home *** CD-ROM | disk | FTP | other *** search
open in:
MacOS 8.1
|
Win98
|
DOS
browse contents |
view JSON data
|
view as text
This file was processed as: LaTeX Document
(document/latex).
Confidence | Program | Detection | Match Type | Support
|
---|
100%
| dexvert
| LaTeX Document (document/latex)
| magic
| Supported |
1%
| dexvert
| Corel 10 Texture (image/corel10Texture)
| ext
| Unsupported |
1%
| dexvert
| Croteam texture file (image/croteamTextureFile)
| ext
| Unsupported |
1%
| dexvert
| Text File (text/txt)
| fallback
| Supported |
100%
| file
| LaTeX document text
| default
| |
99%
| file
| LaTeX document, ASCII text, with CRLF line terminators
| default
| |
100%
| checkBytes
| Printable ASCII
| default
| |
100%
| perlTextCheck
| Likely Text (Perl)
| default
| |
100%
| siegfried
| fmt/281 LaTeX (Subdocument)
| default
| |
100%
| detectItEasy
| Format: plain text[CRLF]
| default (weak)
|
|
hex view+--------+-------------------------+-------------------------+--------+--------+
|00000000| 5c 63 68 61 70 74 65 72 | 7b 49 6e 74 72 6f 64 75 |\chapter|{Introdu|
|00000010| 63 74 69 6f 6e 7d 0d 0a | 5c 73 65 63 74 69 6f 6e |ction}..|\section|
|00000020| 7b 57 68 61 74 20 69 73 | 20 74 68 65 20 50 72 6f |{What is| the Pro|
|00000030| 62 6c 65 6d 3f 7d 0d 0a | 49 20 77 72 6f 74 65 20 |blem?}..|I wrote |
|00000040| 74 68 69 73 20 63 6f 64 | 65 20 74 6f 20 73 6f 6c |this cod|e to sol|
|00000050| 76 65 20 73 6f 6d 65 20 | 6f 66 20 6d 79 20 6d 61 |ve some |of my ma|
|00000060| 74 72 69 78 20 61 6c 67 | 65 62 72 61 0d 0a 70 72 |trix alg|ebra..pr|
|00000070| 6f 62 6c 65 6d 73 20 66 | 6f 72 20 6d 79 20 72 65 |oblems f|or my re|
|00000080| 73 65 61 72 63 68 2e 20 | 49 20 6e 65 65 64 65 64 |search. |I needed|
|00000090| 20 61 20 67 65 6e 65 72 | 69 63 20 70 72 6f 67 72 | a gener|ic progr|
|000000a0| 61 6d 20 74 68 61 74 0d | 0a 63 6f 75 6c 64 20 68 |am that.|.could h|
|000000b0| 61 6e 64 6c 65 20 6d 61 | 74 72 69 78 20 61 6c 67 |andle ma|trix alg|
|000000c0| 65 62 72 61 20 6f 6e 20 | 61 20 50 43 2c 20 66 6f |ebra on |a PC, fo|
|000000d0| 72 20 69 6e 73 74 61 6e | 63 65 20 0d 0a 24 24 0d |r instan|ce ..$$.|
|000000e0| 0a 5c 68 61 74 20 5c 62 | 65 74 61 20 3d 20 28 58 |.\hat \b|eta = (X|
|000000f0| 27 58 29 5e 7b 2d 31 7d | 58 27 79 0d 0a 24 24 20 |'X)^{-1}|X'y..$$ |
|00000100| 0d 0a 54 68 65 20 63 6f | 6d 70 75 74 65 72 20 6c |..The co|mputer l|
|00000110| 61 6e 67 75 61 67 65 73 | 20 49 27 76 65 20 75 73 |anguages| I've us|
|00000120| 65 64 20 66 6f 72 20 74 | 68 69 73 20 70 72 6f 62 |ed for t|his prob|
|00000130| 6c 65 6d 20 6a 75 73 74 | 20 77 65 72 65 0d 0a 6e |lem just| were..n|
|00000140| 6f 74 20 61 67 69 6c 65 | 20 65 6e 6f 75 67 68 20 |ot agile| enough |
|00000150| 74 6f 20 68 61 6e 64 6c | 65 20 61 6c 6f 74 20 6f |to handl|e alot o|
|00000160| 66 20 6d 61 74 72 69 78 | 20 61 6c 67 65 62 72 61 |f matrix| algebra|
|00000170| 20 28 46 4f 52 54 52 41 | 4e 2c 0d 0a 50 61 73 63 | (FORTRA|N,..Pasc|
|00000180| 61 6c 20 61 6e 64 20 43 | 29 2e 20 54 68 69 73 20 |al and C|). This |
|00000190| 43 2b 2b 20 69 6e 63 61 | 72 6e 61 74 69 6f 6e 20 |C++ inca|rnation |
|000001a0| 6f 66 20 6d 79 20 61 74 | 74 65 6d 70 74 73 20 74 |of my at|tempts t|
|000001b0| 6f 20 73 6f 6c 76 65 0d | 0a 74 68 69 73 20 70 72 |o solve.|.this pr|
|000001c0| 6f 62 6c 65 6d 20 77 6f | 75 6c 64 20 63 6f 64 65 |oblem wo|uld code|
|000001d0| 20 74 68 65 20 65 71 75 | 61 74 69 6f 6e 20 61 62 | the equ|ation ab|
|000001e0| 6f 76 65 20 61 73 0d 0a | 0d 0a 5c 62 65 67 69 6e |ove as..|..\begin|
|000001f0| 7b 76 65 72 62 61 74 69 | 6d 7d 0d 0a 20 20 20 20 |{verbati|m}.. |
|00000200| 20 20 56 4d 61 74 72 69 | 78 20 62 2c 58 2c 59 3b | VMatri|x b,X,Y;|
|00000210| 0d 0a 20 20 20 20 20 20 | 2f 2f 20 63 6f 64 65 20 |.. |// code |
|00000220| 74 6f 20 72 65 61 64 20 | 69 6e 20 58 20 61 6e 64 |to read |in X and|
|00000230| 20 59 0d 0a 20 20 20 20 | 20 20 20 20 20 20 20 2e | Y.. | .|
|00000240| 0d 0a 20 20 20 20 20 20 | 20 20 20 20 20 2e 0d 0a |.. | ...|
