home *** CD-ROM | disk | FTP | other *** search
/ Chip: Special Computer Graphics & Animation / Chip-Special-Computergrafik.bin / programs / povray / povsrc.sea / POVSRC / SOURCE / VECT.C < prev    next >
MacBinary  |  1993-10-03  |  24.6 KB  |  [TEXT/MPS ]
open in: MacOS 8.1     |     Win98     |     DOS

browse contents    |     view JSON data     |     view as text


This file was processed as: MacBinary (archive/macBinary).

ConfidenceProgramDetectionMatch TypeSupport
66% dexvert Compact Compressed (Unix) (archive/compact) ext Supported
10% dexvert MacBinary (archive/macBinary) fallback Supported
10% dexvert Jesper Olsen Module (music/jesperOlsen) magic Supported
1% dexvert Text File (text/txt) fallback Supported
100% file MacBinary II, inited, Sun Oct 3 12:25:56 1993, modified Sun Oct 3 12:25:56 1993, creator 'MPS ', type ASCII, 24522 bytes "VECT.C" , at 0x604a 428 bytes resource default (weak)
99% file data default
49% TrID Macintosh plain text (MacBinary) default
33% TrID TTComp archive compressed (bin-4K) default (weak)
16% TrID MacBinary 2 default (weak)
100% lsar MacBinary default


id metadata
keyvalue
macFileType[TEXT]
macFileCreator[MPS ]


hex view
+--------+-------------------------+-------------------------+--------+--------+
|00000000| 00 06 56 45 43 54 2e 43 | 00 00 00 00 00 00 00 00 |..VECT.C|........|
|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 53 | 20 01 00 00 00 00 00 00 |.TEXTMPS| .......|
|00000050| 00 00 00 00 00 5f ca 00 | 00 01 ac a8 d4 ad 94 a8 |....._..|........|
|00000060| d4 ad 94 00 00 08 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00000070| 00 00 00 00 00 00 00 00 | 00 00 81 81 63 60 00 00 |........|....c`..|
|00000080| 2f 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |/*******|********|
|00000090| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |********|********|
|000000a0| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |********|********|
|000000b0| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |********|********|
|000000c0| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 0d 2a 20 |********|*****.* |
|000000d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 76 |        |       v|
|000000e0| 65 63 74 2e 63 0d 2a 0d | 2a 20 20 54 68 69 73 20 |ect.c.*.|*  This |
|000000f0| 66 69 6c 65 20 77 61 73 | 20 77 72 69 74 74 65 6e |file was| written|
|00000100| 20 62 79 20 41 6c 65 78 | 61 6e 64 65 72 20 45 6e | by Alex|ander En|
|00000110| 7a 6d 61 6e 6e 2e 20 20 | 48 65 20 77 72 6f 74 65 |zmann.  |He wrote|
|00000120| 20 74 68 65 20 63 6f 64 | 65 20 66 6f 72 0d 2a 20 | the cod|e for.* |
|00000130| 20 34 74 68 2d 36 74 68 | 20 6f 72 64 65 72 20 73 | 4th-6th| order s|
|00000140| 68 61 70 65 73 20 61 6e | 64 20 67 65 6e 65 72 6f |hapes an|d genero|
|00000150| 75 73 6c 79 20 70 72 6f | 76 69 64 65 64 20 75 73 |usly pro|vided us|
|00000160| 20 74 68 65 73 65 20 65 | 6e 68 61 6e 63 65 6d 65 | these e|nhanceme|
|00000170| 6e 74 73 2e 0d 2a 0d 2a | 20 20 66 72 6f 6d 20 50 |nts..*.*|  from P|
|00000180| 65 72 73 69 73 74 65 6e | 63 65 20 6f 66 20 56 69 |ersisten|ce of Vi|
|00000190| 73 69 6f 6e 20 52 61 79 | 74 72 61 63 65 72 0d 2a |sion Ray|tracer.*|
|000001a0| 20 20 43 6f 70 79 72 69 | 67 68 74 20 31 39 39 33 |  Copyri|ght 1993|
|000001b0| 20 50 65 72 73 69 73 74 | 65 6e 63 65 20 6f 66 20 | Persist|ence of |
|000001c0| 56 69 73 69 6f 6e 20 54 | 65 61 6d 0d 2a 2d 2d 2d |Vision T|eam.*---|
|000001d0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|000001e0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|000001f0| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000200| 2d 2d 2d 2d 2d 2d 2d 2d | 2d 2d 2d 2d 2d 2d 2d 2d |--------|--------|
|00000210| 2d 2d 2d 2d 2d 2d 2d 2d | 0d 2a 20 20 4e 4f 54 49 |--------|.*  NOTI|
|00000220| 43 45 3a 20 54 68 69 73 | 20 73 6f 75 72 63 65 20 |CE: This| source |
|00000230| 63 6f 64 65 20 66 69 6c | 65 20 69 73 20 70 72 6f |code fil|e is pro|
|00000240| 76 69 64 65 64 20 73 6f | 20 74 68 61 74 20 75 73 |vided so| that us|
|00000250| 65 72 73 20 6d 61 79 20 | 65 78 70 65 72 69 6d 65 |ers may |experime|
|00000260| 6e 74 0d 2a 20 20 77 69 | 74 68 20 65 6e 68 61 6e |nt.*  wi|th enhan|
|00000270| 63 65 6d 65 6e 74 73 20 | 74 6f 20 50 4f 56 2d 52 |cements |to POV-R|
|00000280| 61 79 20 61 6e 64 20 74 | 6f 20 70 6f 72 74 20 74 |ay and t|o port t|
|00000290| 68 65 20 73 6f 66 74 77 | 61 72 65 20 74 6f 20 70 |he softw|are to p|
|000002a0| 6c 61 74 66 6f 72 6d 73 | 20 6f 74 68 65 72 20 0d |latforms| other .|
|000002b0| 2a 20 20 74 68 61 6e 20 | 74 68 6f 73 65 20 73 75 |*  than |those su|
|000002c0| 70 70 6f 72 74 65 64 20 | 62 79 20 74 68 65 20 50 |pported |by the P|
|000002d0| 4f 56 2d 52 61 79 20 54 | 65 61 6d 2e 20 20 54 68 |OV-Ray T|eam.  Th|
|000002e0| 65 72 65 20 61 72 65 20 | 73 74 72 69 63 74 20 72 |ere are |strict r|
|000002f0| 75 6c 65 73 20 75 6e 64 | 65 72 0d 2a 20 20 77 68 |ules und|er.*  wh|
|00000300| 69 63 68 20 79 6f 75 20 | 61 72 65 20 70 65 72 6d |ich you |are perm|
|00000310| 69 74 74 65 64 20 74 6f | 20 75 73 65 20 74 68 69 |itted to| use thi|
|00000320| 73 20 66 69 6c 65 2e 20 | 20 54 68 65 20 72 75 6c |s file. | The rul|
|00000330| 65 73 20 61 72 65 20 69 | 6e 20 74 68 65 20 66 69 |es are i|n the fi|
|00000340| 6c 65 0d 2a 20 20 6e 61 | 6d 65 64 20 50 4f 56 4c |le.*  na|med POVL|
|00000350| 45 47 41 4c 2e 44 4f 43 | 20 77 68 69 63 68 20 73 |EGAL.DOC| which s|
|00000360| 68 6f 75 6c 64 20 62 65 | 20 64 69 73 74 72 69 62 |hould be| distrib|
|00000370| 75 74 65 64 20 77 69 74 | 68 20 74 68 69 73 20 66 |uted wit|h this f|
|00000380| 69 6c 65 2e 20 49 66 20 | 0d 2a 20 20 50 4f 56 4c |ile. If |.*  POVL|
|00000390| 45 47 41 4c 2e 44 4f 43 | 20 69 73 20 6e 6f 74 20 |EGAL.DOC| is not |
|000003a0| 61 76 61 69 6c 61 62 6c | 65 20 6f 72 20 66 6f 72 |availabl|e or for|
|000003b0| 20 6d 6f 72 65 20 69 6e | 66 6f 20 70 6c 65 61 73 | more in|fo pleas|
|000003c0| 65 20 63 6f 6e 74 61 63 | 74 20 74 68 65 20 50 4f |e contac|t the PO|
|000003d0| 56 2d 52 61 79 0d 2a 20 | 20 54 65 61 6d 20 43 6f |V-Ray.* | Team Co|
|000003e0| 6f 72 64 69 6e 61 74 6f | 72 20 62 79 20 6c 65 61 |ordinato|r by lea|
|000003f0| 76 69 6e 67 20 61 20 6d | 65 73 73 61 67 65 20 69 |ving a m|essage i|
|00000400| 6e 20 43 6f 6d 70 75 53 | 65 72 76 65 27 73 20 47 |n CompuS|erve's G|
|00000410| 72 61 70 68 69 63 73 20 | 44 65 76 65 6c 6f 70 65 |raphics |Develope|
|00000420| 72 27 73 0d 2a 20 20 46 | 6f 72 75 6d 2e 20 20 54 |r's.*  F|orum.  T|
|00000430| 68 65 20 6c 61 74 65 73 | 74 20 76 65 72 73 69 6f |he lates|t versio|
|00000440| 6e 20 6f 66 20 50 4f 56 | 2d 52 61 79 20 6d 61 79 |n of POV|-Ray may|
|00000450| 20 62 65 20 66 6f 75 6e | 64 20 74 68 65 72 65 20 | be foun|d there |
|00000460| 61 73 20 77 65 6c 6c 2e | 0d 2a 0d 2a 20 54 68 69 |as well.|.*.* Thi|
|00000470| 73 20 70 72 6f 67 72 61 | 6d 20 69 73 20 62 61 73 |s progra|m is bas|
|00000480| 65 64 20 6f 6e 20 74 68 | 65 20 70 6f 70 75 6c 61 |ed on th|e popula|
|00000490| 72 20 44 4b 42 20 72 61 | 79 74 72 61 63 65 72 20 |r DKB ra|ytracer |
|000004a0| 76 65 72 73 69 6f 6e 20 | 32 2e 31 32 2e 0d 2a 20 |version |2.12..* |
|000004b0| 44 4b 42 54 72 61 63 65 | 20 77 61 73 20 6f 72 69 |DKBTrace| was ori|
|000004c0| 67 69 6e 61 6c 6c 79 20 | 77 72 69 74 74 65 6e 20 |ginally |written |
|000004d0| 62 79 20 44 61 76 69 64 | 20 4b 2e 20 42 75 63 6b |by David| K. Buck|
|000004e0| 2e 0d 2a 20 44 4b 42 54 | 72 61 63 65 20 56 65 72 |..* DKBT|race Ver|
|000004f0| 20 32 2e 30 2d 32 2e 31 | 32 20 77 65 72 65 20 77 | 2.0-2.1|2 were w|
|00000500| 72 69 74 74 65 6e 20 62 | 79 20 44 61 76 69 64 20 |ritten b|y David |
|00000510| 4b 2e 20 42 75 63 6b 20 | 26 20 41 61 72 6f 6e 20 |K. Buck |& Aaron |
|00000520| 41 2e 20 43 6f 6c 6c 69 | 6e 73 2e 0d 2a 0d 2a 2a |A. Colli|ns..*.**|
|00000530| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |********|********|
|00000540| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |********|********|
|00000550| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |********|********|
|00000560| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |********|********|
|00000570| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2f 0d 0d 23 69 |********|***/..#i|
|00000580| 6e 63 6c 75 64 65 20 22 | 66 72 61 6d 65 2e 68 22 |nclude "|frame.h"|
|00000590| 0d 23 69 6e 63 6c 75 64 | 65 20 22 70 6f 76 70 72 |.#includ|e "povpr|
|000005a0| 6f 74 6f 2e 68 22 0d 23 | 69 6e 63 6c 75 64 65 20 |oto.h".#|include |
|000005b0| 22 76 65 63 74 6f 72 2e | 68 22 0d 0d 23 69 66 64 |"vector.|h"..#ifd|
|000005c0| 65 66 20 41 42 53 0d 23 | 75 6e 64 65 66 20 41 42 |ef ABS.#|undef AB|
|000005d0| 53 0d 23 65 6e 64 69 66 | 0d 0d 23 69 66 64 65 66 |S.#endif|..#ifdef|
|000005e0| 20 4d 41 58 0d 23 75 6e | 64 65 66 20 4d 41 58 0d | MAX.#un|def MAX.|
|000005f0| 23 65 6e 64 69 66 0d 0d | 65 78 74 65 72 6e 20 69 |#endif..|extern i|
|00000600| 6e 74 20 53 68 61 64 6f | 77 5f 54 65 73 74 5f 46 |nt Shado|w_Test_F|
|00000610| 6c 61 67 3b 0d 23 75 6e | 64 65 66 20 45 50 53 49 |lag;.#un|def EPSI|
|00000620| 4c 4f 4e 0d 23 64 65 66 | 69 6e 65 20 45 50 53 49 |LON.#def|ine EPSI|
|00000630| 4c 4f 4e 20 31 2e 30 65 | 2d 31 30 0d 23 64 65 66 |LON 1.0e|-10.#def|
|00000640| 69 6e 65 20 43 4f 45 46 | 46 5f 4c 49 4d 49 54 20 |ine COEF|F_LIMIT |
|00000650| 31 2e 30 65 2d 32 30 0d | 0d 2f 2a 20 20 20 20 20 |1.0e-20.|./*     |
|00000660| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 57 41 52 |        |     WAR|
|00000670| 4e 49 4e 47 20 20 20 20 | 20 57 41 52 4e 49 4e 47 |NING    | WARNING|
|00000680| 20 20 20 20 57 41 52 4e | 49 4e 47 0d 0d 20 20 20 |    WARN|ING..   |
|00000690| 54 68 65 20 66 6f 6c 6c | 6f 77 69 6e 67 20 74 68 |The foll|owing th|
|000006a0| 72 65 65 20 63 6f 6e 73 | 74 61 6e 74 73 20 68 61 |ree cons|tants ha|
|000006b0| 76 65 20 62 65 65 6e 20 | 64 65 66 69 6e 65 64 20 |ve been |defined |
|000006c0| 73 6f 20 74 68 61 74 20 | 71 75 61 72 74 69 63 20 |so that |quartic |
|000006d0| 65 71 75 61 74 69 6f 6e | 73 0d 20 20 20 77 69 6c |equation|s.   wil|
|000006e0| 6c 20 70 72 6f 70 65 72 | 6c 79 20 72 65 6e 64 65 |l proper|ly rende|
|000006f0| 72 20 6f 6e 20 66 70 75 | 2f 63 6f 6d 70 69 6c 65 |r on fpu|/compile|
|00000700| 72 20 63 6f 6d 62 69 6e | 61 74 69 6f 6e 73 20 74 |r combin|ations t|
|00000710| 68 61 74 20 6f 6e 6c 79 | 20 68 61 76 65 20 36 34 |hat only| have 64|
|00000720| 20 62 69 74 0d 20 20 20 | 49 45 45 45 20 70 72 65 | bit.   |IEEE pre|
|00000730| 63 69 73 69 6f 6e 2e 20 | 20 44 6f 20 6e 6f 74 20 |cision. | Do not |
|00000740| 61 72 62 69 74 72 61 72 | 69 6c 79 20 63 68 61 6e |arbitrar|ily chan|
|00000750| 67 65 20 61 6e 79 20 6f | 66 20 74 68 65 73 65 20 |ge any o|f these |
|00000760| 76 61 6c 75 65 73 2e 0d | 0d 20 20 20 49 66 20 79 |values..|.   If y|
|00000770| 6f 75 20 68 61 76 65 20 | 61 20 6d 61 63 68 69 6e |ou have |a machin|
|00000780| 65 20 77 69 74 68 20 61 | 20 70 72 6f 70 65 72 20 |e with a| proper |
|00000790| 66 70 75 2f 63 6f 6d 70 | 69 6c 65 72 20 63 6f 6d |fpu/comp|iler com|
|000007a0| 62 6f 20 2d 20 6c 69 6b | 65 20 61 20 4d 61 63 20 |bo - lik|e a Mac |
|000007b0| 77 69 74 68 0d 20 20 20 | 61 20 36 38 38 38 31 2c |with.   |a 68881,|
|000007c0| 20 74 68 65 6e 20 75 73 | 65 20 74 68 65 20 6e 61 | then us|e the na|
|000007d0| 74 69 76 65 20 66 6c 6f | 61 74 69 6e 67 20 66 6f |tive flo|ating fo|
|000007e0| 72 6d 61 74 20 28 39 36 | 20 62 69 74 73 29 20 61 |rmat (96| bits) a|
|000007f0| 6e 64 20 74 75 6e 65 20 | 74 68 65 0d 20 20 20 76 |nd tune |the.   v|
|00000800| 61 6c 75 65 73 20 66 6f | 72 20 61 20 6c 69 74 74 |alues fo|r a litt|
|00000810| 6c 65 20 6c 65 73 73 20 | 74 6f 6c 65 72 61 6e 63 |le less |toleranc|
|00000820| 65 3a 20 73 6f 6d 65 74 | 68 69 6e 67 20 6c 69 6b |e: somet|hing lik|
|00000830| 65 3a 20 66 61 63 74 6f | 72 31 20 3d 20 31 2e 30 |e: facto|r1 = 1.0|
|00000840| 65 31 35 2c 0d 20 20 20 | 66 61 63 74 6f 72 32 20 |e15,.   |factor2 |
|00000850| 3d 20 2d 31 2e 30 65 2d | 37 2c 20 66 61 63 74 6f |= -1.0e-|7, facto|
|00000860| 72 33 20 3d 20 31 2e 30 | 65 2d 31 30 2e 0d 0d 20 |r3 = 1.0|e-10... |
|00000870| 20 20 54 68 65 20 6d 65 | 61 6e 69 6e 67 20 6f 66 |  The me|aning of|
|00000880| 20 74 68 65 20 74 68 72 | 65 65 20 63 6f 6e 73 74 | the thr|ee const|
|00000890| 61 6e 74 73 20 61 72 65 | 3a 0d 20 20 20 20 20 20 |ants are|:.      |
|000008a0| 66 61 63 74 6f 72 31 20 | 2d 20 74 68 65 20 6d 61 |factor1 |- the ma|
|000008b0| 67 6e 69 74 75 64 65 20 | 6f 66 20 64 69 66 66 65 |gnitude |of diffe|
|000008c0| 72 65 6e 63 65 20 62 65 | 74 77 65 65 6e 20 63 6f |rence be|tween co|
|000008d0| 65 66 66 69 63 69 65 6e | 74 73 20 69 6e 20 61 0d |efficien|ts in a.|
|000008e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000008f0| 71 75 61 72 74 69 63 20 | 65 71 75 61 74 69 6f 6e |quartic |equation|
|00000900| 20 61 74 20 77 68 69 63 | 68 20 74 68 65 20 53 74 | at whic|h the St|
|00000910| 75 72 6d 69 61 6e 20 72 | 6f 6f 74 20 73 6f 6c 76 |urmian r|oot solv|
|00000920| 65 72 20 77 69 6c 6c 0d | 09 09 6b 69 63 6b 20 69 |er will.|..kick i|
|00000930| 6e 2e 20 20 54 68 65 20 | 53 74 75 72 6d 20 73 6f |n.  The |Sturm so|
|00000940| 6c 76 65 72 20 69 73 20 | 71 75 69 74 65 20 61 20 |lver is |quite a |
|00000950| 62 69 74 20 73 6c 6f 77 | 65 72 20 74 68 61 6e 0d |bit slow|er than.|
|00000960| 09 09 74 68 65 20 61 6c | 67 65 62 72 61 69 63 20 |..the al|gebraic |
|00000970| 73 6f 6c 76 65 72 2c 20 | 73 6f 20 74 68 65 20 62 |solver, |so the b|
|00000980| 69 67 67 65 72 20 74 68 | 69 73 20 69 73 2c 20 74 |igger th|is is, t|
|00000990| 68 65 20 66 61 73 74 65 | 72 0d 09 09 74 68 65 20 |he faste|r...the |
|000009a0| 72 6f 6f 74 20 73 6f 6c | 76 69 6e 67 20 77 69 6c |root sol|ving wil|
|000009b0| 6c 20 67 6f 2e 20 20 54 | 68 65 20 61 6c 67 65 62 |l go.  T|he algeb|
|000009c0| 72 61 69 63 20 73 6f 6c | 76 65 72 20 69 73 20 6c |raic sol|ver is l|
|000009d0| 65 73 73 0d 09 09 61 63 | 63 75 72 61 74 65 20 73 |ess...ac|curate s|
|000009e0| 6f 20 74 68 65 20 62 69 | 67 67 65 72 20 74 68 69 |o the bi|gger thi|
|000009f0| 73 20 69 73 2c 20 74 68 | 65 20 6d 6f 72 65 20 6c |s is, th|e more l|
|00000a00| 69 6b 65 6c 79 20 79 6f | 75 20 77 69 6c 6c 0d 09 |ikely yo|u will..|
|00000a10| 09 67 65 74 20 62 61 64 | 20 72 6f 6f 74 73 2e 0d |.get bad| roots..|
|00000a20| 0d 20 20 20 20 20 20 66 | 61 63 74 6f 72 32 20 2d |.      f|actor2 -|
|00000a30| 20 54 6f 6c 65 72 61 6e | 63 65 20 76 61 6c 75 65 | Toleran|ce value|
|00000a40| 20 74 68 61 74 20 64 65 | 66 69 6e 65 73 20 68 6f | that de|fines ho|
|00000a50| 77 20 63 6c 6f 73 65 20 | 74 68 65 20 71 75 61 72 |w close |the quar|
|00000a60| 74 69 63 20 65 71 75 61 | 74 69 6f 6e 0d 09 09 69 |tic equa|tion...i|
|00000a70| 73 20 74 6f 20 62 65 69 | 6e 67 20 61 20 73 71 75 |s to bei|ng a squ|
|00000a80| 61 72 65 20 6f 66 20 61 | 20 71 75 61 64 72 61 74 |are of a| quadrat|
|00000a90| 69 63 2e 20 20 54 68 65 | 20 63 6c 6f 73 65 72 20 |ic.  The| closer |
|00000aa0| 74 68 69 73 20 63 61 6e | 0d 09 09 67 65 74 20 74 |this can|...get t|
|00000ab0| 6f 20 7a 65 72 6f 20 62 | 65 66 6f 72 65 20 72 6f |o zero b|efore ro|
|00000ac0| 6f 74 73 20 64 69 73 61 | 70 70 65 61 72 2c 20 74 |ots disa|ppear, t|
|00000ad0| 68 65 20 6c 65 73 73 20 | 74 68 65 20 63 68 61 6e |he less |the chan|
|00000ae0| 63 65 0d 09 09 79 6f 75 | 20 77 69 6c 6c 20 67 65 |ce...you| will ge|
|00000af0| 74 20 73 70 75 72 69 6f | 75 73 20 72 6f 6f 74 73 |t spurio|us roots|
|00000b00| 2e 0d 0d 20 20 20 20 20 | 20 66 61 63 74 6f 72 33 |...     | factor3|
|00000b10| 20 2d 20 53 69 6d 69 6c | 61 72 20 74 6f 20 66 61 | - Simil|ar to fa|
|00000b20| 63 74 6f 72 32 20 61 74 | 20 61 20 6c 61 74 65 72 |ctor2 at| a later|
|00000b30| 20 73 74 61 67 65 20 6f | 66 20 74 68 65 20 61 6c | stage o|f the al|
|00000b40| 67 65 62 72 61 69 63 20 | 73 6f 6c 76 65 72 2e 0d |gebraic |solver..|
|00000b50| 2a 2f 0d 23 64 65 66 69 | 6e 65 20 46 55 44 47 45 |*/.#defi|ne FUDGE|
|00000b60| 5f 46 41 43 54 4f 52 31 | 20 31 2e 30 65 31 32 0d |_FACTOR1| 1.0e12.|
|00000b70| 23 64 65 66 69 6e 65 20 | 46 55 44 47 45 5f 46 41 |#define |FUDGE_FA|
|00000b80| 43 54 4f 52 32 20 2d 31 | 2e 30 65 2d 35 0d 23 64 |CTOR2 -1|.0e-5.#d|
|00000b90| 65 66 69 6e 65 20 46 55 | 44 47 45 5f 46 41 43 54 |efine FU|DGE_FACT|
