6-Ε-019 - ΜΑΘΗΜΑΤΙΚΗ ΑΝΑΛΥΣΗ ΙΙ - ΠΕΡΙΕΧΟΜΕΝΑ

¶∂ƒπ∂Ã√ª∂¡∞ ∞ÊȤڈÛË ....................................................................................................................5 µÈÔÁÚ·ÊÈο Û˘ÁÁڷʤˆÓ ..........................................................................................6 ¶ÚfiÏÔÁÔ˜ ....................................................................................................................9 ∂ÏÏËÓÈÎfi Û‡ÛÙËÌ· ·Ú›ıÌËÛ˘................................................................................10 ¶ÂÚȯfiÌÂÓ· ............................................................................................................12 ª∂ƒ√™ ∞' – ™˘Ó·ÚÙ‹ÛÂȘ ¶ÔÏÏÒÓ ªÂÙ·‚ÏËÙÒÓ ..............................................23 1. ªÂÚÈ΋ ¶·Ú·ÁÒÁÈÛË........................................................................................25 1.1 ∏ ¤ÓÓÔÈ· Ù˘ ÌÂÚÈ΋˜ ·Ú·ÁÒÁÈÛ˘ ..........................................................25 1.2 ªÂÚÈΤ˜ ·Ú¿ÁˆÁÔÈ ·ÓÒÙÂÚ˘ Ù¿Í˘........................................................26 1.3 ∏ ¤ÓÓÔÈ· ÙÔ˘ ÔÏÈÎÔ‡ ‰È·ÊÔÚÈÎÔ‡ ..............................................................28 1.3.1 √ÏÈÎfi ‰È·ÊÔÚÈÎfi ·ÓÒÙÂÚ˘ Ù¿Í˘ ................................................................30

1.4 1.5 1.6 1.7 1.8 1.9

°ÂˆÌÂÙÚÈ΋ ÂÚÌËÓ›· ÙÔ˘ ÔÏÈÎÔ‡ ‰È·ÊÔÚÈÎÔ‡ ........................................31 §˘Ì¤Ó· ·Ú·‰Â›ÁÌ·Ù· ................................................................................35 ∂Ê·ÚÌÔÁ¤˜ ÙˆÓ ÌÂÚÈÎÒÓ ·Ú·ÁÒÁˆÓ ......................................................38 ª¤ÁÈÛÙ·-ÂÏ¿¯ÈÛÙ· Ì ‰‡Ô ‹ ÂÚÈÛÛfiÙÂÚ˜ ·ÓÂÍ¿ÚÙËÙ˜ ÌÂÙ·‚ÏËÙ¤˜....41 ∞ÎÚfiٷٷ ˘fi Û˘Óı‹Î˜ ............................................................................45 ∞Û΋ÛÂȘ ......................................................................................................50

2. ¶ÔÏÏ·Ï‹ √ÏÔÎÏ‹ÚˆÛË ....................................................................................51 2.1 ¢ÈÏ¿ ÔÏÔÎÏËÚÒÌ·Ù· ..................................................................................51 2.2 ∆ÚÈÏ¿ ÔÏÔÎÏËÚÒÌ·Ù· ................................................................................53 2.3 ªÂÙ·Û¯ËÌ·ÙÈÛÌÔ› ÙˆÓ ÔÏÏ·ÏÒÓ ÔÏÔÎËÚˆÌ¿ÙˆÓ ................................54 2.4 §˘Ì¤Ó˜ ·Û΋ÛÂȘ........................................................................................55 2.5 ÕÏ˘Ù˜ ·Û΋ÛÂȘ ..........................................................................................66 3. ∂Ê·ÚÌÔÁ¤˜ ÙˆÓ √ÏÔÎÏËڈ̿وÓ......................................................................69 3.1 ∂Ê·ÚÌÔÁ¤˜ ÙˆÓ ·ÏÒÓ ÔÏÔÎÏËÚˆÌ¿ÙˆÓ ÛÙÔÓ ˘ÔÏÔÁÈÛÌfi ÂÌ‚·‰ÒÓ-fiÁΈÓ............................................................................................69 3.1.1 ∂Ì‚·‰fi ÔÚÈ˙fiÌÂÓÔ ·fi Û˘Ó¿ÚÙËÛË ..............................................................69