|00000250| 20 20 20 20 20 20 20 20 | 20 20 20 2e 0d 0a 20 20 | | ... |
|00000260| 20 20 20 20 62 20 3d 20 | 49 6e 76 28 54 72 61 6e | b = |Inv(Tran|
|00000270| 28 58 29 2a 58 29 2a 54 | 72 61 6e 28 58 29 2a 59 |(X)*X)*T|ran(X)*Y|
|00000280| 3b 0d 0a 5c 65 6e 64 7b | 76 65 72 62 61 74 69 6d |;..\end{|verbatim|
|00000290| 7d 0d 0a 0d 0a 5c 6e 6f | 69 6e 64 65 6e 74 20 77 |}....\no|indent w|
|000002a0| 68 65 72 65 20 62 2c 20 | 58 2c 20 61 6e 64 20 59 |here b, |X, and Y|
|000002b0| 20 61 72 65 20 6d 61 74 | 72 69 63 65 73 2e 20 54 | are mat|rices. T|
|000002c0| 68 69 73 20 6e 6f 74 20 | 71 75 69 74 65 20 61 73 |his not |quite as|
|000002d0| 0d 0a 70 6f 77 65 72 66 | 75 6c 20 61 73 20 73 6f |..powerf|ul as so|
|000002e0| 6d 65 20 6d 61 74 72 69 | 78 20 69 6e 74 65 72 70 |me matri|x interp|
|000002f0| 72 65 74 65 72 73 2c 20 | 62 75 74 20 69 74 20 77 |reters, |but it w|
|00000300| 6f 72 6b 73 2e 20 43 2b | 2b 20 69 73 0d 0a 70 6f |orks. C+|+ is..po|
|00000310| 77 65 72 66 75 6c 20 65 | 6e 6f 75 67 68 20 66 6f |werful e|nough fo|
|00000320| 72 20 74 68 69 73 20 70 | 72 6f 62 6c 65 6d 20 73 |r this p|roblem s|
|00000330| 69 6e 63 65 20 69 74 20 | 63 61 6e 20 6f 76 65 72 |ince it |can over|
|00000340| 6c 6f 61 64 0d 0a 66 75 | 6e 63 74 69 6f 6e 73 20 |load..fu|nctions |
|00000350| 66 6f 72 20 6f 62 6a 65 | 63 74 73 2e 20 49 74 20 |for obje|cts. It |
|00000360| 61 6c 73 6f 20 66 6c 65 | 78 69 62 6c 65 20 65 6e |also fle|xible en|
|00000370| 6f 75 67 68 20 74 6f 20 | 68 61 6e 64 6c 65 0d 0a |ough to |handle..|
|00000380| 74 68 65 20 72 65 63 75 | 72 73 69 76 65 20 61 6e |the recu|rsive an|
|00000390| 64 20 6e 65 73 74 65 64 | 20 66 75 6e 63 74 69 6f |d nested| functio|
|000003a0| 6e 20 63 61 6c 6c 73 20 | 6e 65 65 64 65 64 20 66 |n calls |needed f|
|000003b0| 6f 72 20 6d 61 74 72 69 | 78 0d 0a 61 6c 67 65 62 |or matri|x..algeb|
|000003c0| 72 61 2e 0d 0a 0d 0a 54 | 68 65 20 6e 65 78 74 20 |ra.....T|he next |
|000003d0| 70 72 6f 62 6c 65 6d 20 | 77 61 73 20 74 6f 20 6d |problem |was to m|
|000003e0| 69 6e 69 6d 69 7a 65 20 | 74 68 65 20 6f 76 65 72 |inimize |the over|
|000003f0| 68 65 61 64 20 6f 66 0d | 0a 6d 61 6e 69 70 75 6c |head of.|.manipul|
|00000400| 61 74 69 6e 67 20 74 68 | 65 20 64 61 74 61 20 73 |ating th|e data s|
|00000410| 74 72 75 63 74 75 72 65 | 73 20 72 65 71 75 69 72 |tructure|s requir|
|00000420| 65 64 20 66 6f 72 20 6b | 65 65 70 69 6e 67 20 74 |ed for k|eeping t|
|00000430| 72 61 63 6b 0d 0a 6f 66 | 20 61 6c 6c 20 6f 66 20 |rack..of| all of |
|00000440| 74 68 65 20 6d 61 74 72 | 69 63 65 73 2e 20 54 68 |the matr|ices. Th|
|00000450| 65 20 61 62 69 6c 69 74 | 79 20 74 6f 20 62 69 6e |e abilit|y to bin|
|00000460| 64 20 66 75 6e 63 74 69 | 6f 6e 73 20 74 6f 0d 0a |d functi|ons to..|
|00000470| 64 61 74 61 20 73 74 72 | 75 63 74 75 72 65 73 20 |data str|uctures |
|00000480| 75 73 69 6e 67 20 6f 62 | 6a 65 63 74 73 20 73 6f |using ob|jects so|
|00000490| 6c 76 65 64 20 74 68 69 | 73 20 70 72 6f 62 6c 65 |lved thi|s proble|
|000004a0| 6d 2e 20 49 74 20 77 61 | 73 0d 0a 72 65 61 73 6f |m. It wa|s..reaso|
|000004b0| 6e 61 62 6c 79 20 65 61 | 73 79 20 74 6f 20 77 72 |nably ea|sy to wr|
|000004c0| 69 74 65 20 61 20 73 74 | 61 63 6b 20 6f 66 20 6d |ite a st|ack of m|
|000004d0| 61 74 72 69 63 65 73 20 | 74 6f 20 6b 65 65 70 20 |atrices |to keep |
|000004e0| 74 72 61 63 6b 0d 0a 6f | 66 20 74 68 65 20 69 6e |track..o|f the in|
|000004f0| 74 65 72 6d 65 64 69 61 | 74 65 20 63 61 6c 63 75 |termedia|te calcu|
|00000500| 6c 61 74 69 6f 6e 73 2c | 20 61 6e 64 20 74 68 65 |lations,| and the|
|00000510| 6e 20 64 65 6c 65 74 65 | 20 74 68 65 0d 0a 75 6e |n delete| the..un|
|00000520| 6e 65 63 65 73 73 61 72 | 79 20 6d 61 74 72 69 63 |necessar|y matric|
|00000530| 65 73 2e 20 49 74 20 61 | 6c 73 6f 20 6d 61 64 65 |es. It a|lso made|
|00000540| 20 69 74 20 72 65 61 73 | 6f 6e 61 62 6c 79 20 65 | it reas|onably e|
|00000550| 61 73 79 20 74 6f 0d 0a | 61 73 73 69 67 6e 20 6d |asy to..|assign m|
