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|00000000| 00 0d 43 6f 6d 70 6c 65 | 78 20 52 6f 6f 74 73 00 |..Comple|x Roots.|
|00000010| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000020| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000030| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000040| 00 54 45 58 54 4d 50 41 | 44 00 00 00 00 00 00 00 |.TEXTMPA|D.......|
|00000050| 00 00 00 00 00 14 02 00 | 00 01 56 a8 c1 94 a2 a8 |........|..V.....|
|00000060| c1 94 a2 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000070| 00 00 00 00 00 00 00 00 | 00 00 81 81 01 b8 00 00 |........|........|
|00000080| 2d 2d 20 54 68 69 73 20 | 65 78 61 6d 70 6c 65 20 |-- This |example |
|00000090| 67 69 76 65 73 20 66 6f | 72 6d 75 6c 61 73 20 66 |gives fo|rmulas f|
|000000a0| 6f 72 20 71 75 61 64 72 | 61 74 69 63 20 61 6e 64 |or quadr|atic and|
|000000b0| 20 63 75 62 69 63 20 72 | 6f 6f 74 73 20 61 6e 64 | cubic r|oots and|
|000000c0| 20 75 73 65 73 20 74 68 | 65 20 69 6d 61 67 65 20 | uses th|e image |
|000000d0| 63 6f 6d 6d 61 6e 64 20 | 74 6f 20 76 69 73 75 61 |command |to visua|
|000000e0| 6c 69 7a 65 20 61 20 63 | 6f 6d 70 6c 65 78 20 66 |lize a c|omplex f|
|000000f0| 75 6e 63 74 69 6f 6e 2e | 20 55 74 69 6c 69 74 79 |unction.| Utility|
|00000100| 20 66 75 6e 63 74 69 6f | 6e 73 20 66 6f 72 20 63 | functio|ns for c|
|00000110| 6f 6d 70 6c 65 78 20 6e | 75 6d 62 65 72 73 20 61 |omplex n|umbers a|
|00000120| 72 65 20 61 74 20 74 68 | 65 20 65 6e 64 20 6f 66 |re at th|e end of|
|00000130| 20 74 68 65 20 64 6f 63 | 75 6d 65 6e 74 2e 0d 0d | the doc|ument...|
|00000140| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000150| 2d 2d 2d 20 71 75 61 64 | 72 61 74 69 63 20 72 6f |--- quad|ratic ro|
|00000160| 6f 74 73 20 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |ots ----|--------|
|00000170| 2d 2d 2d 0d 2d 2d 20 41 | 6c 67 6f 72 69 74 68 6d |---.-- A|lgorithm|
|00000180| 20 66 6f 72 20 72 65 61 | 6c 20 70 61 72 74 73 20 | for rea|l parts |
|00000190| 6f 66 20 72 6f 6f 74 73 | 20 69 73 20 66 72 6f 6d |of roots| is from|
|000001a0| 20 62 79 20 57 2e 48 2e | 20 50 72 65 73 73 2c 20 | by W.H.| Press, |
|000001b0| 53 2e 20 54 65 75 6b 6f | 6c 73 6b 79 20 65 74 20 |S. Teuko|lsky et |
|000001c0| 61 6c 2c 22 4e 75 6d 65 | 72 69 63 61 6c 20 52 65 |al,"Nume|rical Re|
|000001d0| 63 69 70 65 73 22 2e 0d | 2d 2d 20 4f 63 63 61 73 |cipes"..|-- Occas|
|000001e0| 69 6f 6e 61 6c 6c 79 20 | 34 61 63 20 3c 3c 20 62 |ionally |4ac << b|
|000001f0| 2c 20 73 6f 20 6f 6e 65 | 20 6f 66 20 74 68 65 20 |, so one| of the |
|00000200| 72 6f 6f 74 73 20 69 73 | 20 28 65 72 72 6f 6e 65 |roots is| (errone|
|00000210| 6f 75 73 6c 79 29 20 63 | 61 6c 6c 65 64 20 30 2e |ously) c|alled 0.|
|00000220| 0d 2d 2d 20 54 68 69 73 | 20 66 6f 72 6d 75 6c 61 |.-- This| formula|
|00000230| 74 69 6f 6e 20 61 76 6f | 69 64 73 20 74 68 65 20 |tion avo|ids the |
|00000240| 70 72 6f 62 6c 65 6d 2e | 0d 2d 2d 20 69 6d 70 6c |problem.|.-- impl|
|00000250| 65 6d 65 6e 74 65 64 20 | 62 79 20 44 61 76 69 64 |emented |by David|
|00000260| 20 44 65 72 62 65 73 20 | 66 6f 72 20 4d 61 74 68 | Derbes |for Math|
|00000270| 50 61 64 0d 0d 2d 2d 20 | 47 69 76 65 6e 20 61 20 |Pad..-- |Given a |
|00000280| 71 75 61 64 72 61 74 69 | 63 20 6f 66 20 74 68 65 |quadrati|c of the|
|00000290| 20 66 6f 72 6d 0d 0d 2d | 2d 20 20 20 20 20 20 20 | form..-|-       |
|000002a0| 20 20 20 61 2a 78 5e 32 | 20 2b 20 62 2a 78 20 2b |   a*x^2| + b*x +|
|000002b0| 20 63 20 3d 20 30 0d 20 | 0d 2d 2d 20 77 69 74 68 | c = 0. |.-- with|
|000002c0| 20 72 65 61 6c 20 63 6f | 65 66 66 69 63 69 65 6e | real co|efficien|