|00000ba0| 4f 52 33 20 31 2e 30 65 | 2d 37 0d 0d 23 64 65 66 |OR3 1.0e|-7..#def|
|00000bb0| 69 6e 65 20 41 42 53 28 | 78 29 20 28 28 78 29 20 |ine ABS(|x) ((x) |
|00000bc0| 3c 20 30 2e 30 20 3f 20 | 28 30 2e 30 20 2d 20 78 |< 0.0 ? |(0.0 - x|
|00000bd0| 29 20 3a 20 28 78 29 29 | 0d 23 64 65 66 69 6e 65 |) : (x))|.#define|
|00000be0| 20 4d 41 58 28 78 2c 79 | 29 20 28 78 3c 79 3f 79 | MAX(x,y|) (x<y?y|
|00000bf0| 3a 78 29 0d 23 64 65 66 | 69 6e 65 20 54 57 4f 5f |:x).#def|ine TWO_|
|00000c00| 50 49 20 36 2e 32 38 33 | 31 38 35 32 30 37 31 37 |PI 6.283|18520717|
|00000c10| 39 35 38 36 34 37 36 39 | 32 35 32 38 36 37 36 36 |95864769|25286766|
|00000c20| 35 36 30 0d 23 64 65 66 | 69 6e 65 20 54 57 4f 5f |560.#def|ine TWO_|
|00000c30| 50 49 5f 33 20 20 32 2e | 30 39 34 33 39 35 31 30 |PI_3  2.|09439510|
|00000c40| 32 33 39 33 31 39 35 34 | 39 32 33 30 38 34 0d 23 |23931954|923084.#|
|00000c50| 64 65 66 69 6e 65 20 54 | 57 4f 5f 50 49 5f 34 33 |define T|WO_PI_43|
|00000c60| 20 34 2e 31 38 38 37 39 | 30 32 30 34 37 38 36 33 | 4.18879|02047863|
|00000c70| 39 30 39 38 34 36 31 36 | 38 0d 23 64 65 66 69 6e |90984616|8.#defin|
|00000c80| 65 20 4d 41 58 5f 49 54 | 45 52 41 54 49 4f 4e 53 |e MAX_IT|ERATIONS|
|00000c90| 20 35 30 0d 23 64 65 66 | 69 6e 65 20 4d 41 58 50 | 50.#def|ine MAXP|
|00000ca0| 4f 57 20 33 32 0d 0d 20 | 20 74 79 70 65 64 65 66 |OW 32.. | typedef|
|00000cb0| 20 73 74 72 75 63 74 20 | 70 20 7b 0d 20 20 69 6e | struct |p {.  in|
|00000cc0| 74 20 6f 72 64 3b 0d 20 | 20 44 42 4c 20 63 6f 65 |t ord;. | DBL coe|
|00000cd0| 66 5b 4d 41 58 5f 4f 52 | 44 45 52 2b 31 5d 3b 0d |f[MAX_OR|DER+1];.|
|00000ce0| 20 20 7d 20 0d 70 6f 6c | 79 6e 6f 6d 69 61 6c 3b |  } .pol|ynomial;|
|00000cf0| 0d 0d 73 74 61 74 69 63 | 20 69 6e 74 20 6d 6f 64 |..static| int mod|
|00000d00| 70 20 50 41 52 41 4d 53 | 28 28 70 6f 6c 79 6e 6f |p PARAMS|((polyno|
|00000d10| 6d 69 61 6c 20 2a 75 2c | 20 70 6f 6c 79 6e 6f 6d |mial *u,| polynom|
|00000d20| 69 61 6c 20 2a 76 2c 20 | 70 6f 6c 79 6e 6f 6d 69 |ial *v, |polynomi|
|00000d30| 61 6c 20 2a 72 29 29 3b | 0d 69 6e 74 20 72 65 67 |al *r));|.int reg|
|00000d40| 75 6c 61 5f 66 61 6c 73 | 61 20 50 41 52 41 4d 53 |ula_fals|a PARAMS|
|00000d50| 28 28 69 6e 74 20 6f 72 | 64 65 72 2c 20 44 42 4c |((int or|der, DBL|
|00000d60| 20 2a 63 6f 65 66 2c 20 | 44 42 4c 20 61 2c 20 44 | *coef, |DBL a, D|
|00000d70| 42 4c 20 62 2c 20 44 42 | 4c 20 2a 76 61 6c 29 29 |BL b, DB|L *val))|
|00000d80| 3b 0d 73 74 61 74 69 63 | 20 69 6e 74 20 73 62 69 |;.static| int sbi|
|00000d90| 73 65 63 74 20 50 41 52 | 41 4d 53 28 28 69 6e 74 |sect PAR|AMS((int|
|00000da0| 20 6e 70 2c 20 70 6f 6c | 79 6e 6f 6d 69 61 6c 20 | np, pol|ynomial |
|00000db0| 2a 73 73 65 71 2c 20 44 | 42 4c 20 6d 69 6e 2c 20 |*sseq, D|BL min, |
|00000dc0| 44 42 4c 20 6d 61 78 2c | 20 69 6e 74 20 61 74 6d |DBL max,| int atm|
|00000dd0| 69 6e 2c 20 69 6e 74 20 | 61 74 6d 61 78 2c 20 44 |in, int |atmax, D|
|00000de0| 42 4c 20 2a 72 6f 6f 74 | 73 29 29 3b 0d 69 6e 74 |BL *root|s));.int|
|00000df0| 20 6e 75 6d 63 68 61 6e | 67 65 73 20 50 41 52 41 | numchan|ges PARA|
|00000e00| 4d 53 28 28 69 6e 74 20 | 6e 70 2c 20 70 6f 6c 79 |MS((int |np, poly|
|00000e10| 6e 6f 6d 69 61 6c 20 2a | 73 73 65 71 2c 20 44 42 |nomial *|sseq, DB|
|00000e20| 4c 20 61 29 29 3b 0d 44 | 42 4c 20 70 6f 6c 79 65 |L a));.D|BL polye|
|00000e30| 76 61 6c 20 50 41 52 41 | 4d 53 28 28 44 42 4c 20 |val PARA|MS((DBL |
|00000e40| 78 2c 20 69 6e 74 20 6e | 2c 20 44 42 4c 20 2a 43 |x, int n|, DBL *C|
|00000e50| 6f 65 66 66 73 29 29 3b | 0d 69 6e 74 20 62 75 69 |oeffs));|.int bui|
|00000e60| 6c 64 73 74 75 72 6d 20 | 50 41 52 41 4d 53 28 28 |ldsturm |PARAMS((|
|00000e70| 69 6e 74 20 6f 72 64 2c | 20 70 6f 6c 79 6e 6f 6d |int ord,| polynom|
|00000e80| 69 61 6c 20 2a 73 73 65 | 71 29 29 3b 0d 69 6e 74 |ial *sse|q));.int|
|00000e90| 20 76 69 73 69 62 6c 65 | 5f 72 6f 6f 74 73 20 50 | visible|_roots P|
|00000ea0| 41 52 41 4d 53 28 28 69 | 6e 74 20 6e 70 2c 20 70 |ARAMS((i|nt np, p|
|00000eb0| 6f 6c 79 6e 6f 6d 69 61 | 6c 20 2a 73 73 65 71 2c |olynomia|l *sseq,|
|00000ec0| 20 69 6e 74 20 2a 61 74 | 6e 65 67 2c 20 69 6e 74 | int *at|neg, int|
|00000ed0| 20 2a 61 74 70 6f 73 29 | 29 3b 0d 73 74 61 74 69 | *atpos)|);.stati|
|00000ee0| 63 20 69 6e 74 20 64 69 | 66 66 69 63 75 6c 74 5f |c int di|fficult_|
|00000ef0| 63 6f 65 66 66 73 20 50 | 41 52 41 4d 53 28 28 69 |coeffs P|ARAMS((i|
|00000f00| 6e 74 20 6e 2c 20 44 42 | 4c 20 2a 78 29 29 3b 0d |nt n, DB|L *x));.|
|00000f10| 0d 2f 2a 0d 20 2a 20 6d | 6f 64 70 0d 20 2a 0d 20 |./*. * m|odp. *. |
|00000f20| 2a 20 20 20 63 61 6c 63 | 75 6c 61 74 65 73 20 74 |*   calc|ulates t|
|00000f30| 68 65 20 6d 6f 64 75 6c | 75 73 20 6f 66 20 75 28 |he modul|us of u(|
|00000f40| 78 29 20 2f 20 76 28 78 | 29 20 6c 65 61 76 69 6e |x) / v(x|) leavin|
|00000f50| 67 20 69 74 20 69 6e 20 | 72 2c 20 69 74 0d 20 2a |g it in |r, it. *|
|00000f60| 20 20 72 65 74 75 72 6e | 73 20 30 20 69 66 20 72 |  return|s 0 if r|
|00000f70| 28 78 29 20 69 73 20 61 | 20 63 6f 6e 73 74 61 6e |(x) is a| constan|
|00000f80| 74 2e 0d 20 2a 20 20 6e | 6f 74 65 3a 20 74 68 69 |t.. *  n|ote: thi|
|00000f90| 73 20 66 75 6e 63 74 69 | 6f 6e 20 61 73 73 75 6d |s functi|on assum|
|00000fa0| 65 73 20 74 68 65 20 6c | 65 61 64 69 6e 67 20 63 |es the l|eading c|
|00000fb0| 6f 65 66 66 69 63 69 65 | 6e 74 20 6f 66 20 76 20 |oefficie|nt of v |
|00000fc0| 0d 20 2a 20 20 20 69 73 | 20 31 20 6f 72 20 2d 31 |. *   is| 1 or -1|
|00000fd0| 0d 20 2a 2f 0d 73 74 61 | 74 69 63 20 69 6e 74 20 |. */.sta|tic int |
|00000fe0| 6d 6f 64 70 28 75 2c 20 | 76 2c 20 72 29 0d 70 6f |modp(u, |v, r).po|
|00000ff0| 6c 79 6e 6f 6d 69 61 6c | 20 20 20 2a 75 2c 20 2a |lynomial|   *u, *|
|00001000| 76 2c 20 2a 72 3b 0d 20 | 20 7b 0d 20 20 69 6e 74 |v, *r;. | {.  int|
|00001010| 20 20 20 20 69 2c 20 6b | 2c 20 6a 3b 0d 20 20 66 |    i, k|, j;.  f|
|00001020| 6f 72 20 28 69 3d 30 3b | 69 3c 75 2d 3e 6f 72 64 |or (i=0;|i<u->ord|
|00001030| 3b 69 2b 2b 29 0d 20 20 | 20 20 72 5b 69 5d 20 3d |;i++).  |  r[i] =|
|00001040| 20 75 5b 69 5d 3b 0d 0d | 20 20 69 66 20 28 76 2d | u[i];..|  if (v-|
|00001050| 3e 63 6f 65 66 5b 76 2d | 3e 6f 72 64 5d 20 3c 20 |>coef[v-|>ord] < |
|00001060| 30 2e 30 29 20 0d 20 20 | 20 20 7b 0d 20 20 20 20 |0.0) .  |  {.    |
|00001070| 66 6f 72 20 28 6b 20 3d | 20 75 2d 3e 6f 72 64 20 |for (k =| u->ord |
|00001080| 2d 20 76 2d 3e 6f 72 64 | 20 2d 20 31 3b 20 6b 20 |- v->ord| - 1; k |
|00001090| 3e 3d 20 30 3b 20 6b 20 | 2d 3d 20 32 29 0d 20 20 |>= 0; k |-= 2).  |
|000010a0| 20 20 20 20 72 2d 3e 63 | 6f 65 66 5b 6b 5d 20 3d |    r->c|oef[k] =|
|000010b0| 20 2d 72 2d 3e 63 6f 65 | 66 5b 6b 5d 3b 0d 20 20 | -r->coe|f[k];.  |
|000010c0| 20 20 66 6f 72 20 28 6b | 20 3d 20 75 2d 3e 6f 72 |  for (k| = u->or|
|000010d0| 64 20 2d 20 76 2d 3e 6f | 72 64 3b 20 6b 20 3e 3d |d - v->o|rd; k >=|
|000010e0| 20 30 3b 20 6b 2d 2d 29 | 0d 20 20 20 20 20 20 66 | 0; k--)|.      f|
|000010f0| 6f 72 20 28 6a 20 3d 20 | 76 2d 3e 6f 72 64 20 2b |or (j = |v->ord +|
|00001100| 20 6b 20 2d 20 31 3b 20 | 6a 20 3e 3d 20 6b 3b 20 | k - 1; |j >= k; |
|00001110| 6a 2d 2d 29 0d 20 20 20 | 20 20 20 72 2d 3e 63 6f |j--).   |   r->co|
|00001120| 65 66 5b 6a 5d 20 3d 20 | 2d 72 2d 3e 63 6f 65 66 |ef[j] = |-r->coef|
|00001130| 5b 6a 5d 20 2d 20 72 2d | 3e 63 6f 65 66 5b 76 2d |[j] - r-|>coef[v-|
|00001140| 3e 6f 72 64 20 2b 20 6b | 5d 20 2a 20 76 2d 3e 63 |>ord + k|] * v->c|
|00001150| 6f 65 66 5b 6a 20 2d 20 | 6b 5d 3b 0d 20 20 20 20 |oef[j - |k];.    |
|00001160| 7d 0d 20 20 65 6c 73 65 | 20 0d 20 20 20 20 7b 0d |}.  else| .    {.|
|00001170| 20 20 20 20 66 6f 72 20 | 28 6b 20 3d 20 75 2d 3e |    for |(k = u->|
|00001180| 6f 72 64 20 2d 20 76 2d | 3e 6f 72 64 3b 20 6b 20 |ord - v-|>ord; k |
|00001190| 3e 3d 20 30 3b 20 6b 2d | 2d 29 0d 20 20 20 20 20 |>= 0; k-|-).     |
|000011a0| 20 66 6f 72 20 28 6a 20 | 3d 20 76 2d 3e 6f 72 64 | for (j |= v->ord|
|000011b0| 20 2b 20 6b 20 2d 20 31 | 3b 20 6a 20 3e 3d 20 6b | + k - 1|; j >= k|
|000011c0| 3b 20 6a 2d 2d 29 0d 20 | 20 20 20 20 20 72 2d 3e |; j--). |     r->|
|000011d0| 63 6f 65 66 5b 6a 5d 20 | 2d 3d 20 72 2d 3e 63 6f |coef[j] |-= r->co|
|000011e0| 65 66 5b 76 2d 3e 6f 72 | 64 20 2b 20 6b 5d 20 2a |ef[v->or|d + k] *|
|000011f0| 20 76 2d 3e 63 6f 65 66 | 5b 6a 20 2d 20 6b 5d 3b | v->coef|[j - k];|
|00001200| 0d 20 20 20 20 7d 0d 0d | 20 20 6b 20 3d 20 76 2d |.    }..|  k = v-|
|00001210| 3e 6f 72 64 20 2d 20 31 | 3b 0d 20 20 77 68 69 6c |>ord - 1|;.  whil|
|00001220| 65 20 28 6b 20 3e 3d 20 | 30 20 26 26 20 66 61 62 |e (k >= |0 && fab|
|00001230| 73 28 72 2d 3e 63 6f 65 | 66 5b 6b 5d 29 20 3c 20 |s(r->coe|f[k]) < |
|00001240| 43 4f 45 46 46 5f 4c 49 | 4d 49 54 29 20 0d 20 20 |COEFF_LI|MIT) .  |
|00001250| 20 20 7b 0d 20 20 20 20 | 72 2d 3e 63 6f 65 66 5b |  {.    |r->coef[|
|00001260| 6b 5d 20 3d 20 30 2e 30 | 3b 0d 20 20 20 20 6b 2d |k] = 0.0|;.    k-|
|00001270| 2d 3b 0d 20 20 20 20 7d | 0d 20 20 72 2d 3e 6f 72 |-;.    }|.  r->or|
|00001280| 64 20 3d 20 28 6b 20 3c | 20 30 29 20 3f 20 30 20 |d = (k <| 0) ? 0 |
|00001290| 3a 20 6b 3b 0d 20 20 72 | 65 74 75 72 6e 28 72 2d |: k;.  r|eturn(r-|
|000012a0| 3e 6f 72 64 29 3b 0d 20 | 20 7d 0d 0d 2f 2a 20 42 |>ord);. | }../* B|
|000012b0| 75 69 6c 64 20 74 68 65 | 20 73 74 75 72 6d 69 61 |uild the| sturmia|
|000012c0| 6e 20 73 65 71 75 65 6e | 63 65 20 66 6f 72 20 61 |n sequen|ce for a|
|000012d0| 20 70 6f 6c 79 6e 6f 6d | 69 61 6c 20 2a 2f 0d 69 | polynom|ial */.i|
|000012e0| 6e 74 20 62 75 69 6c 64 | 73 74 75 72 6d 28 6f 72 |nt build|sturm(or|
|000012f0| 64 2c 20 73 73 65 71 29 | 0d 69 6e 74 20 20 20 20 |d, sseq)|.int    |
|00001300| 20 20 6f 72 64 3b 0d 70 | 6f 6c 79 6e 6f 6d 69 61 |  ord;.p|olynomia|
|00001310| 6c 20 20 20 2a 73 73 65 | 71 3b 0d 20 20 7b 0d 20 |l   *sse|q;.  {. |
|00001320| 20 69 6e 74 20 69 3b 0d | 20 20 44 42 4c 20 66 2c | int i;.|  DBL f,|
|00001330| 20 2a 66 70 2c 20 2a 66 | 63 3b 0d 20 20 70 6f 6c | *fp, *f|c;.  pol|
|00001340| 79 6e 6f 6d 69 61 6c 20 | 2a 73 70 3b 0d 0d 20 20 |ynomial |*sp;..  |
|00001350| 73 73 65 71 5b 30 5d 2e | 6f 72 64 20 3d 20 6f 72 |sseq[0].|ord = or|
|00001360| 64 3b 0d 20 20 73 73 65 | 71 5b 31 5d 2e 6f 72 64 |d;.  sse|q[1].ord|
|00001370| 20 3d 20 6f 72 64 20 2d | 20 31 3b 0d 0d 20 20 2f | = ord -| 1;..  /|
|00001380| 2a 20 63 61 6c 63 75 6c | 61 74 65 20 74 68 65 20 |* calcul|ate the |
|00001390| 64 65 72 69 76 61 74 69 | 76 65 20 61 6e 64 20 6e |derivati|ve and n|
|000013a0| 6f 72 6d 61 6c 69 7a 65 | 20 74 68 65 20 6c 65 61 |ormalize| the lea|
|000013b0| 64 69 6e 67 20 63 6f 65 | 66 66 69 63 69 65 6e 74 |ding coe|fficient|
|000013c0| 2e 20 2a 2f 0d 20 20 66 | 20 3d 20 66 61 62 73 28 |. */.  f| = fabs(|
|000013d0| 73 73 65 71 5b 30 5d 2e | 63 6f 65 66 5b 6f 72 64 |sseq[0].|coef[ord|
|000013e0| 5d 20 2a 20 6f 72 64 29 | 3b 0d 20 20 66 70 20 3d |] * ord)|;.  fp =|
|000013f0| 20 73 73 65 71 5b 31 5d | 2e 63 6f 65 66 3b 0d 20 | sseq[1]|.coef;. |
|00001400| 20 66 63 20 3d 20 73 73 | 65 71 5b 30 5d 2e 63 6f | fc = ss|eq[0].co|
|00001410| 65 66 20 2b 20 31 3b 0d | 20 20 66 6f 72 20 28 69 |ef + 1;.|  for (i|
|00001420| 20 3d 20 31 3b 20 69 20 | 3c 3d 20 6f 72 64 3b 20 | = 1; i |<= ord; |
|00001430| 69 2b 2b 29 0d 20 20 20 | 20 2a 66 70 2b 2b 20 3d |i++).   | *fp++ =|
|00001440| 20 2a 66 63 2b 2b 20 2a | 20 69 20 2f 20 66 3b 0d | *fc++ *| i / f;.|
|00001450| 0d 20 20 2f 2a 20 63 6f | 6e 73 74 72 75 63 74 20 |.  /* co|nstruct |
|00001460| 74 68 65 20 72 65 73 74 | 20 6f 66 20 74 68 65 20 |the rest| of the |
|00001470| 53 74 75 72 6d 20 73 65 | 71 75 65 6e 63 65 20 2a |Sturm se|quence *|
|00001480| 2f 0d 20 20 66 6f 72 20 | 28 73 70 20 3d 20 73 73 |/.  for |(sp = ss|
|00001490| 65 71 20 2b 20 32 3b 6d | 6f 64 70 28 73 70 20 2d |eq + 2;m|odp(sp -|
|000014a0| 20 32 2c 20 73 70 20 2d | 20 31 2c 20 73 70 29 3b | 2, sp -| 1, sp);|
|000014b0| 20 73 70 2b 2b 29 20 0d | 20 20 20 20 7b 0d 20 20 | sp++) .|    {.  |
|000014c0| 20 20 2f 2a 20 72 65 76 | 65 72 73 65 20 74 68 65 |  /* rev|erse the|
|000014d0| 20 73 69 67 6e 20 61 6e | 64 20 6e 6f 72 6d 61 6c | sign an|d normal|
|000014e0| 69 7a 65 20 2a 2f 0d 20 | 20 20 20 66 20 3d 20 2d |ize */. |   f = -|
|000014f0| 66 61 62 73 28 73 70 2d | 3e 63 6f 65 66 5b 73 70 |fabs(sp-|>coef[sp|
|00001500| 2d 3e 6f 72 64 5d 29 3b | 0d 20 20 20 20 66 6f 72 |->ord]);|.    for|
|00001510| 20 28 66 70 20 3d 20 26 | 73 70 2d 3e 63 6f 65 66 | (fp = &|sp->coef|
|00001520| 5b 73 70 2d 3e 6f 72 64 | 5d 3b 20 66 70 20 3e 3d |[sp->ord|]; fp >=|
|00001530| 20 73 70 2d 3e 63 6f 65 | 66 3b 20 66 70 2d 2d 29 | sp->coe|f; fp--)|
|00001540| 0d 20 20 20 20 20 20 2a | 66 70 20 2f 3d 20 66 3b |.      *|fp /= f;|
|00001550| 0d 20 20 20 20 7d 0d 20 | 20 73 70 2d 3e 63 6f 65 |.    }. | sp->coe|
|00001560| 66 5b 30 5d 20 3d 20 2d | 73 70 2d 3e 63 6f 65 66 |f[0] = -|sp->coef|
|00001570| 5b 30 5d 3b 20 20 20 2f | 2a 20 72 65 76 65 72 73 |[0];   /|* revers|
|00001580| 65 20 74 68 65 20 73 69 | 67 6e 20 2a 2f 0d 20 20 |e the si|gn */.  |
|00001590| 72 65 74 75 72 6e 28 73 | 70 20 2d 20 73 73 65 71 |return(s|p - sseq|
|000015a0| 29 3b 0d 20 20 7d 0d 0d | 2f 2a 20 46 69 6e 64 20 |);.  }..|/* Find |
|000015b0| 6f 75 74 20 68 6f 77 20 | 6d 61 6e 79 20 76 69 73 |out how |many vis|
|000015c0| 69 62 6c 65 20 69 6e 74 | 65 72 73 65 63 74 69 6f |ible int|ersectio|
|000015d0| 6e 73 20 74 68 65 72 65 | 20 61 72 65 20 2a 2f 0d |ns there| are */.|
|000015e0| 69 6e 74 20 76 69 73 69 | 62 6c 65 5f 72 6f 6f 74 |int visi|ble_root|
|000015f0| 73 28 6e 70 2c 20 73 73 | 65 71 2c 20 61 74 7a 65 |s(np, ss|eq, atze|
|00001600| 72 2c 20 61 74 70 6f 73 | 29 0d 69 6e 74 20 6e 70 |r, atpos|).int np|
|00001610| 3b 0d 70 6f 6c 79 6e 6f | 6d 69 61 6c 20 2a 73 73 |;.polyno|mial *ss|
|00001620| 65 71 3b 0d 69 6e 74 20 | 2a 61 74 7a 65 72 2c 20 |eq;.int |*atzer, |
|00001630| 2a 61 74 70 6f 73 3b 0d | 20 20 7b 0d 20 20 69 6e |*atpos;.|  {.  in|
|00001640| 74 20 61 74 70 6f 73 69 | 6e 66 2c 20 61 74 7a 65 |t atposi|nf, atze|
|00001650| 72 6f 3b 0d 20 20 70 6f | 6c 79 6e 6f 6d 69 61 6c |ro;.  po|lynomial|
|00001660| 20 2a 73 3b 0d 20 20 44 | 42 4c 20 66 2c 20 6c 66 | *s;.  D|BL f, lf|
|00001670| 3b 0d 0d 20 20 61 74 70 | 6f 73 69 6e 66 20 3d 20 |;..  atp|osinf = |
|00001680| 61 74 7a 65 72 6f 20 3d | 20 30 3b 0d 20 20 2f 2a |atzero =| 0;.  /*|
|00001690| 20 63 68 61 6e 67 65 73 | 20 61 74 20 70 6f 73 69 | changes| at posi|
|000016a0| 74 76 65 20 69 6e 66 69 | 6e 69 74 79 20 2a 2f 0d |tve infi|nity */.|
|000016b0| 20 20 6c 66 20 3d 20 73 | 73 65 71 5b 30 5d 2e 63 |  lf = s|seq[0].c|
|000016c0| 6f 65 66 5b 73 73 65 71 | 5b 30 5d 2e 6f 72 64 5d |oef[sseq|[0].ord]|
|000016d0| 3b 0d 20 20 66 6f 72 20 | 28 73 20 3d 20 73 73 65 |;.  for |(s = sse|
|000016e0| 71 20 2b 20 31 3b 20 73 | 20 3c 3d 20 73 73 65 71 |q + 1; s| <= sseq|
|000016f0| 20 2b 20 6e 70 3b 20 73 | 2b 2b 29 20 0d 20 20 20 | + np; s|++) .   |
|00001700| 20 7b 0d 20 20 20 20 66 | 20 3d 20 73 2d 3e 63 6f | {.    f| = s->co|
|00001710| 65 66 5b 73 2d 3e 6f 72 | 64 5d 3b 0d 20 20 20 20 |ef[s->or|d];.    |
|00001720| 69 66 20 28 6c 66 20 3d | 3d 20 30 2e 30 20 7c 7c |if (lf =|= 0.0 |||
|00001730| 20 6c 66 20 2a 20 66 20 | 3c 20 30 29 0d 20 20 20 | lf * f |< 0).   |
|00001740| 20 20 20 61 74 70 6f 73 | 69 6e 66 2b 2b 3b 0d 20 |   atpos|inf++;. |
|00001750| 20 20 20 6c 66 20 3d 20 | 66 3b 0d 20 20 20 20 7d |   lf = |f;.    }|
|00001760| 0d 0d 20 20 2f 2a 20 43 | 68 61 6e 67 65 73 20 61 |..  /* C|hanges a|
|00001770| 74 20 7a 65 72 6f 20 2a | 2f 0d 20 20 6c 66 20 3d |t zero *|/.  lf =|
|00001780| 20 73 73 65 71 5b 30 5d | 2e 63 6f 65 66 5b 30 5d | sseq[0]|.coef[0]|
|00001790| 3b 0d 20 20 66 6f 72 20 | 28 73 20 3d 20 73 73 65 |;.  for |(s = sse|
|000017a0| 71 2b 31 3b 20 73 20 3c | 3d 20 73 73 65 71 20 2b |q+1; s <|= sseq +|
|000017b0| 20 6e 70 3b 20 73 2b 2b | 29 20 0d 20 20 20 20 7b | np; s++|) .    {|
|000017c0| 0d 20 20 20 20 66 20 3d | 20 73 2d 3e 63 6f 65 66 |.    f =| s->coef|
|000017d0| 5b 30 5d 3b 0d 20 20 20 | 20 69 66 20 28 6c 66 20 |[0];.   | if (lf |
|000017e0| 3d 3d 20 30 2e 30 20 7c | 7c 20 6c 66 20 2a 20 66 |== 0.0 ||| lf * f|
|000017f0| 20 3c 20 30 29 0d 20 20 | 20 20 20 20 61 74 7a 65 | < 0).  |    atze|