|00000560| 61 74 72 69 63 65 73 20 | 75 73 69 6e 67 20 74 68 |atrices |using th|
|00000570| 65 20 65 71 75 61 6c 73 | 20 6f 70 65 72 61 74 6f |e equals| operato|
|00000580| 72 2e 0d 0a 0d 0a 54 68 | 65 20 6e 65 78 74 20 70 |r.....Th|e next p|
|00000590| 72 6f 62 6c 65 6d 20 77 | 61 73 20 74 68 65 20 36 |roblem w|as the 6|
|000005a0| 34 30 4b 20 62 61 72 72 | 69 65 72 20 6f 6e 20 74 |40K barr|ier on t|
|000005b0| 68 65 20 50 43 20 63 6c | 61 73 73 0d 0a 6d 61 63 |he PC cl|ass..mac|
|000005c0| 68 69 6e 65 73 2e 20 4d | 6f 73 74 20 6f 66 20 6d |hines. M|ost of m|
|000005d0| 79 20 70 72 6f 62 6c 65 | 6d 73 20 61 72 65 20 74 |y proble|ms are t|
|000005e0| 6f 6f 20 62 69 67 20 74 | 6f 20 6b 65 65 70 20 61 |oo big t|o keep a|
|000005f0| 6c 6c 20 6f 66 0d 0a 74 | 68 65 20 64 61 74 61 20 |ll of..t|he data |
|00000600| 69 6e 20 52 41 4d 2c 20 | 73 6f 20 49 20 68 61 76 |in RAM, |so I hav|
|00000610| 65 20 74 6f 20 72 65 73 | 6f 72 74 20 74 6f 20 73 |e to res|ort to s|
|00000620| 70 65 63 69 61 6c 20 64 | 61 74 61 20 49 4f 0d 0a |pecial d|ata IO..|
|00000630| 74 72 69 63 6b 73 2e 20 | 54 68 69 73 20 69 73 20 |tricks. |This is |
|00000640| 66 69 6e 65 20 66 6f 72 | 20 61 6e 79 20 73 69 6e |fine for| any sin|
|00000650| 67 6c 65 20 70 72 6f 62 | 6c 65 6d 2c 20 62 75 74 |gle prob|lem, but|
|00000660| 20 69 74 20 64 6f 65 73 | 0d 0a 6e 6f 74 20 67 65 | it does|..not ge|
|00000670| 6e 65 72 61 6c 69 7a 65 | 20 76 65 72 79 20 77 65 |neralize| very we|
|00000680| 6c 6c 2e 20 49 20 66 6f | 75 6e 64 20 61 20 76 69 |ll. I fo|und a vi|
|00000690| 72 74 75 61 6c 20 6d 65 | 6d 6f 72 79 20 61 6c 6c |rtual me|mory all|
|000006a0| 6f 63 61 74 6f 72 0d 0a | 74 6f 20 73 6f 6c 76 65 |ocator..|to solve|
|000006b0| 20 74 68 69 73 20 70 72 | 6f 62 6c 65 6d 2e 20 49 | this pr|oblem. I|
|000006c0| 74 20 77 72 69 74 65 73 | 20 74 6f 20 61 20 64 69 |t writes| to a di|
|000006d0| 73 6b 20 66 69 6c 65 2e | 20 54 68 65 20 6e 65 77 |sk file.| The new|
|000006e0| 0d 0a 63 6c 61 73 73 20 | 6f 66 20 33 38 36 20 6d |..class |of 386 m|
|000006f0| 61 63 68 69 6e 65 73 20 | 63 61 6e 20 6b 65 65 70 |achines |can keep|
|00000700| 20 52 41 4d 20 64 69 73 | 6b 73 20 69 6e 20 65 78 | RAM dis|ks in ex|
|00000710| 74 65 6e 64 65 64 0d 0a | 6d 65 6d 6f 72 79 2c 20 |tended..|memory, |
|00000720| 73 6f 20 74 68 65 72 65 | 20 69 73 20 6f 6e 6c 79 |so there| is only|
|00000730| 20 61 20 6d 6f 64 65 73 | 74 20 70 65 72 66 6f 72 | a modes|t perfor|
|00000740| 6d 61 6e 63 65 20 70 65 | 6e 61 6c 74 79 20 66 6f |mance pe|nalty fo|
|00000750| 72 0d 0a 61 63 63 65 73 | 73 69 6e 67 20 74 68 65 |r..acces|sing the|
|00000760| 20 64 61 74 61 20 69 6e | 20 65 6c 65 63 74 72 6f | data in| electro|
|00000770| 6e 69 63 20 6d 65 6d 6f | 72 79 2e 20 20 54 68 65 |nic memo|ry. The|
|00000780| 20 76 69 72 74 75 61 6c | 0d 0a 6d 65 6d 6f 72 79 | virtual|..memory|
|00000790| 20 73 63 68 65 6d 65 20 | 6f 6e 6c 79 20 6b 65 65 | scheme |only kee|
|000007a0| 70 73 20 61 62 6f 75 74 | 20 36 34 20 65 6c 65 6d |ps about| 64 elem|
|000007b0| 65 6e 74 73 20 69 6e 20 | 52 41 4d 2c 20 73 6f 0d |ents in |RAM, so.|
|000007c0| 0a 66 69 6e 64 69 6e 67 | 20 65 6c 65 6d 65 6e 74 |.finding| element|
|000007d0| 73 20 69 6e 20 61 20 6c | 61 72 67 65 20 6d 61 74 |s in a l|arge mat|
|000007e0| 72 69 78 20 69 73 20 73 | 6c 6f 77 65 72 20 74 68 |rix is s|lower th|
|000007f0| 61 6e 20 64 69 72 65 63 | 74 0d 0a 61 63 63 65 73 |an direc|t..acces|
|00000800| 73 20 74 6f 20 74 68 65 | 20 68 65 61 70 2e 20 54 |s to the| heap. T|
|00000810| 68 69 73 20 74 72 61 64 | 65 6f 66 66 20 73 75 69 |his trad|eoff sui|
|00000820| 74 65 64 20 6d 65 20 73 | 69 6e 63 65 20 49 20 77 |ted me s|ince I w|
|00000830| 6f 75 6c 64 0d 0a 72 61 | 74 68 65 72 20 62 65 20 |ould..ra|ther be |
|00000840| 61 62 6c 65 20 74 6f 20 | 75 73 65 20 6d 61 74 72 |able to |use matr|
|00000850| 69 63 65 73 20 74 68 61 | 74 20 61 72 65 20 70 6f |ices tha|t are po|
|00000860| 74 65 6e 74 69 61 6c 6c | 79 20 6c 61 72 67 65 72 |tentiall|y larger|
|00000870| 0d 0a 74 68 61 6e 20 36 | 34 30 4b 2e 20 41 20 33 |..than 6|40K. A 3|