¶∂ƒπ∂Ã√ª∂¡∞

13

3.1.2 ∂Ì‚·‰fi ÂÚÈÔ¯‹˜ Ô˘ ÔÚ›˙ÂÙ·È ·fi ‰‡Ô Û˘Ó·ÚÙ‹ÛÂȘ ..............................73 3.1.3 ŸÁÎÔÈ ÛÙÂÚÂÒÓ ÂÎ ÂÚÈÛÙÚÔÊ‹˜ ..................................................................75 3.1.4 ª¤ÙÚËÛË ÙÔ˘ Ì‹ÎÔ˘˜ ÌÈ·˜ Â›‰˘ ηÌ‡Ï˘............................................77 3.1.5 ∂Ê·ÚÌÔÁ¤˜ ÛÙË º˘ÛÈ΋ ..................................................................................80 3.1.6 ∂Ê·ÚÌÔÁ‹ ÛÙË ıˆڛ· Èı·ÓÔÙ‹ÙˆÓ ............................................................83

3.2 ∂Ê·ÚÌÔÁ¤˜ ÙˆÓ ‰ÈÏÒÓ-ÙÚÈÏÒÓ ÔÏÔÎÏËÚˆÌ¿ÙˆÓ ..................................85 3.2.1 ÀÔÏÔÁÈÛÌfi˜ ÂÌ‚·‰ÒÓ-fiÁÎˆÓ ......................................................................85 3.2.2 ÀÔÏÔÁÈÛÌfi˜ ÙÔ˘ ΤÓÙÚÔ˘ ‚¿ÚÔ˘˜ ÔÌÔÈÔÁÂÓÒÓ ˘ÏÈÎÒÓ Â›‰ˆÓ ÂÈÊ·ÓÂÈÒÓ ....................................................................................87 3.2.3 ÕÏ˘Ù˜ ·Û΋ÛÂȘ..............................................................................................89

3.3 ƒÔ‹ ·‰Ú¿ÓÂÈ·˜ ..........................................................................................90 3.3.1 ƒÔ¤˜ ·‰Ú¿ÓÂÈ·˜ ˘ÏÈÎÒÓ ÁÚ·ÌÌÒÓ ..............................................................93 3.3.2 ƒÔ¤˜ ·‰Ú¿ÓÂÈ·˜ ˘ÏÈÎÒÓ ÂÈÊ·ÓÂÈÒÓ ..........................................................95 3.3.3 ƒÔ¤˜ ·‰Ú¿ÓÂÈ·˜ ˘ÏÈÎÒÓ fiÁΈÓ..................................................................100

ª∂ƒ√™ µ' – ¢È·ÊÔÚÈΤ˜ ∂ÍÈÛÒÛÂȘ....................................................................105 4. µ·ÛÈΤ˜ ŒÓÓÔȘ ..............................................................................................107 4.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................107 4.2 √ÚÈÛÌfi˜ ......................................................................................................108 5

√ÏÔÎÏËÚˆÙÈΤ˜ °Ú·Ì̤˜ Î·È ¶Â‰›· ¢È¢ı‡ÓÛÂˆÓ ..........................................111 5.1 ªÔÓÔ·Ú·ÌÂÙÚÈ΋ ÔÈÎÔÁ¤ÓÂÈ· ηÌ˘ÏÒÓ ..............................................111 5.2 ¶Â‰›Ô ‰È¢ı‡ÓÛˆÓ....................................................................................111 5.3 πÛÔÎÏÈÓ›˜ ηÌ‡Ï˜ ..................................................................................114 5.4 º˘ÛÈ΋ ÂÚÌËÓ›· ÙˆÓ ‰È·ÊÔÚÈÎÒÓ ÂÍÈÛÒÛÂˆÓ ......................................115 5.5 ¢È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ ¯ˆÚÈ˙fiÌÂÓˆÓ ÌÂÙ·‚ÏËÙÒÓ ................................115 5.6 ¶ÚÔÙÂÈÓfiÌÂÓ˜ ·Û΋ÛÂȘ ..........................................................................124

6. √ÌÔÁÂÓ›˜ ¢È·ÊÔÚÈΤ˜ ∂ÍÈÛÒÛÂȘ (√.¢.∂.) ....................................................127 6.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................127 6.2 ¢È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ Ì ÁÚ·ÌÌÈÎÔ‡˜ ·ÏÏ¿ ÌË ÔÌÔÁÂÓ›˜ Û˘ÓÙÂÏÂÛÙ¤˜ ........................................................................128 6.3 ¶ÚÔÙÂÈÓfiÌÂÓ˜ ·Û΋ÛÂȘ ..........................................................................133 7. °Ú·ÌÌÈΤ˜ ¢È·ÊÔÚÈΤ˜ ∂ÍÈÛÒÛÂȘ ....................................................................135 7.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................135 7.2 °Ú·ÌÌÈΤ˜ ‰È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ ¯ˆÚ›˜ ‰Â‡ÙÂÚÔ Ì¤ÏÔ˜ ......................137 7.3 °Ú·ÌÌÈΤ˜ ‰È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ Ì ÛÙ·ıÂÚÔ‡˜ Û˘ÓÙÂÏÂÛÙ¤˜ Î·È Ì ÛÙ·ıÂÚfi ‰Â‡ÙÂÚÔ Ì¤ÏÔ˜ ................................................................137

