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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 61 | 64 01 00 00 00 00 00 00 |.TEXTMPa|d.......|
|00000050| 00 00 00 00 00 10 9b 00 | 00 01 56 aa 23 b4 95 aa |........|..V.#...|
|00000060| 23 b4 95 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 59 5b 00 00 |........|....Y[..|
|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 0d | 0d 2d 2d 2d 2d 2d 2d 2d |unction.|.-------|
|00000100| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 20 71 75 61 |--------|---- qua|
|00000110| 64 72 61 74 69 63 20 72 | 6f 6f 74 73 20 2d 2d 2d |dratic r|oots ---|
|00000120| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 0d 2d 2d 20 |--------|----.-- |
|00000130| 41 6c 67 6f 72 69 74 68 | 6d 20 66 6f 72 20 72 65 |Algorith|m for re|
|00000140| 61 6c 20 70 61 72 74 73 | 20 6f 66 20 72 6f 6f 74 |al parts| of root|
|00000150| 73 20 69 73 20 66 72 6f | 6d 20 62 79 20 57 2e 48 |s is fro|m by W.H|
|00000160| 2e 20 50 72 65 73 73 2c | 20 53 2e 20 54 65 75 6b |. Press,| S. Teuk|
|00000170| 6f 6c 73 6b 79 20 65 74 | 20 61 6c 2c 22 4e 75 6d |olsky et| al,"Num|
|00000180| 65 72 69 63 61 6c 20 52 | 65 63 69 70 65 73 22 2e |erical R|ecipes".|
|00000190| 0d 2d 2d 20 4f 63 63 61 | 73 69 6f 6e 61 6c 6c 79 |.-- Occa|sionally|
|000001a0| 20 34 61 63 20 3c 3c 20 | 62 2c 20 73 6f 20 6f 6e | 4ac << |b, so on|
|000001b0| 65 20 6f 66 20 74 68 65 | 20 72 6f 6f 74 73 20 69 |e of the| roots i|
|000001c0| 73 20 28 65 72 72 6f 6e | 65 6f 75 73 6c 79 29 20 |s (erron|eously) |
|000001d0| 63 61 6c 6c 65 64 20 30 | 2e 0d 2d 2d 20 54 68 69 |called 0|..-- Thi|
|000001e0| 73 20 66 6f 72 6d 75 6c | 61 74 69 6f 6e 20 61 76 |s formul|ation av|
|000001f0| 6f 69 64 73 20 74 68 65 | 20 70 72 6f 62 6c 65 6d |oids the| problem|
|00000200| 2e 0d 2d 2d 20 69 6d 70 | 6c 65 6d 65 6e 74 65 64 |..-- imp|lemented|
|00000210| 20 62 79 20 44 61 76 69 | 64 20 44 65 72 62 65 73 | by Davi|d Derbes|
|00000220| 20 28 64 65 72 62 65 73 | 40 75 68 75 72 75 2e 75 | (derbes|@uhuru.u|
|00000230| 63 68 69 63 61 67 6f 2e | 65 64 75 29 20 66 6f 72 |chicago.|edu) for|
|00000240| 20 4d 61 74 68 50 61 64 | 0d 0d 2d 2d 20 47 69 76 | MathPad|..-- Giv|
|00000250| 65 6e 20 61 20 71 75 61 | 64 72 61 74 69 63 20 6f |en a qua|dratic o|
|00000260| 66 20 74 68 65 20 66 6f | 72 6d 0d 0d 2d 2d 20 20 |f the fo|rm..--  |
|00000270| 20 20 20 20 20 20 20 20 | 61 2a 78 5e 32 20 2b 20 |        |a*x^2 + |
|00000280| 62 2a 78 20 2b 20 63 20 | 3d 20 30 0d 20 0d 2d 2d |b*x + c |= 0. .--|
|00000290| 20 77 69 74 68 20 72 65 | 61 6c 20 63 6f 65 66 66 | with re|al coeff|
|000002a0| 69 63 69 65 6e 74 73 2c | 20 66 69 6e 64 20 74 68 |icients,| find th|
|000002b0| 65 20 28 70 6f 73 73 69 | 62 6c 79 20 63 6f 6d 70 |e (possi|bly comp|
|000002c0| 6c 65 78 29 20 72 6f 6f | 74 73 2e 0d 2d 2d 20 52 |lex) roo|ts..-- R|
|000002d0| 6f 6f 74 73 20 61 72 65 | 20 67 69 76 65 6e 20 69 |oots are| given i|
|000002e0| 6e 20 74 68 65 20 66 6f | 72 6d 20 78 20 2b 20 69 |n the fo|rm x + i|