|000002d0| 74 73 2c 20 66 69 6e 64 | 20 74 68 65 20 28 70 6f |ts, find| the (po|
|000002e0| 73 73 69 62 6c 79 20 63 | 6f 6d 70 6c 65 78 29 20 |ssibly c|omplex) |
|000002f0| 72 6f 6f 74 73 2e 0d 2d | 2d 20 52 6f 6f 74 73 20 |roots..-|- Roots |
|00000300| 61 72 65 20 67 69 76 65 | 6e 20 69 6e 20 74 68 65 |are give|n in the|
|00000310| 20 66 6f 72 6d 20 78 20 | 2b 20 69 79 2e 0d 7e 0d | form x |+ iy..~.|
|00000320| 73 67 6e 28 78 29 20 3d | 20 31 20 77 68 65 6e 20 |sgn(x) =| 1 when |
|00000330| 78 20 3e 3d 20 30 2c 2d | 31 20 6f 74 68 65 72 77 |x >= 0,-|1 otherw|
|00000340| 69 73 65 0d 44 20 3d 20 | 28 62 2a 62 20 2d 20 34 |ise.D = |(b*b - 4|
|00000350| 2a 61 2a 63 29 09 2d 2d | 20 64 69 73 63 72 69 6d |*a*c).--| discrim|
|00000360| 69 6e 61 6e 74 0d 78 31 | 20 3d 20 2d 28 62 20 2b |inant.x1| = -(b +|
|00000370| 20 73 67 6e 28 62 29 2a | 73 71 72 74 28 44 29 29 | sgn(b)*|sqrt(D))|
|00000380| 2f 28 32 2a 61 29 20 77 | 68 65 6e 20 44 20 3e 3d |/(2*a) w|hen D >=|
|00000390| 20 30 2c 20 2d 62 2f 28 | 32 2a 61 29 20 6f 74 68 | 0, -b/(|2*a) oth|
|000003a0| 65 72 77 69 73 65 0d 78 | 32 20 3d 20 63 2f 78 31 |erwise.x|2 = c/x1|
|000003b0| 20 77 68 65 6e 20 44 20 | 3e 3d 20 30 2c 20 2d 62 | when D |>= 0, -b|
|000003c0| 2f 28 32 2a 61 29 20 6f | 74 68 65 72 77 69 73 65 |/(2*a) o|therwise|
|000003d0| 0d 79 31 20 3d 20 30 20 | 77 68 65 6e 20 44 20 3e |.y1 = 0 |when D >|
|000003e0| 3d 20 30 2c 20 73 71 72 | 74 28 2d 44 29 2f 28 32 |= 0, sqr|t(-D)/(2|
|000003f0| 2a 61 29 20 6f 74 68 65 | 72 77 69 73 65 0d 79 32 |*a) othe|rwise.y2|
|00000400| 20 3d 20 2d 79 31 0d 7e | 0d 0d 2d 2d 2d 2d 2d 2d | = -y1.~|..------|
|00000410| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 20 63 75 |--------|----- cu|
|00000420| 62 69 63 20 72 6f 6f 74 | 73 20 2d 2d 2d 2d 2d 2d |bic root|s ------|
|00000430| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 0d 2d 2d |--------|-----.--|
|00000440| 20 54 61 72 74 61 67 6c | 69 61 27 73 20 26 20 43 | Tartagl|ia's & C|
|00000450| 61 72 64 61 6e 6f 27 73 | 20 66 6f 72 6d 75 6c 61 |ardano's| formula|
|00000460| 65 20 66 6f 72 20 74 68 | 65 20 72 6f 6f 74 73 20 |e for th|e roots |
|00000470| 6f 66 20 61 20 63 75 62 | 69 63 0d 2d 2d 20 66 72 |of a cub|ic.-- fr|
|00000480| 6f 6d 20 55 6e 69 76 65 | 72 73 61 6c 20 45 6e 63 |om Unive|rsal Enc|
|00000490| 79 63 6c 6f 70 61 65 64 | 69 61 20 6f 66 20 4d 61 |yclopaed|ia of Ma|
|000004a0| 74 68 65 6d 61 74 69 63 | 73 0d 2d 2d 20 69 6d 70 |thematic|s.-- imp|
|000004b0| 6c 65 6d 65 6e 74 65 64 | 20 62 79 20 44 61 76 69 |lemented| by Davi|
|000004c0| 64 20 44 65 72 62 65 73 | 20 66 6f 72 20 4d 61 74 |d Derbes| for Mat|
|000004d0| 68 50 61 64 2c 20 38 20 | 53 65 70 74 20 31 39 39 |hPad, 8 |Sept 199|
|000004e0| 33 0d 2d 2d 20 47 69 76 | 65 6e 20 61 20 63 75 62 |3.-- Giv|en a cub|
|000004f0| 69 63 20 6f 66 20 74 68 | 65 20 66 6f 72 6d 0d 2d |ic of th|e form.-|
|00000500| 2d 0d 2d 2d 20 20 20 20 | 20 20 20 20 20 20 20 20 |-.--    |        |
|00000510| 61 30 2a 78 5e 33 20 2b | 20 61 31 2a 78 5e 32 20 |a0*x^3 +| a1*x^2 |
|00000520| 2b 20 61 32 2a 78 20 2b | 20 61 33 20 3d 20 30 0d |+ a2*x +| a3 = 0.|
|00000530| 2d 2d 20 0d 2d 2d 20 77 | 69 74 68 20 72 65 61 6c |-- .-- w|ith real|
|00000540| 20 63 6f 65 66 66 69 63 | 69 65 6e 74 73 2c 20 66 | coeffic|ients, f|
|00000550| 69 6e 64 20 74 68 65 20 | 28 70 6f 73 73 69 62 6c |ind the |(possibl|
|00000560| 79 20 63 6f 6d 70 6c 65 | 78 29 20 72 6f 6f 74 73 |y comple|x) roots|
|00000570| 2e 0d 0d 63 31 20 3d 20 | 61 31 2f 61 30 20 20 2d |...c1 = |a1/a0  -|
|00000580| 2d 20 22 6e 6f 72 6d 61 | 6c 69 7a 65 22 2c 20 69 |- "norma|lize", i|
|00000590| 2e 65 2e 20 6d 61 6b 65 | 20 6c 65 61 64 69 6e 67 |.e. make| leading|