14

ª ∞£∏ª∞∆π∫√™ § √°π™ª√™ ππ

7.4 °Ú·ÌÌÈΤ˜ ‰È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ Ì ‰Â‡ÙÂÚÔ Ì¤ÏÔ˜ ÌÂÙ·‚ÏËÙ‹ Û˘Ó¿ÚÙËÛË f(x) ·ÏÏ¿ fi¯È ÙÔ˘ y ..............................................138 7.5 ¶ÚÔÙÂÈÓfiÌÂÓ˜ ·Û΋ÛÂȘ ..........................................................................143 8. °Ú·ÌÌÈΤ˜ ¢È·ÊÔÚÈΤ˜ ∂ÍÈÛÒÛÂȘ µ' ∆¿Í˘ Ì ™Ù·ıÂÚÔ‡˜ ™˘ÓÙÂÏÂÛÙ¤˜........145 8.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................145 8.2 ∂ÏÏÂÈ›˜ ‰È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ..............................................................145 8.3 ¶Ï‹ÚÂȘ ‰È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ µ' Ù¿Í˘ - √ÌÔÁÂÓ›˜ ..........................149 8.3.1 ÷ڷÎÙËÚÈÛÙÈ΋ Â͛ۈÛË ............................................................................149 8.3.2 ÷ڷÎÙËÚÈÛÙÈ΋ Â͛ۈÛË Ì Ú·ÁÌ·ÙÈΤ˜ Î·È ¿ÓÈÛ˜ Ú›˙˜ ..................150 8.3.3 ÷ڷÎÙËÚÈÛÙÈ΋ Â͛ۈÛË Ì Ú·ÁÌ·ÙÈΤ˜ ›Û˜ Ú›˙˜ ..............................152 8.3.4 ÷ڷÎÙËÚÈÛÙÈ΋ Â͛ۈÛË Ì ÌÈÁ·‰ÈΤ˜ Ú›˙˜............................................153

8.4 ¶Ï‹ÚÂȘ ÁÚ·ÌÌÈΤ˜ ‰È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ µ' Ù¿Í˘ "ÌË ÔÌÔÁÂÓ›˜" ............................................................................................155 9. ¶Ï‹ÚÂȘ ¢È·ÊÔÚÈΤ˜ ∂ÍÈÛÒÛÂȘ........................................................................159 9.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................159 9.2 ∂͛ۈÛË Bernoulli ....................................................................................160 9.3 ∂͛ۈÛË Riccati ........................................................................................162 9.4 ∂ÍÈÛÒÛÂȘ Lagrange Î·È Clairaut ............................................................163 ª∂ƒ√™ °' – ªÂÙ·Û¯ËÌ·ÙÈÛÌÔ› Laplace - °Ú·ÌÌÈο ™˘ÛÙ‹Ì·Ù· ¢È·ÊÔÚÈÎÒÓ ∂ÍÈÛÒÛÂˆÓ ................................................................167 10. ªÂÙ·Û¯ËÌ·ÙÈÛÌÔ› Laplace ..............................................................................169 10.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................169 10.2 µ·ÛÈÎÔ› ÌÂÙ·Û¯ËÌ·ÙÈÛÌÔ› Laplace ..........................................................170 10.3 π‰ÈfiÙËÙ˜ ÙÔ˘ ÌÂÙ·Û¯ËÌ·ÙÈÛÌÔ‡ Laplace ................................................171 10.4 ∞Û΋ÛÂȘ ....................................................................................................176 10.5 ∞ÓÙ›ÛÙÚÔÊÔ˜ ÌÂÙ·Û¯ËÌ·ÙÈÛÌfi˜ Laplace..................................................177 10.6 π‰ÈfiÙËÙ˜ ·ÓÙ›ÛÙÚÔÊÔ˘ ÌÂÙ·Û¯ËÌ·ÙÈÛÌÔ‡ Laplace ................................178 10.7 ÀÔÏÔÁÈÛÌfi˜ ·ÓÙ›ÛÙÚÔÊÔ˘ ÌÂÙ·Û¯ËÌ·ÙÈÛÌÔ‡ Laplace ........................179 11. °Ú·ÌÌÈο ™˘ÛÙ‹Ì·Ù· ¢È·ÊÔÚÈÎÒÓ ∂ÍÈÛÒÛÂˆÓ ..............................................191 11.1 ∂›Ï˘ÛË ÁÚ·ÌÌÈÎÒÓ Û˘ÛÙËÌ¿ÙˆÓ ‰È·ÊÔÚÈÎÒÓ ÂÍÈÛÒÛÂˆÓ Ì ÙË ‚Ô‹ıÂÈ· ÙÔ˘ ÌÂÙ·Û¯ËÌ·ÙÈÛÌÔ‡ Laplace..............................................191 ª∂ƒ√™ ¢' – ¢È·Ó˘ÛÌ·ÙÈÎfi˜ ¢È·ÊÔÚÈÎfi˜ §ÔÁÈÛÌfi˜ ........................................195 12. ¢È·Ó˘ÛÌ·ÙÈÎfi˜ ¢È·ÊÔÚÈÎfi˜ §ÔÁÈÛÌfi˜ ............................................................197 12.1 µ·ı̈ٿ-‰È·Ó˘ÛÌ·ÙÈο ‰›· ................................................................197