|00000880| 20 6f 72 20 34 20 6d 65 | 67 20 52 41 4d 20 64 69 | or 4 me|g RAM di|
|00000890| 73 6b 20 77 6f 75 6c 64 | 20 73 75 69 74 20 6d 6f |sk would| suit mo|
|000008a0| 73 74 20 6f 66 20 6d 79 | 0d 0a 6e 65 65 64 73 2e |st of my|..needs.|
|000008b0| 20 54 68 65 20 68 61 72 | 64 20 64 72 69 76 65 20 | The har|d drive |
|000008c0| 63 61 6e 20 62 65 20 75 | 73 65 64 20 74 6f 6f 2e |can be u|sed too.|
|000008d0| 20 41 67 61 69 6e 2c 20 | 69 74 20 61 69 6e 27 74 | Again, |it ain't|
|000008e0| 0d 0a 66 61 73 74 2c 20 | 62 75 74 20 69 74 20 77 |..fast, |but it w|
|000008f0| 6f 72 6b 73 2e 0d 0a 0d | 0a 54 68 65 20 70 72 6f |orks....|.The pro|
|00000900| 70 65 72 74 69 65 73 20 | 6f 66 20 43 2b 2b 20 68 |perties |of C++ h|
|00000910| 65 6c 70 65 64 20 73 6f | 6c 76 65 20 74 68 65 73 |elped so|lve thes|
|00000920| 65 20 70 72 6f 62 6c 65 | 6d 73 2e 20 54 68 69 73 |e proble|ms. This|
|00000930| 0d 0a 64 6f 63 75 6d 65 | 6e 74 20 77 69 6c 6c 20 |..docume|nt will |
|00000940| 64 65 76 65 6c 6f 70 20 | 61 20 6d 61 74 72 69 78 |develop |a matrix|
|00000950| 20 63 6c 61 73 73 20 66 | 72 6f 6d 20 6c 6f 77 65 | class f|rom lowe|
|00000960| 72 20 6c 65 76 65 6c 0d | 0a 66 75 6e 63 74 69 6f |r level.|.functio|
|00000970| 6e 73 2e 20 54 68 65 20 | 66 69 72 73 74 20 73 74 |ns. The |first st|
|00000980| 65 70 20 69 73 20 74 6f | 20 65 78 70 6c 61 69 6e |ep is to| explain|
|00000990| 20 74 68 65 20 76 69 72 | 74 75 61 6c 20 6d 65 6d | the vir|tual mem|
|000009a0| 6f 72 79 0d 0a 61 6c 6c | 6f 63 61 74 6f 72 2e 20 |ory..all|ocator. |
|000009b0| 54 68 65 20 6e 65 78 74 | 20 73 74 65 70 20 69 73 |The next| step is|
|000009c0| 20 74 6f 20 65 78 70 6c | 61 69 6e 20 74 68 65 20 | to expl|ain the |
|000009d0| 73 74 72 69 6e 67 0d 0a | 66 75 6e 63 74 69 6f 6e |string..|function|
|000009e0| 73 2e 20 54 68 65 20 74 | 68 69 72 64 20 73 74 65 |s. The t|hird ste|
|000009f0| 70 20 69 73 20 74 6f 20 | 64 65 76 65 6c 6f 70 20 |p is to |develop |
|00000a00| 61 20 76 65 63 74 6f 72 | 20 63 6c 61 73 73 0d 0a |a vector| class..|
|00000a10| 62 61 73 65 64 20 6f 6e | 20 74 68 65 20 76 69 72 |based on| the vir|
|00000a20| 74 75 61 6c 20 6d 65 6d | 6f 72 79 20 61 6c 6c 6f |tual mem|ory allo|
|00000a30| 63 61 74 6f 72 2e 20 54 | 68 65 20 6e 65 78 74 20 |cator. T|he next |
|00000a40| 73 74 65 70 20 69 73 20 | 74 6f 0d 0a 65 78 70 6c |step is |to..expl|
|00000a50| 61 69 6e 20 74 68 65 20 | 6d 61 74 72 69 78 20 63 |ain the |matrix c|
|00000a60| 6c 61 73 73 20 61 6e 64 | 20 6d 61 74 72 69 78 20 |lass and| matrix |
|00000a70| 73 74 61 63 6b 20 63 6c | 61 73 73 65 73 2e 20 54 |stack cl|asses. T|
|00000a80| 68 65 0d 0a 76 69 72 74 | 75 61 6c 20 76 65 63 74 |he..virt|ual vect|
|00000a90| 6f 72 20 63 6c 61 73 73 | 20 69 73 20 74 68 65 20 |or class| is the |
|00000aa0| 62 61 73 65 20 63 6c 61 | 73 73 20 66 6f 72 20 74 |base cla|ss for t|
|00000ab0| 68 65 20 6d 61 74 72 69 | 78 0d 0a 63 6c 61 73 73 |he matri|x..class|
|00000ac0| 2e 20 54 68 65 20 6d 61 | 74 72 69 78 20 73 74 61 |. The ma|trix sta|
|00000ad0| 63 6b 20 69 73 20 64 65 | 72 69 76 65 64 20 66 72 |ck is de|rived fr|
|00000ae0| 6f 6d 20 74 68 65 20 6d | 61 74 72 69 78 20 63 6c |om the m|atrix cl|
|00000af0| 61 73 73 2e 0d 0a 44 65 | 76 65 6c 6f 70 69 6e 67 |ass...De|veloping|
|00000b00| 20 61 20 67 65 6e 65 72 | 61 6c 20 6d 61 74 72 69 | a gener|al matri|
|00000b10| 78 20 70 72 6f 67 72 61 | 6d 20 69 73 20 68 61 72 |x progra|m is har|
|00000b20| 64 20 77 69 74 68 6f 75 | 74 0d 0a 65 6e 63 61 70 |d withou|t..encap|
|00000b30| 73 75 6c 61 74 69 6f 6e | 20 61 6e 64 20 69 6e 68 |sulation| and inh|
|00000b40| 65 72 69 74 61 6e 63 65 | 2e 0d 0a 0d 0a 0d 0a 0d |eritance|........|
|00000b50| 0a 5c 73 65 63 74 69 6f | 6e 7b 57 68 6f 20 4f 77 |.\sectio|n{Who Ow|
|00000b60| 6e 73 20 74 68 65 20 43 | 6f 64 65 3f 7d 0d 0a 4d |ns the C|ode?}..M|
|00000b70| 79 20 70 68 69 6c 6f 73 | 6f 70 68 79 20 61 62 6f |y philos|ophy abo|
|00000b80| 75 74 20 74 68 65 20 63 | 6f 64 65 20 69 73 20 74 |ut the c|ode is t|