|00001800| 72 6f 2b 2b 3b 0d 20 20 | 20 20 6c 66 20 3d 20 66 |ro++;.  |  lf = f|
|00001810| 3b 0d 20 20 20 20 7d 0d | 0d 20 20 2a 61 74 7a 65 |;.    }.|.  *atze|
|00001820| 72 20 3d 20 61 74 7a 65 | 72 6f 3b 0d 20 20 2a 61 |r = atze|ro;.  *a|
|00001830| 74 70 6f 73 20 3d 20 61 | 74 70 6f 73 69 6e 66 3b |tpos = a|tposinf;|
|00001840| 0d 20 20 72 65 74 75 72 | 6e 28 61 74 7a 65 72 6f |.  retur|n(atzero|
|00001850| 20 2d 20 61 74 70 6f 73 | 69 6e 66 29 3b 0d 20 20 | - atpos|inf);.  |
|00001860| 7d 0d 0d 2f 2a 0d 20 2a | 20 6e 75 6d 63 68 61 6e |}../*. *| numchan|
|00001870| 67 65 73 0d 20 2a 0d 20 | 2a 20 20 20 72 65 74 75 |ges. *. |*   retu|
|00001880| 72 6e 20 74 68 65 20 6e | 75 6d 62 65 72 20 6f 66 |rn the n|umber of|
|00001890| 20 73 69 67 6e 20 63 68 | 61 6e 67 65 73 20 69 6e | sign ch|anges in|
|000018a0| 20 74 68 65 20 53 74 75 | 72 6d 20 73 65 71 75 65 | the Stu|rm seque|
|000018b0| 6e 63 65 20 69 6e 0d 20 | 2a 20 73 73 65 71 20 61 |nce in. |* sseq a|
|000018c0| 74 20 74 68 65 20 76 61 | 6c 75 65 20 61 2e 0d 20 |t the va|lue a.. |
|000018d0| 2a 2f 0d 69 6e 74 20 6e | 75 6d 63 68 61 6e 67 65 |*/.int n|umchange|
|000018e0| 73 28 6e 70 2c 20 73 73 | 65 71 2c 20 61 29 0d 69 |s(np, ss|eq, a).i|
|000018f0| 6e 74 20 20 20 20 20 20 | 6e 70 3b 0d 70 6f 6c 79 |nt      |np;.poly|
|00001900| 6e 6f 6d 69 61 6c 20 20 | 20 2a 73 73 65 71 3b 0d |nomial  | *sseq;.|
|00001910| 44 42 4c 20 20 20 61 3b | 0d 0d 20 20 7b 0d 20 20 |DBL   a;|..  {.  |
|00001920| 69 6e 74 20 20 20 20 20 | 20 63 68 61 6e 67 65 73 |int     | changes|
|00001930| 3b 0d 20 20 44 42 4c 20 | 20 20 66 2c 20 6c 66 3b |;.  DBL |  f, lf;|
|00001940| 0d 20 20 70 6f 6c 79 6e | 6f 6d 69 61 6c 20 20 20 |.  polyn|omial   |
|00001950| 2a 73 3b 0d 20 20 63 68 | 61 6e 67 65 73 20 3d 20 |*s;.  ch|anges = |
|00001960| 30 3b 0d 20 20 43 4f 4f | 50 45 52 41 54 45 0d 20 |0;.  COO|PERATE. |
|00001970| 20 6c 66 20 3d 20 70 6f | 6c 79 65 76 61 6c 28 61 | lf = po|lyeval(a|
|00001980| 2c 20 73 73 65 71 5b 30 | 5d 2e 6f 72 64 2c 20 73 |, sseq[0|].ord, s|
|00001990| 73 65 71 5b 30 5d 2e 63 | 6f 65 66 29 3b 0d 20 20 |seq[0].c|oef);.  |
|000019a0| 66 6f 72 20 28 73 20 3d | 20 73 73 65 71 20 2b 20 |for (s =| sseq + |
|000019b0| 31 3b 20 73 20 3c 3d 20 | 73 73 65 71 20 2b 20 6e |1; s <= |sseq + n|
|000019c0| 70 3b 20 73 2b 2b 29 20 | 0d 20 20 20 20 7b 0d 20 |p; s++) |.    {. |
|000019d0| 20 20 20 66 20 3d 20 70 | 6f 6c 79 65 76 61 6c 28 |   f = p|olyeval(|
|000019e0| 61 2c 20 73 2d 3e 6f 72 | 64 2c 20 73 2d 3e 63 6f |a, s->or|d, s->co|
|000019f0| 65 66 29 3b 0d 20 20 20 | 20 69 66 20 28 6c 66 20 |ef);.   | if (lf |
|00001a00| 3d 3d 20 30 2e 30 20 7c | 7c 20 6c 66 20 2a 20 66 |== 0.0 ||| lf * f|
|00001a10| 20 3c 20 30 29 0d 20 20 | 20 20 20 20 63 68 61 6e | < 0).  |    chan|
|00001a20| 67 65 73 2b 2b 3b 0d 20 | 20 20 20 6c 66 20 3d 20 |ges++;. |   lf = |
|00001a30| 66 3b 0d 20 20 20 20 7d | 0d 20 20 72 65 74 75 72 |f;.    }|.  retur|
|00001a40| 6e 28 63 68 61 6e 67 65 | 73 29 3b 0d 20 20 7d 0d |n(change|s);.  }.|
|00001a50| 0d 2f 2a 0d 20 2a 20 73 | 62 69 73 65 63 74 0d 20 |./*. * s|bisect. |
|00001a60| 2a 0d 20 2a 20 20 20 75 | 73 65 73 20 61 20 62 69 |*. *   u|ses a bi|
|00001a70| 73 65 63 74 69 6f 6e 20 | 62 61 73 65 64 20 6f 6e |section |based on|
|00001a80| 20 74 68 65 20 73 74 75 | 72 6d 20 73 65 71 75 65 | the stu|rm seque|
|00001a90| 6e 63 65 20 66 6f 72 20 | 74 68 65 20 70 6f 6c 79 |nce for |the poly|
|00001aa0| 6e 6f 6d 69 61 6c 0d 20 | 2a 20 64 65 73 63 72 69 |nomial. |* descri|
|00001ab0| 62 65 64 20 69 6e 20 73 | 73 65 71 20 74 6f 20 69 |bed in s|seq to i|
|00001ac0| 73 6f 6c 61 74 65 20 69 | 6e 74 65 72 76 61 6c 73 |solate i|ntervals|
|00001ad0| 20 69 6e 20 77 68 69 63 | 68 20 72 6f 6f 74 73 20 | in whic|h roots |
|00001ae0| 6f 63 63 75 72 2c 0d 20 | 2a 20 74 68 65 20 72 6f |occur,. |* the ro|
|00001af0| 6f 74 73 20 61 72 65 20 | 72 65 74 75 72 6e 65 64 |ots are |returned|
|00001b00| 20 69 6e 20 74 68 65 20 | 72 6f 6f 74 73 20 61 72 | in the |roots ar|
|00001b10| 72 61 79 20 69 6e 20 6f | 72 64 65 72 20 6f 66 20 |ray in o|rder of |
|00001b20| 6d 61 67 6e 69 74 75 64 | 65 2e 0d 0d 4e 6f 74 65 |magnitud|e...Note|
|00001b30| 3a 20 54 68 69 73 20 72 | 6f 75 74 69 6e 65 20 68 |: This r|outine h|
|00001b40| 61 73 20 6f 6e 65 20 73 | 65 76 65 72 65 20 62 75 |as one s|evere bu|
|00001b50| 67 3a 20 57 68 65 6e 20 | 74 68 65 20 69 6e 74 65 |g: When |the inte|
|00001b60| 72 76 61 6c 20 63 6f 6e | 74 61 69 6e 69 6e 67 20 |rval con|taining |
|00001b70| 74 68 65 0d 20 20 20 20 | 20 20 72 6f 6f 74 20 5b |the.    |  root [|
|00001b80| 6d 69 6e 2c 20 6d 61 78 | 5d 20 68 61 73 20 61 20 |min, max|] has a |
|00001b90| 72 6f 6f 74 20 61 74 20 | 6f 6e 65 20 6f 66 20 69 |root at |one of i|
|00001ba0| 74 73 20 65 6e 64 70 6f | 69 6e 74 73 2c 20 61 73 |ts endpo|ints, as|
|00001bb0| 20 77 65 6c 6c 20 61 73 | 20 6f 6e 65 0d 20 20 20 | well as| one.   |
|00001bc0| 20 20 20 77 69 74 68 69 | 6e 20 74 68 65 20 69 6e |   withi|n the in|
|00001bd0| 74 65 72 76 61 6c 2c 20 | 74 68 65 20 72 6f 6f 74 |terval, |the root|
|00001be0| 20 61 74 20 74 68 65 20 | 65 6e 64 70 6f 69 6e 74 | at the |endpoint|
|00001bf0| 20 77 69 6c 6c 20 62 65 | 20 72 65 74 75 72 6e 65 | will be| returne|
|00001c00| 64 20 72 61 74 68 65 72 | 0d 20 20 20 20 20 20 74 |d rather|.      t|
|00001c10| 68 61 6e 20 74 68 65 20 | 6f 6e 65 20 69 6e 73 69 |han the |one insi|
|00001c20| 64 65 2e 0d 0d 20 2a 2f | 0d 73 74 61 74 69 63 20 |de... */|.static |
|00001c30| 69 6e 74 0d 73 62 69 73 | 65 63 74 28 6e 70 2c 20 |int.sbis|ect(np, |
|00001c40| 73 73 65 71 2c 20 6d 69 | 6e 5f 76 61 6c 75 65 2c |sseq, mi|n_value,|
|00001c50| 20 6d 61 78 5f 76 61 6c | 75 65 2c 20 61 74 6d 69 | max_val|ue, atmi|
|00001c60| 6e 2c 20 61 74 6d 61 78 | 2c 20 72 6f 6f 74 73 29 |n, atmax|, roots)|
|00001c70| 0d 69 6e 74 20 6e 70 3b | 0d 70 6f 6c 79 6e 6f 6d |.int np;|.polynom|
|00001c80| 69 61 6c 20 2a 73 73 65 | 71 3b 0d 44 42 4c 20 6d |ial *sse|q;.DBL m|
|00001c90| 69 6e 5f 76 61 6c 75 65 | 2c 20 6d 61 78 5f 76 61 |in_value|, max_va|
|00001ca0| 6c 75 65 3b 0d 69 6e 74 | 20 61 74 6d 69 6e 2c 20 |lue;.int| atmin, |
|00001cb0| 61 74 6d 61 78 3b 0d 44 | 42 4c 20 2a 72 6f 6f 74 |atmax;.D|BL *root|
|00001cc0| 73 3b 0d 20 20 7b 0d 20 | 20 44 42 4c 20 20 6d 69 |s;.  {. | DBL  mi|
|00001cd0| 64 3b 0d 20 20 69 6e 74 | 20 20 6e 31 2c 20 6e 32 |d;.  int|  n1, n2|
|00001ce0| 2c 20 69 74 73 2c 20 61 | 74 6d 69 64 3b 0d 0d 20 |, its, a|tmid;.. |
|00001cf0| 20 69 66 20 28 28 61 74 | 6d 69 6e 20 2d 20 61 74 | if ((at|min - at|
|00001d00| 6d 61 78 29 20 3d 3d 20 | 31 29 20 0d 20 20 20 20 |max) == |1) .    |
|00001d10| 7b 0d 20 20 20 20 2f 2a | 20 66 69 72 73 74 20 74 |{.    /*| first t|
|00001d20| 72 79 20 75 73 69 6e 67 | 20 72 65 67 75 6c 61 2d |ry using| regula-|
|00001d30| 66 61 6c 73 61 20 74 6f | 20 66 69 6e 64 20 74 68 |falsa to| find th|
|00001d40| 65 20 72 6f 6f 74 2e 20 | 20 2a 2f 0d 20 20 20 20 |e root. | */.    |
|00001d50| 69 66 20 28 72 65 67 75 | 6c 61 5f 66 61 6c 73 61 |if (regu|la_falsa|
|00001d60| 28 73 73 65 71 2d 3e 6f | 72 64 2c 20 73 73 65 71 |(sseq->o|rd, sseq|
|00001d70| 2d 3e 63 6f 65 66 2c 20 | 6d 69 6e 5f 76 61 6c 75 |->coef, |min_valu|
|00001d80| 65 2c 20 6d 61 78 5f 76 | 61 6c 75 65 2c 20 72 6f |e, max_v|alue, ro|
|00001d90| 6f 74 73 29 29 0d 20 20 | 20 20 20 20 72 65 74 75 |ots)).  |    retu|
|00001da0| 72 6e 20 31 3b 0d 20 20 | 20 20 65 6c 73 65 20 0d |rn 1;.  |  else .|
|00001db0| 20 20 20 20 20 20 7b 0d | 20 20 20 20 20 20 2f 2a |      {.|      /*|
|00001dc0| 20 54 68 61 74 20 66 61 | 69 6c 65 64 2c 20 73 6f | That fa|iled, so|
|00001dd0| 20 6e 6f 77 20 66 69 6e | 64 20 69 74 20 62 79 20 | now fin|d it by |
|00001de0| 62 69 73 65 63 74 69 6f | 6e 20 2a 2f 0d 20 20 20 |bisectio|n */.   |
|00001df0| 20 20 20 20 20 66 6f 72 | 20 28 69 74 73 20 3d 20 |     for| (its = |
|00001e00| 30 3b 20 69 74 73 20 3c | 20 4d 41 58 5f 49 54 45 |0; its <| MAX_ITE|
|00001e10| 52 41 54 49 4f 4e 53 3b | 20 69 74 73 2b 2b 29 20 |RATIONS;| its++) |
|00001e20| 0d 20 20 20 20 20 20 20 | 20 7b 0d 20 20 20 20 20 |.       | {.     |
|00001e30| 20 20 20 6d 69 64 20 3d | 20 28 6d 69 6e 5f 76 61 |   mid =| (min_va|
|00001e40| 6c 75 65 20 2b 20 6d 61 | 78 5f 76 61 6c 75 65 29 |lue + ma|x_value)|
|00001e50| 20 2f 20 32 3b 0d 20 20 | 20 20 20 20 20 20 61 74 | / 2;.  |      at|
|00001e60| 6d 69 64 20 3d 20 6e 75 | 6d 63 68 61 6e 67 65 73 |mid = nu|mchanges|
|00001e70| 28 6e 70 2c 20 73 73 65 | 71 2c 20 6d 69 64 29 3b |(np, sse|q, mid);|
|00001e80| 0d 20 20 20 20 20 20 20 | 20 69 66 20 28 66 61 62 |.       | if (fab|
|00001e90| 73 28 6d 69 64 29 20 3e | 20 45 50 53 49 4c 4f 4e |s(mid) >| EPSILON|
|00001ea0| 29 20 0d 20 20 20 20 20 | 20 20 20 20 20 7b 0d 20 |) .     |     {. |
|00001eb0| 20 20 20 20 20 20 20 20 | 20 69 66 20 28 66 61 62 |        | if (fab|
|00001ec0| 73 28 28 6d 61 78 5f 76 | 61 6c 75 65 20 2d 20 6d |s((max_v|alue - m|
|00001ed0| 69 6e 5f 76 61 6c 75 65 | 29 20 2f 20 6d 69 64 29 |in_value|) / mid)|
|00001ee0| 20 3c 20 45 50 53 49 4c | 4f 4e 29 20 0d 20 20 20 | < EPSIL|ON) .   |
|00001ef0| 20 20 20 20 20 20 20 20 | 20 7b 0d 20 20 20 20 20 |        | {.     |
|00001f00| 20 20 20 20 20 20 20 72 | 6f 6f 74 73 5b 30 5d 20 |       r|oots[0] |
|00001f10| 3d 20 6d 69 64 3b 0d 20 | 20 20 20 20 20 20 20 20 |= mid;. |        |
|00001f20| 20 20 20 72 65 74 75 72 | 6e 20 31 3b 0d 20 20 20 |   retur|n 1;.   |
|00001f30| 20 20 20 20 20 20 20 20 | 20 7d 0d 20 20 20 20 20 |        | }.     |
|00001f40| 20 20 20 20 20 7d 0d 20 | 20 20 20 20 20 20 20 65 |     }. |       e|
|00001f50| 6c 73 65 20 69 66 20 28 | 66 61 62 73 28 6d 61 78 |lse if (|fabs(max|
|00001f60| 5f 76 61 6c 75 65 20 2d | 20 6d 69 6e 5f 76 61 6c |_value -| min_val|
|00001f70| 75 65 29 20 3c 20 45 50 | 53 49 4c 4f 4e 29 20 0d |ue) < EP|SILON) .|
|00001f80| 20 20 20 20 20 20 20 20 | 20 20 7b 0d 20 20 20 20 |        |  {.    |
|00001f90| 20 20 20 20 20 20 72 6f | 6f 74 73 5b 30 5d 20 3d |      ro|ots[0] =|
|00001fa0| 20 6d 69 64 3b 0d 20 20 | 20 20 20 20 20 20 20 20 | mid;.  |        |
|00001fb0| 72 65 74 75 72 6e 20 31 | 3b 0d 20 20 20 20 20 20 |return 1|;.      |
|00001fc0| 20 20 20 20 7d 0d 20 20 | 20 20 20 20 20 20 65 6c |    }.  |      el|
|00001fd0| 73 65 20 69 66 20 28 28 | 61 74 6d 69 6e 20 2d 20 |se if ((|atmin - |
|00001fe0| 61 74 6d 69 64 29 20 3d | 3d 20 30 29 0d 20 20 20 |atmid) =|= 0).   |
|00001ff0| 20 20 20 20 20 20 20 6d | 69 6e 5f 76 61 6c 75 65 |       m|in_value|
|00002000| 20 3d 20 6d 69 64 3b 0d | 20 20 20 20 20 20 20 20 | = mid;.|        |
|00002010| 65 6c 73 65 0d 20 20 20 | 20 20 20 20 20 20 20 6d |else.   |       m|
|00002020| 61 78 5f 76 61 6c 75 65 | 20 3d 20 6d 69 64 3b 0d |ax_value| = mid;.|
|00002030| 20 20 20 20 20 20 20 20 | 7d 0d 20 20 20 20 20 20 |        |}.      |
|00002040| 2f 2a 20 42 69 73 65 63 | 74 69 6f 6e 20 74 6f 6f |/* Bisec|tion too|
|00002050| 6b 20 74 6f 6f 20 6c 6f | 6e 67 20 2d 20 6a 75 73 |k too lo|ng - jus|
|00002060| 74 20 72 65 74 75 72 6e | 20 77 68 61 74 20 77 65 |t return| what we|
|00002070| 20 67 6f 74 20 2a 2f 0d | 20 20 20 20 20 20 72 6f | got */.|      ro|
|00002080| 6f 74 73 5b 30 5d 20 3d | 20 6d 69 64 3b 0d 20 20 |ots[0] =| mid;.  |
|00002090| 20 20 20 20 72 65 74 75 | 72 6e 20 31 3b 0d 20 20 |    retu|rn 1;.  |
|000020a0| 20 20 20 20 7d 0d 20 20 | 20 20 7d 0d 0d 20 20 2f |    }.  |  }..  /|
|000020b0| 2a 20 54 68 65 72 65 20 | 69 73 20 6d 6f 72 65 20 |* There |is more |
|000020c0| 74 68 61 6e 20 6f 6e 65 | 20 72 6f 6f 74 20 69 6e |than one| root in|
|000020d0| 20 74 68 65 20 69 6e 74 | 65 72 76 61 6c 2e 0d 20 | the int|erval.. |
|000020e0| 20 20 20 20 20 42 69 73 | 65 63 74 20 74 6f 20 66 |     Bis|ect to f|
|000020f0| 69 6e 64 20 6e 65 77 20 | 69 6e 74 65 72 76 61 6c |ind new |interval|
|00002100| 73 20 2a 2f 0d 20 20 66 | 6f 72 20 28 69 74 73 20 |s */.  f|or (its |
|00002110| 3d 20 30 3b 20 69 74 73 | 20 3c 20 4d 41 58 5f 49 |= 0; its| < MAX_I|
|00002120| 54 45 52 41 54 49 4f 4e | 53 3b 20 69 74 73 2b 2b |TERATION|S; its++|
|00002130| 29 20 0d 20 20 20 20 7b | 0d 20 20 20 20 6d 69 64 |) .    {|.    mid|
|00002140| 20 3d 20 28 6d 69 6e 5f | 76 61 6c 75 65 20 2b 20 | = (min_|value + |
|00002150| 6d 61 78 5f 76 61 6c 75 | 65 29 20 2f 20 32 3b 0d |max_valu|e) / 2;.|
|00002160| 20 20 20 20 61 74 6d 69 | 64 20 3d 20 6e 75 6d 63 |    atmi|d = numc|
|00002170| 68 61 6e 67 65 73 28 6e | 70 2c 20 73 73 65 71 2c |hanges(n|p, sseq,|
|00002180| 20 6d 69 64 29 3b 0d 20 | 20 20 20 6e 31 20 3d 20 | mid);. |   n1 = |
|00002190| 61 74 6d 69 6e 20 2d 20 | 61 74 6d 69 64 3b 0d 20 |atmin - |atmid;. |
|000021a0| 20 20 20 6e 32 20 3d 20 | 61 74 6d 69 64 20 2d 20 |   n2 = |atmid - |
|000021b0| 61 74 6d 61 78 3b 0d 20 | 20 20 20 69 66 20 28 6e |atmax;. |   if (n|
|000021c0| 31 20 21 3d 20 30 20 26 | 26 20 6e 32 20 21 3d 20 |1 != 0 &|& n2 != |
|000021d0| 30 29 20 0d 20 20 20 20 | 20 20 7b 0d 20 20 20 20 |0) .    |  {.    |
|000021e0| 20 20 6e 31 20 3d 20 73 | 62 69 73 65 63 74 28 6e |  n1 = s|bisect(n|
|000021f0| 70 2c 20 73 73 65 71 2c | 20 6d 69 6e 5f 76 61 6c |p, sseq,| min_val|
|00002200| 75 65 2c 20 6d 69 64 2c | 20 61 74 6d 69 6e 2c 20 |ue, mid,| atmin, |
|00002210| 61 74 6d 69 64 2c 20 72 | 6f 6f 74 73 29 3b 0d 20 |atmid, r|oots);. |
|00002220| 20 20 20 20 20 6e 32 20 | 3d 20 73 62 69 73 65 63 |     n2 |= sbisec|
|00002230| 74 28 6e 70 2c 20 73 73 | 65 71 2c 20 6d 69 64 2c |t(np, ss|eq, mid,|
|00002240| 20 6d 61 78 5f 76 61 6c | 75 65 2c 20 61 74 6d 69 | max_val|ue, atmi|
|00002250| 64 2c 20 61 74 6d 61 78 | 2c 20 26 72 6f 6f 74 73 |d, atmax|, &roots|
|00002260| 5b 6e 31 5d 29 3b 0d 20 | 20 20 20 20 20 72 65 74 |[n1]);. |     ret|
|00002270| 75 72 6e 20 6e 31 20 2b | 20 6e 32 3b 0d 20 20 20 |urn n1 +| n2;.   |
|00002280| 20 20 20 7d 0d 20 20 20 | 20 69 66 20 28 6e 31 20 |   }.   | if (n1 |
|00002290| 3d 3d 20 30 29 0d 20 20 | 20 20 20 20 6d 69 6e 5f |== 0).  |    min_|
|000022a0| 76 61 6c 75 65 20 3d 20 | 6d 69 64 3b 0d 20 20 20 |value = |mid;.   |
|000022b0| 20 65 6c 73 65 0d 20 20 | 20 20 20 20 6d 61 78 5f | else.  |    max_|
|000022c0| 76 61 6c 75 65 20 3d 20 | 6d 69 64 3b 0d 20 20 20 |value = |mid;.   |
|000022d0| 20 7d 0d 0d 20 20 2f 2a | 20 54 6f 6f 6b 20 74 6f | }..  /*| Took to|
|000022e0| 6f 20 6c 6f 6e 67 20 74 | 6f 20 62 69 73 65 63 74 |o long t|o bisect|
|000022f0| 2e 20 20 4a 75 73 74 20 | 72 65 74 75 72 6e 20 77 |.  Just |return w|
|00002300| 68 61 74 20 77 65 20 67 | 6f 74 2e 20 2a 2f 0d 20 |hat we g|ot. */. |
|00002310| 20 72 6f 6f 74 73 5b 30 | 5d 20 3d 20 6d 69 64 3b | roots[0|] = mid;|
|00002320| 0d 20 20 72 65 74 75 72 | 6e 20 31 3b 0d 20 20 7d |.  retur|n 1;.  }|
|00002330| 0d 0d 44 42 4c 20 70 6f | 6c 79 65 76 61 6c 28 78 |..DBL po|lyeval(x|
|00002340| 2c 20 6e 2c 20 43 6f 65 | 66 66 73 29 0d 44 42 4c |, n, Coe|ffs).DBL|
|00002350| 20 78 2c 20 2a 43 6f 65 | 66 66 73 3b 0d 69 6e 74 | x, *Coe|ffs;.int|
|00002360| 20 6e 3b 0d 20 20 7b 0d | 20 20 72 65 67 69 73 74 | n;.  {.|  regist|
|00002370| 65 72 20 69 6e 74 20 69 | 3b 0d 20 20 44 42 4c 20 |er int i|;.  DBL |
|00002380| 76 61 6c 3b 0d 20 20 76 | 61 6c 20 3d 20 43 6f 65 |val;.  v|al = Coe|
|00002390| 66 66 73 5b 6e 5d 3b 0d | 20 20 66 6f 72 20 28 69 |ffs[n];.|  for (i|
|000023a0| 3d 6e 2d 31 3b 69 3e 3d | 30 3b 69 2d 2d 29 20 76 |=n-1;i>=|0;i--) v|
|000023b0| 61 6c 20 3d 20 76 61 6c | 20 2a 20 78 20 2b 20 43 |al = val| * x + C|
|000023c0| 6f 65 66 66 73 5b 69 5d | 3b 0d 20 20 72 65 74 75 |oeffs[i]|;.  retu|
|000023d0| 72 6e 20 76 61 6c 3b 0d | 20 20 7d 0d 0d 2f 2a 20 |rn val;.|  }../* |
|000023e0| 43 6c 6f 73 65 20 69 6e | 20 6f 6e 20 61 20 72 6f |Close in| on a ro|
|000023f0| 6f 74 20 62 79 20 75 73 | 69 6e 67 20 72 65 67 75 |ot by us|ing regu|
|00002400| 6c 61 2d 66 61 6c 73 61 | 20 2a 2f 0d 69 6e 74 20 |la-falsa| */.int |
|00002410| 72 65 67 75 6c 61 5f 66 | 61 6c 73 61 28 6f 72 64 |regula_f|alsa(ord|
|00002420| 65 72 2c 20 63 6f 65 66 | 2c 20 61 2c 20 62 2c 20 |er, coef|, a, b, |
|00002430| 76 61 6c 29 0d 69 6e 74 | 20 6f 72 64 65 72 3b 0d |val).int| order;.|
|00002440| 44 42 4c 20 2a 63 6f 65 | 66 3b 0d 44 42 4c 20 61 |DBL *coe|f;.DBL a|
|00002450| 2c 20 62 2c 20 2a 76 61 | 6c 3b 0d 20 20 7b 0d 20 |, b, *va|l;.  {. |
|00002460| 20 69 6e 74 20 69 74 73 | 3b 0d 20 20 44 42 4c 20 | int its|;.  DBL |