|000005a0| 20 63 6f 65 66 66 69 63 | 69 65 6e 74 20 3d 20 31 | coeffic|ient = 1|
|000005b0| 0d 63 32 20 3d 20 61 32 | 2f 61 30 0d 63 33 20 3d |.c2 = a2|/a0.c3 =|
|000005c0| 20 61 33 2f 61 30 0d 0d | 2d 2d 20 64 69 73 63 72 | a3/a0..|-- discr|
|000005d0| 69 6d 69 6e 61 6e 74 20 | 44 3b 20 69 66 20 44 20 |iminant |D; if D |
|000005e0| 3e 20 30 2c 20 6f 6e 65 | 20 72 65 61 6c 20 72 6f |> 0, one| real ro|
|000005f0| 6f 74 3b 20 65 6c 73 65 | 20 74 68 72 65 65 20 72 |ot; else| three r|
|00000600| 65 61 6c 20 72 6f 6f 74 | 73 0d 2d 2d 20 28 61 74 |eal root|s.-- (at|
|00000610| 20 6d 6f 73 74 20 74 77 | 6f 20 64 69 73 74 69 6e | most tw|o distin|
|00000620| 63 74 20 72 65 61 6c 20 | 69 66 20 44 20 3d 20 30 |ct real |if D = 0|
|00000630| 29 0d 0d 61 20 3d 20 63 | 32 20 2d 20 28 63 31 2a |)..a = c|2 - (c1*|
|00000640| 63 31 29 2f 33 2e 30 0d | 62 20 3d 20 28 28 28 32 |c1)/3.0.|b = (((2|
|00000650| 2e 30 2a 63 31 2a 63 31 | 2a 63 31 29 20 2d 20 28 |.0*c1*c1|*c1) - (|
|00000660| 39 2e 30 2a 63 31 2a 63 | 32 29 29 2f 32 37 2e 30 |9.0*c1*c|2))/27.0|
|00000670| 29 20 2b 20 63 33 0d 44 | 20 3d 20 28 62 2a 62 2f |) + c3.D| = (b*b/|
|00000680| 34 2e 30 29 20 2b 20 28 | 61 2a 61 2a 61 2f 32 37 |4.0) + (|a*a*a/27|
|00000690| 2e 30 29 0d 0d 6e 75 6d | 52 65 61 6c 52 6f 6f 74 |.0)..num|RealRoot|
|000006a0| 73 20 3d 20 33 20 77 68 | 65 6e 20 44 20 3c 20 30 |s = 3 wh|en D < 0|
|000006b0| 2c 20 31 20 6f 74 68 65 | 72 77 69 73 65 0d 0d 64 |, 1 othe|rwise..d|
|000006c0| 48 61 6c 66 20 3d 20 73 | 71 72 74 28 61 62 73 28 |Half = s|qrt(abs(|
|000006d0| 44 29 29 0d 0d 73 67 6e | 70 20 3d 20 2d 31 20 77 |D))..sgn|p = -1 w|
|000006e0| 68 65 6e 20 28 28 2d 62 | 2f 32 2e 30 29 20 2b 20 |hen ((-b|/2.0) + |
|000006f0| 64 48 61 6c 66 29 20 3c | 20 30 2c 20 31 20 6f 74 |dHalf) <| 0, 1 ot|
|00000700| 68 65 72 77 69 73 65 20 | 20 20 20 0d 73 67 6e 71 |herwise |   .sgnq|
|00000710| 20 3d 20 2d 31 20 77 68 | 65 6e 20 28 28 2d 62 2f | = -1 wh|en ((-b/|
|00000720| 32 2e 30 29 20 2d 20 64 | 48 61 6c 66 29 20 3c 20 |2.0) - d|Half) < |
|00000730| 30 2c 20 31 20 6f 74 68 | 65 72 77 69 73 65 0d 0d |0, 1 oth|erwise..|
|00000740| 70 20 3d 20 30 2e 30 20 | 77 68 65 6e 20 28 28 2d |p = 0.0 |when ((-|
|00000750| 62 2f 32 2e 30 29 20 2b | 20 64 48 61 6c 66 29 20 |b/2.0) +| dHalf) |
|00000760| 3d 20 30 2c 0d 20 20 20 | 20 20 73 67 6e 70 2a 28 |= 0,.   |  sgnp*(|
|00000770| 61 62 73 28 28 2d 62 2f | 32 2e 30 29 20 2b 20 64 |abs((-b/|2.0) + d|
|00000780| 48 61 6c 66 29 29 5e 28 | 31 2e 30 2f 33 2e 30 29 |Half))^(|1.0/3.0)|
|00000790| 20 6f 74 68 65 72 77 69 | 73 65 0d 71 20 3d 20 30 | otherwi|se.q = 0|
|000007a0| 2e 30 20 77 68 65 6e 20 | 28 28 2d 62 2f 32 2e 30 |.0 when |((-b/2.0|
|000007b0| 29 20 2d 20 64 48 61 6c | 66 29 20 3d 20 30 2c 0d |) - dHal|f) = 0,.|
|000007c0| 20 20 20 20 20 73 67 6e | 71 2a 28 61 62 73 28 28 |     sgn|q*(abs((|
|000007d0| 2d 62 2f 32 2e 30 29 20 | 2d 20 64 48 61 6c 66 29 |-b/2.0) |- dHalf)|
|000007e0| 29 5e 28 31 2e 30 2f 33 | 2e 30 29 20 6f 74 68 65 |)^(1.0/3|.0) othe|
|000007f0| 72 77 69 73 65 0d 0d 73 | 20 3d 20 28 2d 62 2f 32 |rwise..s| = (-b/2|
|00000800| 2e 30 29 2f 73 71 72 74 | 28 2d 61 2a 61 2a 61 2f |.0)/sqrt|(-a*a*a/|
|00000810| 32 37 2e 30 29 3b 20 20 | 74 68 65 74 61 20 3d 20 |27.0);  |theta = |
|00000820| 61 63 6f 73 28 73 29 0d | 0d 2d 2d 20 72 6f 6f 74 |acos(s).|.-- root|
|00000830| 73 20 6f 66 20 74 68 65 | 20 66 6f 72 6d 20 78 20 |s of the| form x |
|00000840| 2b 20 69 79 0d 0d 78 31 | 20 3d 20 32 2e 30 2a 70 |+ iy..x1| = 2.0*p|
|00000850| 20 2d 20 28 63 31 2f 33 | 2e 30 29 20 77 68 65 6e | - (c1/3|.0) when|