|000002f0| 79 2e 0d 7e 0d 73 67 6e | 28 78 29 20 3d 20 31 20 |y..~.sgn|(x) = 1 |
|00000300| 77 68 65 6e 20 78 20 3e | 3d 20 30 2c 2d 31 20 6f |when x >|= 0,-1 o|
|00000310| 74 68 65 72 77 69 73 65 | 0d 44 20 3d 20 28 62 2a |therwise|.D = (b*|
|00000320| 62 20 2d 20 34 2a 61 2a | 63 29 09 2d 2d 20 64 69 |b - 4*a*|c).-- di|
|00000330| 73 63 72 69 6d 69 6e 61 | 6e 74 0d 78 31 20 3d 20 |scrimina|nt.x1 = |
|00000340| 2d 28 62 20 2b 20 73 67 | 6e 28 62 29 2a 73 71 72 |-(b + sg|n(b)*sqr|
|00000350| 74 28 44 29 29 2f 28 32 | 2a 61 29 20 77 68 65 6e |t(D))/(2|*a) when|
|00000360| 20 44 20 3e 3d 20 30 2c | 20 2d 62 2f 28 32 2a 61 | D >= 0,| -b/(2*a|
|00000370| 29 20 6f 74 68 65 72 77 | 69 73 65 0d 78 32 20 3d |) otherw|ise.x2 =|
|00000380| 20 63 2f 78 31 20 77 68 | 65 6e 20 44 20 3e 3d 20 | c/x1 wh|en D >= |
|00000390| 30 2c 20 2d 62 2f 28 32 | 2a 61 29 20 6f 74 68 65 |0, -b/(2|*a) othe|
|000003a0| 72 77 69 73 65 0d 79 31 | 20 3d 20 30 20 77 68 65 |rwise.y1| = 0 whe|
|000003b0| 6e 20 44 20 3e 3d 20 30 | 2c 20 73 71 72 74 28 2d |n D >= 0|, sqrt(-|
|000003c0| 44 29 2f 28 32 2a 61 29 | 20 6f 74 68 65 72 77 69 |D)/(2*a)| otherwi|
|000003d0| 73 65 0d 79 32 20 3d 20 | 2d 79 31 0d 7e 0d 0d 2d |se.y2 = |-y1.~..-|
|000003e0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|000003f0| 2d 2d 20 63 75 62 69 63 | 20 72 6f 6f 74 73 20 2d |-- cubic| roots -|
|00000400| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000410| 2d 2d 0d 2d 2d 20 54 61 | 72 74 61 67 6c 69 61 27 |--.-- Ta|rtaglia'|
|00000420| 73 20 26 20 43 61 72 64 | 61 6e 6f 27 73 20 66 6f |s & Card|ano's fo|
|00000430| 72 6d 75 6c 61 65 20 66 | 6f 72 20 74 68 65 20 72 |rmulae f|or the r|
|00000440| 6f 6f 74 73 20 6f 66 20 | 61 20 63 75 62 69 63 0d |oots of |a cubic.|
|00000450| 2d 2d 20 66 72 6f 6d 20 | 55 6e 69 76 65 72 73 61 |-- from |Universa|
|00000460| 6c 20 45 6e 63 79 63 6c | 6f 70 61 65 64 69 61 20 |l Encycl|opaedia |
|00000470| 6f 66 20 4d 61 74 68 65 | 6d 61 74 69 63 73 0d 2d |of Mathe|matics.-|
|00000480| 2d 20 69 6d 70 6c 65 6d | 65 6e 74 65 64 20 62 79 |- implem|ented by|
|00000490| 20 44 61 76 69 64 20 44 | 65 72 62 65 73 20 66 6f | David D|erbes fo|
|000004a0| 72 20 4d 61 74 68 50 61 | 64 2c 20 38 20 53 65 70 |r MathPa|d, 8 Sep|
|000004b0| 74 20 31 39 39 33 0d 2d | 2d 20 47 69 76 65 6e 20 |t 1993.-|- Given |
|000004c0| 61 20 63 75 62 69 63 20 | 6f 66 20 74 68 65 20 66 |a cubic |of the f|
|000004d0| 6f 72 6d 0d 2d 2d 0d 2d | 2d 20 20 20 20 20 20 20 |orm.--.-|-       |
|000004e0| 20 20 20 20 20 61 30 2a | 78 5e 33 20 2b 20 61 31 |     a0*|x^3 + a1|
|000004f0| 2a 78 5e 32 20 2b 20 61 | 32 2a 78 20 2b 20 61 33 |*x^2 + a|2*x + a3|
|00000500| 20 3d 20 30 0d 2d 2d 20 | 0d 2d 2d 20 77 69 74 68 | = 0.-- |.-- with|
|00000510| 20 72 65 61 6c 20 63 6f | 65 66 66 69 63 69 65 6e | real co|efficien|
|00000520| 74 73 2c 20 66 69 6e 64 | 20 74 68 65 20 28 70 6f |ts, find| the (po|
|00000530| 73 73 69 62 6c 79 20 63 | 6f 6d 70 6c 65 78 29 20 |ssibly c|omplex) |