|00002470| 66 61 2c 20 66 62 2c 20 | 78 2c 20 66 78 2c 20 6c |fa, fb, |x, fx, l|
|00002480| 66 78 3b 0d 0d 20 20 66 | 61 20 3d 20 70 6f 6c 79 |fx;..  f|a = poly|
|00002490| 65 76 61 6c 28 61 2c 20 | 6f 72 64 65 72 2c 20 63 |eval(a, |order, c|
|000024a0| 6f 65 66 29 3b 0d 20 20 | 66 62 20 3d 20 70 6f 6c |oef);.  |fb = pol|
|000024b0| 79 65 76 61 6c 28 62 2c | 20 6f 72 64 65 72 2c 20 |yeval(b,| order, |
|000024c0| 63 6f 65 66 29 3b 0d 0d | 20 20 69 66 20 28 66 61 |coef);..|  if (fa|
|000024d0| 20 2a 20 66 62 20 3e 20 | 30 2e 30 29 0d 20 20 20 | * fb > |0.0).   |
|000024e0| 20 72 65 74 75 72 6e 20 | 30 3b 0d 0d 20 20 69 66 | return |0;..  if|
|000024f0| 20 28 66 61 62 73 28 66 | 61 29 20 3c 20 43 4f 45 | (fabs(f|a) < COE|
|00002500| 46 46 5f 4c 49 4d 49 54 | 29 20 0d 20 20 20 20 7b |FF_LIMIT|) .    {|
|00002510| 0d 20 20 20 20 2a 76 61 | 6c 20 3d 20 61 3b 0d 20 |.    *va|l = a;. |
|00002520| 20 20 20 72 65 74 75 72 | 6e 20 31 3b 0d 20 20 20 |   retur|n 1;.   |
|00002530| 20 7d 0d 0d 20 20 69 66 | 20 28 66 61 62 73 28 66 | }..  if| (fabs(f|
|00002540| 62 29 20 3c 20 43 4f 45 | 46 46 5f 4c 49 4d 49 54 |b) < COE|FF_LIMIT|
|00002550| 29 20 0d 20 20 20 20 7b | 0d 20 20 20 20 2a 76 61 |) .    {|.    *va|
|00002560| 6c 20 3d 20 62 3b 0d 20 | 20 20 20 72 65 74 75 72 |l = b;. |   retur|
|00002570| 6e 20 31 3b 0d 20 20 20 | 20 7d 0d 0d 20 20 6c 66 |n 1;.   | }..  lf|
|00002580| 78 20 3d 20 66 61 3b 0d | 0d 20 20 43 4f 4f 50 45 |x = fa;.|.  COOPE|
|00002590| 52 41 54 45 0d 20 20 66 | 6f 72 20 28 69 74 73 20 |RATE.  f|or (its |
|000025a0| 3d 20 30 3b 20 69 74 73 | 20 3c 20 4d 41 58 5f 49 |= 0; its| < MAX_I|
|000025b0| 54 45 52 41 54 49 4f 4e | 53 3b 20 69 74 73 2b 2b |TERATION|S; its++|
|000025c0| 29 20 0d 20 20 20 20 7b | 0d 20 20 20 20 78 20 3d |) .    {|.    x =|
|000025d0| 20 28 66 62 20 2a 20 61 | 20 2d 20 66 61 20 2a 20 | (fb * a| - fa * |
|000025e0| 62 29 20 2f 20 28 66 62 | 20 2d 20 66 61 29 3b 0d |b) / (fb| - fa);.|
|000025f0| 20 20 20 20 66 78 20 3d | 20 70 6f 6c 79 65 76 61 |    fx =| polyeva|
|00002600| 6c 28 78 2c 20 6f 72 64 | 65 72 2c 20 63 6f 65 66 |l(x, ord|er, coef|
|00002610| 29 3b 0d 0d 20 20 20 20 | 69 66 20 28 66 61 62 73 |);..    |if (fabs|
|00002620| 28 78 29 20 3e 20 45 50 | 53 49 4c 4f 4e 29 20 0d |(x) > EP|SILON) .|
|00002630| 20 20 20 20 20 20 7b 0d | 20 20 20 20 20 20 69 66 |      {.|      if|
|00002640| 20 28 66 61 62 73 28 66 | 78 20 2f 20 78 29 20 3c | (fabs(f|x / x) <|
|00002650| 20 45 50 53 49 4c 4f 4e | 29 20 0d 20 20 20 20 20 | EPSILON|) .     |
|00002660| 20 20 20 7b 0d 20 20 20 | 20 20 20 20 20 2a 76 61 |   {.   |     *va|
|00002670| 6c 20 3d 20 78 3b 0d 20 | 20 20 20 20 20 20 20 72 |l = x;. |       r|
|00002680| 65 74 75 72 6e 20 31 3b | 0d 20 20 20 20 20 20 20 |eturn 1;|.       |
|00002690| 20 7d 0d 20 20 20 20 20 | 20 7d 0d 20 20 20 20 65 | }.     | }.    e|
|000026a0| 6c 73 65 20 69 66 20 28 | 66 61 62 73 28 66 78 29 |lse if (|fabs(fx)|
|000026b0| 20 3c 20 45 50 53 49 4c | 4f 4e 29 20 0d 20 20 20 | < EPSIL|ON) .   |
|000026c0| 20 20 20 7b 0d 20 20 20 | 20 20 20 2a 76 61 6c 20 |   {.   |   *val |
|000026d0| 3d 20 78 3b 0d 20 20 20 | 20 20 20 72 65 74 75 72 |= x;.   |   retur|
|000026e0| 6e 20 31 3b 0d 20 20 20 | 20 20 20 7d 0d 0d 20 20 |n 1;.   |   }..  |
|000026f0| 20 20 69 66 20 28 66 61 | 20 3c 20 30 29 0d 20 20 |  if (fa| < 0).  |
|00002700| 20 20 20 20 69 66 20 28 | 66 78 20 3c 20 30 29 20 |    if (|fx < 0) |
|00002710| 0d 20 20 20 20 20 20 7b | 0d 20 20 20 20 20 20 61 |.      {|.      a|
|00002720| 20 3d 20 78 3b 0d 20 20 | 20 20 20 20 66 61 20 3d | = x;.  |    fa =|
|00002730| 20 66 78 3b 0d 20 20 20 | 20 20 20 69 66 20 28 28 | fx;.   |   if ((|
|00002740| 6c 66 78 20 2a 20 66 78 | 29 20 3e 20 30 29 0d 20 |lfx * fx|) > 0). |
|00002750| 20 20 20 20 20 20 20 66 | 62 20 2f 3d 20 32 3b 0d |       f|b /= 2;.|
|00002760| 20 20 20 20 20 20 7d 0d | 20 20 20 20 20 20 65 6c |      }.|      el|
|00002770| 73 65 20 0d 20 20 20 20 | 20 20 7b 0d 20 20 20 20 |se .    |  {.    |
|00002780| 20 20 62 20 3d 20 78 3b | 0d 20 20 20 20 20 20 66 |  b = x;|.      f|
|00002790| 62 20 3d 20 66 78 3b 0d | 20 20 20 20 20 20 69 66 |b = fx;.|      if|
|000027a0| 20 28 28 6c 66 78 20 2a | 20 66 78 29 20 3e 20 30 | ((lfx *| fx) > 0|
|000027b0| 29 0d 20 20 20 20 20 20 | 20 20 66 61 20 2f 3d 20 |).      |  fa /= |
|000027c0| 32 3b 0d 20 20 20 20 20 | 20 7d 0d 20 20 20 20 65 |2;.     | }.    e|
|000027d0| 6c 73 65 20 69 66 20 28 | 66 78 20 3c 20 30 29 20 |lse if (|fx < 0) |
|000027e0| 0d 20 20 20 20 20 20 7b | 0d 20 20 20 20 20 20 62 |.      {|.      b|
|000027f0| 20 3d 20 78 3b 0d 20 20 | 20 20 20 20 66 62 20 3d | = x;.  |    fb =|
|00002800| 20 66 78 3b 0d 20 20 20 | 20 20 20 69 66 20 28 28 | fx;.   |   if ((|
|00002810| 6c 66 78 20 2a 20 66 78 | 29 20 3e 20 30 29 0d 20 |lfx * fx|) > 0). |
|00002820| 20 20 20 20 20 20 20 66 | 61 20 2f 3d 20 32 3b 0d |       f|a /= 2;.|
|00002830| 20 20 20 20 20 20 7d 0d | 20 20 20 20 65 6c 73 65 |      }.|    else|
|00002840| 20 0d 20 20 20 20 20 20 | 7b 0d 20 20 20 20 20 20 | .      |{.      |
|00002850| 61 20 3d 20 78 3b 0d 20 | 20 20 20 20 20 66 61 20 |a = x;. |     fa |
|00002860| 3d 20 66 78 3b 0d 20 20 | 20 20 20 20 69 66 20 28 |= fx;.  |    if (|
|00002870| 28 6c 66 78 20 2a 20 66 | 78 29 20 3e 20 30 29 0d |(lfx * f|x) > 0).|
|00002880| 20 20 20 20 20 20 20 20 | 66 62 20 2f 3d 20 32 3b |        |fb /= 2;|
|00002890| 0d 20 20 20 20 20 20 7d | 0d 20 20 20 20 69 66 20 |.      }|.    if |
|000028a0| 28 66 61 62 73 28 62 2d | 61 29 20 3c 20 45 50 53 |(fabs(b-|a) < EPS|
|000028b0| 49 4c 4f 4e 29 20 0d 20 | 20 20 20 20 20 7b 0d 20 |ILON) . |     {. |
|000028c0| 20 20 20 20 20 2f 2a 20 | 43 68 65 63 6b 20 66 6f |     /* |Check fo|
|000028d0| 72 20 75 6e 64 65 72 66 | 6c 6f 77 20 69 6e 20 74 |r underf|low in t|
|000028e0| 68 65 20 64 6f 6d 61 69 | 6e 20 2a 2f 0d 20 20 20 |he domai|n */.   |
|000028f0| 20 20 20 2a 76 61 6c 20 | 3d 20 78 3b 0d 20 20 20 |   *val |= x;.   |
|00002900| 20 20 20 72 65 74 75 72 | 6e 20 31 3b 0d 20 20 20 |   retur|n 1;.   |
|00002910| 20 20 20 7d 0d 20 20 20 | 20 6c 66 78 20 3d 20 66 |   }.   | lfx = f|
|00002920| 78 3b 0d 20 20 20 20 7d | 0d 20 20 72 65 74 75 72 |x;.    }|.  retur|
|00002930| 6e 20 30 3b 0d 20 20 7d | 0d 0d 2f 2a 0d 20 20 20 |n 0;.  }|../*.   |
|00002940| 53 6f 6c 76 65 20 74 68 | 65 20 71 75 61 64 72 61 |Solve th|e quadra|
|00002950| 74 69 63 20 65 71 75 61 | 74 69 6f 6e 3a 0d 20 20 |tic equa|tion:.  |
|00002960| 20 20 20 20 09 09 78 5b | 30 5d 20 2a 20 78 5e 32 |    ..x[|0] * x^2|
|00002970| 20 2b 20 78 5b 31 5d 20 | 2a 20 78 20 2b 20 78 5b | + x[1] |* x + x[|
|00002980| 32 5d 20 3d 20 30 2e 0d | 0d 20 20 20 54 68 65 20 |2] = 0..|.   The |
|00002990| 76 61 6c 75 65 20 72 65 | 74 75 72 6e 65 64 20 62 |value re|turned b|
|000029a0| 79 20 74 68 69 73 20 66 | 75 6e 63 74 69 6f 6e 20 |y this f|unction |
|000029b0| 69 73 20 74 68 65 20 6e | 75 6d 62 65 72 20 6f 66 |is the n|umber of|
|000029c0| 20 72 65 61 6c 20 72 6f | 6f 74 73 2e 0d 20 20 20 | real ro|ots..   |
|000029d0| 54 68 65 20 72 6f 6f 74 | 73 20 74 68 65 6d 73 65 |The root|s themse|
|000029e0| 6c 76 65 73 20 61 72 65 | 20 72 65 74 75 72 6e 65 |lves are| returne|
|000029f0| 64 20 69 6e 20 79 5b 30 | 5d 2c 20 79 5b 31 5d 2e |d in y[0|], y[1].|
|00002a00| 0d 2a 2f 0d 69 6e 74 20 | 73 6f 6c 76 65 5f 71 75 |.*/.int |solve_qu|
|00002a10| 61 64 72 61 74 69 63 28 | 78 2c 20 79 29 0d 44 42 |adratic(|x, y).DB|
|00002a20| 4c 20 2a 78 2c 20 2a 79 | 3b 0d 20 20 7b 0d 20 20 |L *x, *y|;.  {.  |
|00002a30| 44 42 4c 20 64 2c 20 74 | 2c 20 61 2c 20 62 2c 20 |DBL d, t|, a, b, |
|00002a40| 63 3b 0d 20 20 61 20 3d | 20 78 5b 30 5d 3b 0d 20 |c;.  a =| x[0];. |
|00002a50| 20 62 20 3d 20 2d 78 5b | 31 5d 3b 0d 20 20 63 20 | b = -x[|1];.  c |
|00002a60| 3d 20 78 5b 32 5d 3b 0d | 20 20 69 66 20 28 61 20 |= x[2];.|  if (a |
|00002a70| 3d 3d 20 30 2e 30 29 20 | 0d 20 20 20 20 7b 0d 20 |== 0.0) |.    {. |
|00002a80| 20 20 20 69 66 20 28 62 | 20 3d 3d 20 30 2e 30 29 |   if (b| == 0.0)|
|00002a90| 0d 20 20 20 20 20 20 72 | 65 74 75 72 6e 20 30 3b |.      r|eturn 0;|
|00002aa0| 0d 20 20 20 20 79 5b 30 | 5d 20 3d 20 63 20 2f 20 |.    y[0|] = c / |
|00002ab0| 62 3b 0d 20 20 20 20 72 | 65 74 75 72 6e 20 31 3b |b;.    r|eturn 1;|
|00002ac0| 0d 20 20 20 20 7d 0d 20 | 20 64 20 3d 20 62 20 2a |.    }. | d = b *|
|00002ad0| 20 62 20 2d 20 34 2e 30 | 20 2a 20 61 20 2a 20 63 | b - 4.0| * a * c|
|00002ae0| 3b 0d 20 20 69 66 20 28 | 64 20 3c 20 30 2e 30 29 |;.  if (|d < 0.0)|
|00002af0| 0d 20 20 20 20 72 65 74 | 75 72 6e 20 30 3b 0d 20 |.    ret|urn 0;. |
|00002b00| 20 65 6c 73 65 20 69 66 | 20 28 66 61 62 73 28 64 | else if| (fabs(d|
|00002b10| 29 20 3c 20 43 4f 45 46 | 46 5f 4c 49 4d 49 54 29 |) < COEF|F_LIMIT)|
|00002b20| 20 0d 20 20 20 20 7b 0d | 20 20 20 20 79 5b 30 5d | .    {.|    y[0]|
|00002b30| 20 3d 20 30 2e 35 20 2a | 20 62 20 2f 20 61 3b 0d | = 0.5 *| b / a;.|
|00002b40| 20 20 20 20 72 65 74 75 | 72 6e 20 31 3b 0d 20 20 |    retu|rn 1;.  |
|00002b50| 20 20 7d 0d 20 20 64 20 | 3d 20 73 71 72 74 28 64 |  }.  d |= sqrt(d|
|00002b60| 29 3b 0d 20 20 74 20 3d | 20 32 2e 30 20 2a 20 61 |);.  t =| 2.0 * a|
|00002b70| 3b 0d 20 20 79 5b 30 5d | 20 3d 20 28 62 20 2b 20 |;.  y[0]| = (b + |
|00002b80| 64 29 20 2f 20 74 3b 0d | 20 20 79 5b 31 5d 20 3d |d) / t;.|  y[1] =|
|00002b90| 20 28 62 20 2d 20 64 29 | 20 2f 20 74 3b 0d 20 20 | (b - d)| / t;.  |
|00002ba0| 72 65 74 75 72 6e 20 32 | 3b 0d 20 20 7d 0d 0d 2f |return 2|;.  }../|
|00002bb0| 2a 0d 20 20 20 53 6f 6c | 76 65 20 74 68 65 20 63 |*.   Sol|ve the c|
|00002bc0| 75 62 69 63 20 65 71 75 | 61 74 69 6f 6e 3a 0d 0d |ubic equ|ation:..|
|00002bd0| 20 20 20 20 20 20 78 5b | 30 5d 20 2a 20 78 5e 33 |      x[|0] * x^3|
|00002be0| 20 2b 20 78 5b 31 5d 20 | 2a 20 78 5e 32 20 2b 20 | + x[1] |* x^2 + |
|00002bf0| 78 5b 32 5d 20 2a 20 78 | 20 2b 20 78 5b 33 5d 20 |x[2] * x| + x[3] |
|00002c00| 3d 20 30 2e 0d 0d 20 20 | 20 54 68 65 20 72 65 73 |= 0...  | The res|
|00002c10| 75 6c 74 20 6f 66 20 74 | 68 69 73 20 66 75 6e 63 |ult of t|his func|
|00002c20| 74 69 6f 6e 20 69 73 20 | 61 6e 20 69 6e 74 65 67 |tion is |an integ|
|00002c30| 65 72 20 74 68 61 74 20 | 74 65 6c 6c 73 20 68 6f |er that |tells ho|
|00002c40| 77 20 6d 61 6e 79 20 72 | 65 61 6c 0d 20 20 20 72 |w many r|eal.   r|
|00002c50| 6f 6f 74 73 20 65 78 69 | 73 74 2e 20 20 44 65 74 |oots exi|st.  Det|
|00002c60| 65 72 6d 69 6e 61 74 69 | 6f 6e 20 6f 66 20 68 6f |erminati|on of ho|
|00002c70| 77 20 6d 61 6e 79 20 61 | 72 65 20 64 69 73 74 69 |w many a|re disti|
|00002c80| 6e 63 74 20 69 73 20 75 | 70 20 74 6f 20 74 68 65 |nct is u|p to the|
|00002c90| 0d 20 20 20 70 72 6f 63 | 65 73 73 20 74 68 61 74 |.   proc|ess that|
|00002ca0| 20 63 61 6c 6c 73 20 74 | 68 69 73 20 72 6f 75 74 | calls t|his rout|
|00002cb0| 69 6e 65 2e 20 20 54 68 | 65 20 72 6f 6f 74 73 20 |ine.  Th|e roots |
|00002cc0| 74 68 61 74 20 65 78 69 | 73 74 20 61 72 65 20 73 |that exi|st are s|
|00002cd0| 74 6f 72 65 64 0d 20 20 | 20 69 6e 20 28 79 5b 30 |tored.  | in (y[0|
|00002ce0| 5d 2c 20 79 5b 31 5d 2c | 20 79 5b 32 5d 29 2e 0d |], y[1],| y[2])..|
|00002cf0| 0d 20 20 20 4e 6f 74 65 | 3a 20 74 68 69 73 20 66 |.   Note|: this f|
|00002d00| 75 6e 63 74 69 6f 6e 20 | 72 65 6c 69 65 73 20 76 |unction |relies v|
|00002d10| 65 72 79 20 68 65 61 76 | 69 6c 79 20 6f 6e 20 74 |ery heav|ily on t|
|00002d20| 72 69 67 6f 6e 6f 6d 65 | 74 72 69 63 20 66 75 6e |rigonome|tric fun|
|00002d30| 63 74 69 6f 6e 73 20 61 | 6e 64 0d 20 20 20 74 68 |ctions a|nd.   th|
|00002d40| 65 20 73 71 75 61 72 65 | 20 72 6f 6f 74 20 66 75 |e square| root fu|
|00002d50| 6e 63 74 69 6f 6e 2e 20 | 20 49 66 20 61 6e 20 61 |nction. | If an a|
|00002d60| 6c 74 65 72 6e 61 74 69 | 76 65 20 73 6f 6c 75 74 |lternati|ve solut|
|00002d70| 69 6f 6e 20 69 73 20 66 | 6f 75 6e 64 20 74 68 61 |ion is f|ound tha|
|00002d80| 74 20 64 6f 65 73 0d 20 | 20 20 6e 6f 74 20 72 65 |t does. |  not re|
|00002d90| 6c 79 20 6f 6e 20 74 72 | 61 6e 73 63 65 6e 64 65 |ly on tr|anscende|
|00002da0| 6e 74 61 6c 73 20 74 68 | 69 73 20 63 6f 64 65 20 |ntals th|is code |
|00002db0| 77 69 6c 6c 20 62 65 20 | 72 65 70 6c 61 63 65 64 |will be |replaced|
|00002dc0| 2e 0d 2a 2f 0d 69 6e 74 | 20 73 6f 6c 76 65 5f 63 |..*/.int| solve_c|
|00002dd0| 75 62 69 63 28 78 2c 20 | 79 29 0d 44 42 4c 20 2a |ubic(x, |y).DBL *|
|00002de0| 78 2c 20 2a 79 3b 0d 20 | 20 7b 0d 20 20 44 42 4c |x, *y;. | {.  DBL|
|00002df0| 20 51 2c 20 52 2c 20 51 | 33 2c 20 52 32 2c 20 73 | Q, R, Q|3, R2, s|
|00002e00| 51 2c 20 64 2c 20 61 6e | 2c 20 74 68 65 74 61 3b |Q, d, an|, theta;|
|00002e10| 0d 20 20 44 42 4c 20 41 | 32 2c 20 61 30 2c 20 61 |.  DBL A|2, a0, a|
|00002e20| 31 2c 20 61 32 2c 20 61 | 33 3b 0d 20 20 61 30 20 |1, a2, a|3;.  a0 |
|00002e30| 3d 20 78 5b 30 5d 3b 0d | 20 20 69 66 20 28 61 30 |= x[0];.|  if (a0|
|00002e40| 20 3d 3d 20 30 2e 30 29 | 20 72 65 74 75 72 6e 20 | == 0.0)| return |
|00002e50| 73 6f 6c 76 65 5f 71 75 | 61 64 72 61 74 69 63 28 |solve_qu|adratic(|
|00002e60| 26 78 5b 31 5d 2c 20 79 | 29 3b 0d 20 20 65 6c 73 |&x[1], y|);.  els|
|00002e70| 65 20 69 66 20 28 61 30 | 20 21 3d 20 31 2e 30 29 |e if (a0| != 1.0)|
|00002e80| 20 0d 20 20 20 20 7b 0d | 20 20 20 20 61 31 20 3d | .    {.|    a1 =|
|00002e90| 20 78 5b 31 5d 20 2f 20 | 61 30 3b 0d 20 20 20 20 | x[1] / |a0;.    |
|00002ea0| 61 32 20 3d 20 78 5b 32 | 5d 20 2f 20 61 30 3b 0d |a2 = x[2|] / a0;.|
|00002eb0| 20 20 20 20 61 33 20 3d | 20 78 5b 33 5d 20 2f 20 |    a3 =| x[3] / |
|00002ec0| 61 30 3b 0d 20 20 20 20 | 7d 0d 20 20 65 6c 73 65 |a0;.    |}.  else|
|00002ed0| 20 0d 20 20 20 20 7b 0d | 20 20 20 20 61 31 20 3d | .    {.|    a1 =|
|00002ee0| 20 78 5b 31 5d 3b 0d 20 | 20 20 20 61 32 20 3d 20 | x[1];. |   a2 = |
|00002ef0| 78 5b 32 5d 3b 0d 20 20 | 20 20 61 33 20 3d 20 78 |x[2];.  |  a3 = x|
|00002f00| 5b 33 5d 3b 0d 20 20 20 | 20 7d 0d 20 20 41 32 20 |[3];.   | }.  A2 |
|00002f10| 3d 20 61 31 20 2a 20 61 | 31 3b 0d 20 20 51 20 3d |= a1 * a|1;.  Q =|
|00002f20| 20 28 41 32 20 2d 20 33 | 2e 30 20 2a 20 61 32 29 | (A2 - 3|.0 * a2)|
|00002f30| 20 2f 20 39 2e 30 3b 0d | 20 20 52 20 3d 20 28 32 | / 9.0;.|  R = (2|
|00002f40| 2e 30 20 2a 20 41 32 20 | 2a 20 61 31 20 2d 20 39 |.0 * A2 |* a1 - 9|
|00002f50| 2e 30 20 2a 20 61 31 20 | 2a 20 61 32 20 2b 20 32 |.0 * a1 |* a2 + 2|
|00002f60| 37 2e 30 20 2a 20 61 33 | 29 20 2f 20 35 34 2e 30 |7.0 * a3|) / 54.0|
|00002f70| 3b 0d 20 20 51 33 20 3d | 20 51 20 2a 20 51 20 2a |;.  Q3 =| Q * Q *|
|00002f80| 20 51 3b 0d 20 20 52 32 | 20 3d 20 52 20 2a 20 52 | Q;.  R2| = R * R|
|00002f90| 3b 0d 20 20 64 20 3d 20 | 51 33 20 2d 20 52 32 3b |;.  d = |Q3 - R2;|
|00002fa0| 0d 20 20 61 6e 20 3d 20 | 61 31 20 2f 20 33 2e 30 |.  an = |a1 / 3.0|
|00002fb0| 3b 0d 20 20 69 66 20 28 | 64 20 3e 3d 20 30 2e 30 |;.  if (|d >= 0.0|
|00002fc0| 29 20 0d 20 20 20 20 7b | 0d 20 20 20 20 2f 2a 20 |) .    {|.    /* |
|00002fd0| 54 68 72 65 65 20 72 65 | 61 6c 20 72 6f 6f 74 73 |Three re|al roots|
|00002fe0| 2e 20 2a 2f 0d 20 20 20 | 20 64 20 3d 20 52 20 2f |. */.   | d = R /|
|00002ff0| 20 73 71 72 74 28 51 33 | 29 3b 0d 20 20 20 20 74 | sqrt(Q3|);.    t|
|00003000| 68 65 74 61 20 3d 20 61 | 63 6f 73 28 64 29 20 2f |heta = a|cos(d) /|
|00003010| 20 33 2e 30 3b 0d 20 20 | 20 20 73 51 20 3d 20 2d | 3.0;.  |  sQ = -|
|00003020| 32 2e 30 20 2a 20 73 71 | 72 74 28 51 29 3b 0d 20 |2.0 * sq|rt(Q);. |
|00003030| 20 20 20 79 5b 30 5d 20 | 3d 20 73 51 20 2a 20 63 |   y[0] |= sQ * c|
|00003040| 6f 73 28 74 68 65 74 61 | 29 20 2d 20 61 6e 3b 0d |os(theta|) - an;.|
|00003050| 20 20 20 20 79 5b 31 5d | 20 3d 20 73 51 20 2a 20 |    y[1]| = sQ * |
|00003060| 63 6f 73 28 74 68 65 74 | 61 20 2b 20 54 57 4f 5f |cos(thet|a + TWO_|
|00003070| 50 49 5f 33 29 20 2d 20 | 61 6e 3b 0d 20 20 20 20 |PI_3) - |an;.    |
|00003080| 79 5b 32 5d 20 3d 20 73 | 51 20 2a 20 63 6f 73 28 |y[2] = s|Q * cos(|
|00003090| 74 68 65 74 61 20 2b 20 | 54 57 4f 5f 50 49 5f 34 |theta + |TWO_PI_4|
|000030a0| 33 29 20 2d 20 61 6e 3b | 0d 20 20 20 20 72 65 74 |3) - an;|.    ret|
|000030b0| 75 72 6e 20 33 3b 0d 20 | 20 20 20 7d 0d 20 20 65 |urn 3;. |   }.  e|
|000030c0| 6c 73 65 20 0d 20 20 20 | 20 7b 0d 20 20 20 20 73 |lse .   | {.    s|