|00000860| 20 6e 75 6d 52 65 61 6c | 52 6f 6f 74 73 20 3d 20 | numReal|Roots = |
|00000870| 31 20 61 6e 64 20 61 62 | 73 28 44 29 20 3c 20 31 |1 and ab|s(D) < 1|
|00000880| 2e 30 65 2d 31 30 2c 0d | 20 20 20 20 20 28 70 2b |.0e-10,.|     (p+|
|00000890| 71 29 20 2d 20 28 63 31 | 2f 33 2e 30 29 20 77 68 |q) - (c1|/3.0) wh|
|000008a0| 65 6e 20 6e 75 6d 52 65 | 61 6c 52 6f 6f 74 73 20 |en numRe|alRoots |
|000008b0| 3d 20 31 20 61 6e 64 20 | 61 62 73 28 44 29 20 3e |= 1 and |abs(D) >|
|000008c0| 20 31 2e 30 65 2d 31 30 | 2c 0d 20 20 20 20 20 32 | 1.0e-10|,.     2|
|000008d0| 2e 30 2a 73 71 72 74 28 | 2d 61 2f 33 2e 30 29 2a |.0*sqrt(|-a/3.0)*|
|000008e0| 63 6f 73 28 74 68 65 74 | 61 2f 33 2e 30 29 20 2d |cos(thet|a/3.0) -|
|000008f0| 20 28 63 31 2f 33 2e 30 | 29 20 6f 74 68 65 72 77 | (c1/3.0|) otherw|
|00000900| 69 73 65 0d 0d 78 32 20 | 3d 20 2d 70 20 2d 20 28 |ise..x2 |= -p - (|
|00000910| 63 31 2f 33 2e 30 29 20 | 77 68 65 6e 20 6e 75 6d |c1/3.0) |when num|
|00000920| 52 65 61 6c 52 6f 6f 74 | 73 20 3d 20 31 20 61 6e |RealRoot|s = 1 an|
|00000930| 64 20 61 62 73 28 44 29 | 20 3c 20 31 2e 30 65 2d |d abs(D)| < 1.0e-|
|00000940| 31 30 2c 0d 20 20 20 20 | 20 2d 28 70 2b 71 29 2f |10,.    | -(p+q)/|
|00000950| 32 2e 30 20 2d 20 28 63 | 31 2f 33 2e 30 29 20 77 |2.0 - (c|1/3.0) w|
|00000960| 68 65 6e 20 6e 75 6d 52 | 65 61 6c 52 6f 6f 74 73 |hen numR|ealRoots|
|00000970| 20 3d 20 31 20 61 6e 64 | 20 61 62 73 28 44 29 20 | = 1 and| abs(D) |
|00000980| 3e 20 31 2e 30 65 2d 31 | 30 2c 0d 20 20 20 20 20 |> 1.0e-1|0,.     |
|00000990| 32 2e 30 2a 73 71 72 74 | 28 2d 61 2f 33 2e 30 29 |2.0*sqrt|(-a/3.0)|
|000009a0| 2a 63 6f 73 28 28 74 68 | 65 74 61 2f 33 2e 30 29 |*cos((th|eta/3.0)|
|000009b0| 20 2b 20 31 32 30 29 20 | 2d 20 63 31 2f 33 2e 30 | + 120) |- c1/3.0|
|000009c0| 20 6f 74 68 65 72 77 69 | 73 65 20 0d 0d 78 33 20 | otherwi|se ..x3 |
|000009d0| 3d 20 78 32 20 77 68 65 | 6e 20 6e 75 6d 52 65 61 |= x2 whe|n numRea|
|000009e0| 6c 52 6f 6f 74 73 20 3d | 20 31 2c 20 20 20 20 20 |lRoots =| 1,     |
|000009f0| 20 0d 20 20 20 20 20 32 | 2e 30 2a 73 71 72 74 28 | .     2|.0*sqrt(|
|00000a00| 2d 61 2f 33 2e 30 29 2a | 63 6f 73 28 28 74 68 65 |-a/3.0)*|cos((the|
|00000a10| 74 61 2f 33 2e 30 29 20 | 2b 20 32 34 30 29 20 2d |ta/3.0) |+ 240) -|
|00000a20| 20 63 31 2f 33 2e 30 20 | 6f 74 68 65 72 77 69 73 | c1/3.0 |otherwis|
|00000a30| 65 20 0d 0d 79 31 20 3d | 20 30 2e 30 20 20 2d 2d |e ..y1 =| 0.0  --|
|00000a40| 20 6e 6f 20 6d 61 74 74 | 65 72 20 77 68 61 74 2c | no matt|er what,|
|00000a50| 20 6d 75 73 74 20 68 61 | 76 65 20 61 74 20 6c 65 | must ha|ve at le|
|00000a60| 61 73 74 20 6f 6e 65 20 | 72 65 61 6c 20 72 6f 6f |ast one |real roo|
|00000a70| 74 0d 0d 79 32 20 3d 20 | 28 70 2d 71 29 2a 73 71 |t..y2 = |(p-q)*sq|
|00000a80| 72 74 28 33 2e 30 29 2f | 32 2e 30 20 77 68 65 6e |rt(3.0)/|2.0 when|
|00000a90| 20 6e 75 6d 52 65 61 6c | 52 6f 6f 74 73 20 3d 20 | numReal|Roots = |
|00000aa0| 31 20 61 6e 64 20 61 62 | 73 28 44 29 20 3e 20 31 |1 and ab|s(D) > 1|
|00000ab0| 2e 30 65 2d 31 30 2c 0d | 20 20 20 20 20 30 2e 30 |.0e-10,.|     0.0|
|00000ac0| 20 6f 74 68 65 72 77 69 | 73 65 0d 0d 79 33 20 3d | otherwi|se..y3 =|
|00000ad0| 20 2d 79 32 20 77 68 65 | 6e 20 6e 75 6d 52 65 61 | -y2 whe|n numRea|
|00000ae0| 6c 52 6f 6f 74 73 20 3d | 20 31 20 61 6e 64 20 61 |lRoots =| 1 and a|
|00000af0| 62 73 28 44 29 20 3e 20 | 31 2e 30 65 2d 31 30 2c |bs(D) > |1.0e-10,|
|00000b00| 0d 20 20 20 20 20 30 2e | 30 20 6f 74 68 65 72 77 |.     0.|0 otherw|
|00000b10| 69 73 65 0d 0d 72 6f 6f | 74 31 20 3a 3d 20 7b 78 |ise..roo|t1 := {x|
|00000b20| 31 2c 79 31 7d 3a 3b 20 | 20 20 20 20 72 6f 6f 74 |1,y1}:; |    root|