|00000540| 72 6f 6f 74 73 2e 0d 0d | 63 31 20 3d 20 61 31 2f |roots...|c1 = a1/|
|00000550| 61 30 20 20 2d 2d 20 22 | 6e 6f 72 6d 61 6c 69 7a |a0  -- "|normaliz|
|00000560| 65 22 2c 20 69 2e 65 2e | 20 6d 61 6b 65 20 6c 65 |e", i.e.| make le|
|00000570| 61 64 69 6e 67 20 63 6f | 65 66 66 69 63 69 65 6e |ading co|efficien|
|00000580| 74 20 3d 20 31 0d 63 32 | 20 3d 20 61 32 2f 61 30 |t = 1.c2| = a2/a0|
|00000590| 0d 63 33 20 3d 20 61 33 | 2f 61 30 0d 0d 2d 2d 20 |.c3 = a3|/a0..-- |
|000005a0| 64 69 73 63 72 69 6d 69 | 6e 61 6e 74 20 44 3b 20 |discrimi|nant D; |
|000005b0| 69 66 20 44 20 3e 20 30 | 2c 20 6f 6e 65 20 72 65 |if D > 0|, one re|
|000005c0| 61 6c 20 72 6f 6f 74 3b | 20 65 6c 73 65 20 74 68 |al root;| else th|
|000005d0| 72 65 65 20 72 65 61 6c | 20 72 6f 6f 74 73 0d 2d |ree real| roots.-|
|000005e0| 2d 20 28 61 74 20 6d 6f | 73 74 20 74 77 6f 20 64 |- (at mo|st two d|
|000005f0| 69 73 74 69 6e 63 74 20 | 72 65 61 6c 20 69 66 20 |istinct |real if |
|00000600| 44 20 3d 20 30 29 0d 0d | 61 20 3d 20 63 32 20 2d |D = 0)..|a = c2 -|
|00000610| 20 28 63 31 2a 63 31 29 | 2f 33 2e 30 0d 62 20 3d | (c1*c1)|/3.0.b =|
|00000620| 20 28 28 28 32 2e 30 2a | 63 31 2a 63 31 2a 63 31 | (((2.0*|c1*c1*c1|
|00000630| 29 20 2d 20 28 39 2e 30 | 2a 63 31 2a 63 32 29 29 |) - (9.0|*c1*c2))|
|00000640| 2f 32 37 2e 30 29 20 2b | 20 63 33 0d 44 20 3d 20 |/27.0) +| c3.D = |
|00000650| 28 62 2a 62 2f 34 2e 30 | 29 20 2b 20 28 61 2a 61 |(b*b/4.0|) + (a*a|
|00000660| 2a 61 2f 32 37 2e 30 29 | 0d 0d 6e 75 6d 52 65 61 |*a/27.0)|..numRea|
|00000670| 6c 52 6f 6f 74 73 20 3d | 20 33 20 77 68 65 6e 20 |lRoots =| 3 when |
|00000680| 44 20 3c 20 30 2c 20 31 | 20 6f 74 68 65 72 77 69 |D < 0, 1| otherwi|
|00000690| 73 65 0d 0d 64 48 61 6c | 66 20 3d 20 73 71 72 74 |se..dHal|f = sqrt|
|000006a0| 28 61 62 73 28 44 29 29 | 0d 0d 73 67 6e 70 20 3d |(abs(D))|..sgnp =|
|000006b0| 20 2d 31 20 77 68 65 6e | 20 28 28 2d 62 2f 32 2e | -1 when| ((-b/2.|
|000006c0| 30 29 20 2b 20 64 48 61 | 6c 66 29 20 3c 20 30 2c |0) + dHa|lf) < 0,|
|000006d0| 20 31 20 6f 74 68 65 72 | 77 69 73 65 20 20 20 20 | 1 other|wise    |
|000006e0| 0d 73 67 6e 71 20 3d 20 | 2d 31 20 77 68 65 6e 20 |.sgnq = |-1 when |
|000006f0| 28 28 2d 62 2f 32 2e 30 | 29 20 2d 20 64 48 61 6c |((-b/2.0|) - dHal|
|00000700| 66 29 20 3c 20 30 2c 20 | 31 20 6f 74 68 65 72 77 |f) < 0, |1 otherw|
|00000710| 69 73 65 0d 0d 70 20 3d | 20 30 2e 30 20 77 68 65 |ise..p =| 0.0 whe|
|00000720| 6e 20 28 28 2d 62 2f 32 | 2e 30 29 20 2b 20 64 48 |n ((-b/2|.0) + dH|
|00000730| 61 6c 66 29 20 3d 20 30 | 2c 0d 20 20 20 20 20 73 |alf) = 0|,.     s|
|00000740| 67 6e 70 2a 28 61 62 73 | 28 28 2d 62 2f 32 2e 30 |gnp*(abs|((-b/2.0|
|00000750| 29 20 2b 20 64 48 61 6c | 66 29 29 5e 28 31 2e 30 |) + dHal|f))^(1.0|
|00000760| 2f 33 2e 30 29 20 6f 74 | 68 65 72 77 69 73 65 0d |/3.0) ot|herwise.|
|00000770| 71 20 3d 20 30 2e 30 20 | 77 68 65 6e 20 28 28 2d |q = 0.0 |when ((-|