|00000b90| 68 61 74 20 69 74 20 73 | 68 6f 75 6c 64 20 70 72 |hat it s|hould pr|
|00000ba0| 6f 76 69 64 65 20 61 0d | 0a 62 61 73 69 63 20 64 |ovide a.|.basic d|
|00000bb0| 61 74 61 20 73 74 72 75 | 63 74 75 72 65 20 66 6f |ata stru|cture fo|
|00000bc0| 72 20 74 68 65 20 75 73 | 65 72 20 74 6f 20 64 65 |r the us|er to de|
|00000bd0| 76 65 6c 6f 70 2e 20 59 | 6f 75 20 63 61 6e 20 61 |velop. Y|ou can a|
|00000be0| 64 64 0d 0a 66 75 6e 63 | 74 69 6f 6e 73 20 61 73 |dd..func|tions as|
|00000bf0| 20 79 6f 75 20 6e 65 65 | 64 20 74 68 65 6d 2e 20 | you nee|d them. |
|00000c00| 4d 61 6e 79 20 6f 66 20 | 79 6f 75 20 61 72 65 20 |Many of |you are |
|00000c10| 66 61 72 20 62 65 74 74 | 65 72 0d 0a 6e 75 6d 65 |far bett|er..nume|
|00000c20| 72 69 63 61 6c 20 61 6e | 61 6c 79 73 74 73 2c 20 |rical an|alysts, |
|00000c30| 73 6f 20 79 6f 75 20 77 | 69 6c 6c 20 77 61 6e 74 |so you w|ill want|
|00000c40| 20 74 6f 20 75 73 65 20 | 79 6f 75 72 20 6f 77 6e | to use |your own|
|00000c50| 0d 0a 66 75 6e 63 74 69 | 6f 6e 73 2e 20 59 6f 75 |..functi|ons. You|
|00000c60| 20 63 61 6e 20 68 61 76 | 65 20 79 6f 75 72 20 63 | can hav|e your c|
|00000c70| 6f 70 79 20 6f 66 20 74 | 68 65 20 73 6f 75 72 63 |opy of t|he sourc|
|00000c80| 65 20 63 6f 64 65 2e 20 | 49 0d 0a 73 74 69 6c 6c |e code. |I..still|
|00000c90| 20 6f 77 6e 20 74 68 65 | 20 63 6f 70 79 77 72 69 | own the| copywri|
|00000ca0| 74 65 2e 20 59 6f 75 20 | 6d 61 79 20 64 69 73 74 |te. You |may dist|
|00000cb0| 72 69 62 75 74 65 20 74 | 68 65 20 63 6f 64 65 2c |ribute t|he code,|
|00000cc0| 0d 0a 70 72 6f 76 69 64 | 65 64 20 79 6f 75 20 64 |..provid|ed you d|
|00000cd0| 6f 6e 27 74 20 63 68 61 | 6e 67 65 20 69 74 2e 20 |on't cha|nge it. |
|00000ce0| 0d 0a 0d 0a 4e 65 76 65 | 72 74 68 65 6c 65 73 73 |....Neve|rtheless|
|00000cf0| 2c 20 74 68 65 20 73 6f | 75 72 63 65 20 69 73 20 |, the so|urce is |
|00000d00| 69 6e 20 74 68 65 20 70 | 75 62 6c 69 63 20 64 6f |in the p|ublic do|
|00000d10| 6d 61 69 6e 2e 20 50 65 | 72 68 61 70 73 0d 0a 74 |main. Pe|rhaps..t|
|00000d20| 68 69 73 20 69 73 20 61 | 20 6e 61 69 76 65 20 61 |his is a| naive a|
|00000d30| 74 74 69 74 75 64 65 2c | 20 62 75 74 20 49 20 64 |ttitude,| but I d|
|00000d40| 6f 20 6e 6f 74 20 74 68 | 69 6e 6b 20 6f 66 20 74 |o not th|ink of t|
|00000d50| 68 69 73 20 63 6f 64 65 | 0d 0a 61 73 20 61 20 74 |his code|..as a t|
|00000d60| 68 69 6e 67 20 74 6f 20 | 62 65 20 6f 77 6e 65 64 |hing to |be owned|
|00000d70| 2c 20 62 75 74 20 61 73 | 20 6d 6f 72 65 20 6f 66 |, but as| more of|
|00000d80| 20 74 68 6f 75 67 68 74 | 20 74 6f 20 62 65 20 73 | thought| to be s|
|00000d90| 70 72 65 61 64 0d 0a 61 | 62 6f 75 74 2e 20 49 66 |pread..a|bout. If|
|00000da0| 20 79 6f 75 20 77 61 6e | 74 20 74 6f 20 73 65 6e | you wan|t to sen|
|00000db0| 64 20 6d 65 20 73 6f 6d | 65 20 6d 6f 6e 65 79 20 |d me som|e money |
|00000dc0| 62 65 63 61 75 73 65 20 | 49 20 68 61 76 65 0d 0a |because |I have..|
|00000dd0| 73 61 76 65 64 20 79 6f | 75 20 24 78 24 2d 6d 61 |saved yo|u $x$-ma|
|00000de0| 6e 68 6f 75 72 73 2c 20 | 74 68 65 6e 20 66 65 65 |nhours, |then fee|
|00000df0| 6c 20 66 72 65 65 20 74 | 6f 20 64 6f 20 73 6f 2e |l free t|o do so.|
|00000e00| 20 46 6f 72 20 6c 61 63 | 6b 0d 0a 6f 66 20 61 20 | For lac|k..of a |
|00000e10| 62 65 74 74 65 72 20 77 | 6f 72 64 2c 20 63 61 6c |better w|ord, cal|
|00000e20| 6c 20 69 74 20 7b 5c 65 | 6d 20 62 65 67 67 65 72 |l it {\e|m begger|
|00000e30| 2d 77 61 72 65 7d 2e 20 | 49 20 6b 69 6e 64 20 6f |-ware}. |I kind o|
|00000e40| 66 20 73 65 65 0d 0a 6d | 79 73 65 6c 66 20 6f 6e |f see..m|yself on|
|00000e50| 20 61 20 73 74 72 65 65 | 74 20 63 6f 72 6e 65 72 | a stree|t corner|
|00000e60| 2c 20 70 6c 61 79 69 6e | 67 20 6d 79 20 67 75 69 |, playin|g my gui|
|00000e70| 74 61 72 2c 20 73 69 6e | 67 69 6e 67 20 61 0d 0a |tar, sin|ging a..|
|00000e80| 73 6f 6e 67 2c 20 61 6e | 64 20 6d 79 20 63 61 73 |song, an|d my cas|
|00000e90| 65 20 77 69 74 68 20 61 | 20 6c 69 74 74 6c 65 20 |e with a| little |