|00000b30| 32 20 3a 3d 20 7b 78 32 | 2c 79 32 7d 3a 3b 20 20 |2 := {x2|,y2}:;  |
|00000b40| 20 72 6f 6f 74 33 20 3a | 3d 20 7b 78 33 2c 79 33 | root3 :|= {x3,y3|
|00000b50| 7d 3a 0d 0d 2d 2d 2d 2d | 2d 2d 2d 20 45 6e 74 65 |}:..----|--- Ente|
|00000b60| 72 20 74 68 65 20 63 6f | 65 66 66 69 63 69 65 6e |r the co|efficien|
|00000b70| 74 73 20 66 6f 72 20 74 | 68 65 20 63 75 62 69 63 |ts for t|he cubic|
|00000b80| 20 68 65 72 65 20 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d | here --|--------|
|00000b90| 2d 2d 2d 2d 0d 2d 2d 20 | 20 20 20 20 20 20 20 20 |----.-- |        |
|00000ba0| 20 20 20 61 30 2a 78 5e | 33 20 2b 20 61 31 2a 78 |   a0*x^|3 + a1*x|
|00000bb0| 5e 32 20 2b 20 61 32 2a | 78 20 2b 20 61 33 20 3d |^2 + a2*|x + a3 =|
|00000bc0| 20 30 0d 0d 61 30 20 3d | 20 33 35 3b 20 20 20 20 | 0..a0 =| 35;    |
|00000bd0| 61 31 20 3d 20 31 35 3b | 20 20 20 20 61 32 20 3d |a1 = 15;|    a2 =|
|00000be0| 20 2d 35 3b 20 20 20 20 | 61 33 20 3d 20 2d 32 30 | -5;    |a3 = -20|
|00000bf0| 0d 0d 72 6f 6f 74 31 3a | 7b 30 2e 37 36 2c 30 2e |..root1:|{0.76,0.|
|00000c00| 30 30 7d 0d 72 6f 6f 74 | 32 3a 7b 2d 30 2e 35 39 |00}.root|2:{-0.59|
|00000c10| 2c 30 2e 36 34 7d 0d 72 | 6f 6f 74 33 3a 7b 2d 30 |,0.64}.r|oot3:{-0|
|00000c20| 2e 35 39 2c 2d 30 2e 36 | 34 7d 0d 0d 2d 2d 2d 2d |.59,-0.6|4}..----|
|00000c30| 2d 2d 2d 2d 2d 2d 2d 20 | 43 68 65 63 6b 20 74 68 |------- |Check th|
|00000c40| 65 20 73 6f 6c 75 74 69 | 6f 6e 2d 2d 2d 2d 2d 2d |e soluti|on------|
|00000c50| 2d 2d 2d 2d 2d 2d 0d 7a | 28 78 29 20 3d 20 61 30 |------.z|(x) = a0|
|00000c60| 2a 43 63 75 62 65 28 78 | 29 20 2b 20 61 31 2a 43 |*Ccube(x|) + a1*C|
|00000c70| 73 71 72 28 78 29 20 2b | 20 61 32 2a 78 20 2b 20 |sqr(x) +| a2*x + |
|00000c80| 7b 61 33 2c 30 7d 20 2d | 2d 20 54 68 65 20 63 6f |{a3,0} -|- The co|
|00000c90| 6d 70 6c 65 78 20 63 75 | 62 69 63 0d 0d 2d 2d 20 |mplex cu|bic..-- |
|00000ca0| 63 6f 6e 66 69 72 6d 20 | 74 68 61 74 20 7a 28 78 |confirm |that z(x|
|00000cb0| 29 20 69 73 20 7a 65 72 | 6f 20 61 74 20 74 68 65 |) is zer|o at the|
|00000cc0| 20 72 6f 6f 74 73 0d 7a | 28 72 6f 6f 74 31 29 3a | roots.z|(root1):|
|00000cd0| 7b 30 2e 30 30 2c 30 2e | 30 30 7d 0d 7a 28 72 6f |{0.00,0.|00}.z(ro|
|00000ce0| 6f 74 32 29 3a 7b 30 2e | 30 30 2c 30 2e 30 30 7d |ot2):{0.|00,0.00}|
|00000cf0| 0d 7a 28 72 6f 6f 74 33 | 29 3a 7b 30 2e 30 30 2c |.z(root3|):{0.00,|
|00000d00| 30 2e 30 30 7d 0d 0d 2d | 2d 2d 2d 2d 2d 2d 2d 20 |0.00}..-|------- |
|00000d10| 63 68 65 63 6b 20 74 68 | 65 20 73 6f 6c 75 74 69 |check th|e soluti|
|00000d20| 6f 6e 20 67 72 61 70 68 | 69 63 61 6c 6c 79 0d 2d |on graph|ically.-|
|00000d30| 2d 20 64 65 66 69 6e 65 | 20 61 6e 20 61 72 72 61 |- define| an arra|
|00000d40| 79 20 74 68 61 74 20 73 | 61 6d 70 6c 65 73 20 70 |y that s|amples p|
|00000d50| 6f 69 6e 74 73 20 6f 6e | 20 74 68 65 20 73 75 72 |oints on| the sur|
|00000d60| 66 61 63 65 20 6f 66 3a | 0d 2d 2d 20 20 20 20 5a |face of:|.--    Z|
|00000d70| 20 3d 20 61 62 73 28 7a | 28 78 29 29 20 76 73 2e | = abs(z|(x)) vs.|
|00000d80| 20 58 20 3d 20 72 65 61 | 6c 28 78 29 20 2c 20 59 | X = rea|l(x) , Y|
|00000d90| 20 3d 20 69 6d 61 67 69 | 6e 61 72 79 28 78 29 0d | = imagi|nary(x).|
|00000da0| 2d 2d 20 54 68 69 73 20 | 73 75 72 66 61 63 65 20 |-- This |surface |
|00000db0| 73 68 6f 75 6c 64 20 64 | 69 70 20 74 6f 20 7a 65 |should d|ip to ze|
|00000dc0| 72 6f 20 61 74 20 74 68 | 65 20 72 6f 6f 74 73 2e |ro at th|e roots.|
|00000dd0| 20 28 54 68 65 20 73 61 | 6d 70 6c 65 64 20 73 75 | (The sa|mpled su|
|00000de0| 72 66 61 63 65 20 77 69 | 6c 6c 20 64 69 70 20 6e |rface wi|ll dip n|
|00000df0| 65 61 72 20 7a 65 72 6f | 20 62 75 74 20 69 6e 20 |ear zero| but in |