¶∂ƒπ∂Ã√ª∂¡∞

15

12.2 °ÂˆÌÂÙÚÈ΋ ·ÂÈÎfiÓÈÛË ‰È·Ó˘ÛÌ·ÙÈÎÔ‡ ‰›Ô˘ ....................................199 12.3 ƒÔ‹ ‰È·Ó˘ÛÌ·ÙÈÎÔ‡ ‰›Ô˘ ......................................................................200 12.4 ™ÙÚÔ‚ÈÏÈÛÌfi˜ ‰È·Ó˘ÛÌ·ÙÈÎÔ‡ ‰›Ô˘ ......................................................201 12.5 ∆Ô ·Ó¿‰ÂÏÙ· ..............................................................................................201 12.6 ∫Ï›ÛË Î·È ·fiÎÏÈÛË ‰È·Ó˘ÛÌ·ÙÈ΋˜ Û˘Ó¿ÚÙËÛ˘ ..................................202 12.7 ¢›Ó˜ Û˘Ó¿ÚÙËÛ˘ ......................................................................................202 12.8 ™ÙÚÔ‚ÈÏfiÙËÙ· Û˘Ó¿ÚÙËÛ˘ ........................................................................202 12.9 ∆Ô ·Ó¿‰ÂÏÙ· Û ‰‡Ó·ÌË ............................................................................204 12.10 ∆ÂÏÂÛÙ‹˜ Laplace ....................................................................................204 12.11 ¢È·ÊÔÚÈΤ˜ ÂÍÈÛÒÛÂȘ Ì ÌÂÚÈΤ˜ ·Ú·ÁÒÁÔ˘˜ (¢∂ª¶) ..................204 ª∂ƒ√™ ∂' – ∂ÈηÌ‡ÏÈ· Î·È ∂ÈÊ·ÓÂȷο √ÏÔÎÏËÚÒÌ·Ù· ........................207 13. ∂ÈηÌ‡ÏÈ· ÔÏÔÎÏËÚÒÌ·Ù· ..........................................................................209 13.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................209 13.2 ∂ÈηÌ‡ÏÈÔ ÔÏÔÎϋڈ̷ ·' ›‰Ô˘˜ ......................................................209 13.3 π‰ÈfiÙËÙ˜ ÙÔ˘ ÂÈηÌ‡ÏÈÔ˘ ÔÏÔÎÏËÚÒÌ·ÙÔ˜ ·' ›‰Ô˘˜ ........................211 13.4 ∂ÈηÌ‡ÏÈÔ ÔÏÔÎϋڈ̷ ‚' ›‰Ô˘˜ ......................................................214 14. ∂ÈÊ·ÓÂȷο √ÏÔÎÏËÚÒÌ·Ù· ..........................................................................219 14.1 ∂ÈÊ·ÓÂȷο ÔÏÔÎÏËÚÒÌ·Ù· Û˘Ó·ÚÙ‹ÛÂˆÓ ÔÏÏÒÓ ÌÂÙ·‚ÏËÙÒÓ ......219 14.2 ∆‡Ô˜ Stokes - ƒÔ‹ ....................................................................................221 14.3 £ÂÒÚËÌ· ·fiÎÏÈÛ˘ (Gauss - Ostrogradsky) ........................................222 14.4 £ÂÒÚËÌ· Green ........................................................................................223 14.5 £ÂÒÚËÌ· ÙÔ˘ Stokes..................................................................................223 14.6 ∂͛ۈÛÂȘ Maxwell ....................................................................................225 ª∂ƒ√™ ˜' – ¶Èı·ÓfiÙËÙ˜ ....................................................................................229 15. £ÂˆÚ›· ¶Èı·ÓÔÙ‹ÙˆÓ ......................................................................................231 15.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................231 16. ∆˘¯·ÈfiÙËÙ· Î·È ¶Èı·ÓfiÙËÙ· ........................................................................235 16.1 ŒÓÓÔÈ· Èı·ÓfiÙËÙ·˜ - ¶Â›Ú·Ì· Ù‡¯Ë˜ ....................................................235 16.2 µ·ÛÈΤ˜ Ú¿ÍÂȘ ÛÙ· ÛËÌÂÈÔÛ‡ÓÔÏ·........................................................237 16.3 µ·ÛÈΤ˜ ·Ú¯¤˜ ··Ú›ıÌËÛ˘ ....................................................................240 16.3.1 ¶ÚÔÛıÂÙÈ΋ ·Ú¯‹ ........................................................................................240 16.3.2 ¶ÔÏÏ·Ï·ÛÈ·ÛÙÈ΋ ·Ú¯‹ ............................................................................240 16.3.3 ªÂÙ·ı¤ÛÂȘ ..................................................................................................247

