74 68 65 20 70 |used,.wi|th the p|
|000001e0| 65 72 6d 75 74 61 74 69 | 6f 6e 20 6d 61 74 72 69 |ermutati|on matri|
|000001f0| 78 20 72 65 74 75 72 6e | 65 64 20 69 6e 20 74 68 |x return|ed in th|
|00000200| 65 20 6c 69 73 74 20 72 | 65 73 75 6c 74 20 61 73 |e list r|esult as|
|00000210| 20 5c 76 65 72 62 2b 70 | 2b 2c 20 73 75 63 68 0d | \verb+p|+, such.|
|00000220| 74 68 61 74 20 66 6f 72 | 20 61 20 70 65 72 6d 75 |that for| a permu|
|00000230| 74 61 74 69 6f 6e 20 6d | 61 74 72 69 78 20 24 50 |tation m|atrix $P|
|00000240| 24 2c 0d 0d 5c 5b 20 41 | 20 50 20 3d 20 51 20 52 |$,..\[ A| P = Q R|
|00000250| 20 5c 5d 0d 0d 24 51 24 | 20 77 69 6c 6c 20 68 61 | \]..$Q$| will ha|
|00000260| 76 65 20 6f 72 74 68 6f | 67 6f 6e 61 6c 20 63 6f |ve ortho|gonal co|
|00000270| 6c 75 6d 6e 73 2c 20 61 | 6e 64 20 24 52 24 20 77 |lumns, a|nd $R$ w|
|00000280| 69 6c 6c 20 62 65 20 75 | 70 70 65 72 2d 74 72 69 |ill be u|pper-tri|
|00000290| 61 6e 67 75 6c 61 72 2e | 0d 0d 5c 70 61 72 61 67 |angular.|..\parag|
|000002a0| 72 61 70 68 7b 45 78 61 | 6d 70 6c 65 7d 0d 5c 69 |raph{Exa|mple}.\i|
|000002b0| 6e 64 65 78 7b 65 78 61 | 6d 70 6c 65 20 6f 66 20 |ndex{exa|mple of |
|000002c0| 71 72 28 29 7d 0d 5c 69 | 6e 64 65 78 7b 71 72 28 |qr()}.\i|ndex{qr(|
|000002d0| 29 21 65 78 61 6d 70 6c | 65 7d 0d 5c 62 65 67 69 |)!exampl|e}.\begi|
|000002e0| 6e 7b 76 65 72 62 61 74 | 69 6d 7d 0d 3e 20 61 20 |n{verbat|im}.> a |
|000002f0| 3d 20 72 61 6e 64 28 35 | 2c 35 29 0d 20 61 20 3d |= rand(5|,5). a =|
|00000300| 0d 20 20 20 20 20 30 2e | 38 31 20 20 20 20 20 20 |.     0.|81      |
|00000310| 30 2e 36 32 31 20 20 20 | 20 20 20 30 2e 33 32 38 |0.621   |   0.328|
|00000320| 20 20 20 20 20 30 2e 30 | 37 36 37 20 20 20 20 20 |     0.0|767     |
|00000330| 20 30 2e 38 31 38 20 20 | 0d 20 20 20 20 30 2e 39 | 0.818  |.    0.9|
|00000340| 30 39 20 20 20 20 20 20 | 30 2e 38 34 35 20 20 20 |09      |0.845   |
|00000350| 20 20 20 30 2e 37 39 39 | 20 20 20 20 20 20 30 2e |   0.799|      0.|
|00000360| 36 34 37 20 20 20 20 20 | 20 30 2e 33 36 37 20 20 |647     | 0.367  |
|00000370| 0d 20 20 20 20 30 2e 32 | 39 33 20 20 20 20 20 20 |.    0.2|93      |
|00000380| 30 2e 31 33 32 20 20 20 | 20 20 30 2e 30 33 39 38 |0.132   |  0.0398|
|00000390| 20 20 20 20 20 30 2e 30 | 32 32 31 20 20 20 20 20 |     0.0|221     |