|00000ea0| 73 70 61 72 65 20 63 68 | 61 6e 67 65 20 69 6e 20 |spare ch|ange in |
|00000eb0| 69 74 2e 20 49 66 20 79 | 6f 75 0d 0a 6c 69 6b 65 |it. If y|ou..like|
|00000ec0| 20 74 68 65 20 73 6f 6e | 67 2c 20 79 6f 75 20 63 | the son|g, you c|
|00000ed0| 61 6e 20 74 6f 73 73 20 | 69 6e 20 61 20 62 75 63 |an toss |in a buc|
|00000ee0| 6b 20 6f 72 20 74 77 6f | 2e 20 47 6f 64 20 6f 6e |k or two|. God on|
|00000ef0| 6c 79 0d 0a 6b 6e 6f 77 | 73 20 62 75 74 20 46 72 |ly..know|s but Fr|
|00000f00| 61 6e 6b 20 42 6f 72 6c | 61 6e 64 20 63 6f 75 6c |ank Borl|and coul|
|00000f10| 64 20 74 72 65 6b 20 62 | 79 20 77 69 74 68 20 68 |d trek b|y with h|
|00000f20| 69 73 20 62 75 72 72 6f | 20 61 6e 64 0d 0a 68 69 |is burro| and..hi|
|00000f30| 73 20 64 6f 67 73 20 6f | 6e 20 68 69 73 20 77 61 |s dogs o|n his wa|
|00000f40| 79 20 74 6f 20 74 68 65 | 20 48 69 67 68 20 53 69 |y to the| High Si|
|00000f50| 65 72 72 61 73 20 66 6f | 72 0d 0a 5c 69 6e 64 65 |erras fo|r..\inde|
|00000f60| 78 7b 59 41 4d 50 20 7d | 7b 5c 65 6d 20 59 65 74 |x{YAMP }|{\em Yet|
|00000f70| 20 41 6e 6f 74 68 65 72 | 20 4d 61 74 72 69 78 20 | Another| Matrix |
|00000f80| 50 72 6f 67 72 61 6d 7d | 2d 2d 6d 69 6e 67 20 63 |Program}|--ming c|
|00000f90| 61 6d 70 20 6f 75 74 2e | 0d 0a 0d 0a 49 20 64 6f |amp out.|....I do|
|00000fa0| 20 6e 6f 74 20 70 6c 61 | 6e 20 74 6f 20 73 75 70 | not pla|n to sup|
|00000fb0| 70 6f 72 74 20 74 68 65 | 20 63 6f 64 65 20 76 65 |port the| code ve|
|00000fc0| 72 79 20 6d 75 63 68 2e | 20 20 49 74 20 63 6f 73 |ry much.| It cos|
|00000fd0| 74 73 20 74 6f 6f 0d 0a | 6d 75 63 68 20 74 69 6d |ts too..|much tim|
|00000fe0| 65 20 61 6e 64 20 6d 6f | 6e 65 79 2e 20 59 6f 75 |e and mo|ney. You|
|00000ff0| 20 63 61 6e 20 77 72 69 | 74 65 20 6d 65 20 69 66 | can wri|te me if|
|00001000| 20 79 6f 75 20 68 61 76 | 65 20 61 6e 79 0d 0a 70 | you hav|e any..p|
|00001010| 72 6f 62 6c 65 6d 73 2e | 20 49 20 77 69 6c 6c 20 |roblems.| I will |
|00001020| 6b 65 65 70 20 61 20 6c | 69 73 74 20 6f 66 20 6e |keep a l|ist of n|
|00001030| 61 6d 65 73 20 61 6e 64 | 20 61 64 64 72 65 73 73 |ames and| address|
|00001040| 65 73 20 6f 66 20 74 68 | 65 0d 0a 70 65 6f 70 6c |es of th|e..peopl|
|00001050| 65 20 77 68 6f 20 63 6f | 6e 74 61 63 74 20 6d 65 |e who co|ntact me|
|00001060| 20 61 62 6f 75 74 20 74 | 68 65 20 63 6f 64 65 2e | about t|he code.|
|00001070| 20 20 49 74 20 6d 61 79 | 20 62 65 63 6f 6d 65 0d | It may| become.|
|00001080| 0a 61 64 76 61 6e 74 61 | 67 65 6f 75 73 20 74 6f |.advanta|geous to|
|00001090| 20 63 68 61 6e 67 65 20 | 74 68 69 73 20 73 75 70 | change |this sup|
|000010a0| 70 6f 72 74 20 70 6f 6c | 69 63 79 2e 20 42 75 74 |port pol|icy. But|
|000010b0| 20 61 73 20 6f 66 20 6e | 6f 77 0d 0a 49 27 6d 20 | as of n|ow..I'm |
|000010c0| 6e 6f 74 20 6f 75 74 20 | 74 6f 20 73 74 61 72 74 |not out |to start|
|000010d0| 20 73 6f 6d 65 20 63 6f | 6d 70 61 6e 79 20 74 68 | some co|mpany th|
|000010e0| 61 74 20 65 61 74 73 20 | 69 6e 74 6f 20 6d 79 0d |at eats |into my.|
|000010f0| 0a 6f 74 68 65 72 77 69 | 73 65 20 62 75 73 79 20 |.otherwi|se busy |
|00001100| 6e 6f 72 6d 61 6c 20 6c | 69 66 65 2e 20 49 20 61 |normal l|ife. I a|
|00001110| 6c 72 65 61 64 79 20 68 | 61 76 65 20 61 20 67 6f |lready h|ave a go|
|00001120| 6f 64 20 6a 6f 62 2e 0d | 0a 0d 0a 49 20 61 73 73 |od job..|...I ass|
|00001130| 75 6d 65 20 74 68 61 74 | 20 61 6e 79 6f 6e 65 20 |ume that| anyone |
|00001140| 77 68 6f 20 75 73 65 73 | 20 74 68 69 73 20 63 6f |who uses| this co|
|00001150| 64 65 20 69 73 20 73 6d | 61 72 74 20 65 6e 6f 75 |de is sm|art enou|
|00001160| 67 68 20 74 6f 0d 0a 75 | 73 65 20 69 74 20 77 65 |gh to..u|se it we|
|00001170| 6c 6c 2c 20 73 6f 20 49 | 20 68 61 76 65 20 6d 69 |ll, so I| have mi|
|00001180| 6e 69 6d 69 7a 65 64 20 | 74 68 65 20 61 6d 6f 75 |nimized |the amou|
|00001190| 6e 74 20 6f 66 20 63 6f | 6d 6d 65 6e 74 69 6e 67 |nt of co|mmenting|
|000011a0| 0d 0a 69 6e 20 69 74 2e | 20 20 53 6f 6d 65 20 6f |..in it.| Some o|