|00000e00| 67 65 6e 65 72 61 6c 20 | 69 73 20 6e 6f 74 20 73 |general |is not s|
|00000e10| 61 6d 70 6c 65 64 20 65 | 78 61 63 74 6c 79 20 61 |ampled e|xactly a|
|00000e20| 74 20 61 20 72 6f 6f 74 | 29 2e 0d 0d 73 75 72 66 |t a root|)...surf|
|00000e30| 5b 69 78 2c 69 79 5d 20 | 3d 20 20 78 3a 3d 43 73 |[ix,iy] |=  x:=Cs|
|00000e40| 63 61 6c 65 28 69 78 2c | 69 79 29 2c 20 43 61 62 |cale(ix,|iy), Cab|
|00000e50| 73 28 7a 28 78 29 29 20 | 64 69 6d 5b 6d 2c 6d 5d |s(z(x)) |dim[m,m]|
|00000e60| 0d 0d 2d 2d 20 54 68 65 | 20 69 6e 64 65 78 20 66 |..-- The| index f|
|00000e70| 6f 72 20 73 75 72 66 5b | 5d 20 72 75 6e 73 20 66 |or surf[|] runs f|
|00000e80| 72 6f 6d 20 31 20 74 6f | 20 6d 2e 20 53 63 61 6c |rom 1 to| m. Scal|
|00000e90| 65 20 74 68 65 20 69 6e | 64 65 78 20 74 6f 20 67 |e the in|dex to g|
|00000ea0| 65 74 20 72 65 61 6c 20 | 61 6e 64 20 69 6d 61 67 |et real |and imag|
|00000eb0| 69 6e 61 72 79 20 70 61 | 72 74 73 20 72 61 6e 67 |inary pa|rts rang|
|00000ec0| 69 6e 67 20 66 72 6f 6d | 20 58 6d 69 6e 20 74 6f |ing from| Xmin to|
|00000ed0| 20 58 6d 61 78 20 61 6e | 64 20 59 6d 69 6e 20 74 | Xmax an|d Ymin t|
|00000ee0| 6f 20 59 6d 61 78 0d 73 | 63 61 6c 65 28 69 2c 6d |o Ymax.s|cale(i,m|
|00000ef0| 69 6e 2c 6d 61 78 2c 6e | 73 74 65 70 73 29 20 3d |in,max,n|steps) =|
|00000f00| 20 28 69 2d 2e 35 29 2a | 28 6d 61 78 2d 6d 69 6e | (i-.5)*|(max-min|
|00000f10| 29 2f 6e 73 74 65 70 73 | 2b 6d 69 6e 0d 43 73 63 |)/nsteps|+min.Csc|
|00000f20| 61 6c 65 28 69 78 2c 69 | 79 29 20 3d 20 7b 73 63 |ale(ix,i|y) = {sc|
|00000f30| 61 6c 65 28 69 78 2c 58 | 6d 69 6e 2c 58 6d 61 78 |ale(ix,X|min,Xmax|
|00000f40| 2c 6d 29 2c 20 73 63 61 | 6c 65 28 69 79 2c 59 6d |,m), sca|le(iy,Ym|
|00000f50| 69 6e 2c 59 6d 61 78 2c | 6d 29 7d 0d 0d 6d 3d 31 |in,Ymax,|m)}..m=1|
|00000f60| 32 3b 20 20 20 2d 2d 20 | 73 75 72 66 61 63 65 20 |2;   -- |surface |
|00000f70| 69 73 20 73 61 6d 70 6c | 65 64 20 6f 6e 20 61 6e |is sampl|ed on an|
|00000f80| 20 6d 20 62 79 20 6d 20 | 67 72 69 64 0d 58 6d 69 | m by m |grid.Xmi|
|00000f90| 6e 20 3d 20 2d 31 3b 20 | 58 6d 61 78 20 3d 20 31 |n = -1; |Xmax = 1|
|00000fa0| 0d 59 6d 69 6e 20 3d 20 | 2d 31 3b 20 59 6d 61 78 |.Ymin = |-1; Ymax|
|00000fb0| 20 3d 20 31 0d 5a 6d 69 | 6e 20 3d 20 20 30 3b 20 | = 1.Zmi|n =  0; |
|00000fc0| 5a 6d 61 78 20 3d 20 35 | 30 0d 69 6d 61 67 65 20 |Zmax = 5|0.image |
|00000fd0| 73 75 72 66 20 20 20 20 | 20 20 20 20 20 20 20 20 |surf    |        |
|00000fe0| 20 20 20 20 20 20 2d 2d | 20 73 68 6f 77 20 69 6d |      --| show im|
|00000ff0| 61 67 65 20 6f 66 20 74 | 68 65 20 73 75 72 66 61 |age of t|he surfa|
|00001000| 63 65 0d 70 6c 6f 74 20 | 7b 72 6f 6f 74 31 2c 72 |ce.plot |{root1,r|
|00001010| 6f 6f 74 32 2c 72 6f 6f | 74 33 7d 20 20 20 20 2d |oot2,roo|t3}    -|
|00001020| 2d 20 73 68 6f 77 20 6c | 6f 63 61 74 69 6f 6e 73 |- show l|ocations|
|00001030| 20 6f 66 20 72 6f 6f 74 | 73 0d 0d 70 6c 6f 74 20 | of root|s..plot |
|00001040| 7a 28 7b 58 2c 30 7d 29 | 5b 31 5d 2f 5a 6d 61 78 |z({X,0})|[1]/Zmax|
|00001050| 20 20 20 20 20 2d 2d 20 | 70 6c 6f 74 20 74 68 65 |     -- |plot the|
|00001060| 20 72 65 61 6c 20 70 61 | 72 74 20 6f 66 20 7a 20 | real pa|rt of z |
|00001070| 66 6f 72 20 72 65 61 6c | 20 78 0d 2d 2d 20 54 68 |for real| x.-- Th|
|00001080| 69 73 20 73 68 6f 75 6c | 64 20 62 65 20 7a 65 72 |is shoul|d be zer|
|00001090| 6f 20 61 74 20 72 65 61 | 6c 20 72 6f 6f 74 73 2e |o at rea|l roots.|
|000010a0| 20 4f 6e 20 74 68 65 20 | 70 6c 6f 74 74 65 64 20 | On the |plotted |
|000010b0| 73 75 72 66 61 63 65 2c | 20 72 65 61 6c 20 72 6f |surface,| real ro|
|000010c0| 6f 74 73 20 61 72 65 20 | 6c 6f 63 61 74 65 64 20 |ots are |located |