|00000780| 62 2f 32 2e 30 29 20 2d | 20 64 48 61 6c 66 29 20 |b/2.0) -| dHalf) |
|00000790| 3d 20 30 2c 0d 20 20 20 | 20 20 73 67 6e 71 2a 28 |= 0,.   |  sgnq*(|
|000007a0| 61 62 73 28 28 2d 62 2f | 32 2e 30 29 20 2d 20 64 |abs((-b/|2.0) - d|
|000007b0| 48 61 6c 66 29 29 5e 28 | 31 2e 30 2f 33 2e 30 29 |Half))^(|1.0/3.0)|
|000007c0| 20 6f 74 68 65 72 77 69 | 73 65 0d 0d 73 20 3d 20 | otherwi|se..s = |
|000007d0| 28 2d 62 2f 32 2e 30 29 | 2f 73 71 72 74 28 2d 61 |(-b/2.0)|/sqrt(-a|
|000007e0| 2a 61 2a 61 2f 32 37 2e | 30 29 3b 20 20 74 68 65 |*a*a/27.|0);  the|
|000007f0| 74 61 20 3d 20 61 63 6f | 73 28 73 29 0d 0d 2d 2d |ta = aco|s(s)..--|
|00000800| 20 72 6f 6f 74 73 20 6f | 66 20 74 68 65 20 66 6f | roots o|f the fo|
|00000810| 72 6d 20 78 20 2b 20 69 | 79 0d 0d 78 31 20 3d 20 |rm x + i|y..x1 = |
|00000820| 32 2e 30 2a 70 20 2d 20 | 28 63 31 2f 33 2e 30 29 |2.0*p - |(c1/3.0)|
|00000830| 20 77 68 65 6e 20 6e 75 | 6d 52 65 61 6c 52 6f 6f | when nu|mRealRoo|
|00000840| 74 73 20 3d 20 31 20 61 | 6e 64 20 61 62 73 28 44 |ts = 1 a|nd abs(D|
|00000850| 29 20 3c 20 31 2e 30 65 | 2d 31 30 2c 0d 20 20 20 |) < 1.0e|-10,.   |
|00000860| 20 20 28 70 2b 71 29 20 | 2d 20 28 63 31 2f 33 2e |  (p+q) |- (c1/3.|
|00000870| 30 29 20 77 68 65 6e 20 | 6e 75 6d 52 65 61 6c 52 |0) when |numRealR|
|00000880| 6f 6f 74 73 20 3d 20 31 | 20 61 6e 64 20 61 62 73 |oots = 1| and abs|
|00000890| 28 44 29 20 3e 20 31 2e | 30 65 2d 31 30 2c 0d 20 |(D) > 1.|0e-10,. |
|000008a0| 20 20 20 20 32 2e 30 2a | 73 71 72 74 28 2d 61 2f |    2.0*|sqrt(-a/|
|000008b0| 33 2e 30 29 2a 63 6f 73 | 28 74 68 65 74 61 2f 33 |3.0)*cos|(theta/3|
|000008c0| 2e 30 29 20 2d 20 28 63 | 31 2f 33 2e 30 29 20 6f |.0) - (c|1/3.0) o|
|000008d0| 74 68 65 72 77 69 73 65 | 0d 0d 78 32 20 3d 20 2d |therwise|..x2 = -|
|000008e0| 70 20 2d 20 28 63 31 2f | 33 2e 30 29 20 77 68 65 |p - (c1/|3.0) whe|
|000008f0| 6e 20 6e 75 6d 52 65 61 | 6c 52 6f 6f 74 73 20 3d |n numRea|lRoots =|
|00000900| 20 31 20 61 6e 64 20 61 | 62 73 28 44 29 20 3c 20 | 1 and a|bs(D) < |
|00000910| 31 2e 30 65 2d 31 30 2c | 0d 20 20 20 20 20 2d 28 |1.0e-10,|.     -(|
|00000920| 70 2b 71 29 2f 32 2e 30 | 20 2d 20 28 63 31 2f 33 |p+q)/2.0| - (c1/3|
|00000930| 2e 30 29 20 77 68 65 6e | 20 6e 75 6d 52 65 61 6c |.0) when| numReal|
|00000940| 52 6f 6f 74 73 20 3d 20 | 31 20 61 6e 64 20 61 62 |Roots = |1 and ab|
|00000950| 73 28 44 29 20 3e 20 31 | 2e 30 65 2d 31 30 2c 0d |s(D) > 1|.0e-10,.|
|00000960| 20 20 20 20 20 32 2e 30 | 2a 73 71 72 74 28 2d 61 |     2.0|*sqrt(-a|
|00000970| 2f 33 2e 30 29 2a 63 6f | 73 28 28 74 68 65 74 61 |/3.0)*co|s((theta|
|00000980| 2f 33 2e 30 29 20 2b 20 | 31 32 30 29 20 2d 20 63 |/3.0) + |120) - c|
|00000990| 31 2f 33 2e 30 20 6f 74 | 68 65 72 77 69 73 65 20 |1/3.0 ot|herwise |
|000009a0| 0d 0d 78 33 20 3d 20 78 | 32 20 77 68 65 6e 20 6e |..x3 = x|2 when n|
|000009b0| 75 6d 52 65 61 6c 52 6f | 6f 74 73 20 3d 20 31 2c |umRealRo|ots = 1,|
|000009c0| 20 20 20 20 20 20 0d 20 | 20 20 20 20 32 2e 30 2a |      . |    2.0*|