|000011b0| 66 20 74 68 65 20 63 6f | 64 65 20 69 73 20 76 65 |f the co|de is ve|
|000011c0| 72 79 20 66 72 61 67 69 | 6c 65 2c 20 73 6f 20 63 |ry fragi|le, so c|
|000011d0| 68 61 6e 67 65 20 69 74 | 20 61 74 0d 0a 79 6f 75 |hange it| at..you|
|000011e0| 72 20 6f 77 6e 20 72 69 | 73 6b 2e 20 59 6f 75 72 |r own ri|sk. Your|
|000011f0| 20 62 65 73 74 20 62 65 | 74 20 69 73 20 74 6f 20 | best be|t is to |
|00001200| 73 74 75 64 79 20 69 74 | 20 61 6e 64 20 6a 75 73 |study it| and jus|
|00001210| 74 20 61 64 64 0d 0a 66 | 75 6e 63 74 69 6f 6e 73 |t add..f|unctions|
|00001220| 2e 20 49 20 77 69 6c 6c | 20 67 69 76 65 20 72 65 |. I will| give re|
|00001230| 66 65 72 65 6e 63 65 73 | 20 66 6f 72 20 74 68 6f |ferences| for tho|
|00001240| 73 65 20 6f 66 20 79 6f | 75 20 77 68 6f 20 77 61 |se of yo|u who wa|
|00001250| 6e 74 0d 0a 74 6f 20 72 | 65 61 6c 6c 79 20 64 69 |nt..to r|eally di|
|00001260| 67 20 69 6e 74 6f 20 74 | 68 65 20 67 75 74 73 2e |g into t|he guts.|
|00001270| 20 41 6c 73 6f 20 72 65 | 61 64 20 74 68 65 20 74 | Also re|ad the t|
|00001280| 65 73 74 20 66 69 6c 65 | 2e 20 49 0d 0a 72 65 61 |est file|. I..rea|
|00001290| 6c 6c 79 20 74 68 69 6e | 6b 20 74 68 69 73 20 64 |lly thin|k this d|
|000012a0| 6f 63 75 6d 65 6e 74 61 | 74 69 6f 6e 20 77 69 6c |ocumenta|tion wil|
|000012b0| 6c 20 62 65 20 61 20 73 | 75 66 66 69 63 69 65 6e |l be a s|ufficien|
|000012c0| 74 0d 0a 65 78 70 6c 61 | 6e 61 74 69 6f 6e 20 6f |t..expla|nation o|
|000012d0| 66 20 74 68 65 20 64 65 | 74 61 69 6c 73 20 79 6f |f the de|tails yo|
|000012e0| 75 20 63 61 6e 20 66 6f | 6f 6c 20 77 69 74 68 20 |u can fo|ol with |
|000012f0| 73 61 66 65 6c 79 2e 20 | 28 54 68 69 73 0d 0a 64 |safely. |(This..d|
|00001300| 6f 63 75 6d 65 6e 74 20 | 69 73 20 70 72 65 70 61 |ocument |is prepa|
|00001310| 72 65 64 20 69 6e 20 5c | 4c 61 54 65 58 20 20 62 |red in \|LaTeX b|
|00001320| 75 74 20 69 74 20 73 68 | 6f 75 6c 64 20 62 65 20 |ut it sh|ould be |
|00001330| 72 65 61 64 61 62 6c 65 | 0d 0a 61 73 20 70 75 72 |readable|..as pur|
|00001340| 65 20 74 65 78 74 29 0d | 0a 0d 0a 41 6c 73 6f 20 |e text).|...Also |
|00001350| 70 6c 65 61 73 65 20 73 | 65 6e 64 20 6d 65 20 79 |please s|end me y|
|00001360| 6f 75 72 20 6e 61 6d 65 | 20 61 6e 64 20 61 64 64 |our name| and add|
|00001370| 72 65 73 73 20 69 66 20 | 79 6f 75 20 61 72 65 20 |ress if |you are |
|00001380| 75 73 69 6e 67 20 79 61 | 6d 70 2e 0d 0a 49 20 6e |using ya|mp...I n|
|00001390| 65 65 64 20 74 6f 20 63 | 6f 6e 76 69 6e 63 65 20 |eed to c|onvince |
|000013a0| 73 6f 6d 65 20 70 65 6f | 70 6c 65 20 74 68 61 74 |some peo|ple that|
|000013b0| 20 69 74 20 68 61 73 20 | 61 20 75 73 65 72 20 62 | it has |a user b|
|000013c0| 61 73 65 2e 20 41 73 20 | 6f 66 0d 0a 74 68 69 73 |ase. As |of..this|
|000013d0| 20 72 65 6c 65 61 73 65 | 2c 20 74 68 65 72 65 20 | release|, there |
|000013e0| 68 61 76 65 20 62 65 65 | 6e 20 61 62 6f 75 74 20 |have bee|n about |
|000013f0| 38 30 30 20 64 6f 77 6e | 6c 6f 61 64 73 20 66 72 |800 down|loads fr|
|00001400| 6f 6d 0d 0a 43 6f 6d 70 | 75 73 65 72 76 65 2c 20 |om..Comp|userve, |
|00001410| 77 69 74 68 20 61 62 6f | 75 74 20 35 30 20 6f 6e |with abo|ut 50 on|
|00001420| 74 6f 20 6c 6f 63 61 6c | 20 62 75 6c 6c 65 74 69 |to local| bulleti|
|00001430| 6e 20 62 6f 61 72 64 73 | 20 61 6e 64 20 70 75 62 |n boards| and pub|
|00001440| 6c 69 63 0d 0a 64 6f 6d | 61 69 6e 20 73 6f 66 74 |lic..dom|ain soft|
|00001450| 77 61 72 65 20 64 69 73 | 74 72 69 62 75 74 6f 72 |ware dis|tributor|
|00001460| 73 2e 20 42 79 20 6e 6f | 77 20 49 20 61 6d 20 67 |s. By no|w I am g|
|00001470| 75 65 73 73 69 6e 67 20 | 74 68 61 74 20 69 74 20 |uessing |that it |
|00001480| 69 73 0d 0a 62 65 69 6e | 67 20 75 73 65 64 20 61 |is..bein|g used a|
|00001490| 72 6f 75 6e 64 20 74 68 | 65 20 77 6f 72 6c 64 2e |round th|e world.|
|000014a0| 20 54 68 61 74 27 73 20 | 6e 65 61 74 2e 0d 0a 0d | That's |neat....|
|000014b0| 0a 0d 0a 49 20 68 61 76 | 65 20 74 65 73 74 65 64 |...I hav|e tested|