|000003a0| 20 30 2e 31 39 31 20 20 | 0d 20 20 20 20 30 2e 37 | 0.191  |.    0.7|
|000003b0| 34 33 20 20 20 20 20 30 | 2e 30 33 37 36 20 20 20 |43     0|.0376   |
|000003c0| 20 20 20 30 2e 33 39 38 | 20 20 20 20 20 20 30 2e |   0.398|      0.|
|000003d0| 32 32 37 20 20 20 20 20 | 20 30 2e 38 34 35 20 20 |227     | 0.845  |
|000003e0| 0d 20 20 20 20 30 2e 37 | 30 35 20 20 20 20 20 20 |.    0.7|05      |
|000003f0| 30 2e 38 31 31 20 20 20 | 20 20 20 30 2e 32 36 33 |0.811   |   0.263|
|00000400| 20 20 20 20 20 20 30 2e | 34 33 31 20 20 20 20 20 |      0.|431     |
|00000410| 20 30 2e 34 30 31 20 20 | 0d 3e 20 72 65 73 20 3d | 0.401  |.> res =|
|00000420| 20 71 72 28 61 29 0d 20 | 72 65 73 20 3d 0d 20 20 | qr(a). |res =.  |
|00000430| 20 71 20 20 20 20 20 20 | 20 20 20 20 20 20 72 20 | q      |      r |
|00000440| 20 20 20 20 20 20 20 20 | 20 20 20 0d 3e 20 61 20 |        |   .> a |
|00000450| 2d 20 72 65 73 2e 71 2a | 72 65 73 2e 72 0d 2d 31 |- res.q*|res.r.-1|
|00000460| 2e 31 31 65 2d 31 36 20 | 20 2d 33 2e 33 33 65 2d |.11e-16 | -3.33e-|
|00000470| 31 36 20 20 2d 33 2e 38 | 39 65 2d 31 36 20 20 2d |16  -3.8|9e-16  -|
|00000480| 31 2e 31 31 65 2d 31 36 | 20 20 2d 32 2e 32 32 65 |1.11e-16|  -2.22e|
|00000490| 2d 31 36 20 20 0d 20 20 | 20 20 20 20 20 20 30 20 |-16  .  |      0 |
|000004a0| 20 20 20 20 20 20 20 20 | 20 30 20 20 20 31 2e 31 |        | 0   1.1|
|000004b0| 31 65 2d 31 36 20 20 20 | 31 2e 31 31 65 2d 31 36 |1e-16   |1.11e-16|
|000004c0| 20 20 2d 31 2e 36 37 65 | 2d 31 36 20 20 0d 20 20 |  -1.67e|-16  .  |
|000004d0| 20 20 20 20 20 20 30 20 | 20 2d 35 2e 35 35 65 2d |      0 | -5.55e-|
|000004e0| 31 37 20 20 2d 39 2e 37 | 31 65 2d 31 37 20 20 2d |17  -9.7|1e-17  -|
|000004f0| 31 2e 34 32 65 2d 31 36 | 20 20 2d 32 2e 37 38 65 |1.42e-16|  -2.78e|
|00000500| 2d 31 37 20 20 0d 20 20 | 20 20 20 20 20 20 30 20 |-17  .  |      0 |
|00000510| 20 2d 36 2e 39 34 65 2d | 31 37 20 20 2d 32 2e 32 | -6.94e-|17  -2.2|
|00000520| 32 65 2d 31 36 20 20 2d | 35 2e 35 35 65 2d 31 37 |2e-16  -|5.55e-17|
|00000530| 20 20 20 20 20 20 20 20 | 20 20 30 20 20 0d 20 20 |        |  0  .  |
|00000540| 20 20 20 20 20 20 30 20 | 20 20 20 20 20 20 20 20 |      0 |        |
|00000550| 20 30 20 20 2d 35 2e 35 | 35 65 2d 31 37 20 20 2d | 0  -5.5|5e-17  -|