|000009d0| 73 71 72 74 28 2d 61 2f | 33 2e 30 29 2a 63 6f 73 |sqrt(-a/|3.0)*cos|
|000009e0| 28 28 74 68 65 74 61 2f | 33 2e 30 29 20 2b 20 32 |((theta/|3.0) + 2|
|000009f0| 34 30 29 20 2d 20 63 31 | 2f 33 2e 30 20 6f 74 68 |40) - c1|/3.0 oth|
|00000a00| 65 72 77 69 73 65 20 0d | 0d 79 31 20 3d 20 30 2e |erwise .|.y1 = 0.|
|00000a10| 30 20 20 2d 2d 20 6e 6f | 20 6d 61 74 74 65 72 20 |0  -- no| matter |
|00000a20| 77 68 61 74 2c 20 6d 75 | 73 74 20 68 61 76 65 20 |what, mu|st have |
|00000a30| 61 74 20 6c 65 61 73 74 | 20 6f 6e 65 20 72 65 61 |at least| one rea|
|00000a40| 6c 20 72 6f 6f 74 0d 0d | 79 32 20 3d 20 28 70 2d |l root..|y2 = (p-|
|00000a50| 71 29 2a 73 71 72 74 28 | 33 2e 30 29 2f 32 2e 30 |q)*sqrt(|3.0)/2.0|
|00000a60| 20 77 68 65 6e 20 6e 75 | 6d 52 65 61 6c 52 6f 6f | when nu|mRealRoo|
|00000a70| 74 73 20 3d 20 31 20 61 | 6e 64 20 61 62 73 28 44 |ts = 1 a|nd abs(D|
|00000a80| 29 20 3e 20 31 2e 30 65 | 2d 31 30 2c 0d 20 20 20 |) > 1.0e|-10,.   |
|00000a90| 20 20 30 2e 30 20 6f 74 | 68 65 72 77 69 73 65 0d |  0.0 ot|herwise.|
|00000aa0| 0d 79 33 20 3d 20 2d 79 | 32 20 77 68 65 6e 20 6e |.y3 = -y|2 when n|
|00000ab0| 75 6d 52 65 61 6c 52 6f | 6f 74 73 20 3d 20 31 20 |umRealRo|ots = 1 |
|00000ac0| 61 6e 64 20 61 62 73 28 | 44 29 20 3e 20 31 2e 30 |and abs(|D) > 1.0|
|00000ad0| 65 2d 31 30 2c 0d 20 20 | 20 20 20 30 2e 30 20 6f |e-10,.  |   0.0 o|
|00000ae0| 74 68 65 72 77 69 73 65 | 0d 0d 72 6f 6f 74 31 20 |therwise|..root1 |
|00000af0| 3a 3d 20 7b 78 31 2c 79 | 31 7d 3a 3b 20 20 20 20 |:= {x1,y|1}:;    |
|00000b00| 20 72 6f 6f 74 32 20 3a | 3d 20 7b 78 32 2c 79 32 | root2 :|= {x2,y2|
|00000b10| 7d 3a 3b 20 20 20 72 6f | 6f 74 33 20 3a 3d 20 7b |}:;   ro|ot3 := {|
|00000b20| 78 33 2c 79 33 7d 3a 0d | 0d 2d 2d 2d 2d 2d 2d 2d |x3,y3}:.|.-------|
|00000b30| 20 45 6e 74 65 72 20 74 | 68 65 20 63 6f 65 66 66 | Enter t|he coeff|
|00000b40| 69 63 69 65 6e 74 73 20 | 66 6f 72 20 74 68 65 20 |icients |for the |
|00000b50| 63 75 62 69 63 20 68 65 | 72 65 20 2d 2d 2d 2d 2d |cubic he|re -----|
|00000b60| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 0d 2d 2d 20 20 20 20 |--------|-.--    |
|00000b70| 20 20 20 20 20 20 20 20 | 61 30 2a 78 5e 33 20 2b |        |a0*x^3 +|
|00000b80| 20 61 31 2a 78 5e 32 20 | 2b 20 61 32 2a 78 20 2b | a1*x^2 |+ a2*x +|
|00000b90| 20 61 33 20 3d 20 30 0d | 0d 61 30 20 3d 20 33 35 | a3 = 0.|.a0 = 35|
|00000ba0| 3b 20 20 20 20 61 31 20 | 3d 20 31 35 3b 20 20 20 |;    a1 |= 15;   |
|00000bb0| 20 61 32 20 3d 20 2d 35 | 3b 20 20 20 20 61 33 20 | a2 = -5|;    a3 |
|00000bc0| 3d 20 2d 32 30 0d 0d 72 | 6f 6f 74 31 3a 7b 30 2e |= -20..r|oot1:{0.|
|00000bd0| 37 36 2c 30 2e 30 30 7d | 0d 72 6f 6f 74 32 3a 7b |76,0.00}|.root2:{|
|00000be0| 2d 30 2e 35 39 2c 30 2e | 36 34 7d 0d 72 6f 6f 74 |-0.59,0.|64}.root|
|00000bf0| 33 3a 7b 2d 30 2e 35 39 | 2c 2d 30 2e 36 34 7d 0d |3:{-0.59|,-0.64}.|
|00000c00| 0d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 20 43 68 65 |.-------|---- Che|