|000014c0| 20 74 68 65 20 70 72 6f | 67 72 61 6d 2c 20 61 6e | the pro|gram, an|
|000014d0| 64 20 69 74 20 64 6f 65 | 73 20 77 68 61 74 20 49 |d it doe|s what I|
|000014e0| 20 74 65 6c 6c 20 69 74 | 20 74 6f 20 64 6f 0d 0a | tell it| to do..|
|000014f0| 28 61 6e 64 20 77 65 20 | 61 6c 6c 20 6b 6e 6f 77 |(and we |all know|
|00001500| 20 68 6f 77 20 6d 75 63 | 68 20 6f 66 20 61 20 70 | how muc|h of a p|
|00001510| 72 6f 62 6c 65 6d 20 74 | 68 61 74 20 63 61 6e 20 |roblem t|hat can |
|00001520| 62 65 29 2e 20 49 20 68 | 61 76 65 0d 0a 6f 6e 6c |be). I h|ave..onl|
|00001530| 79 20 74 65 73 74 65 64 | 20 69 74 20 69 6e 20 54 |y tested| it in T|
|00001540| 75 72 62 6f 20 43 2b 2b | 2e 20 49 74 20 6d 61 79 |urbo C++|. It may|
|00001550| 20 6e 6f 74 20 77 6f 72 | 6b 20 69 6e 20 61 6e 79 | not wor|k in any|
|00001560| 20 6f 74 68 65 72 0d 0a | 6c 61 6e 67 75 61 67 65 | other..|language|
|00001570| 2e 20 49 74 20 77 6f 72 | 6b 73 20 6f 6e 20 6d 79 |. It wor|ks on my|
|00001580| 20 58 54 2c 20 41 54 2c | 20 61 6e 64 20 33 38 36 | XT, AT,| and 386|
|00001590| 20 63 6c 6f 6e 65 73 2c | 20 73 6f 20 49 0d 0a 61 | clones,| so I..a|
|000015a0| 73 73 75 6d 65 20 69 74 | 20 77 69 6c 6c 20 77 6f |ssume it| will wo|
|000015b0| 72 6b 20 6f 6e 20 74 68 | 65 20 73 61 6d 65 20 6d |rk on th|e same m|
|000015c0| 61 63 68 69 6e 65 73 20 | 77 68 65 72 65 20 54 75 |achines |where Tu|
|000015d0| 72 62 6f 20 43 2b 2b 0d | 0a 77 6f 72 6b 73 20 77 |rbo C++.|.works w|
|000015e0| 65 6c 6c 2e 20 20 54 68 | 65 72 65 20 61 72 65 20 |ell. Th|ere are |
|000015f0| 70 6c 61 63 65 73 20 77 | 68 65 72 65 20 69 74 73 |places w|here its|
|00001600| 20 70 65 72 66 6f 72 6d | 61 6e 63 65 20 63 61 6e | perform|ance can|
|00001610| 20 62 65 0d 0a 69 6d 70 | 72 6f 76 65 64 2e 20 49 | be..imp|roved. I|
|00001620| 20 6a 75 73 74 20 77 61 | 6e 74 65 64 20 73 6f 6d | just wa|nted som|
|00001630| 65 74 68 69 6e 67 20 74 | 68 61 74 20 77 6f 72 6b |ething t|hat work|
|00001640| 73 2e 20 20 49 27 6d 20 | 6e 6f 74 20 61 0d 0a 70 |s. I'm |not a..p|
|00001650| 72 6f 66 65 73 73 69 6f | 6e 61 6c 20 63 6f 6d 70 |rofessio|nal comp|
|00001660| 75 74 65 72 20 73 63 69 | 65 6e 74 69 73 74 2c 20 |uter sci|entist, |
|00001670| 73 6f 20 69 6d 70 72 6f | 76 69 6e 67 20 74 68 65 |so impro|ving the|
|00001680| 0d 0a 61 6c 67 6f 72 69 | 74 68 6d 73 20 77 69 6c |..algori|thms wil|
|00001690| 6c 20 70 72 6f 76 69 64 | 65 20 73 6f 6d 65 20 63 |l provid|e some c|
|000016a0| 68 61 6c 6c 65 6e 67 65 | 73 20 74 6f 20 61 20 70 |hallenge|s to a p|
|000016b0| 72 6f 66 65 73 73 69 6f | 6e 61 6c 2e 0d 0a 49 20 |rofessio|nal...I |
|000016c0| 61 6c 73 6f 20 68 6f 70 | 65 20 69 74 20 73 70 61 |also hop|e it spa|
|000016d0| 72 6b 73 20 73 6f 6d 65 | 20 69 6e 74 65 72 65 73 |rks some| interes|
|000016e0| 74 20 69 6e 20 43 2b 2b | 20 61 6d 6f 6e 67 0d 0a |t in C++| among..|
|000016f0| 6d 61 74 68 65 6d 61 74 | 69 63 69 61 6e 73 2e 0d |mathemat|icians..|
|00001700| 0a 0d 0a 4d 61 79 62 65 | 20 74 68 65 20 6f 6e 6c |...Maybe| the onl|
|00001710| 79 20 72 65 61 73 6f 6e | 20 49 27 6d 20 63 69 72 |y reason| I'm cir|
|00001720| 63 75 6c 61 74 69 6e 67 | 20 74 68 65 20 63 6f 64 |culating| the cod|
|00001730| 65 20 62 65 63 61 75 73 | 65 20 49 27 6d 0d 0a 61 |e becaus|e I'm..a|
|00001740| 6d 61 7a 65 64 20 74 68 | 61 74 20 69 74 20 77 6f |mazed th|at it wo|
|00001750| 72 6b 73 20 61 74 20 61 | 6c 6c 2e 20 49 20 67 75 |rks at a|ll. I gu|
|00001760| 65 73 73 20 74 68 61 74 | 20 6f 74 68 65 72 20 70 |ess that| other p|
|00001770| 65 6f 70 6c 65 20 68 61 | 76 65 0d 0a 62 65 65 6e |eople ha|ve..been|
|00001780| 20 66 69 73 68 69 6e 67 | 20 61 72 6f 75 6e 64 20 | fishing| around |
|00001790| 66 6f 72 20 61 6e 20 61 | 6e 73 77 65 72 20 74 6f |for an a|nswer to|
|000017a0| 20 74 68 65 20 73 61 6d | 65 20 70 72 6f 62 6c 65 | the sam|e proble|
|000017b0| 6d 73 2e 0d 0a 48 65 72 | 65 20 69 73 20 6d 79 20 |ms...Her|e is my |
|000017c0| 61 6e 73 77 65 72 2e | |answer. | |
+--------+-------------------------+-------------------------+--------+--------+