|000030d0| 51 20 3d 20 70 6f 77 28 | 73 71 72 74 28 52 32 20 |Q = pow(|sqrt(R2 |
|000030e0| 2d 20 51 33 29 20 2b 20 | 41 42 53 28 52 29 2c 20 |- Q3) + |ABS(R), |
|000030f0| 31 2e 30 20 2f 20 33 2e | 30 29 3b 0d 20 20 20 20 |1.0 / 3.|0);.    |
|00003100| 69 66 20 28 52 20 3c 20 | 30 29 0d 20 20 20 20 20 |if (R < |0).     |
|00003110| 20 79 5b 30 5d 20 3d 20 | 28 73 51 20 2b 20 51 20 | y[0] = |(sQ + Q |
|00003120| 2f 20 73 51 29 20 2d 20 | 61 6e 3b 0d 20 20 20 20 |/ sQ) - |an;.    |
|00003130| 65 6c 73 65 0d 20 20 20 | 20 20 20 79 5b 30 5d 20 |else.   |   y[0] |
|00003140| 3d 20 2d 28 73 51 20 2b | 20 51 20 2f 20 73 51 29 |= -(sQ +| Q / sQ)|
|00003150| 20 2d 20 61 6e 3b 0d 20 | 20 20 20 72 65 74 75 72 | - an;. |   retur|
|00003160| 6e 20 31 3b 0d 20 20 20 | 20 7d 0d 20 20 7d 0d 0d |n 1;.   | }.  }..|
|00003170| 2f 2a 20 54 65 73 74 20 | 74 6f 20 73 65 65 20 69 |/* Test |to see i|
|00003180| 66 20 61 6e 79 20 63 6f | 65 66 66 73 20 61 72 65 |f any co|effs are|
|00003190| 20 6d 6f 72 65 20 74 68 | 61 6e 20 36 20 6f 72 64 | more th|an 6 ord|
|000031a0| 65 72 73 20 6f 66 20 6d | 61 67 6e 69 74 75 64 65 |ers of m|agnitude|
|000031b0| 0d 20 20 20 6c 61 72 67 | 65 72 20 74 68 61 6e 20 |.   larg|er than |
|000031c0| 74 68 65 20 73 6d 61 6c | 6c 65 73 74 20 2a 2f 0d |the smal|lest */.|
|000031d0| 73 74 61 74 69 63 20 69 | 6e 74 0d 64 69 66 66 69 |static i|nt.diffi|
|000031e0| 63 75 6c 74 5f 63 6f 65 | 66 66 73 28 6e 2c 20 78 |cult_coe|ffs(n, x|
|000031f0| 29 0d 69 6e 74 20 6e 3b | 0d 44 42 4c 20 2a 78 3b |).int n;|.DBL *x;|
|00003200| 0d 20 20 7b 0d 20 20 69 | 6e 74 20 69 2c 20 66 6c |.  {.  i|nt i, fl|
|00003210| 61 67 20 3d 20 30 3b 0d | 20 20 44 42 4c 20 74 2c |ag = 0;.|  DBL t,|
|00003220| 20 62 69 67 67 65 73 74 | 3b 0d 0d 20 20 62 69 67 | biggest|;..  big|
|00003230| 67 65 73 74 20 3d 20 66 | 61 62 73 28 78 5b 30 5d |gest = f|abs(x[0]|
|00003240| 29 3b 0d 20 20 66 6f 72 | 20 28 69 3d 31 3b 69 3c |);.  for| (i=1;i<|
|00003250| 3d 6e 3b 69 2b 2b 29 20 | 0d 20 20 20 20 7b 0d 20 |=n;i++) |.    {. |
|00003260| 20 20 20 74 20 3d 20 66 | 61 62 73 28 78 5b 69 5d |   t = f|abs(x[i]|
|00003270| 29 3b 0d 20 20 20 20 69 | 66 20 28 74 20 3e 20 62 |);.    i|f (t > b|
|00003280| 69 67 67 65 73 74 29 0d | 20 20 20 20 20 20 62 69 |iggest).|      bi|
|00003290| 67 67 65 73 74 20 3d 20 | 74 3b 0d 20 20 20 20 7d |ggest = |t;.    }|
|000032a0| 0d 0d 20 20 2f 2a 20 45 | 76 65 72 79 74 68 69 6e |..  /* E|verythin|
|000032b0| 67 20 69 73 20 7a 65 72 | 6f 20 6e 6f 20 73 65 6e |g is zer|o no sen|
|000032c0| 73 65 20 69 6e 20 64 6f | 69 6e 67 20 61 6e 79 20 |se in do|ing any |
|000032d0| 6d 6f 72 65 20 2a 2f 0d | 20 20 69 66 20 28 62 69 |more */.|  if (bi|
|000032e0| 67 67 65 73 74 20 3d 3d | 20 30 2e 30 29 20 72 65 |ggest ==| 0.0) re|
|000032f0| 74 75 72 6e 20 30 3b 0d | 0d 20 20 66 6f 72 20 28 |turn 0;.|.  for (|
|00003300| 69 3d 30 3b 69 3c 3d 6e | 3b 69 2b 2b 29 0d 20 20 |i=0;i<=n|;i++).  |
|00003310| 20 20 69 66 20 28 78 5b | 69 5d 20 21 3d 20 30 2e |  if (x[|i] != 0.|
|00003320| 30 29 0d 20 20 20 20 20 | 20 69 66 20 28 66 61 62 |0).     | if (fab|
|00003330| 73 28 62 69 67 67 65 73 | 74 20 2f 20 78 5b 69 5d |s(bigges|t / x[i]|
|00003340| 29 20 3e 20 46 55 44 47 | 45 5f 46 41 43 54 4f 52 |) > FUDG|E_FACTOR|
|00003350| 31 29 20 0d 20 20 20 20 | 20 20 7b 0d 20 20 20 20 |1) .    |  {.    |
|00003360| 20 20 78 5b 69 5d 20 3d | 20 30 2e 30 3b 0d 20 20 |  x[i] =| 0.0;.  |
|00003370| 20 20 20 20 66 6c 61 67 | 20 3d 20 31 3b 0d 20 20 |    flag| = 1;.  |
|00003380| 20 20 20 20 7d 0d 0d 20 | 20 72 65 74 75 72 6e 20 |    }.. | return |
|00003390| 66 6c 61 67 3b 0d 20 20 | 7d 0d 0d 23 69 66 64 65 |flag;.  |}..#ifde|
|000033a0| 66 20 54 45 53 54 5f 53 | 4f 4c 56 45 52 0d 2f 2a |f TEST_S|OLVER./*|
|000033b0| 20 54 68 65 20 6f 6c 64 | 20 77 61 79 20 6f 66 20 | The old| way of |
|000033c0| 73 6f 6c 76 69 6e 67 20 | 71 75 61 72 74 69 63 73 |solving |quartics|
|000033d0| 20 61 6c 67 65 62 72 61 | 69 63 61 6c 6c 79 20 2a | algebra|ically *|
|000033e0| 2f 0d 2f 2a 20 54 68 69 | 73 20 69 73 20 61 6e 20 |/./* Thi|s is an |
|000033f0| 61 64 61 70 74 61 74 69 | 6f 6e 20 6f 66 20 74 68 |adaptati|on of th|
|00003400| 65 20 6d 65 74 68 6f 64 | 20 6f 66 20 4c 6f 64 6f |e method| of Lodo|
|00003410| 76 69 63 6f 20 46 65 72 | 72 61 72 69 20 28 43 69 |vico Fer|rari (Ci|
|00003420| 72 63 61 20 31 37 33 31 | 29 2e 20 2a 2f 0d 69 6e |rca 1731|). */.in|
|00003430| 74 20 73 6f 6c 76 65 5f | 71 75 61 72 74 69 63 28 |t solve_|quartic(|
|00003440| 78 2c 20 72 65 73 75 6c | 74 73 29 0d 44 42 4c 20 |x, resul|ts).DBL |
|00003450| 2a 78 2c 20 2a 72 65 73 | 75 6c 74 73 3b 0d 20 20 |*x, *res|ults;.  |
|00003460| 7b 0d 20 20 44 42 4c 20 | 63 75 62 69 63 5b 34 5d |{.  DBL |cubic[4]|
|00003470| 2c 20 72 6f 6f 74 73 5b | 33 5d 3b 0d 20 20 44 42 |, roots[|3];.  DB|
|00003480| 4c 20 61 30 2c 20 61 31 | 2c 20 79 2c 20 64 31 2c |L a0, a1|, y, d1,|
|00003490| 20 78 31 2c 20 74 31 2c | 20 74 32 3b 0d 20 20 44 | x1, t1,| t2;.  D|
|000034a0| 42 4c 20 63 30 2c 20 63 | 31 2c 20 63 32 2c 20 63 |BL c0, c|1, c2, c|
|000034b0| 33 2c 20 63 34 2c 20 64 | 32 2c 20 71 31 2c 20 71 |3, c4, d|2, q1, q|
|000034c0| 32 3b 0d 20 20 69 6e 74 | 20 69 3b 0d 0d 20 20 2f |2;.  int| i;..  /|
|000034d0| 2a 20 46 69 67 75 72 65 | 20 6f 75 74 20 74 68 65 |* Figure| out the|
|000034e0| 20 73 69 7a 65 20 64 69 | 66 66 65 72 65 6e 63 65 | size di|fference|
|000034f0| 20 62 65 74 77 65 65 6e | 20 63 6f 65 66 66 69 63 | between| coeffic|
|00003500| 69 65 6e 74 73 20 2a 2f | 0d 20 20 69 66 20 28 64 |ients */|.  if (d|
|00003510| 69 66 66 69 63 75 6c 74 | 5f 63 6f 65 66 66 73 28 |ifficult|_coeffs(|
|00003520| 34 2c 20 78 29 29 20 0d | 20 20 20 20 7b 0d 20 20 |4, x)) .|    {.  |
|00003530| 20 20 69 66 20 28 66 61 | 62 73 28 78 5b 30 5d 29 |  if (fa|bs(x[0])|
|00003540| 20 3c 20 43 4f 45 46 46 | 5f 4c 49 4d 49 54 29 0d | < COEFF|_LIMIT).|
|00003550| 20 20 20 20 20 20 69 66 | 20 28 66 61 62 73 28 78 |      if| (fabs(x|
|00003560| 5b 31 5d 29 20 3c 20 43 | 4f 45 46 46 5f 4c 49 4d |[1]) < C|OEFF_LIM|
|00003570| 49 54 29 0d 20 20 20 20 | 20 20 20 20 72 65 74 75 |IT).    |    retu|
|00003580| 72 6e 20 73 6f 6c 76 65 | 5f 71 75 61 64 72 61 74 |rn solve|_quadrat|
|00003590| 69 63 28 26 78 5b 32 5d | 2c 20 72 65 73 75 6c 74 |ic(&x[2]|, result|
|000035a0| 73 29 3b 0d 20 20 20 20 | 20 20 65 6c 73 65 0d 20 |s);.    |  else. |
|000035b0| 20 20 20 20 20 20 20 72 | 65 74 75 72 6e 20 73 6f |       r|eturn so|
|000035c0| 6c 76 65 5f 63 75 62 69 | 63 28 26 78 5b 31 5d 2c |lve_cubi|c(&x[1],|
|000035d0| 20 72 65 73 75 6c 74 73 | 29 3b 0d 20 20 20 20 65 | results|);.    e|
|000035e0| 6c 73 65 0d 20 20 20 20 | 20 20 72 65 74 75 72 6e |lse.    |  return|
|000035f0| 20 70 6f 6c 79 73 6f 6c | 76 65 28 34 2c 20 78 2c | polysol|ve(4, x,|
|00003600| 20 72 65 73 75 6c 74 73 | 29 3b 0d 20 20 20 20 7d | results|);.    }|
|00003610| 0d 0d 20 20 63 30 20 3d | 20 78 5b 30 5d 3b 0d 20 |..  c0 =| x[0];. |
|00003620| 20 69 66 20 28 66 61 62 | 73 28 63 30 29 20 3c 20 | if (fab|s(c0) < |
|00003630| 43 4f 45 46 46 5f 4c 49 | 4d 49 54 29 0d 20 20 20 |COEFF_LI|MIT).   |
|00003640| 20 72 65 74 75 72 6e 20 | 73 6f 6c 76 65 5f 63 75 | return |solve_cu|
|00003650| 62 69 63 28 26 78 5b 31 | 5d 2c 20 72 65 73 75 6c |bic(&x[1|], resul|
|00003660| 74 73 29 3b 0d 20 20 65 | 6c 73 65 20 69 66 20 28 |ts);.  e|lse if (|
|00003670| 66 61 62 73 28 78 5b 34 | 5d 29 20 3c 20 43 4f 45 |fabs(x[4|]) < COE|
|00003680| 46 46 5f 4c 49 4d 49 54 | 29 20 0d 20 20 20 20 7b |FF_LIMIT|) .    {|
|00003690| 0d 20 20 20 20 72 65 74 | 75 72 6e 20 73 6f 6c 76 |.    ret|urn solv|
|000036a0| 65 5f 63 75 62 69 63 28 | 78 2c 20 72 65 73 75 6c |e_cubic(|x, resul|
|000036b0| 74 73 29 3b 0d 20 20 20 | 20 7d 0d 20 20 65 6c 73 |ts);.   | }.  els|
|000036c0| 65 20 69 66 20 28 63 30 | 20 21 3d 20 31 2e 30 29 |e if (c0| != 1.0)|
|000036d0| 20 0d 20 20 20 20 7b 0d | 20 20 20 20 63 31 20 3d | .    {.|    c1 =|
|000036e0| 20 78 5b 31 5d 20 2f 20 | 63 30 3b 0d 20 20 20 20 | x[1] / |c0;.    |
|000036f0| 63 32 20 3d 20 78 5b 32 | 5d 20 2f 20 63 30 3b 0d |c2 = x[2|] / c0;.|
|00003700| 20 20 20 20 63 33 20 3d | 20 78 5b 33 5d 20 2f 20 |    c3 =| x[3] / |
|00003710| 63 30 3b 0d 20 20 20 20 | 63 34 20 3d 20 78 5b 34 |c0;.    |c4 = x[4|
|00003720| 5d 20 2f 20 63 30 3b 0d | 20 20 20 20 7d 0d 20 20 |] / c0;.|    }.  |
|00003730| 65 6c 73 65 20 0d 20 20 | 20 20 7b 0d 20 20 20 20 |else .  |  {.    |
|00003740| 63 31 20 3d 20 78 5b 31 | 5d 3b 0d 20 20 20 20 63 |c1 = x[1|];.    c|
|00003750| 32 20 3d 20 78 5b 32 5d | 3b 0d 20 20 20 20 63 33 |2 = x[2]|;.    c3|
|00003760| 20 3d 20 78 5b 33 5d 3b | 0d 20 20 20 20 63 34 20 | = x[3];|.    c4 |
|00003770| 3d 20 78 5b 34 5d 3b 0d | 20 20 20 20 7d 0d 0d 20 |= x[4];.|    }.. |
|00003780| 20 20 20 2f 2a 20 54 68 | 65 20 66 69 72 73 74 20 |   /* Th|e first |
|00003790| 73 74 65 70 20 69 73 20 | 74 6f 20 74 61 6b 65 20 |step is |to take |
|000037a0| 74 68 65 20 6f 72 69 67 | 69 6e 61 6c 20 65 71 75 |the orig|inal equ|
|000037b0| 61 74 69 6f 6e 3a 0d 0d | 09 20 78 5e 34 20 2b 20 |ation:..|. x^4 + |
|000037c0| 62 2a 78 5e 33 20 2b 20 | 63 2a 78 5e 32 20 2b 20 |b*x^3 + |c*x^2 + |
|000037d0| 64 2a 78 20 2b 20 65 20 | 3d 20 30 0d 0d 20 20 20 |d*x + e |= 0..   |
|000037e0| 20 20 20 61 6e 64 20 72 | 65 77 72 69 74 65 20 69 |   and r|ewrite i|
|000037f0| 74 20 61 73 3a 0d 0d 09 | 20 78 5e 34 20 2b 20 62 |t as:...| x^4 + b|
|00003800| 2a 78 5e 33 20 3d 20 2d | 63 2a 78 5e 32 20 2d 20 |*x^3 = -|c*x^2 - |
|00003810| 64 2a 78 20 2d 20 65 2c | 0d 0d 20 20 20 20 20 20 |d*x - e,|..      |
|00003820| 61 64 64 69 6e 67 20 28 | 62 2a 78 2f 32 29 5e 32 |adding (|b*x/2)^2|
|00003830| 20 2b 20 28 78 5e 32 20 | 2b 20 62 2a 78 2f 32 29 | + (x^2 |+ b*x/2)|
|00003840| 79 20 2b 20 79 5e 32 2f | 34 20 74 6f 20 65 61 63 |y + y^2/|4 to eac|
|00003850| 68 20 73 69 64 65 20 67 | 69 76 65 73 20 61 0d 20 |h side g|ives a. |
|00003860| 20 20 20 20 20 70 65 72 | 66 65 63 74 20 73 71 75 |     per|fect squ|
|00003870| 61 72 65 20 6f 6e 20 74 | 68 65 20 6c 68 73 3a 0d |are on t|he lhs:.|
|00003880| 0d 09 20 28 78 5e 32 20 | 2b 20 62 2a 78 2f 32 20 |.. (x^2 |+ b*x/2 |
|00003890| 2b 20 79 2f 32 29 5e 32 | 20 3d 20 28 62 5e 32 2f |+ y/2)^2| = (b^2/|
|000038a0| 34 20 2d 20 63 20 2b 20 | 79 29 78 5e 32 20 2b 20 |4 - c + |y)x^2 + |
|000038b0| 28 62 2a 79 2f 32 20 2d | 20 64 29 78 20 2b 20 79 |(b*y/2 -| d)x + y|
|000038c0| 5e 32 2f 34 20 2d 20 65 | 0d 0d 20 20 20 20 20 20 |^2/4 - e|..      |
|000038d0| 42 79 20 63 68 6f 6f 73 | 69 6e 67 20 74 68 65 20 |By choos|ing the |
|000038e0| 61 70 70 72 6f 70 72 69 | 61 74 65 20 76 61 6c 75 |appropri|ate valu|
|000038f0| 65 20 66 6f 72 20 79 2c | 20 74 68 65 20 72 68 73 |e for y,| the rhs|
|00003900| 20 63 61 6e 20 62 65 20 | 6d 61 64 65 20 61 20 70 | can be |made a p|
|00003910| 65 72 66 65 63 74 0d 20 | 20 20 20 20 20 73 71 75 |erfect. |     squ|
|00003920| 61 72 65 20 61 6c 73 6f | 2e 20 20 54 68 69 73 20 |are also|.  This |
|00003930| 76 61 6c 75 65 20 69 73 | 20 66 6f 75 6e 64 20 77 |value is| found w|
|00003940| 68 65 6e 20 74 68 65 20 | 72 68 73 20 69 73 20 74 |hen the |rhs is t|
|00003950| 72 65 61 74 65 64 20 61 | 73 20 61 20 71 75 61 64 |reated a|s a quad|
|00003960| 72 61 74 69 63 0d 20 20 | 20 20 20 20 69 6e 20 78 |ratic.  |    in x|
|00003970| 20 77 69 74 68 20 74 68 | 65 20 64 69 73 63 72 69 | with th|e discri|
|00003980| 6d 69 6e 61 6e 74 20 65 | 71 75 61 6c 20 74 6f 20 |minant e|qual to |
|00003990| 30 2e 20 20 54 68 69 73 | 20 77 69 6c 6c 20 62 65 |0.  This| will be|
|000039a0| 20 74 72 75 65 20 77 68 | 65 6e 3a 0d 0d 09 20 28 | true wh|en:... (|
|000039b0| 62 2a 79 2f 32 20 2d 20 | 64 29 5e 32 20 2d 20 34 |b*y/2 - |d)^2 - 4|
|000039c0| 2e 30 20 2a 20 28 62 5e | 32 2f 34 20 2d 20 63 2a |.0 * (b^|2/4 - c*|
|000039d0| 79 29 2a 28 79 5e 32 2f | 34 20 2d 20 65 29 20 3d |y)*(y^2/|4 - e) =|
|000039e0| 20 30 2c 20 6f 72 0d 0d | 09 20 79 5e 33 20 2d 20 | 0, or..|. y^3 - |
|000039f0| 63 2a 79 5e 32 20 2b 20 | 28 62 2a 64 20 2d 20 34 |c*y^2 + |(b*d - 4|
|00003a00| 2a 65 29 2a 79 20 2d 20 | 62 5e 32 2a 65 20 2b 20 |*e)*y - |b^2*e + |
|00003a10| 34 2a 63 2a 65 20 2d 20 | 64 5e 32 20 3d 20 30 2e |4*c*e - |d^2 = 0.|
|00003a20| 0d 0d 20 20 20 20 20 20 | 54 68 69 73 20 69 73 20 |..      |This is |
|00003a30| 63 61 6c 6c 65 64 20 74 | 68 65 20 72 65 73 6f 6c |called t|he resol|
|00003a40| 76 65 6e 74 20 6f 66 20 | 74 68 65 20 71 75 61 72 |vent of |the quar|
|00003a50| 74 69 63 20 65 71 75 61 | 74 69 6f 6e 2e 20 20 2a |tic equa|tion.  *|
|00003a60| 2f 0d 0d 20 20 20 20 61 | 30 20 3d 20 34 2e 30 20 |/..    a|0 = 4.0 |
|00003a70| 2a 20 63 34 3b 0d 20 20 | 63 75 62 69 63 5b 30 5d |* c4;.  |cubic[0]|
|00003a80| 20 3d 20 31 2e 30 3b 0d | 20 20 63 75 62 69 63 5b | = 1.0;.|  cubic[|
|00003a90| 31 5d 20 3d 20 2d 31 2e | 30 20 2a 20 63 32 3b 0d |1] = -1.|0 * c2;.|
|00003aa0| 20 20 63 75 62 69 63 5b | 32 5d 20 3d 20 63 31 20 |  cubic[|2] = c1 |
|00003ab0| 2a 20 63 33 20 2d 20 61 | 30 3b 0d 20 20 63 75 62 |* c3 - a|0;.  cub|
|00003ac0| 69 63 5b 33 5d 20 3d 20 | 61 30 20 2a 20 63 32 20 |ic[3] = |a0 * c2 |
|00003ad0| 2d 20 63 31 20 2a 20 63 | 31 20 2a 20 63 34 20 2d |- c1 * c|1 * c4 -|
|00003ae0| 20 63 33 20 2a 20 63 33 | 3b 0d 20 20 69 20 3d 20 | c3 * c3|;.  i = |
|00003af0| 73 6f 6c 76 65 5f 63 75 | 62 69 63 28 26 63 75 62 |solve_cu|bic(&cub|
|00003b00| 69 63 5b 30 5d 2c 20 26 | 72 6f 6f 74 73 5b 30 5d |ic[0], &|roots[0]|
|00003b10| 29 3b 0d 20 20 69 66 20 | 28 69 20 3e 20 30 29 0d |);.  if |(i > 0).|
|00003b20| 20 20 20 20 79 20 3d 20 | 72 6f 6f 74 73 5b 30 5d |    y = |roots[0]|
|00003b30| 3b 0d 20 20 65 6c 73 65 | 0d 20 20 20 20 72 65 74 |;.  else|.    ret|
|00003b40| 75 72 6e 20 30 3b 0d 0d | 20 20 2f 2a 20 57 68 61 |urn 0;..|  /* Wha|
|00003b50| 74 20 77 65 20 61 72 65 | 20 6c 65 66 74 20 77 69 |t we are| left wi|
|00003b60| 74 68 20 69 73 20 61 20 | 71 75 61 64 72 61 74 69 |th is a |quadrati|
|00003b70| 63 20 73 71 75 61 72 65 | 64 20 6f 6e 20 74 68 65 |c square|d on the|
|00003b80| 20 6c 68 73 20 61 6e 64 | 20 61 0d 20 20 20 20 20 | lhs and| a.     |
|00003b90| 20 6c 69 6e 65 61 72 20 | 74 65 72 6d 20 6f 6e 20 | linear |term on |
|00003ba0| 74 68 65 20 72 69 67 68 | 74 2e 20 20 54 68 65 20 |the righ|t.  The |
|00003bb0| 6c 69 6e 65 61 72 20 74 | 65 72 6d 20 68 61 73 20 |linear t|erm has |
|00003bc0| 6f 6e 65 20 6f 66 20 74 | 77 6f 20 73 69 67 6e 73 |one of t|wo signs|
|00003bd0| 2c 0d 20 20 20 20 20 20 | 74 61 6b 65 20 65 61 63 |,.      |take eac|
|00003be0| 68 20 61 6e 64 20 61 64 | 64 20 69 74 20 74 6f 20 |h and ad|d it to |
|00003bf0| 74 68 65 20 6c 68 73 2e | 20 20 54 68 65 20 66 6f |the lhs.|  The fo|
|00003c00| 72 6d 20 6f 66 20 74 68 | 65 20 71 75 61 72 74 69 |rm of th|e quarti|
|00003c10| 63 20 69 73 20 6e 6f 77 | 3a 0d 0d 09 20 61 27 20 |c is now|:... a' |
|00003c20| 3d 20 62 5e 32 2f 34 20 | 2d 20 63 20 2b 20 79 2c |= b^2/4 |- c + y,|
|00003c30| 20 20 20 20 62 27 20 3d | 20 62 2a 79 2f 32 20 2d |    b' =| b*y/2 -|
|00003c40| 20 64 2c 20 28 66 72 6f | 6d 20 72 68 73 20 71 75 | d, (fro|m rhs qu|
|00003c50| 61 64 72 69 74 69 63 20 | 61 62 6f 76 65 29 0d 0d |adritic |above)..|
|00003c60| 09 20 28 78 5e 32 20 2b | 20 62 2a 78 2f 32 20 2b |. (x^2 +| b*x/2 +|
|00003c70| 20 79 2f 32 29 20 3d 20 | 2b 73 71 72 74 28 61 27 | y/2) = |+sqrt(a'|
|00003c80| 2a 28 78 20 2b 20 31 2f | 32 20 2a 20 62 27 2f 61 |*(x + 1/|2 * b'/a|
|00003c90| 27 29 5e 32 29 2c 20 61 | 6e 64 0d 09 20 28 78 5e |')^2), a|nd.. (x^|
|00003ca0| 32 20 2b 20 62 2a 78 2f | 32 20 2b 20 79 2f 32 29 |2 + b*x/|2 + y/2)|
|00003cb0| 20 3d 20 2d 73 71 72 74 | 28 61 27 2a 28 78 20 2b | = -sqrt|(a'*(x +|
|00003cc0| 20 31 2f 32 20 2a 20 62 | 27 2f 61 27 29 5e 32 29 | 1/2 * b|'/a')^2)|
|00003cd0| 2e 0d 0d 20 20 20 20 20 | 20 42 79 20 74 61 6b 69 |...     | By taki|
|00003ce0| 6e 67 20 74 68 65 20 6c | 69 6e 65 61 72 20 74 65 |ng the l|inear te|
|00003cf0| 72 6d 20 66 72 6f 6d 20 | 65 61 63 68 20 6f 66 20 |rm from |each of |
|00003d00| 74 68 65 20 72 69 67 68 | 74 20 68 61 6e 64 20 73 |the righ|t hand s|
|00003d10| 69 64 65 73 20 61 6e 64 | 0d 20 20 20 20 20 20 61 |ides and|.      a|
|00003d20| 64 64 69 6e 67 20 74 6f | 20 74 68 65 20 61 70 70 |dding to| the app|