|000010d0| 61 6c 6f 6e 67 20 79 3d | 30 20 73 6f 20 74 68 65 |along y=|0 so the|
|000010e0| 20 72 65 61 6c 20 63 75 | 62 69 63 20 70 6c 6f 74 | real cu|bic plot|
|000010f0| 74 65 64 20 69 6e 20 74 | 68 69 73 20 77 61 79 20 |ted in t|his way |
|00001100| 73 68 6f 75 6c 64 20 70 | 61 73 73 20 74 68 6f 75 |should p|ass thou|
|00001110| 67 68 20 69 74 73 20 72 | 65 61 6c 20 72 6f 6f 74 |gh its r|eal root|
|00001120| 73 2e 0d 70 6c 6f 74 20 | 30 0d 0d 2d 2d 2d 2d 2d |s..plot |0..-----|
|00001130| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00001140| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00001150| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00001160| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 0d |--------|-------.|
|00001170| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 20 75 74 69 6c |--------|--- util|
|00001180| 69 74 79 20 66 75 6e 63 | 74 69 6f 6e 73 20 66 6f |ity func|tions fo|
|00001190| 72 20 63 6f 6d 70 6c 65 | 78 20 6e 75 6d 62 65 72 |r comple|x number|
|000011a0| 73 20 2d 2d 2d 2d 2d 2d | 0d 2d 2d 20 41 72 72 61 |s ------|.-- Arra|
|000011b0| 79 73 20 63 61 6e 20 62 | 65 20 75 73 65 64 20 74 |ys can b|e used t|
|000011c0| 6f 20 72 65 70 72 65 73 | 65 6e 74 20 63 6f 6d 70 |o repres|ent comp|
|000011d0| 6c 65 78 20 6e 75 6d 62 | 65 72 73 2e 20 41 64 64 |lex numb|ers. Add|
|000011e0| 69 74 69 6f 6e 2c 20 73 | 75 62 74 72 61 63 74 69 |ition, s|ubtracti|
|000011f0| 6f 6e 20 61 6e 64 20 6d | 75 6c 74 69 70 6c 69 63 |on and m|ultiplic|
|00001200| 61 74 69 6f 6e 20 62 79 | 20 61 20 72 65 61 6c 20 |ation by| a real |
|00001210| 63 61 6e 20 62 65 20 64 | 6f 6e 65 20 64 69 72 65 |can be d|one dire|
|00001220| 63 74 6c 79 2e 20 54 68 | 65 20 66 6f 6c 6c 6f 77 |ctly. Th|e follow|
|00001230| 69 6e 67 20 66 75 6e 63 | 74 69 6f 6e 73 20 69 6d |ing func|tions im|
|00001240| 70 6c 65 6d 65 6e 74 20 | 6f 74 68 65 72 20 62 61 |plement |other ba|
|00001250| 73 69 63 20 6f 70 65 72 | 61 74 69 6f 6e 73 20 6f |sic oper|ations o|
|00001260| 6e 20 63 6f 6d 70 6c 65 | 78 20 6e 75 6d 62 65 72 |n comple|x number|
|00001270| 73 2e 20 4e 6f 74 65 20 | 74 68 61 74 20 61 20 72 |s. Note |that a r|
|00001280| 65 61 6c 20 6e 75 6d 62 | 65 72 20 72 20 6d 75 73 |eal numb|er r mus|
|00001290| 74 20 62 65 20 77 72 69 | 74 74 65 6e 20 61 73 20 |t be wri|tten as |
|000012a0| 7b 72 2c 30 7d 2e 0d 0d | 43 61 62 73 28 41 29 20 |{r,0}...|Cabs(A) |
|000012b0| 3d 20 73 71 72 74 28 41 | 5b 31 5d 5e 32 2b 41 5b |= sqrt(A|[1]^2+A[|
|000012c0| 32 5d 5e 32 29 0d 43 6d | 75 6c 74 28 41 2c 42 29 |2]^2).Cm|ult(A,B)|
|000012d0| 20 3d 20 7b 41 5b 31 5d | 2a 42 5b 31 5d 2d 41 5b | = {A[1]|*B[1]-A[|
|000012e0| 32 5d 2a 42 5b 32 5d 2c | 41 5b 32 5d 2a 42 5b 31 |2]*B[2],|A[2]*B[1|
|000012f0| 5d 2b 41 5b 31 5d 2a 42 | 5b 32 5d 7d 0d 43 73 71 |]+A[1]*B|[2]}.Csq|
|00001300| 72 28 41 29 20 3d 20 7b | 41 5b 31 5d 2a 41 5b 31 |r(A) = {|A[1]*A[1|
|00001310| 5d 2d 41 5b 32 5d 2a 41 | 5b 32 5d 2c 32 2a 41 5b |]-A[2]*A|[2],2*A[|
|00001320| 31 5d 2a 41 5b 32 5d 7d | 0d 43 63 75 62 65 28 41 |1]*A[2]}|.Ccube(A|
|00001330| 29 20 3d 20 43 6d 75 6c | 74 28 41 2c 43 73 71 72 |) = Cmul|t(A,Csqr|
|00001340| 28 41 29 29 0d 2d 2d 20 | 28 74 68 65 20 66 6f 6c |(A)).-- |(the fol|
|00001350| 6c 6f 77 69 6e 67 20 77 | 65 72 65 20 6e 6f 74 20 |lowing w|ere not |
|00001360| 75 73 65 64 20 69 6e 20 | 74 68 69 73 20 65 78 61 |used in |this exa|
|00001370| 6d 70 6c 65 29 0d 43 64 | 69 76 28 41 2c 42 29 3d |mple).Cd|iv(A,B)=|
|00001380| 7b 28 42 5b 31 5d 2a 41 | 5b 31 5d 2b 42 5b 32 5d |{(B[1]*A|[1]+B[2]|
|00001390| 2a 41 5b 32 5d 29 2f 28 | 42 5b 31 5d 5e 32 2b 42 |*A[2])/(|B[1]^2+B|