|00000560| 35 2e 35 35 65 2d 31 37 | 20 20 2d 35 2e 35 35 65 |5.55e-17|  -5.55e|
|00000570| 2d 31 37 20 20 0d 3e 20 | 72 65 73 32 20 3d 20 71 |-17  .> |res2 = q|
|00000580| 72 28 61 2c 22 70 22 29 | 0d 20 72 65 73 32 20 3d |r(a,"p")|. res2 =|
|00000590| 0d 20 20 20 70 20 20 20 | 20 20 20 20 20 20 20 20 |.   p   |        |
|000005a0| 20 71 20 20 20 20 20 20 | 20 20 20 20 20 20 72 20 | q      |      r |
|000005b0| 20 20 20 20 20 20 20 20 | 20 20 20 0d 3e 20 61 2a |        |   .> a*|
|000005c0| 72 65 73 32 2e 70 20 2d | 20 72 65 73 32 2e 71 2a |res2.p -| res2.q*|
|000005d0| 72 65 73 2e 72 0d 2d 31 | 2e 31 31 65 2d 31 36 20 |res.r.-1|.11e-16 |
|000005e0| 20 2d 33 2e 33 33 65 2d | 31 36 20 20 2d 33 2e 38 | -3.33e-|16  -3.8|
|000005f0| 39 65 2d 31 36 20 20 2d | 31 2e 31 31 65 2d 31 36 |9e-16  -|1.11e-16|
|00000600| 20 20 2d 32 2e 32 32 65 | 2d 31 36 20 20 0d 20 20 |  -2.22e|-16  .  |
|00000610| 20 20 20 20 20 20 30 20 | 20 20 20 20 20 20 20 20 |      0 |        |
|00000620| 20 30 20 20 20 31 2e 31 | 31 65 2d 31 36 20 20 20 | 0   1.1|1e-16   |
|00000630| 31 2e 31 31 65 2d 31 36 | 20 20 2d 31 2e 36 37 65 |1.11e-16|  -1.67e|
|00000640| 2d 31 36 20 20 0d 20 20 | 20 20 20 20 20 20 30 20 |-16  .  |      0 |
|00000650| 20 2d 35 2e 35 35 65 2d | 31 37 20 20 2d 39 2e 37 | -5.55e-|17  -9.7|
|00000660| 31 65 2d 31 37 20 20 2d | 31 2e 34 32 65 2d 31 36 |1e-17  -|1.42e-16|
|00000670| 20 20 2d 32 2e 37 38 65 | 2d 31 37 20 20 0d 20 20 |  -2.78e|-17  .  |
|00000680| 20 20 20 20 20 20 30 20 | 20 2d 36 2e 39 34 65 2d |      0 | -6.94e-|
|00000690| 31 37 20 20 2d 32 2e 32 | 32 65 2d 31 36 20 20 2d |17  -2.2|2e-16  -|
|000006a0| 35 2e 35 35 65 2d 31 37 | 20 20 20 20 20 20 20 20 |5.55e-17|        |
|000006b0| 20 20 30 20 20 0d 20 20 | 20 20 20 20 20 20 30 20 |  0  .  |      0 |
|000006c0| 20 20 20 20 20 20 20 20 | 20 30 20 20 2d 35 2e 35 |        | 0  -5.5|
|000006d0| 35 65 2d 31 37 20 20 2d | 35 2e 35 35 65 2d 31 37 |5e-17  -|5.55e-17|
|000006e0| 20 20 2d 35 2e 35 35 65 | 2d 31 37 20 20 0d 5c 65 |  -5.55e|-17  .\e|
|000006f0| 6e 64 7b 76 65 72 62 61 | 74 69 6d 7d 0d 0d 5c 6e |nd{verba|tim}..\n|
|00000700| 65 77 70 61 67 65 0d 0d | 0d 0d                   |ewpage..|..      |
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