|00000c10| 63 6b 20 74 68 65 20 73 | 6f 6c 75 74 69 6f 6e 2d |ck the s|olution-|
|00000c20| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 0d 69 6e 63 6c |--------|---.incl|
|00000c30| 75 64 65 20 22 3a 69 6e | 63 6c 3a 63 6f 6d 70 6c |ude ":in|cl:compl|
|00000c40| 65 78 20 6f 70 73 22 0d | 7a 28 78 29 20 3d 20 61 |ex ops".|z(x) = a|
|00000c50| 30 2a 43 63 75 62 65 28 | 78 29 20 2b 20 61 31 2a |0*Ccube(|x) + a1*|
|00000c60| 43 73 71 72 28 78 29 20 | 2b 20 61 32 2a 78 20 2b |Csqr(x) |+ a2*x +|
|00000c70| 20 7b 61 33 2c 30 7d 20 | 2d 2d 20 54 68 65 20 63 | {a3,0} |-- The c|
|00000c80| 6f 6d 70 6c 65 78 20 63 | 75 62 69 63 0d 0d 2d 2d |omplex c|ubic..--|
|00000c90| 20 63 6f 6e 66 69 72 6d | 20 74 68 61 74 20 7a 28 | confirm| that z(|
|00000ca0| 78 29 20 69 73 20 7a 65 | 72 6f 20 61 74 20 74 68 |x) is ze|ro at th|
|00000cb0| 65 20 72 6f 6f 74 73 0d | 7a 28 72 6f 6f 74 31 29 |e roots.|z(root1)|
|00000cc0| 3a 7b 30 2e 30 30 2c 30 | 2e 30 30 7d 0d 7a 28 72 |:{0.00,0|.00}.z(r|
|00000cd0| 6f 6f 74 32 29 3a 7b 30 | 2e 30 30 2c 30 2e 30 30 |oot2):{0|.00,0.00|
|00000ce0| 7d 0d 7a 28 72 6f 6f 74 | 33 29 3a 7b 30 2e 30 30 |}.z(root|3):{0.00|
|00000cf0| 2c 30 2e 30 30 7d 0d 0d | 2d 2d 2d 2d 2d 2d 2d 2d |,0.00}..|--------|
|00000d00| 20 63 68 65 63 6b 20 74 | 68 65 20 73 6f 6c 75 74 | check t|he solut|
|00000d10| 69 6f 6e 20 67 72 61 70 | 68 69 63 61 6c 6c 79 0d |ion grap|hically.|
|00000d20| 2d 2d 20 64 65 66 69 6e | 65 20 61 6e 20 61 72 72 |-- defin|e an arr|
|00000d30| 61 79 20 74 68 61 74 20 | 73 61 6d 70 6c 65 73 20 |ay that |samples |
|00000d40| 70 6f 69 6e 74 73 20 6f | 6e 20 74 68 65 20 73 75 |points o|n the su|
|00000d50| 72 66 61 63 65 20 6f 66 | 3a 0d 2d 2d 20 20 20 20 |rface of|:.--    |
|00000d60| 5a 20 3d 20 61 62 73 28 | 7a 28 78 29 29 20 76 73 |Z = abs(|z(x)) vs|
|00000d70| 2e 20 58 20 3d 20 72 65 | 61 6c 28 78 29 20 2c 20 |. X = re|al(x) , |
|00000d80| 59 20 3d 20 69 6d 61 67 | 69 6e 61 72 79 28 78 29 |Y = imag|inary(x)|
|00000d90| 0d 2d 2d 20 54 68 69 73 | 20 73 75 72 66 61 63 65 |.-- This| surface|
|00000da0| 20 73 68 6f 75 6c 64 20 | 64 69 70 20 74 6f 20 7a | should |dip to z|
|00000db0| 65 72 6f 20 61 74 20 74 | 68 65 20 72 6f 6f 74 73 |ero at t|he roots|
|00000dc0| 2e 20 28 54 68 65 20 73 | 61 6d 70 6c 65 64 20 73 |. (The s|ampled s|
|00000dd0| 75 72 66 61 63 65 20 77 | 69 6c 6c 20 64 69 70 20 |urface w|ill dip |
|00000de0| 6e 65 61 72 20 7a 65 72 | 6f 20 62 75 74 20 69 6e |near zer|o but in|
|00000df0| 20 67 65 6e 65 72 61 6c | 20 69 73 20 6e 6f 74 20 | general| is not |
|00000e00| 73 61 6d 70 6c 65 64 20 | 65 78 61 63 74 6c 79 20 |sampled |exactly |
|00000e10| 61 74 20 61 20 72 6f 6f | 74 29 2e 0d 0d 73 75 72 |at a roo|t)...sur|
|00000e20| 66 5b 69 78 2c 69 79 5d | 20 3d 20 20 78 3a 3d 43 |f[ix,iy]| =  x:=C|
|00000e30| 73 63 61 6c 65 28 69 78 | 2c 69 79 29 2c 20 43 61 |scale(ix|,iy), Ca|
|00000e40| 62 73 28 7a 28 78 29 29 | 20 64 69 6d 5b 6d 2c 6d |bs(z(x))| dim[m,m|
|00000e50| 5d 0d 0d 2d 2d 20 54 68 | 65 20 69 6e 64 65 78 20 |]..-- Th|e index |