|00003d30| 72 6f 70 72 69 61 74 65 | 20 70 61 72 74 20 6f 66 |ropriate| part of|
|00003d40| 20 74 68 65 20 6c 65 66 | 74 20 68 61 6e 64 20 73 | the lef|t hand s|
|00003d50| 69 64 65 2c 20 74 77 6f | 20 71 75 61 64 72 61 74 |ide, two| quadrat|
|00003d60| 69 63 0d 20 20 20 20 20 | 20 66 6f 72 6d 75 6c 61 |ic.     | formula|
|00003d70| 73 20 61 72 65 20 63 72 | 65 61 74 65 64 2e 20 20 |s are cr|eated.  |
|00003d80| 42 79 20 73 6f 6c 76 69 | 6e 67 20 65 61 63 68 20 |By solvi|ng each |
|00003d90| 6f 66 20 74 68 65 73 65 | 20 74 68 65 20 66 6f 75 |of these| the fou|
|00003da0| 72 20 72 6f 6f 74 73 20 | 6f 66 0d 20 20 20 20 20 |r roots |of.     |
|00003db0| 20 74 68 65 20 71 75 61 | 72 74 69 63 20 61 72 65 | the qua|rtic are|
|00003dc0| 20 64 65 74 65 72 6d 69 | 6e 65 64 2e 0d 20 20 20 | determi|ned..   |
|00003dd0| 2a 2f 0d 20 20 69 20 3d | 20 30 3b 0d 20 20 61 30 |*/.  i =| 0;.  a0|
|00003de0| 20 3d 20 63 31 20 2f 20 | 32 2e 30 3b 0d 20 20 61 | = c1 / |2.0;.  a|
|00003df0| 31 20 3d 20 79 20 2f 20 | 32 2e 30 3b 0d 0d 20 20 |1 = y / |2.0;..  |
|00003e00| 74 31 20 3d 20 61 30 20 | 2a 20 61 30 20 2d 20 63 |t1 = a0 |* a0 - c|
|00003e10| 32 20 2b 20 79 3b 0d 20 | 20 69 66 20 28 74 31 20 |2 + y;. | if (t1 |
|00003e20| 3c 20 30 2e 30 29 20 0d | 20 20 20 20 7b 0d 20 20 |< 0.0) .|    {.  |
|00003e30| 20 20 69 66 20 28 74 31 | 20 3e 20 46 55 44 47 45 |  if (t1| > FUDGE|
|00003e40| 5f 46 41 43 54 4f 52 32 | 29 0d 20 20 20 20 20 20 |_FACTOR2|).      |
|00003e50| 74 31 20 3d 20 30 2e 30 | 3b 0d 20 20 20 20 65 6c |t1 = 0.0|;.    el|
|00003e60| 73 65 20 0d 20 20 20 20 | 20 20 7b 0d 20 20 20 20 |se .    |  {.    |
|00003e70| 20 20 2f 2a 20 46 69 72 | 73 74 20 53 70 65 63 69 |  /* Fir|st Speci|
|00003e80| 61 6c 20 63 61 73 65 2c | 20 61 27 20 3c 20 30 20 |al case,| a' < 0 |
|00003e90| 6d 65 61 6e 73 20 61 6c | 6c 20 72 6f 6f 74 73 20 |means al|l roots |
|00003ea0| 61 72 65 20 63 6f 6d 70 | 6c 65 78 2e 20 2a 2f 0d |are comp|lex. */.|
|00003eb0| 20 20 20 20 20 20 20 20 | 72 65 74 75 72 6e 20 30 |        |return 0|
|00003ec0| 3b 0d 20 20 20 20 20 20 | 7d 0d 20 20 20 20 7d 0d |;.      |}.    }.|
|00003ed0| 20 20 69 66 20 28 74 31 | 20 3c 20 46 55 44 47 45 |  if (t1| < FUDGE|
|00003ee0| 5f 46 41 43 54 4f 52 33 | 29 20 0d 20 20 20 20 7b |_FACTOR3|) .    {|
|00003ef0| 0d 20 20 20 20 2f 2a 20 | 53 65 63 6f 6e 64 20 73 |.    /* |Second s|
|00003f00| 70 65 63 69 61 6c 20 63 | 61 73 65 2c 20 74 68 65 |pecial c|ase, the|
|00003f10| 20 22 78 22 20 74 65 72 | 6d 20 6f 6e 20 74 68 65 | "x" ter|m on the|
|00003f20| 20 72 69 67 68 74 20 68 | 61 6e 64 20 73 69 64 65 | right h|and side|
|00003f30| 20 61 62 6f 76 65 0d 09 | 20 68 61 73 20 76 61 6e | above..| has van|
|00003f40| 69 73 68 65 64 2e 20 20 | 49 6e 20 74 68 69 73 20 |ished.  |In this |
|00003f50| 63 61 73 65 3a 0d 09 09 | 28 78 5e 32 20 2b 20 62 |case:...|(x^2 + b|
|00003f60| 2a 78 2f 32 20 2b 20 79 | 2f 32 29 20 3d 20 2b 73 |*x/2 + y|/2) = +s|
|00003f70| 71 72 74 28 79 5e 32 2f | 34 20 2d 20 65 29 2c 20 |qrt(y^2/|4 - e), |
|00003f80| 61 6e 64 0d 09 09 28 78 | 5e 32 20 2b 20 62 2a 78 |and...(x|^2 + b*x|
|00003f90| 2f 32 20 2b 20 79 2f 32 | 29 20 3d 20 2d 73 71 72 |/2 + y/2|) = -sqr|
|00003fa0| 74 28 79 5e 32 2f 34 20 | 2d 20 65 29 2e 20 20 2a |t(y^2/4 |- e).  *|
|00003fb0| 2f 0d 20 20 20 20 74 32 | 20 3d 20 61 31 20 2a 20 |/.    t2| = a1 * |
|00003fc0| 61 31 20 2d 20 63 34 3b | 0d 20 20 20 20 69 66 20 |a1 - c4;|.    if |
|00003fd0| 28 74 32 20 3c 20 30 2e | 30 29 20 0d 20 20 20 20 |(t2 < 0.|0) .    |
|00003fe0| 20 20 7b 0d 20 20 20 20 | 20 20 72 65 74 75 72 6e |  {.    |  return|
|00003ff0| 20 30 3b 0d 20 20 20 20 | 20 20 7d 0d 20 20 20 20 | 0;.    |  }.    |
|00004000| 78 31 20 3d 20 30 2e 30 | 3b 0d 20 20 20 20 64 31 |x1 = 0.0|;.    d1|
|00004010| 20 3d 20 73 71 72 74 28 | 74 32 29 3b 0d 20 20 20 | = sqrt(|t2);.   |
|00004020| 20 7d 0d 20 20 65 6c 73 | 65 20 0d 20 20 20 20 7b | }.  els|e .    {|
|00004030| 0d 20 20 20 20 78 31 20 | 3d 20 73 71 72 74 28 74 |.    x1 |= sqrt(t|
|00004040| 31 29 3b 0d 20 20 20 20 | 64 31 20 3d 20 30 2e 35 |1);.    |d1 = 0.5|
|00004050| 20 2a 20 28 61 30 20 2a | 20 79 20 2d 20 63 33 29 | * (a0 *| y - c3)|
|00004060| 20 2f 20 78 31 3b 0d 20 | 20 20 20 7d 0d 20 20 2f | / x1;. |   }.  /|
|00004070| 2a 20 53 6f 6c 76 65 20 | 74 68 65 20 66 69 72 73 |* Solve |the firs|
|00004080| 74 20 71 75 61 64 72 61 | 74 69 63 20 2a 2f 0d 20 |t quadra|tic */. |
|00004090| 20 20 20 71 31 20 3d 20 | 2d 61 30 20 2d 20 78 31 |   q1 = |-a0 - x1|
|000040a0| 3b 0d 20 20 71 32 20 3d | 20 61 31 20 2b 20 64 31 |;.  q2 =| a1 + d1|
|000040b0| 3b 0d 20 20 64 32 20 3d | 20 71 31 20 2a 20 71 31 |;.  d2 =| q1 * q1|
|000040c0| 20 2d 20 34 2e 30 20 2a | 20 71 32 3b 0d 20 20 69 | - 4.0 *| q2;.  i|
|000040d0| 66 20 28 64 32 20 3e 3d | 20 30 2e 30 29 20 0d 20 |f (d2 >=| 0.0) . |
|000040e0| 20 20 20 7b 0d 20 20 20 | 20 64 32 20 3d 20 73 71 |   {.   | d2 = sq|
|000040f0| 72 74 28 64 32 29 3b 0d | 20 20 20 20 72 65 73 75 |rt(d2);.|    resu|
|00004100| 6c 74 73 5b 30 5d 20 3d | 20 30 2e 35 20 2a 20 28 |lts[0] =| 0.5 * (|
|00004110| 71 31 20 2b 20 64 32 29 | 3b 0d 20 20 20 20 72 65 |q1 + d2)|;.    re|
|00004120| 73 75 6c 74 73 5b 31 5d | 20 3d 20 30 2e 35 20 2a |sults[1]| = 0.5 *|
|00004130| 20 28 71 31 20 2d 20 64 | 32 29 3b 0d 20 20 20 20 | (q1 - d|2);.    |
|00004140| 69 20 3d 20 32 3b 0d 20 | 20 20 20 7d 0d 20 20 2f |i = 2;. |   }.  /|
|00004150| 2a 20 53 6f 6c 76 65 20 | 74 68 65 20 73 65 63 6f |* Solve |the seco|
|00004160| 6e 64 20 71 75 61 64 72 | 61 74 69 63 20 2a 2f 0d |nd quadr|atic */.|
|00004170| 20 20 71 31 20 3d 20 71 | 31 20 2b 20 78 31 20 2b |  q1 = q|1 + x1 +|
|00004180| 20 78 31 3b 0d 20 20 71 | 32 20 3d 20 61 31 20 2d | x1;.  q|2 = a1 -|
|00004190| 20 64 31 3b 0d 20 20 64 | 32 20 3d 20 71 31 20 2a | d1;.  d|2 = q1 *|
|000041a0| 20 71 31 20 2d 20 34 2e | 30 20 2a 20 71 32 3b 0d | q1 - 4.|0 * q2;.|
|000041b0| 20 20 69 66 20 28 64 32 | 20 3e 3d 20 30 2e 30 29 |  if (d2| >= 0.0)|
|000041c0| 20 0d 20 20 20 20 7b 0d | 20 20 20 20 64 32 20 3d | .    {.|    d2 =|
|000041d0| 20 73 71 72 74 28 64 32 | 29 3b 0d 20 20 20 20 72 | sqrt(d2|);.    r|
|000041e0| 65 73 75 6c 74 73 5b 69 | 2b 2b 5d 20 3d 20 30 2e |esults[i|++] = 0.|
|000041f0| 35 20 2a 20 28 71 31 20 | 2b 20 64 32 29 3b 0d 20 |5 * (q1 |+ d2);. |
|00004200| 20 20 20 72 65 73 75 6c | 74 73 5b 69 2b 2b 5d 20 |   resul|ts[i++] |
|00004210| 3d 20 30 2e 35 20 2a 20 | 28 71 31 20 2d 20 64 32 |= 0.5 * |(q1 - d2|
|00004220| 29 3b 0d 20 20 20 20 7d | 0d 20 20 72 65 74 75 72 |);.    }|.  retur|
|00004230| 6e 20 69 3b 0d 20 20 7d | 0d 23 65 6c 73 65 0d 20 |n i;.  }|.#else. |
|00004240| 20 2f 2a 20 53 6f 6c 76 | 65 20 61 20 71 75 61 72 | /* Solv|e a quar|
|00004250| 74 69 63 20 75 73 69 6e | 67 20 74 68 65 20 6d 65 |tic usin|g the me|
|00004260| 74 68 6f 64 20 6f 66 20 | 46 72 61 6e 63 6f 69 73 |thod of |Francois|
|00004270| 20 56 69 65 74 61 20 28 | 43 69 72 63 61 20 31 37 | Vieta (|Circa 17|
|00004280| 33 35 29 20 2a 2f 0d 20 | 20 69 6e 74 0d 20 20 73 |35) */. | int.  s|
|00004290| 6f 6c 76 65 5f 71 75 61 | 72 74 69 63 28 78 2c 20 |olve_qua|rtic(x, |
|000042a0| 72 65 73 75 6c 74 73 29 | 0d 20 20 20 20 44 42 4c |results)|.    DBL|
|000042b0| 20 2a 78 2c 20 2a 72 65 | 73 75 6c 74 73 3b 0d 20 | *x, *re|sults;. |
|000042c0| 20 20 20 7b 0d 20 20 20 | 20 44 42 4c 20 63 75 62 |   {.   | DBL cub|
|000042d0| 69 63 5b 34 5d 2c 20 72 | 6f 6f 74 73 5b 33 5d 3b |ic[4], r|oots[3];|
|000042e0| 0d 20 20 20 20 44 42 4c | 20 63 31 32 2c 20 7a 2c |.    DBL| c12, z,|
|000042f0| 20 70 2c 20 71 2c 20 71 | 31 2c 20 71 32 2c 20 72 | p, q, q|1, q2, r|
|00004300| 2c 20 64 31 2c 20 64 32 | 3b 0d 20 20 20 20 44 42 |, d1, d2|;.    DB|
|00004310| 4c 20 63 30 2c 20 63 31 | 2c 20 63 32 2c 20 63 33 |L c0, c1|, c2, c3|
|00004320| 2c 20 63 34 3b 0d 20 20 | 20 20 69 6e 74 20 69 3b |, c4;.  |  int i;|
|00004330| 0d 0d 20 20 20 20 2f 2a | 20 46 69 67 75 72 65 20 |..    /*| Figure |
|00004340| 6f 75 74 20 74 68 65 20 | 73 69 7a 65 20 64 69 66 |out the |size dif|
|00004350| 66 65 72 65 6e 63 65 20 | 62 65 74 77 65 65 6e 20 |ference |between |
|00004360| 63 6f 65 66 66 69 63 69 | 65 6e 74 73 20 2a 2f 0d |coeffici|ents */.|
|00004370| 20 20 20 20 69 66 20 28 | 64 69 66 66 69 63 75 6c |    if (|difficul|
|00004380| 74 5f 63 6f 65 66 66 73 | 28 34 2c 20 78 29 29 20 |t_coeffs|(4, x)) |
|00004390| 0d 20 20 20 20 20 20 7b | 0d 20 20 20 20 20 20 69 |.      {|.      i|
|000043a0| 66 20 28 66 61 62 73 28 | 78 5b 30 5d 29 20 3c 20 |f (fabs(|x[0]) < |
|000043b0| 43 4f 45 46 46 5f 4c 49 | 4d 49 54 29 0d 20 20 20 |COEFF_LI|MIT).   |
|000043c0| 20 20 20 20 20 69 66 20 | 28 66 61 62 73 28 78 5b |     if |(fabs(x[|
|000043d0| 31 5d 29 20 3c 20 43 4f | 45 46 46 5f 4c 49 4d 49 |1]) < CO|EFF_LIMI|
|000043e0| 54 29 0d 20 20 20 20 20 | 20 20 20 20 20 72 65 74 |T).     |     ret|
|000043f0| 75 72 6e 20 73 6f 6c 76 | 65 5f 71 75 61 64 72 61 |urn solv|e_quadra|
|00004400| 74 69 63 28 26 78 5b 32 | 5d 2c 20 72 65 73 75 6c |tic(&x[2|], resul|
|00004410| 74 73 29 3b 0d 20 20 20 | 20 20 20 20 20 65 6c 73 |ts);.   |     els|
|00004420| 65 0d 20 20 20 20 20 20 | 20 20 20 20 72 65 74 75 |e.      |    retu|
|00004430| 72 6e 20 73 6f 6c 76 65 | 5f 63 75 62 69 63 28 26 |rn solve|_cubic(&|
|00004440| 78 5b 31 5d 2c 20 72 65 | 73 75 6c 74 73 29 3b 0d |x[1], re|sults);.|
|00004450| 20 20 20 20 20 20 65 6c | 73 65 0d 20 20 20 20 20 |      el|se.     |
|00004460| 20 20 20 72 65 74 75 72 | 6e 20 70 6f 6c 79 73 6f |   retur|n polyso|
|00004470| 6c 76 65 28 34 2c 20 78 | 2c 20 72 65 73 75 6c 74 |lve(4, x|, result|
|00004480| 73 29 3b 0d 20 20 20 20 | 20 20 7d 0d 0d 20 20 20 |s);.    |  }..   |
|00004490| 20 2f 2a 20 53 65 65 20 | 69 66 20 74 68 65 20 68 | /* See |if the h|
|000044a0| 69 67 68 20 6f 72 64 65 | 72 20 74 65 72 6d 20 68 |igh orde|r term h|
|000044b0| 61 73 20 76 61 6e 69 73 | 68 65 64 20 2a 2f 0d 20 |as vanis|hed */. |
|000044c0| 20 20 20 63 30 20 3d 20 | 78 5b 30 5d 3b 0d 20 20 |   c0 = |x[0];.  |
|000044d0| 20 20 69 66 20 28 66 61 | 62 73 28 63 30 29 20 3c |  if (fa|bs(c0) <|
|000044e0| 20 43 4f 45 46 46 5f 4c | 49 4d 49 54 29 0d 20 20 | COEFF_L|IMIT).  |
|000044f0| 20 20 20 20 72 65 74 75 | 72 6e 20 73 6f 6c 76 65 |    retu|rn solve|
|00004500| 5f 63 75 62 69 63 28 26 | 78 5b 31 5d 2c 20 72 65 |_cubic(&|x[1], re|
|00004510| 73 75 6c 74 73 29 3b 0d | 0d 20 20 20 20 2f 2a 20 |sults);.|.    /* |
|00004520| 53 65 65 20 69 66 20 74 | 68 65 20 63 6f 6e 73 74 |See if t|he const|
|00004530| 61 6e 74 20 74 65 72 6d | 20 68 61 73 20 76 61 6e |ant term| has van|
|00004540| 69 73 68 65 64 20 2a 2f | 0d 20 20 20 20 69 66 20 |ished */|.    if |
|00004550| 28 66 61 62 73 28 78 5b | 34 5d 29 20 3c 20 43 4f |(fabs(x[|4]) < CO|
|00004560| 45 46 46 5f 4c 49 4d 49 | 54 29 0d 20 20 20 20 20 |EFF_LIMI|T).     |
|00004570| 20 72 65 74 75 72 6e 20 | 73 6f 6c 76 65 5f 63 75 | return |solve_cu|
|00004580| 62 69 63 28 78 2c 20 72 | 65 73 75 6c 74 73 29 3b |bic(x, r|esults);|
|00004590| 0d 0d 20 20 20 20 2f 2a | 20 4d 61 6b 65 20 73 75 |..    /*| Make su|
|000045a0| 72 65 20 74 68 65 20 71 | 75 61 72 74 69 63 20 68 |re the q|uartic h|
|000045b0| 61 73 20 61 20 6c 65 61 | 64 69 6e 67 20 63 6f 65 |as a lea|ding coe|
|000045c0| 66 66 69 63 69 65 6e 74 | 20 6f 66 20 31 2e 30 20 |fficient| of 1.0 |
|000045d0| 2a 2f 0d 20 20 20 20 69 | 66 20 28 63 30 20 21 3d |*/.    i|f (c0 !=|
|000045e0| 20 31 2e 30 29 20 0d 20 | 20 20 20 20 20 7b 0d 20 | 1.0) . |     {. |
|000045f0| 20 20 20 20 20 63 31 20 | 3d 20 78 5b 31 5d 20 2f |     c1 |= x[1] /|
|00004600| 20 63 30 3b 0d 20 20 20 | 20 20 20 63 32 20 3d 20 | c0;.   |   c2 = |
|00004610| 78 5b 32 5d 20 2f 20 63 | 30 3b 0d 20 20 20 20 20 |x[2] / c|0;.     |
|00004620| 20 63 33 20 3d 20 78 5b | 33 5d 20 2f 20 63 30 3b | c3 = x[|3] / c0;|
|00004630| 0d 20 20 20 20 20 20 63 | 34 20 3d 20 78 5b 34 5d |.      c|4 = x[4]|
|00004640| 20 2f 20 63 30 3b 0d 20 | 20 20 20 20 20 7d 0d 20 | / c0;. |     }. |
|00004650| 20 20 20 65 6c 73 65 20 | 0d 20 20 20 20 20 20 7b |   else |.      {|
|00004660| 0d 20 20 20 20 20 20 63 | 31 20 3d 20 78 5b 31 5d |.      c|1 = x[1]|
|00004670| 3b 0d 20 20 20 20 20 20 | 63 32 20 3d 20 78 5b 32 |;.      |c2 = x[2|
|00004680| 5d 3b 0d 20 20 20 20 20 | 20 63 33 20 3d 20 78 5b |];.     | c3 = x[|
|00004690| 33 5d 3b 0d 20 20 20 20 | 20 20 63 34 20 3d 20 78 |3];.    |  c4 = x|
|000046a0| 5b 34 5d 3b 0d 20 20 20 | 20 20 20 7d 0d 0d 20 20 |[4];.   |   }..  |
|000046b0| 20 20 20 20 2f 2a 20 43 | 6f 6d 70 75 74 65 20 74 |    /* C|ompute t|
|000046c0| 68 65 20 63 75 62 69 63 | 20 72 65 73 6f 6c 76 61 |he cubic| resolva|
|000046d0| 6e 74 20 2a 2f 0d 20 20 | 20 20 20 20 63 31 32 20 |nt */.  |    c12 |
|000046e0| 3d 20 63 31 20 2a 20 63 | 31 3b 0d 20 20 20 20 70 |= c1 * c|1;.    p|
|000046f0| 20 3d 20 2d 30 2e 33 37 | 35 20 2a 20 63 31 32 20 | = -0.37|5 * c12 |
|00004700| 2b 20 63 32 3b 0d 20 20 | 20 20 71 20 3d 20 30 2e |+ c2;.  |  q = 0.|
|00004710| 31 32 35 20 2a 20 63 31 | 32 20 2a 20 63 31 20 2d |125 * c1|2 * c1 -|
|00004720| 20 30 2e 35 20 2a 20 63 | 31 20 2a 20 63 32 20 2b | 0.5 * c|1 * c2 +|
|00004730| 20 63 33 3b 0d 20 20 20 | 20 72 20 3d 20 2d 30 2e | c3;.   | r = -0.|
|00004740| 30 31 31 37 31 38 37 35 | 20 2a 20 63 31 32 20 2a |01171875| * c12 *|
|00004750| 20 63 31 32 20 2b 20 30 | 2e 30 36 32 35 20 2a 20 | c12 + 0|.0625 * |
|00004760| 63 31 32 20 2a 20 63 32 | 20 2d 20 30 2e 32 35 20 |c12 * c2| - 0.25 |
|00004770| 2a 20 63 31 20 2a 20 63 | 33 20 2b 20 63 34 3b 0d |* c1 * c|3 + c4;.|
|00004780| 0d 20 20 20 20 63 75 62 | 69 63 5b 30 5d 20 3d 20 |.    cub|ic[0] = |
|00004790| 31 2e 30 3b 0d 20 20 20 | 20 63 75 62 69 63 5b 31 |1.0;.   | cubic[1|
|000047a0| 5d 20 3d 20 2d 30 2e 35 | 20 2a 20 70 3b 0d 20 20 |] = -0.5| * p;.  |
|000047b0| 20 20 63 75 62 69 63 5b | 32 5d 20 3d 20 2d 72 3b |  cubic[|2] = -r;|
|000047c0| 0d 20 20 20 20 63 75 62 | 69 63 5b 33 5d 20 3d 20 |.    cub|ic[3] = |
|000047d0| 30 2e 35 20 2a 20 72 20 | 2a 20 70 20 2d 20 30 2e |0.5 * r |* p - 0.|
|000047e0| 31 32 35 20 2a 20 71 20 | 2a 20 71 3b 0d 20 20 20 |125 * q |* q;.   |
|000047f0| 20 69 20 3d 20 73 6f 6c | 76 65 5f 63 75 62 69 63 | i = sol|ve_cubic|
|00004800| 28 63 75 62 69 63 2c 20 | 72 6f 6f 74 73 29 3b 0d |(cubic, |roots);.|
|00004810| 20 20 20 20 69 66 20 28 | 69 20 3e 20 30 29 0d 20 |    if (|i > 0). |
|00004820| 20 20 20 20 20 7a 20 3d | 20 72 6f 6f 74 73 5b 30 |     z =| roots[0|
|00004830| 5d 3b 0d 20 20 20 20 65 | 6c 73 65 0d 20 20 20 20 |];.    e|lse.    |
|00004840| 20 20 72 65 74 75 72 6e | 20 30 3b 0d 0d 20 20 20 |  return| 0;..   |
|00004850| 20 64 31 20 3d 20 32 2e | 30 20 2a 20 7a 20 2d 20 | d1 = 2.|0 * z - |
|00004860| 70 3b 0d 0d 20 20 20 20 | 69 66 20 28 64 31 20 3c |p;..    |if (d1 <|
|00004870| 20 30 2e 30 29 20 0d 20 | 20 20 20 20 20 7b 0d 20 | 0.0) . |     {. |
|00004880| 20 20 20 20 20 69 66 20 | 28 64 31 20 3e 20 2d 45 |     if |(d1 > -E|
|00004890| 50 53 49 4c 4f 4e 29 0d | 20 20 20 20 20 20 20 20 |PSILON).|        |
|000048a0| 64 31 20 3d 20 30 2e 30 | 3b 0d 20 20 20 20 20 20 |d1 = 0.0|;.      |
|000048b0| 65 6c 73 65 0d 20 20 20 | 20 20 20 20 20 72 65 74 |else.   |     ret|
|000048c0| 75 72 6e 20 30 3b 0d 20 | 20 20 20 20 20 7d 0d 20 |urn 0;. |     }. |
|000048d0| 20 20 20 69 66 20 28 64 | 31 20 3c 20 45 50 53 49 |   if (d|1 < EPSI|
|000048e0| 4c 4f 4e 29 20 0d 20 20 | 20 20 20 20 7b 0d 20 20 |LON) .  |    {.  |
|000048f0| 20 20 20 20 64 32 20 3d | 20 7a 20 2a 20 7a 20 2d |    d2 =| z * z -|
|00004900| 20 72 3b 0d 20 20 20 20 | 20 20 69 66 20 28 64 32 | r;.    |  if (d2|
|00004910| 20 3c 20 30 2e 30 29 0d | 20 20 20 20 20 20 20 20 | < 0.0).|        |
|00004920| 72 65 74 75 72 6e 20 30 | 3b 0d 20 20 20 20 20 20 |return 0|;.      |
|00004930| 64 32 20 3d 20 73 71 72 | 74 28 64 32 29 3b 0d 20 |d2 = sqr|t(d2);. |
|00004940| 20 20 20 20 20 7d 0d 20 | 20 20 20 65 6c 73 65 20 |     }. |   else |
|00004950| 0d 20 20 20 20 20 20 7b | 0d 20 20 20 20 20 20 64 |.      {|.      d|
|00004960| 31 20 3d 20 73 71 72 74 | 28 64 31 29 3b 0d 20 20 |1 = sqrt|(d1);.  |
|00004970| 20 20 20 20 64 32 20 3d | 20 30 2e 35 20 2a 20 71 |    d2 =| 0.5 * q|
|00004980| 20 2f 20 64 31 3b 0d 20 | 20 20 20 20 20 7d 0d 0d | / d1;. |     }..|
|00004990| 20 20 20 20 20 20 2f 2a | 20 53 65 74 20 75 70 20 |      /*| Set up |