|000013a0| 5b 32 5d 5e 32 29 2c 0d | 20 20 20 20 20 20 20 20 |[2]^2),.|        |
|000013b0| 20 28 42 5b 31 5d 2a 41 | 5b 32 5d 2d 42 5b 32 5d | (B[1]*A|[2]-B[2]|
|000013c0| 2a 41 5b 31 5d 29 2f 28 | 42 5b 31 5d 5e 32 2b 42 |*A[1])/(|B[1]^2+B|
|000013d0| 5b 32 5d 5e 32 29 7d 0d | 43 6f 6e 6a 28 41 29 20 |[2]^2)}.|Conj(A) |
|000013e0| 3d 20 7b 41 5b 31 5d 2c | 2d 41 5b 32 5d 7d 0d 43 |= {A[1],|-A[2]}.C|
|000013f0| 61 72 67 28 41 29 20 3d | 20 30 20 77 68 65 6e 20 |arg(A) =| 0 when |
|00001400| 41 5b 31 5d 3d 30 20 61 | 6e 64 20 41 5b 32 5d 3d |A[1]=0 a|nd A[2]=|
|00001410| 30 2c 0d 20 20 20 20 20 | 20 20 20 20 61 74 61 6e |0,.     |    atan|
|00001420| 28 41 5b 32 5d 2f 41 5b | 31 5d 29 20 77 68 65 6e |(A[2]/A[|1]) when|
|00001430| 20 41 5b 31 5d 3e 3d 30 | 2c 0d 20 20 20 20 20 20 | A[1]>=0|,.      |
|00001440| 20 20 20 61 74 61 6e 28 | 41 5b 32 5d 2f 41 5b 31 |   atan(|A[2]/A[1|
|00001450| 5d 29 2b 31 38 30 20 77 | 68 65 6e 20 41 5b 32 5d |])+180 w|hen A[2]|
|00001460| 3e 3d 30 2c 0d 20 20 20 | 20 20 20 20 20 20 61 74 |>=0,.   |      at|
|00001470| 61 6e 28 41 5b 32 5d 2f | 41 5b 31 5d 29 2d 31 38 |an(A[2]/|A[1])-18|
|00001480| 30 0d 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |0.......|........|
|00001490| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000014a0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000014b0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000014c0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000014d0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000014e0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000014f0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001500| 00 00 01 00 00 00 01 24 | 00 00 00 24 00 00 00 32 |.......$|...$...2|
|00001510| 77 68 65 6e 20 41 5b 32 | 5d 3e 3d 30 2c 0d 20 20 |when A[2|]>=0,.  |
|00001520| 20 20 20 20 20 20 20 61 | 74 61 6e 28 41 5b 32 5d |       a|tan(A[2]|
|00001530| 0d 43 6f 6d 70 6c 65 78 | 20 52 6f 6f 74 73 02 00 |.Complex| Roots..|
|00001540| 00 00 50 61 72 74 53 49 | 54 78 00 00 00 00 00 00 |..PartSI|Tx......|
|00001550| 00 00 50 61 72 74 53 49 | 54 78 00 00 00 00 00 00 |..PartSI|Tx......|
|00001560| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001570| 00 00 a9 7e bd ee 00 00 | 00 00 00 00 01 56 30 7d |...~....|.....V0}|
|00001580| 0d db 6d b6 db 6d b6 db | 6d b6 db 6d b6 db 6d b6 |..m..m..|m..m..m.|
|00001590| db 6d b6 db 6d b6 db 6d | b6 db 6d b6 db 6d b6 db |.m..m..m|..m..m..|
|000015a0| 6d b6 db 6d b6 db 6d b6 | db 6d b6 db 6d b6 db 6d |m..m..m.|.m..m..m|
|000015b0| b6 db 6d b6 db 6d b6 db | 6d b6 db 6d b6 db 6d b6 |..m..m..|m..m..m.|
|000015c0| db 6d b6 db 6d b6 db 6d | b6 db 6d b6 db 6d b6 db |.m..m..m|..m..m..|
|000015d0| 6d b6 db 6d b6 db 6d b6 | db 6d b6 db 6d b6 db 6d |m..m..m.|.m..m..m|
|000015e0| b6 db 6d b6 db 6d b6 db | 6d b6 db 6d b6 db 6d b6 |..m..m..|m..m..m.|
|000015f0| db 6d b6 db 6d b6 db 6d | b6 db 6d b6 db 6d b6 db |.m..m..m|..m..m..|
|00001600| 00 00 00 20 05 00 00 02 | 00 02 3f f9 8e fa 35 12 |... ....|..?...5.|
|00001610| 94 e9 c8 ae 01 d5 01 2b | 00 03 00 28 01 0d 01 1b |.......+|...(....|
|00001620| 00 c7 00 34 00 00 01 00 | 00 00 01 24 00 00 00 24 |...4....|...$...$|
|00001630| 00 00 00 32 00 40 62 8c | 06 40 00 00 00 1c 00 32 |...2.@b.|.@.....2|
|00001640| 00 00 50 52 65 66 00 00 | 00 0a 00 80 ff ff 00 00 |..PRef..|........|
|00001650| 00 00 00 40 c5 6c 00 00 | 00 00 00 00 00 00 00 00 |...@.l..|........|
|00001660| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001670| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
+--------+-------------------------+-------------------------+--------+--------+