|00000e60| 66 6f 72 20 73 75 72 66 | 5b 5d 20 72 75 6e 73 20 |for surf|[] runs |
|00000e70| 66 72 6f 6d 20 31 20 74 | 6f 20 6d 2e 20 53 63 61 |from 1 t|o m. Sca|
|00000e80| 6c 65 20 74 68 65 20 69 | 6e 64 65 78 20 74 6f 20 |le the i|ndex to |
|00000e90| 67 65 74 20 72 65 61 6c | 20 61 6e 64 20 69 6d 61 |get real| and ima|
|00000ea0| 67 69 6e 61 72 79 20 70 | 61 72 74 73 20 72 61 6e |ginary p|arts ran|
|00000eb0| 67 69 6e 67 20 66 72 6f | 6d 20 58 6d 69 6e 20 74 |ging fro|m Xmin t|
|00000ec0| 6f 20 58 6d 61 78 20 61 | 6e 64 20 59 6d 69 6e 20 |o Xmax a|nd Ymin |
|00000ed0| 74 6f 20 59 6d 61 78 0d | 73 63 61 6c 65 28 69 2c |to Ymax.|scale(i,|
|00000ee0| 6d 69 6e 2c 6d 61 78 2c | 6e 73 74 65 70 73 29 20 |min,max,|nsteps) |
|00000ef0| 3d 20 28 69 2d 2e 35 29 | 2a 28 6d 61 78 2d 6d 69 |= (i-.5)|*(max-mi|
|00000f00| 6e 29 2f 6e 73 74 65 70 | 73 2b 6d 69 6e 0d 43 73 |n)/nstep|s+min.Cs|
|00000f10| 63 61 6c 65 28 69 78 2c | 69 79 29 20 3d 20 7b 73 |cale(ix,|iy) = {s|
|00000f20| 63 61 6c 65 28 69 78 2c | 58 6d 69 6e 2c 58 6d 61 |cale(ix,|Xmin,Xma|
|00000f30| 78 2c 6d 29 2c 20 73 63 | 61 6c 65 28 69 79 2c 59 |x,m), sc|ale(iy,Y|
|00000f40| 6d 69 6e 2c 59 6d 61 78 | 2c 6d 29 7d 0d 0d 6d 3d |min,Ymax|,m)}..m=|
|00000f50| 31 32 3b 20 20 20 2d 2d | 20 73 75 72 66 61 63 65 |12;   --| surface|
|00000f60| 20 69 73 20 73 61 6d 70 | 6c 65 64 20 6f 6e 20 61 | is samp|led on a|
|00000f70| 6e 20 6d 20 62 79 20 6d | 20 67 72 69 64 0d 58 6d |n m by m| grid.Xm|
|00000f80| 69 6e 20 3d 20 2d 31 3b | 20 58 6d 61 78 20 3d 20 |in = -1;| Xmax = |
|00000f90| 31 0d 59 6d 69 6e 20 3d | 20 2d 31 3b 20 59 6d 61 |1.Ymin =| -1; Yma|
|00000fa0| 78 20 3d 20 31 0d 5a 6d | 69 6e 20 3d 20 20 30 3b |x = 1.Zm|in =  0;|
|00000fb0| 20 5a 6d 61 78 20 3d 20 | 35 30 0d 69 6d 61 67 65 | Zmax = |50.image|
|00000fc0| 20 73 75 72 66 20 20 20 | 20 20 20 20 20 20 20 20 | surf   |        |
|00000fd0| 20 20 20 20 20 20 20 2d | 2d 20 73 68 6f 77 20 69 |       -|- show i|
|00000fe0| 6d 61 67 65 20 6f 66 20 | 74 68 65 20 73 75 72 66 |mage of |the surf|
|00000ff0| 61 63 65 0d 70 6c 6f 74 | 20 7b 72 6f 6f 74 31 2c |ace.plot| {root1,|
|00001000| 72 6f 6f 74 32 2c 72 6f | 6f 74 33 7d 20 20 20 20 |root2,ro|ot3}    |
|00001010| 2d 2d 20 73 68 6f 77 20 | 6c 6f 63 61 74 69 6f 6e |-- show |location|
|00001020| 73 20 6f 66 20 72 6f 6f | 74 73 0d 0d 70 6c 6f 74 |s of roo|ts..plot|
|00001030| 20 7a 28 7b 58 2c 30 7d | 29 5b 31 5d 2f 5a 6d 61 | z({X,0}|)[1]/Zma|
|00001040| 78 20 20 20 20 20 2d 2d | 20 70 6c 6f 74 20 74 68 |x     --| plot th|
|00001050| 65 20 72 65 61 6c 20 70 | 61 72 74 20 6f 66 20 7a |e real p|art of z|
|00001060| 20 66 6f 72 20 72 65 61 | 6c 20 78 0d 2d 2d 20 54 | for rea|l x.-- T|
|00001070| 68 69 73 20 73 68 6f 75 | 6c 64 20 62 65 20 7a 65 |his shou|ld be ze|
|00001080| 72 6f 20 61 74 20 72 65 | 61 6c 20 72 6f 6f 74 73 |ro at re|al roots|
|00001090| 2e 20 4f 6e 20 74 68 65 | 20 70 6c 6f 74 74 65 64 |. On the| plotted|