|000049a0| 75 73 65 66 75 6c 20 76 | 61 6c 75 65 73 20 66 6f |useful v|alues fo|
|000049b0| 72 20 74 68 65 20 71 75 | 61 64 72 61 74 69 63 20 |r the qu|adratic |
|000049c0| 66 61 63 74 6f 72 73 20 | 2a 2f 0d 20 20 20 20 20 |factors |*/.     |
|000049d0| 20 71 31 20 3d 20 64 31 | 20 2a 20 64 31 3b 0d 20 | q1 = d1| * d1;. |
|000049e0| 20 20 20 71 32 20 3d 20 | 2d 30 2e 32 35 20 2a 20 |   q2 = |-0.25 * |
|000049f0| 63 31 3b 0d 20 20 20 20 | 69 20 3d 20 30 3b 0d 0d |c1;.    |i = 0;..|
|00004a00| 20 20 20 20 2f 2a 20 53 | 6f 6c 76 65 20 74 68 65 |    /* S|olve the|
|00004a10| 20 66 69 72 73 74 20 71 | 75 61 64 72 61 74 69 63 | first q|uadratic|
|00004a20| 20 2a 2f 0d 20 20 20 20 | 70 20 3d 20 71 31 20 2d | */.    |p = q1 -|
|00004a30| 20 34 2e 30 20 2a 20 28 | 7a 20 2d 20 64 32 29 3b | 4.0 * (|z - d2);|
|00004a40| 0d 20 20 20 20 69 66 20 | 28 70 20 3d 3d 20 30 29 |.    if |(p == 0)|
|00004a50| 0d 20 20 20 20 20 20 72 | 65 73 75 6c 74 73 5b 69 |.      r|esults[i|
|00004a60| 2b 2b 5d 20 3d 20 2d 30 | 2e 35 20 2a 20 64 31 20 |++] = -0|.5 * d1 |
|00004a70| 2d 20 71 32 3b 0d 20 20 | 20 20 65 6c 73 65 20 69 |- q2;.  |  else i|
|00004a80| 66 20 28 70 20 3e 20 30 | 29 20 0d 20 20 20 20 20 |f (p > 0|) .     |
|00004a90| 20 7b 0d 20 20 20 20 20 | 20 70 20 3d 20 73 71 72 | {.     | p = sqr|
|00004aa0| 74 28 70 29 3b 0d 20 20 | 20 20 20 20 72 65 73 75 |t(p);.  |    resu|
|00004ab0| 6c 74 73 5b 69 2b 2b 5d | 20 3d 20 2d 30 2e 35 20 |lts[i++]| = -0.5 |
|00004ac0| 2a 20 28 64 31 20 2b 20 | 70 29 20 2b 20 71 32 3b |* (d1 + |p) + q2;|
|00004ad0| 0d 20 20 20 20 20 20 72 | 65 73 75 6c 74 73 5b 69 |.      r|esults[i|
|00004ae0| 2b 2b 5d 20 3d 20 2d 30 | 2e 35 20 2a 20 28 64 31 |++] = -0|.5 * (d1|
|00004af0| 20 2d 20 70 29 20 2b 20 | 71 32 3b 0d 20 20 20 20 | - p) + |q2;.    |
|00004b00| 20 20 7d 0d 20 20 20 20 | 2f 2a 20 53 6f 6c 76 65 |  }.    |/* Solve|
|00004b10| 20 74 68 65 20 73 65 63 | 6f 6e 64 20 71 75 61 64 | the sec|ond quad|
|00004b20| 72 61 74 69 63 20 2a 2f | 0d 20 20 20 20 70 20 3d |ratic */|.    p =|
|00004b30| 20 71 31 20 2d 20 34 2e | 30 20 2a 20 28 7a 20 2b | q1 - 4.|0 * (z +|
|00004b40| 20 64 32 29 3b 0d 20 20 | 20 20 69 66 20 28 70 20 | d2);.  |  if (p |
|00004b50| 3d 3d 20 30 29 0d 20 20 | 20 20 20 20 72 65 73 75 |== 0).  |    resu|
|00004b60| 6c 74 73 5b 69 2b 2b 5d | 20 3d 20 30 2e 35 20 2a |lts[i++]| = 0.5 *|
|00004b70| 20 64 31 20 2d 20 71 32 | 3b 0d 20 20 20 20 65 6c | d1 - q2|;.    el|
|00004b80| 73 65 20 69 66 20 28 70 | 20 3e 20 30 29 20 0d 20 |se if (p| > 0) . |
|00004b90| 20 20 20 20 20 7b 0d 20 | 20 20 20 20 20 70 20 3d |     {. |     p =|
|00004ba0| 20 73 71 72 74 28 70 29 | 3b 0d 20 20 20 20 20 20 | sqrt(p)|;.      |
|00004bb0| 72 65 73 75 6c 74 73 5b | 69 2b 2b 5d 20 3d 20 30 |results[|i++] = 0|
|00004bc0| 2e 35 20 2a 20 28 64 31 | 20 2b 20 70 29 20 2b 20 |.5 * (d1| + p) + |
|00004bd0| 71 32 3b 0d 20 20 20 20 | 20 20 72 65 73 75 6c 74 |q2;.    |  result|
|00004be0| 73 5b 69 2b 2b 5d 20 3d | 20 30 2e 35 20 2a 20 28 |s[i++] =| 0.5 * (|
|00004bf0| 64 31 20 2d 20 70 29 20 | 2b 20 71 32 3b 0d 20 20 |d1 - p) |+ q2;.  |
|00004c00| 20 20 20 20 7d 0d 20 20 | 20 20 72 65 74 75 72 6e |    }.  |  return|
|00004c10| 20 69 3b 0d 20 20 20 20 | 7d 0d 23 65 6e 64 69 66 | i;.    |}.#endif|
|00004c20| 0d 0d 2f 2a 20 52 6f 6f | 74 20 73 6f 6c 76 65 72 |../* Roo|t solver|
|00004c30| 20 62 61 73 65 64 20 6f | 6e 20 74 68 65 20 53 74 | based o|n the St|
|00004c40| 75 72 6d 20 73 65 71 75 | 65 6e 63 65 73 20 66 6f |urm sequ|ences fo|
|00004c50| 72 20 61 20 70 6f 6c 79 | 6e 6f 6d 69 61 6c 2e 20 |r a poly|nomial. |
|00004c60| 2a 2f 0d 69 6e 74 20 70 | 6f 6c 79 73 6f 6c 76 65 |*/.int p|olysolve|
|00004c70| 28 6f 72 64 65 72 2c 20 | 43 6f 65 66 66 73 2c 20 |(order, |Coeffs, |
|00004c80| 72 6f 6f 74 73 29 0d 69 | 6e 74 20 6f 72 64 65 72 |roots).i|nt order|
|00004c90| 3b 0d 44 42 4c 20 2a 43 | 6f 65 66 66 73 2c 20 2a |;.DBL *C|oeffs, *|
|00004ca0| 72 6f 6f 74 73 3b 0d 20 | 20 7b 0d 20 20 70 6f 6c |roots;. | {.  pol|
|00004cb0| 79 6e 6f 6d 69 61 6c 20 | 73 73 65 71 5b 4d 41 58 |ynomial |sseq[MAX|
|00004cc0| 5f 4f 52 44 45 52 2b 31 | 5d 3b 0d 20 20 44 42 4c |_ORDER+1|];.  DBL|
|00004cd0| 20 6d 69 6e 5f 76 61 6c | 75 65 2c 20 6d 61 78 5f | min_val|ue, max_|
|00004ce0| 76 61 6c 75 65 3b 0d 20 | 20 69 6e 74 20 69 2c 20 |value;. | int i, |
|00004cf0| 6e 72 6f 6f 74 73 2c 20 | 6e 70 2c 20 61 74 6d 69 |nroots, |np, atmi|
|00004d00| 6e 2c 20 61 74 6d 61 78 | 3b 0d 0d 20 20 2f 2a 20 |n, atmax|;..  /* |
|00004d10| 50 75 74 20 74 68 65 20 | 63 6f 65 66 66 69 63 69 |Put the |coeffici|
|00004d20| 65 6e 74 73 20 69 6e 74 | 6f 20 74 68 65 20 74 6f |ents int|o the to|
|00004d30| 70 20 6f 66 20 74 68 65 | 20 73 74 61 63 6b 2e 20 |p of the| stack. |
|00004d40| 2a 2f 0d 20 20 66 6f 72 | 20 28 69 3d 30 3b 69 3c |*/.  for| (i=0;i<|
|00004d50| 3d 6f 72 64 65 72 3b 69 | 2b 2b 29 0d 20 20 20 20 |=order;i|++).    |
|00004d60| 73 73 65 71 5b 30 5d 2e | 63 6f 65 66 5b 6f 72 64 |sseq[0].|coef[ord|
|00004d70| 65 72 2d 69 5d 20 3d 20 | 43 6f 65 66 66 73 5b 69 |er-i] = |Coeffs[i|
|00004d80| 5d 3b 0d 0d 20 20 2f 2a | 20 42 75 69 6c 64 20 74 |];..  /*| Build t|
|00004d90| 68 65 20 53 74 75 72 6d | 20 73 65 71 75 65 6e 63 |he Sturm| sequenc|
|00004da0| 65 20 2a 2f 0d 20 20 6e | 70 20 3d 20 62 75 69 6c |e */.  n|p = buil|
|00004db0| 64 73 74 75 72 6d 28 6f | 72 64 65 72 2c 20 26 73 |dsturm(o|rder, &s|
|00004dc0| 73 65 71 5b 30 5d 29 3b | 0d 0d 20 20 2f 2a 20 47 |seq[0]);|..  /* G|
|00004dd0| 65 74 20 74 68 65 20 74 | 6f 74 61 6c 20 6e 75 6d |et the t|otal num|
|00004de0| 62 65 72 20 6f 66 20 76 | 69 73 69 62 6c 65 20 72 |ber of v|isible r|
|00004df0| 6f 6f 74 73 20 2a 2f 0d | 20 20 69 66 20 28 28 6e |oots */.|  if ((n|
|00004e00| 72 6f 6f 74 73 20 3d 20 | 76 69 73 69 62 6c 65 5f |roots = |visible_|
|00004e10| 72 6f 6f 74 73 28 6e 70 | 2c 20 73 73 65 71 2c 20 |roots(np|, sseq, |
|00004e20| 26 61 74 6d 69 6e 2c 20 | 26 61 74 6d 61 78 29 29 |&atmin, |&atmax))|
|00004e30| 20 3d 3d 20 30 29 0d 20 | 20 20 20 72 65 74 75 72 | == 0). |   retur|
|00004e40| 6e 20 30 3b 0d 0d 20 20 | 2f 2a 20 42 72 61 63 6b |n 0;..  |/* Brack|
|00004e50| 65 74 20 74 68 65 20 72 | 6f 6f 74 73 20 2a 2f 0d |et the r|oots */.|
|00004e60| 20 20 6d 69 6e 5f 76 61 | 6c 75 65 20 3d 20 30 2e |  min_va|lue = 0.|
|00004e70| 30 3b 0d 20 20 6d 61 78 | 5f 76 61 6c 75 65 20 3d |0;.  max|_value =|
|00004e80| 20 4d 61 78 5f 44 69 73 | 74 61 6e 63 65 3b 0d 0d | Max_Dis|tance;..|
|00004e90| 20 20 61 74 6d 69 6e 20 | 3d 20 6e 75 6d 63 68 61 |  atmin |= numcha|
|00004ea0| 6e 67 65 73 28 6e 70 2c | 20 73 73 65 71 2c 20 6d |nges(np,| sseq, m|
|00004eb0| 69 6e 5f 76 61 6c 75 65 | 29 3b 0d 20 20 61 74 6d |in_value|);.  atm|
|00004ec0| 61 78 20 3d 20 6e 75 6d | 63 68 61 6e 67 65 73 28 |ax = num|changes(|
|00004ed0| 6e 70 2c 20 73 73 65 71 | 2c 20 6d 61 78 5f 76 61 |np, sseq|, max_va|
|00004ee0| 6c 75 65 29 3b 0d 20 20 | 6e 72 6f 6f 74 73 20 3d |lue);.  |nroots =|
|00004ef0| 20 61 74 6d 69 6e 20 2d | 20 61 74 6d 61 78 3b 0d | atmin -| atmax;.|
|00004f00| 20 20 69 66 20 28 6e 72 | 6f 6f 74 73 20 3d 3d 20 |  if (nr|oots == |
|00004f10| 30 29 20 72 65 74 75 72 | 6e 20 30 3b 0d 0d 20 20 |0) retur|n 0;..  |
|00004f20| 2f 2a 20 70 65 72 66 6f | 72 6d 20 74 68 65 20 62 |/* perfo|rm the b|
|00004f30| 69 73 65 63 74 69 6f 6e | 2e 20 2a 2f 0d 20 20 72 |isection|. */.  r|
|00004f40| 65 74 75 72 6e 20 73 62 | 69 73 65 63 74 28 6e 70 |eturn sb|isect(np|
|00004f50| 2c 20 73 73 65 71 2c 20 | 6d 69 6e 5f 76 61 6c 75 |, sseq, |min_valu|
|00004f60| 65 2c 20 6d 61 78 5f 76 | 61 6c 75 65 2c 20 61 74 |e, max_v|alue, at|
|00004f70| 6d 69 6e 2c 20 61 74 6d | 61 78 2c 20 72 6f 6f 74 |min, atm|ax, root|
|00004f80| 73 29 3b 0d 20 20 7d 0d | 0d 23 69 66 64 65 66 20 |s);.  }.|.#ifdef |
|00004f90| 54 45 53 54 20 20 20 20 | 20 2f 2a 20 54 68 69 73 |TEST    | /* This|
|00004fa0| 20 69 73 20 6e 6f 74 20 | 75 73 65 64 20 61 6e 79 | is not |used any|
|00004fb0| 77 68 65 72 65 20 6f 72 | 20 74 65 73 74 65 64 2e |where or| tested.|
|00004fc0| 20 20 49 6e 74 65 72 65 | 73 74 69 6e 67 3f 20 2a |  Intere|sting? *|
|00004fd0| 2f 0d 0d 23 64 65 66 69 | 6e 65 20 4d 41 58 5f 50 |/..#defi|ne MAX_P|
|00004fe0| 4f 4c 59 47 4f 4e 5f 53 | 49 44 45 53 20 38 0d 23 |OLYGON_S|IDES 8.#|
|00004ff0| 64 65 66 69 6e 65 20 43 | 72 6f 73 73 69 6e 67 5f |define C|rossing_|
|00005000| 50 6f 69 6e 74 28 78 31 | 2c 20 79 31 2c 20 78 32 |Point(x1|, y1, x2|
|00005010| 2c 20 79 32 29 20 28 78 | 31 20 2d 20 79 31 20 2a |, y2) (x|1 - y1 *|
|00005020| 20 28 78 32 20 2d 20 78 | 31 29 20 2f 20 28 79 32 | (x2 - x|1) / (y2|
|00005030| 20 2d 20 79 31 29 29 0d | 0d 2f 2a 20 53 65 65 20 | - y1)).|./* See |
|00005040| 69 66 20 22 52 61 79 22 | 20 69 6e 74 65 72 73 65 |if "Ray"| interse|
|00005050| 63 74 73 20 74 68 65 20 | 70 6f 6c 79 67 6f 6e 20 |cts the |polygon |
|00005060| 64 65 66 69 6e 65 64 20 | 62 79 20 74 68 65 20 63 |defined |by the c|
|00005070| 6f 6f 72 64 69 6e 61 74 | 65 20 6c 69 73 74 20 22 |oordinat|e list "|
|00005080| 76 65 72 74 69 63 65 73 | 22 2e 20 2a 2f 0d 69 6e |vertices|". */.in|
|00005090| 74 20 49 6e 74 65 72 73 | 65 63 74 5f 50 6f 6c 79 |t Inters|ect_Poly|
|000050a0| 67 6f 6e 28 52 61 79 2c | 20 76 65 72 74 65 78 5f |gon(Ray,| vertex_|
|000050b0| 63 6f 75 6e 74 2c 20 76 | 65 72 74 69 63 65 73 2c |count, v|ertices,|
|000050c0| 20 6e 2c 20 64 2c 20 44 | 65 70 74 68 2c 20 49 6e | n, d, D|epth, In|
|000050d0| 74 65 72 73 65 63 74 5f | 50 6f 69 6e 74 29 0d 52 |tersect_|Point).R|
|000050e0| 41 59 20 2a 52 61 79 3b | 0d 69 6e 74 20 76 65 72 |AY *Ray;|.int ver|
|000050f0| 74 65 78 5f 63 6f 75 6e | 74 3b 0d 56 45 43 54 4f |tex_coun|t;.VECTO|
|00005100| 52 20 2a 76 65 72 74 69 | 63 65 73 2c 20 2a 6e 2c |R *verti|ces, *n,|
|00005110| 20 2a 49 6e 74 65 72 73 | 65 63 74 5f 50 6f 69 6e | *Inters|ect_Poin|
|00005120| 74 3b 0d 44 42 4c 20 64 | 2c 20 2a 44 65 70 74 68 |t;.DBL d|, *Depth|
|00005130| 3b 0d 20 20 7b 0d 20 20 | 44 42 4c 20 73 2c 20 74 |;.  {.  |DBL s, t|
|00005140| 2c 20 78 2c 20 79 2c 20 | 7a 3b 0d 20 20 69 6e 74 |, x, y, |z;.  int|
|00005150| 20 53 69 67 6e 5f 48 6f | 6c 64 65 72 2c 20 4e 65 | Sign_Ho|lder, Ne|
|00005160| 78 74 5f 53 69 67 6e 2c | 20 43 72 6f 73 73 69 6e |xt_Sign,| Crossin|
|00005170| 67 73 3b 0d 20 20 69 6e | 74 20 69 2c 20 74 68 69 |gs;.  in|t i, thi|
|00005180| 73 5f 76 65 72 74 65 78 | 2c 20 6e 65 78 74 5f 76 |s_vertex|, next_v|
|00005190| 65 72 74 65 78 3b 0d 0d | 20 20 73 74 61 74 69 63 |ertex;..|  static|
|000051a0| 20 44 42 4c 20 74 65 6d | 70 5f 78 5b 4d 41 58 5f | DBL tem|p_x[MAX_|
|000051b0| 50 4f 4c 59 47 4f 4e 5f | 53 49 44 45 53 5d 2c 0d |POLYGON_|SIDES],.|
|000051c0| 20 20 74 65 6d 70 5f 79 | 5b 4d 41 58 5f 50 4f 4c |  temp_y|[MAX_POL|
|000051d0| 59 47 4f 4e 5f 53 49 44 | 45 53 5d 3b 0d 0d 20 20 |YGON_SID|ES];..  |
|000051e0| 2f 2a 20 43 61 6c 63 75 | 6c 61 74 65 20 74 68 65 |/* Calcu|late the|
|000051f0| 20 70 6f 69 6e 74 20 6f | 66 20 69 6e 74 65 72 73 | point o|f inters|
|00005200| 65 63 74 69 6f 6e 20 61 | 6e 64 20 74 68 65 20 64 |ection a|nd the d|
|00005210| 65 70 74 68 2e 20 2a 2f | 0d 20 20 56 44 6f 74 28 |epth. */|.  VDot(|
|00005220| 73 2c 20 52 61 79 2d 3e | 44 69 72 65 63 74 69 6f |s, Ray->|Directio|
|00005230| 6e 2c 20 2a 6e 29 3b 0d | 20 20 69 66 20 28 73 20 |n, *n);.|  if (s |
|00005240| 3d 3d 20 30 2e 30 29 0d | 20 20 20 20 72 65 74 75 |== 0.0).|    retu|
|00005250| 72 6e 20 30 3b 0d 20 20 | 56 44 6f 74 28 74 2c 20 |rn 0;.  |VDot(t, |
|00005260| 52 61 79 2d 3e 49 6e 69 | 74 69 61 6c 2c 20 2a 6e |Ray->Ini|tial, *n|
|00005270| 29 3b 0d 20 20 2a 44 65 | 70 74 68 20 3d 20 30 2e |);.  *De|pth = 0.|
|00005280| 30 20 2d 20 28 64 20 2b | 20 74 29 20 2f 20 73 3b |0 - (d +| t) / s;|
|00005290| 0d 20 20 69 66 20 28 2a | 44 65 70 74 68 20 3c 20 |.  if (*|Depth < |
|000052a0| 53 6d 61 6c 6c 5f 54 6f | 6c 65 72 61 6e 63 65 29 |Small_To|lerance)|
|000052b0| 0d 20 20 20 20 72 65 74 | 75 72 6e 20 30 3b 0d 20 |.    ret|urn 0;. |
|000052c0| 20 56 53 63 61 6c 65 28 | 2a 49 6e 74 65 72 73 65 | VScale(|*Interse|
|000052d0| 63 74 5f 50 6f 69 6e 74 | 2c 20 52 61 79 2d 3e 44 |ct_Point|, Ray->D|
|000052e0| 69 72 65 63 74 69 6f 6e | 2c 20 2a 44 65 70 74 68 |irection|, *Depth|
|000052f0| 29 3b 0d 20 20 56 41 64 | 64 28 2a 49 6e 74 65 72 |);.  VAd|d(*Inter|
|00005300| 73 65 63 74 5f 50 6f 69 | 6e 74 2c 20 2a 49 6e 74 |sect_Poi|nt, *Int|
|00005310| 65 72 73 65 63 74 5f 50 | 6f 69 6e 74 2c 20 52 61 |ersect_P|oint, Ra|
|00005320| 79 2d 3e 49 6e 69 74 69 | 61 6c 29 3b 0d 0d 20 20 |y->Initi|al);..  |
|00005330| 2f 2a 20 4d 61 70 20 74 | 68 65 20 69 6e 74 65 72 |/* Map t|he inter|
|00005340| 73 65 63 74 69 6f 6e 20 | 70 6f 69 6e 74 20 61 6e |section |point an|
|00005350| 64 20 74 68 65 20 74 72 | 69 61 6e 67 6c 65 20 6f |d the tr|iangle o|
|00005360| 6e 74 6f 20 61 20 70 6c | 61 6e 65 2e 20 2a 2f 0d |nto a pl|ane. */.|
|00005370| 20 20 78 20 3d 20 41 42 | 53 28 6e 2d 3e 78 29 3b |  x = AB|S(n->x);|
|00005380| 20 79 20 3d 20 41 42 53 | 28 6e 2d 3e 79 29 3b 20 | y = ABS|(n->y); |
|00005390| 7a 20 3d 20 41 42 53 28 | 6e 2d 3e 7a 29 3b 0d 20 |z = ABS(|n->z);. |
|000053a0| 20 69 66 20 28 78 3e 79 | 29 0d 20 20 20 20 69 66 | if (x>y|).    if|
|000053b0| 20 28 78 3e 7a 29 0d 20 | 20 20 20 20 20 66 6f 72 | (x>z). |     for|
|000053c0| 20 28 69 3d 30 3b 69 3c | 76 65 72 74 65 78 5f 63 | (i=0;i<|vertex_c|
|000053d0| 6f 75 6e 74 3b 69 2b 2b | 29 20 0d 20 20 20 20 7b |ount;i++|) .    {|
|000053e0| 0d 20 20 20 20 74 65 6d | 70 5f 78 5b 69 5d 20 3d |.    tem|p_x[i] =|
|000053f0| 20 76 65 72 74 69 63 65 | 73 5b 69 5d 2e 79 20 2d | vertice|s[i].y -|
|00005400| 20 49 6e 74 65 72 73 65 | 63 74 5f 50 6f 69 6e 74 | Interse|ct_Point|
|00005410| 2d 3e 79 3b 0d 20 20 20 | 20 74 65 6d 70 5f 79 5b |->y;.   | temp_y[|
|00005420| 69 5d 20 3d 20 76 65 72 | 74 69 63 65 73 5b 69 5d |i] = ver|tices[i]|
|00005430| 2e 7a 20 2d 20 49 6e 74 | 65 72 73 65 63 74 5f 50 |.z - Int|ersect_P|
|00005440| 6f 69 6e 74 2d 3e 7a 3b | 0d 20 20 20 20 7d 0d 20 |oint->z;|.    }. |
|00005450| 20 65 6c 73 65 0d 20 20 | 20 20 66 6f 72 20 28 69 | else.  |  for (i|
|00005460| 3d 30 3b 69 3c 76 65 72 | 74 65 78 5f 63 6f 75 6e |=0;i<ver|tex_coun|
|00005470| 74 3b 69 2b 2b 29 20 0d | 20 20 20 20 7b 0d 20 20 |t;i++) .|    {.  |
|00005480| 20 20 74 65 6d 70 5f 78 | 5b 69 5d 20 3d 20 76 65 |  temp_x|[i] = ve|
|00005490| 72 74 69 63 65 73 5b 69 | 5d 2e 78 20 2d 20 49 6e |rtices[i|].x - In|
|000054a0| 74 65 72 73 65 63 74 5f | 50 6f 69 6e 74 2d 3e 78 |tersect_|Point->x|
|000054b0| 3b 0d 20 20 20 20 74 65 | 6d 70 5f 79 5b 69 5d 20 |;.    te|mp_y[i] |
|000054c0| 3d 20 76 65 72 74 69 63 | 65 73 5b 69 5d 2e 79 20 |= vertic|es[i].y |
|000054d0| 2d 20 49 6e 74 65 72 73 | 65 63 74 5f 50 6f 69 6e |- Inters|ect_Poin|
|000054e0| 74 2d 3e 79 3b 0d 20 20 | 20 20 7d 0d 20 20 65 6c |t->y;.  |  }.  el|
|000054f0| 73 65 20 69 66 20 28 79 | 3e 7a 29 0d 20 20 20 20 |se if (y|>z).    |
|00005500| 66 6f 72 20 28 69 3d 30 | 3b 69 3c 76 65 72 74 65 |for (i=0|;i<verte|
|00005510| 78 5f 63 6f 75 6e 74 3b | 69 2b 2b 29 20 0d 20 20 |x_count;|i++) .  |
|00005520| 20 20 7b 0d 20 20 20 20 | 74 65 6d 70 5f 78 5b 69 |  {.    |temp_x[i|
|00005530| 5d 20 3d 20 76 65 72 74 | 69 63 65 73 5b 69 5d 2e |] = vert|ices[i].|
|00005540| 78 20 2d 20 49 6e 74 65 | 72 73 65 63 74 5f 50 6f |x - Inte|rsect_Po|
|00005550| 69 6e 74 2d 3e 78 3b 0d | 20 20 20 20 74 65 6d 70 |int->x;.|    temp|
|00005560| 5f 79 5b 69 5d 20 3d 20 | 76 65 72 74 69 63 65 73 |_y[i] = |vertices|
|00005570| 5b 69 5d 2e 7a 20 2d 20 | 49 6e 74 65 72 73 65 63 |[i].z - |Intersec|
|00005580| 74 5f 50 6f 69 6e 74 2d | 3e 7a 3b 0d 20 20 20 20 |t_Point-|>z;.    |
|00005590| 7d 0d 20 20 65 6c 73 65 | 0d 20 20 20 20 66 6f 72 |}.  else|.    for|
|000055a0| 20 28 69 3d 30 3b 69 3c | 76 65 72 74 65 78 5f 63 | (i=0;i<|vertex_c|
|000055b0| 6f 75 6e 74 3b 69 2b 2b | 29 20 0d 20 20 20 20 7b |ount;i++|) .    {|
|000055c0| 0d 20 20 20 20 74 65 6d | 70 5f 78 5b 69 5d 20 3d |.    tem|p_x[i] =|
|000055d0| 20 76 65 72 74 69 63 65 | 73 5b 69 5d 2e 78 20 2d | vertice|s[i].x -|
|000055e0| 20 49 6e 74 65 72 73 65 | 63 74 5f 50 6f 69 6e 74 | Interse|ct_Point|
|000055f0| 2d 3e 78 3b 0d 20 20 20 | 20 74 65 6d 70 5f 79 5b |->x;.   | temp_y[|