|000010a0| 20 73 75 72 66 61 63 65 | 2c 20 72 65 61 6c 20 72 | surface|, real r|
|000010b0| 6f 6f 74 73 20 61 72 65 | 20 6c 6f 63 61 74 65 64 |oots are| located|
|000010c0| 20 61 6c 6f 6e 67 20 79 | 3d 30 20 73 6f 20 74 68 | along y|=0 so th|
|000010d0| 65 20 72 65 61 6c 20 63 | 75 62 69 63 20 70 6c 6f |e real c|ubic plo|
|000010e0| 74 74 65 64 20 69 6e 20 | 74 68 69 73 20 77 61 79 |tted in |this way|
|000010f0| 20 73 68 6f 75 6c 64 20 | 70 61 73 73 20 74 68 6f | should |pass tho|
|00001100| 75 67 68 20 69 74 73 20 | 72 65 61 6c 20 72 6f 6f |ugh its |real roo|
|00001110| 74 73 2e 0d 70 6c 6f 74 | 20 30 0d 00 00 00 00 00 |ts..plot| 0......|
|00001120| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001130| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001140| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001150| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001160| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001170| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00001180| 00 00 01 00 00 00 01 24 | 00 00 00 24 00 00 00 32 |.......$|...$...2|
|00001190| 77 68 65 6e 20 41 5b 32 | 5d 3e 3d 30 2c 0d 20 20 |when A[2|]>=0,.  |
|000011a0| 20 20 20 20 20 20 20 61 | 74 61 6e 28 41 5b 32 5d |       a|tan(A[2]|
|000011b0| 0d 43 6f 6d 70 6c 65 78 | 20 52 6f 6f 74 73 02 00 |.Complex| Roots..|
|000011c0| 00 00 54 45 58 54 4d 50 | 61 64 01 00 00 00 00 00 |..TEXTMP|ad......|
|000011d0| 00 00 54 45 58 54 4d 50 | 61 64 01 00 00 00 00 00 |..TEXTMP|ad......|
|000011e0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000011f0| 00 00 a8 b8 1c 3f 00 00 | 10 9b 00 00 01 56 30 7d |.....?..|.....V0}|
|00001200| 0d db 6d b6 db 6d b6 db | 6d b6 db 6d b6 db 6d b6 |..m..m..|m..m..m.|
|00001210| db 6d b6 db 6d b6 db 6d | b6 db 6d b6 db 6d b6 db |.m..m..m|..m..m..|
|00001220| 6d b6 db 6d b6 db 6d b6 | db 6d b6 db 6d b6 db 6d |m..m..m.|.m..m..m|
|00001230| b6 db 6d b6 db 6d b6 db | 6d b6 db 6d b6 db 6d b6 |..m..m..|m..m..m.|
|00001240| db 6d b6 db 6d b6 db 6d | b6 db 6d b6 db 6d b6 db |.m..m..m|..m..m..|
|00001250| 6d b6 db 6d b6 db 6d b6 | db 6d b6 db 6d b6 db 6d |m..m..m.|.m..m..m|
|00001260| b6 db 6d b6 db 6d b6 db | 6d b6 db 6d b6 db 6d b6 |..m..m..|m..m..m.|
|00001270| db 6d b6 db 6d b6 db 6d | b6 db 6d b6 db 6d b6 db |.m..m..m|..m..m..|
|00001280| 00 00 00 20 05 00 00 02 | 00 02 3f f9 8e fa 35 12 |... ....|..?...5.|
|00001290| 94 e9 c8 ae 01 d5 01 2b | 00 03 00 28 01 0d 01 1b |.......+|...(....|
|000012a0| 00 c7 00 34 00 00 01 00 | 00 00 01 24 00 00 00 24 |...4....|...$...$|
|000012b0| 00 00 00 32 00 6e 20 7c | 07 b8 00 00 00 1c 00 32 |...2.n ||.......2|
|000012c0| 00 00 50 52 65 66 00 00 | 00 0a 00 80 ff ff 00 00 |..PRef..|........|
|000012d0| 00 00 00 6e 1f e0 00 00 | 00 00 00 00 00 00 00 00 |...n....|........|
|000012e0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000012f0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
+--------+-------------------------+-------------------------+--------+--------+