|00005600| 69 5d 20 3d 20 76 65 72 | 74 69 63 65 73 5b 69 5d |i] = ver|tices[i]|
|00005610| 2e 79 20 2d 20 49 6e 74 | 65 72 73 65 63 74 5f 50 |.y - Int|ersect_P|
|00005620| 6f 69 6e 74 2d 3e 79 3b | 0d 20 20 20 20 7d 0d 0d |oint->y;|.    }..|
|00005630| 20 20 2f 2a 20 44 65 74 | 65 72 6d 69 6e 65 20 63 |  /* Det|ermine c|
|00005640| 72 6f 73 73 69 6e 67 20 | 63 6f 75 6e 74 20 2a 2f |rossing |count */|
|00005650| 0d 20 20 43 72 6f 73 73 | 69 6e 67 73 20 3d 20 30 |.  Cross|ings = 0|
|00005660| 3b 0d 20 20 69 66 20 28 | 74 65 6d 70 5f 79 5b 30 |;.  if (|temp_y[0|
|00005670| 5d 20 3c 20 30 29 20 53 | 69 67 6e 5f 48 6f 6c 64 |] < 0) S|ign_Hold|
|00005680| 65 72 20 3d 20 2d 31 3b | 0d 20 20 65 6c 73 65 20 |er = -1;|.  else |
|00005690| 53 69 67 6e 5f 48 6f 6c | 64 65 72 20 3d 20 31 3b |Sign_Hol|der = 1;|
|000056a0| 0d 0d 20 20 20 20 66 6f | 72 20 28 69 3d 30 3b 69 |..    fo|r (i=0;i|
|000056b0| 3c 76 65 72 74 65 78 5f | 63 6f 75 6e 74 3b 69 2b |<vertex_|count;i+|
|000056c0| 2b 29 20 0d 20 20 20 20 | 7b 0d 20 20 20 20 2f 2a |+) .    |{.    /*|
|000056d0| 20 53 74 61 72 74 20 6f | 66 20 77 69 6e 64 69 6e | Start o|f windin|
|000056e0| 67 20 74 65 73 74 73 2c | 20 74 65 73 74 20 74 68 |g tests,| test th|
|000056f0| 65 20 73 65 67 6d 65 6e | 74 20 66 72 6f 6d 20 76 |e segmen|t from v|
|00005700| 31 20 74 6f 20 76 32 20 | 2a 2f 0d 20 20 20 20 74 |1 to v2 |*/.    t|
|00005710| 68 69 73 5f 76 65 72 74 | 65 78 20 3d 20 69 3b 0d |his_vert|ex = i;.|
|00005720| 20 20 20 20 6e 65 78 74 | 5f 76 65 72 74 65 78 20 |    next|_vertex |
|00005730| 3d 20 28 69 20 2b 20 31 | 29 20 25 20 76 65 72 74 |= (i + 1|) % vert|
|00005740| 65 78 5f 63 6f 75 6e 74 | 3b 0d 20 20 20 20 69 66 |ex_count|;.    if|
|00005750| 20 28 74 65 6d 70 5f 79 | 5b 6e 65 78 74 5f 76 65 | (temp_y|[next_ve|
|00005760| 72 74 65 78 5d 20 3c 20 | 30 29 20 4e 65 78 74 5f |rtex] < |0) Next_|
|00005770| 53 69 67 6e 20 3d 20 2d | 31 3b 0d 20 20 20 20 65 |Sign = -|1;.    e|
|00005780| 6c 73 65 20 4e 65 78 74 | 5f 53 69 67 6e 20 3d 20 |lse Next|_Sign = |
|00005790| 31 3b 0d 20 20 20 20 69 | 66 20 28 53 69 67 6e 5f |1;.    i|f (Sign_|
|000057a0| 48 6f 6c 64 65 72 20 21 | 3d 20 4e 65 78 74 5f 53 |Holder !|= Next_S|
|000057b0| 69 67 6e 29 20 0d 20 20 | 20 20 20 20 7b 0d 20 20 |ign) .  |    {.  |
|000057c0| 20 20 20 20 69 66 20 28 | 74 65 6d 70 5f 78 5b 74 |    if (|temp_x[t|
|000057d0| 68 69 73 5f 76 65 72 74 | 65 78 5d 20 3e 20 30 2e |his_vert|ex] > 0.|
|000057e0| 30 29 20 0d 20 20 20 20 | 20 20 20 20 7b 0d 20 20 |0) .    |    {.  |
|000057f0| 20 20 20 20 20 20 69 66 | 20 28 74 65 6d 70 5f 78 |      if| (temp_x|
|00005800| 5b 6e 65 78 74 5f 76 65 | 72 74 65 78 5d 20 3e 20 |[next_ve|rtex] > |
|00005810| 30 2e 30 29 0d 20 20 20 | 20 20 20 20 20 20 20 43 |0.0).   |       C|
|00005820| 72 6f 73 73 69 6e 67 73 | 2b 2b 3b 0d 20 20 20 20 |rossings|++;.    |
|00005830| 20 20 20 20 65 6c 73 65 | 20 69 66 20 28 43 72 6f |    else| if (Cro|
|00005840| 73 73 69 6e 67 5f 50 6f | 69 6e 74 28 74 65 6d 70 |ssing_Po|int(temp|
|00005850| 5f 78 5b 74 68 69 73 5f | 76 65 72 74 65 78 5d 2c |_x[this_|vertex],|
|00005860| 20 74 65 6d 70 5f 79 5b | 74 68 69 73 5f 76 65 72 | temp_y[|this_ver|
|00005870| 74 65 78 5d 2c 0d 20 20 | 20 20 20 20 20 20 20 20 |tex],.  |        |
|00005880| 74 65 6d 70 5f 78 5b 6e | 65 78 74 5f 76 65 72 74 |temp_x[n|ext_vert|
|00005890| 65 78 5d 2c 20 74 65 6d | 70 5f 79 5b 6e 65 78 74 |ex], tem|p_y[next|
|000058a0| 5f 76 65 72 74 65 78 5d | 29 20 3e 20 30 2e 30 29 |_vertex]|) > 0.0)|
|000058b0| 0d 20 20 20 20 20 20 20 | 20 20 20 43 72 6f 73 73 |.       |   Cross|
|000058c0| 69 6e 67 73 2b 2b 3b 0d | 20 20 20 20 20 20 20 20 |ings++;.|        |
|000058d0| 7d 0d 20 20 20 20 20 20 | 65 6c 73 65 20 69 66 20 |}.      |else if |
|000058e0| 28 74 65 6d 70 5f 78 5b | 6e 65 78 74 5f 76 65 72 |(temp_x[|next_ver|
|000058f0| 74 65 78 5d 20 3e 20 30 | 2e 30 29 20 0d 20 20 20 |tex] > 0|.0) .   |
|00005900| 20 20 20 20 20 7b 0d 20 | 20 20 20 20 20 20 20 69 |     {. |       i|
|00005910| 66 20 28 43 72 6f 73 73 | 69 6e 67 5f 50 6f 69 6e |f (Cross|ing_Poin|
|00005920| 74 28 74 65 6d 70 5f 78 | 5b 74 68 69 73 5f 76 65 |t(temp_x|[this_ve|
|00005930| 72 74 65 78 5d 2c 20 74 | 65 6d 70 5f 79 5b 74 68 |rtex], t|emp_y[th|
|00005940| 69 73 5f 76 65 72 74 65 | 78 5d 2c 0d 20 20 20 20 |is_verte|x],.    |
|00005950| 20 20 20 20 20 20 74 65 | 6d 70 5f 78 5b 6e 65 78 |      te|mp_x[nex|
|00005960| 74 5f 76 65 72 74 65 78 | 5d 2c 20 74 65 6d 70 5f |t_vertex|], temp_|
|00005970| 79 5b 6e 65 78 74 5f 76 | 65 72 74 65 78 5d 29 20 |y[next_v|ertex]) |
|00005980| 3e 20 30 2e 30 29 0d 20 | 20 20 20 20 20 20 20 20 |> 0.0). |        |
|00005990| 20 43 72 6f 73 73 69 6e | 67 73 2b 2b 3b 0d 20 20 | Crossin|gs++;.  |
|000059a0| 20 20 20 20 20 20 7d 0d | 20 20 20 20 20 20 53 69 |      }.|      Si|
|000059b0| 67 6e 5f 48 6f 6c 64 65 | 72 20 3d 20 4e 65 78 74 |gn_Holde|r = Next|
|000059c0| 5f 53 69 67 6e 3b 0d 20 | 20 20 20 20 20 7d 0d 20 |_Sign;. |     }. |
|000059d0| 20 20 20 7d 0d 0d 20 20 | 72 65 74 75 72 6e 20 28 |   }..  |return (|
|000059e0| 43 72 6f 73 73 69 6e 67 | 73 26 31 29 3b 20 2f 2a |Crossing|s&1); /*|
|000059f0| 20 49 6e 73 69 64 65 20 | 6f 6e 6c 79 20 69 66 20 | Inside |only if |
|00005a00| 23 20 6f 66 20 63 72 6f | 73 73 69 6e 67 73 20 6f |# of cro|ssings o|
|00005a10| 64 64 2e 20 2a 2f 0d 20 | 20 7d 0d 0d 2f 2a 20 54 |dd. */. | }../* T|
|00005a20| 72 61 6e 73 6c 61 74 65 | 20 73 63 72 65 65 6e 20 |ranslate| screen |
|00005a30| 63 6f 6f 72 64 69 6e 61 | 74 65 73 20 69 6e 74 6f |coordina|tes into|
|00005a40| 20 77 6f 72 6c 64 20 63 | 6f 6f 72 64 69 6e 61 74 | world c|oordinat|
|00005a50| 65 73 2e 20 2a 2f 0d 76 | 6f 69 64 20 57 6f 72 6c |es. */.v|oid Worl|
|00005a60| 64 5f 43 6f 6f 72 64 69 | 6e 61 74 65 20 28 56 69 |d_Coordi|nate (Vi|
|00005a70| 65 77 70 6f 69 6e 74 2c | 20 52 61 79 2c 20 77 69 |ewpoint,| Ray, wi|
|00005a80| 64 74 68 2c 20 68 65 69 | 67 68 74 2c 20 78 2c 20 |dth, hei|ght, x, |
|00005a90| 79 29 0d 56 49 45 57 50 | 4f 49 4e 54 20 2a 56 69 |y).VIEWP|OINT *Vi|
|00005aa0| 65 77 70 6f 69 6e 74 3b | 0d 52 41 59 20 2a 52 61 |ewpoint;|.RAY *Ra|
|00005ab0| 79 3b 0d 69 6e 74 20 77 | 69 64 74 68 2c 20 68 65 |y;.int w|idth, he|
|00005ac0| 69 67 68 74 3b 0d 44 42 | 4c 20 78 2c 20 79 3b 0d |ight;.DB|L x, y;.|
|00005ad0| 20 20 7b 0d 20 20 44 42 | 4c 20 73 63 61 6c 65 78 |  {.  DB|L scalex|
|00005ae0| 2c 20 73 63 61 6c 65 79 | 3b 0d 20 20 56 45 43 54 |, scaley|;.  VECT|
|00005af0| 4f 52 20 56 31 2c 20 56 | 32 3b 0d 0d 20 20 2f 2a |OR V1, V|2;..  /*|
|00005b00| 20 43 6f 6e 76 65 72 74 | 20 74 68 65 20 58 20 43 | Convert| the X C|
|00005b10| 6f 6f 72 64 69 6e 61 74 | 65 20 74 6f 20 62 65 20 |oordinat|e to be |
|00005b20| 61 20 44 42 4c 20 66 72 | 6f 6d 20 2d 30 2e 35 20 |a DBL fr|om -0.5 |
|00005b30| 74 6f 20 30 2e 35 20 2a | 2f 0d 20 20 73 63 61 6c |to 0.5 *|/.  scal|
|00005b40| 65 78 20 3d 20 28 78 20 | 2d 20 28 44 42 4c 29 77 |ex = (x |- (DBL)w|
|00005b50| 69 64 74 68 20 2f 20 32 | 2e 30 29 20 2f 20 28 44 |idth / 2|.0) / (D|
|00005b60| 42 4c 29 77 69 64 74 68 | 3b 0d 20 20 2f 2a 20 43 |BL)width|;.  /* C|
|00005b70| 6f 6e 76 65 72 74 20 74 | 68 65 20 59 20 43 6f 6f |onvert t|he Y Coo|
|00005b80| 72 64 69 6e 61 74 65 20 | 74 6f 20 62 65 20 61 20 |rdinate |to be a |
|00005b90| 44 42 4c 20 66 72 6f 6d | 20 2d 30 2e 35 20 74 6f |DBL from| -0.5 to|
|00005ba0| 20 30 2e 35 20 2a 2f 0d | 20 20 73 63 61 6c 65 79 | 0.5 */.|  scaley|
|00005bb0| 20 3d 20 28 28 28 44 42 | 4c 29 28 68 65 69 67 68 | = (((DB|L)(heigh|
|00005bc0| 74 20 2d 20 31 29 20 2d | 20 79 29 20 2d 20 28 44 |t - 1) -| y) - (D|
|00005bd0| 42 4c 29 68 65 69 67 68 | 74 20 2f 20 32 2e 30 29 |BL)heigh|t / 2.0)|
|00005be0| 20 2f 20 28 44 42 4c 29 | 68 65 69 67 68 74 3b 0d | / (DBL)|height;.|
|00005bf0| 20 20 2f 2a 20 43 6f 6d | 70 75 74 65 20 74 68 65 |  /* Com|pute the|
|00005c00| 20 64 69 72 65 63 74 69 | 6f 6e 20 6f 66 20 74 68 | directi|on of th|
|00005c10| 65 20 73 63 72 65 65 6e | 20 70 6f 69 6e 74 20 66 |e screen| point f|
|00005c20| 72 6f 6d 20 74 68 65 20 | 76 69 65 77 70 6f 69 6e |rom the |viewpoin|
|00005c30| 74 20 2a 2f 0d 20 20 56 | 53 63 61 6c 65 20 28 56 |t */.  V|Scale (V|
|00005c40| 31 2c 20 56 69 65 77 70 | 6f 69 6e 74 2d 3e 55 70 |1, Viewp|oint->Up|
|00005c50| 2c 20 73 63 61 6c 65 79 | 29 3b 0d 20 20 56 53 63 |, scaley|);.  VSc|
|00005c60| 61 6c 65 20 28 56 32 2c | 20 56 69 65 77 70 6f 69 |ale (V2,| Viewpoi|
|00005c70| 6e 74 2d 3e 52 69 67 68 | 74 2c 20 73 63 61 6c 65 |nt->Righ|t, scale|
|00005c80| 78 29 3b 0d 20 20 56 41 | 64 64 20 28 52 61 79 2d |x);.  VA|dd (Ray-|
|00005c90| 3e 44 69 72 65 63 74 69 | 6f 6e 2c 20 56 31 2c 20 |>Directi|on, V1, |
|00005ca0| 56 32 29 3b 0d 20 20 56 | 41 64 64 20 28 52 61 79 |V2);.  V|Add (Ray|
|00005cb0| 2d 3e 44 69 72 65 63 74 | 69 6f 6e 2c 20 52 61 79 |->Direct|ion, Ray|
|00005cc0| 2d 3e 44 69 72 65 63 74 | 69 6f 6e 2c 20 56 69 65 |->Direct|ion, Vie|
|00005cd0| 77 70 6f 69 6e 74 2d 3e | 44 69 72 65 63 74 69 6f |wpoint->|Directio|
|00005ce0| 6e 29 3b 0d 20 20 56 4e | 6f 72 6d 61 6c 69 7a 65 |n);.  VN|ormalize|
|00005cf0| 20 28 52 61 79 2d 3e 44 | 69 72 65 63 74 69 6f 6e | (Ray->D|irection|
|00005d00| 2c 20 52 61 79 2d 3e 44 | 69 72 65 63 74 69 6f 6e |, Ray->D|irection|
|00005d10| 29 3b 0d 20 20 7d 0d 2f | 2a 20 55 6e 63 6f 6d 6d |);.  }./|* Uncomm|
|00005d20| 65 6e 74 20 74 68 65 73 | 65 20 74 77 6f 20 64 65 |ent thes|e two de|
|00005d30| 63 6c 61 72 61 74 69 6f | 6e 73 20 69 66 20 79 6f |claratio|ns if yo|
|00005d40| 75 72 20 63 6f 6d 70 69 | 6c 65 72 20 6e 65 65 64 |ur compi|ler need|
|00005d50| 73 20 74 68 65 6d 20 2a | 2f 0d 2f 2a 20 54 68 65 |s them *|/./* The|
|00005d60| 79 20 67 69 76 65 20 4d | 69 63 72 6f 73 6f 66 74 |y give M|icrosoft|
|00005d70| 20 43 20 61 6e 20 6f 75 | 74 20 6f 66 20 6d 61 63 | C an ou|t of mac|
|00005d80| 72 6f 20 65 78 70 61 6e | 73 69 6f 6e 20 73 70 61 |ro expan|sion spa|
|00005d90| 63 65 20 65 72 72 6f 72 | 20 2a 2f 0d 2f 2a 20 76 |ce error| */./* v|
|00005da0| 6f 69 64 20 73 68 6f 77 | 5f 75 6e 69 76 61 72 69 |oid show|_univari|
|00005db0| 61 74 65 5f 70 6f 6c 79 | 20 50 41 52 41 4d 53 28 |ate_poly| PARAMS(|
|00005dc0| 28 63 68 61 72 20 2a 6c | 61 62 65 6c 2c 20 69 6e |(char *l|abel, in|
|00005dd0| 74 20 6f 72 64 65 72 2c | 44 42 4c 20 2a 43 6f 65 |t order,|DBL *Coe|
|00005de0| 66 66 73 29 29 3b 20 20 | 2a 2f 0d 2f 2a 20 76 6f |ffs));  |*/./* vo|
|00005df0| 69 64 20 73 68 6f 77 5f | 70 6f 69 6e 74 73 20 50 |id show_|points P|
|00005e00| 41 52 41 4d 53 28 28 63 | 68 61 72 20 2a 6c 61 62 |ARAMS((c|har *lab|
|00005e10| 65 6c 2c 69 6e 74 20 63 | 6f 75 6e 74 2c 44 42 4c |el,int c|ount,DBL|
|00005e20| 20 2a 70 6f 69 6e 74 5f | 6c 69 73 74 29 3b 20 20 | *point_|list);  |
|00005e30| 2a 2f 0d 0d 76 6f 69 64 | 20 73 68 6f 77 5f 75 6e |*/..void| show_un|
|00005e40| 69 76 61 72 69 61 74 65 | 5f 70 6f 6c 79 28 6c 61 |ivariate|_poly(la|
|00005e50| 62 65 6c 2c 20 6f 72 64 | 65 72 2c 20 43 6f 65 66 |bel, ord|er, Coef|
|00005e60| 66 73 29 0d 63 68 61 72 | 20 2a 6c 61 62 65 6c 3b |fs).char| *label;|
|00005e70| 0d 69 6e 74 20 6f 72 64 | 65 72 3b 0d 44 42 4c 20 |.int ord|er;.DBL |
|00005e80| 2a 43 6f 65 66 66 73 3b | 0d 20 20 7b 0d 20 20 69 |*Coeffs;|.  {.  i|
|00005e90| 6e 74 20 69 3b 0d 20 20 | 70 72 69 6e 74 66 28 22 |nt i;.  |printf("|
|00005ea0| 25 73 22 2c 20 6c 61 62 | 65 6c 29 3b 0d 20 20 66 |%s", lab|el);.  f|
|00005eb0| 6f 72 20 28 69 3d 30 3b | 69 3c 3d 6f 72 64 65 72 |or (i=0;|i<=order|
|00005ec0| 3b 69 2b 2b 29 20 0d 20 | 20 20 20 7b 0d 20 20 20 |;i++) . |   {.   |
|00005ed0| 20 70 72 69 6e 74 66 28 | 22 25 2e 32 6c 66 20 78 | printf(|"%.2lf x|
|00005ee0| 5e 25 64 22 2c 20 43 6f | 65 66 66 73 5b 69 5d 2c |^%d", Co|effs[i],|
|00005ef0| 20 6f 72 64 65 72 2d 69 | 29 3b 0d 20 20 20 20 69 | order-i|);.    i|
|00005f00| 66 20 28 69 3d 3d 6f 72 | 64 65 72 29 20 70 72 69 |f (i==or|der) pri|
|00005f10| 6e 74 66 28 22 5c 6e 22 | 29 3b 0d 20 20 20 20 65 |ntf("\n"|);.    e|
|00005f20| 6c 73 65 20 70 72 69 6e | 74 66 28 22 20 2b 20 22 |lse prin|tf(" + "|
|00005f30| 29 3b 0d 20 20 20 20 7d | 0d 20 20 7d 0d 0d 20 20 |);.    }|.  }..  |
|00005f40| 76 6f 69 64 20 73 68 6f | 77 5f 70 6f 69 6e 74 73 |void sho|w_points|
|00005f50| 28 6c 61 62 65 6c 2c 20 | 63 6f 75 6e 74 2c 20 70 |(label, |count, p|
|00005f60| 6f 69 6e 74 5f 6c 69 73 | 74 29 0d 20 20 20 20 63 |oint_lis|t).    c|
|00005f70| 68 61 72 20 2a 6c 61 62 | 65 6c 3b 0d 69 6e 74 20 |har *lab|el;.int |
|00005f80| 63 6f 75 6e 74 3b 0d 44 | 42 4c 20 2a 70 6f 69 6e |count;.D|BL *poin|
|00005f90| 74 5f 6c 69 73 74 3b 0d | 20 20 7b 0d 20 20 69 6e |t_list;.|  {.  in|
|00005fa0| 74 20 69 3b 0d 20 20 70 | 72 69 6e 74 66 28 22 25 |t i;.  p|rintf("%|
|00005fb0| 73 22 2c 20 6c 61 62 65 | 6c 29 3b 0d 20 20 66 6f |s", labe|l);.  fo|
|00005fc0| 72 20 28 69 3d 30 3b 69 | 3c 63 6f 75 6e 74 3b 69 |r (i=0;i|<count;i|
|00005fd0| 2b 2b 29 20 0d 20 20 20 | 20 7b 0d 20 20 20 20 70 |++) .   | {.    p|
|00005fe0| 72 69 6e 74 66 28 22 25 | 6c 66 22 2c 20 70 6f 69 |rintf("%|lf", poi|
|00005ff0| 6e 74 5f 6c 69 73 74 5b | 69 5d 29 3b 0d 20 20 20 |nt_list[|i]);.   |
|00006000| 20 69 66 20 28 69 3d 3d | 28 63 6f 75 6e 74 2d 31 | if (i==|(count-1|
|00006010| 29 29 20 70 72 69 6e 74 | 66 28 22 5c 6e 22 29 3b |)) print|f("\n");|
|00006020| 0d 20 20 20 20 65 6c 73 | 65 20 70 72 69 6e 74 66 |.    els|e printf|
|00006030| 28 22 2c 20 22 29 3b 0d | 20 20 20 20 7d 0d 20 20 |(", ");.|    }.  |
|00006040| 7d 0d 0d 23 65 6e 64 69 | 66 0d 00 00 00 00 00 00 |}..#endi|f.......|
|00006050| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00006060| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00006070| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00006080| 00 00 01 00 00 00 01 6e | 00 00 00 6e 00 00 00 3e |.......n|...n...>|
|00006090| 69 61 6e 67 6c 65 2d 3e | 50 33 2e 78 20 2d 20 73 |iangle->|P3.x - s|
|000060a0| 29 2a 28 54 72 69 61 6e | 67 6c 65 2d 3e 50 33 2e |)*(Trian|gle->P3.|
|000060b0| 06 56 45 43 54 2e 43 da | 02 00 00 00 54 45 58 54 |.VECT.C.|....TEXT|
|000060c0| 4d 50 53 20 01 08 ff ff | ff ff 00 00 00 00 16 b9 |MPS ....|........|
|000060d0| 00 00 54 45 58 54 4d 50 | 53 20 01 08 ff ff ff ff |..TEXTMP|S ......|
|000060e0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000060f0| 00 00 a8 d4 67 44 00 00 | 5f ca 00 00 01 ac 32 2e |....gD..|_.....2.|
|00006100| 78 29 29 0d 20 20 20 20 | 20 20 72 65 74 75 72 6e |x)).    |  return|
|00006110| 20 28 46 41 4c 53 45 29 | 3b 0d 0d 20 20 20 20 69 | (FALSE)|;..    i|
|00006120| 66 20 28 28 54 72 69 61 | 6e 67 6c 65 2d 3e 50 31 |f ((Tria|ngle->P1|
|00006130| 2e 78 20 2d 20 73 29 2a | 28 54 72 69 61 6e 67 6c |.x - s)*|(Triangl|
|00006140| 65 2d 3e 50 31 2e 7a 20 | 2d 20 54 72 69 61 6e 67 |e->P1.z |- Triang|
|00006150| 6c 65 2d 3e 50 33 2e 7a | 29 20 3c 0d 20 20 20 20 |le->P3.z|) <.    |
|00006160| 20 20 28 54 72 69 61 6e | 67 6c 65 2d 3e 50 31 2e |  (Trian|gle->P1.|
|00006170| 7a 20 2d 20 74 29 2a 28 | 54 72 69 61 6e 67 6c 65 |z - t)*(|Triangle|
|00006180| 00 00 00 48 00 09 4d 6f | 6e 61 63 6f 00 2a 2a 2a |...H..Mo|naco.***|
|00006190| 2a 2a 2a 2a 2a 2a 2a 2a | 2a 2a 2a 2a 2a 2a 2a 2a |********|********|
|000061a0| 2a 0d 2a 20 20 20 00 06 | 00 04 00 3c 00 24 01 dd |*.*   ..|...<.$..|
|000061b0| 02 3d 00 3c 00 24 01 dd | 02 3d a8 d4 67 44 00 00 |.=.<.$..|.=..gD..|
|000061c0| 00 00 00 00 00 00 00 00 | 00 00 01 00 00 00 00 1e |........|........|
|000061d0| 00 3c 00 24 01 dd 02 3d | 00 3c 00 24 01 dd 02 3d |.<.$...=|.<.$...=|
|000061e0| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|000061f0| 01 00 00 00 01 6e 00 00 | 00 6e 00 00 00 3e 00 8d |.....n..|.n...>..|
|00006200| 3b f8 18 3e 00 00 00 1c | 00 3e 00 00 4d 50 53 52 |;..>....|.>..MPSR|
|00006210| 00 01 00 0a 03 ed ff ff | 00 00 00 00 00 00 00 00 |........|........|
|00006220| 03 f0 ff ff 00 00 00 4c | 00 8f 92 8c 00 00 00 00 |.......L|........|
|00006230| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00006240| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00006250| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00006260| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
|00006270| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........|
+--------+-------------------------+-------------------------+--------+--------+