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ª ∞£∏ª∞∆π∫√™ § √°π™ª√™ ππ 16.3.4 ™˘Ó‰˘·ÛÌÔ› ..................................................................................................247 16.3.5 ∆˘¯·ÈfiÙËÙ· Î·È Èı·ÓfiÙËÙ· ......................................................................249 16.3.6 ªÔÚʤ˜ Èı·ÓfiÙËÙ·˜ ..................................................................................249 16.3.7 ∆˘¯·›Â˜ ÌÂÙ·‚ÏËÙ¤˜ ....................................................................................251 16.3.8 ¢ÂÈÁÌ·ÙÔ¯ÒÚÔ˜ Î·È Èı·ÓfiÙËÙ· ................................................................252 16.3.9 √ÚÈÛÌfi˜ Î·È ¤ÓÓÔÈ· Ù˘ Èı·ÓfiÙËÙ·˜ ......................................................252 16.3.10 ¡fiÌÔÈ ÚfiÛıÂÛ˘ Èı·ÓÔÙ‹ÙˆÓ ..............................................................254 16.3.11 ¢ÂÛÌÂ˘Ì¤ÓË ‹ ˘fi Û˘Óı‹ÎË Èı·ÓfiÙËÙ· - °ÂÓÈÎfi˜ ÓfiÌÔ˜ ÙÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡ ..................................................................258 16.3.12 ∞ÓÂÍ¿ÚÙËÙ· ÂӉ¯fiÌÂÓ· ..........................................................................260 £ÂÒÚËÌ· Bayes ........................................................................................261 16.3.13 ∆˘¯·›· ÌÂÙ·‚ÏËÙ‹ Î·È Î·Ù·ÓÔÌ‹ ............................................................262

17. ∫·Ù·ÓÔ̤˜ ÌÈ·˜ ªÂÙ·‚ÏËÙ‹˜ ..........................................................................263 17.1 ™˘Ó¿ÚÙËÛË Î·Ù·ÓÔÌ‹˜ ..............................................................................263 17.2 ∞·ÚÈıÌËÙ¤˜ ηٷÓÔ̤˜ ..........................................................................265 17.2.1 ∫·Ù·ÓÔÌ‹ ÙÔ˘ Poison..................................................................................270 17.2.2 ¢ÈˆÓ˘ÌÈ΋ ηٷÓÔÌ‹ ..................................................................................274 17.2.3 °ÂˆÌÂÙÚÈ΋ ηٷÓÔÌ‹ ................................................................................278 17.2.4 ÀÂÚÁˆÌÂÙÚÈ΋ ηٷÓÔÌ‹........................................................................281 17.2.5 ∫·Ù·ÓÔÌ‹ Pascal ........................................................................................283

17.3 ™˘Ó¯›˜ ηٷÓÔ̤˜ ..................................................................................284 17.3.1 √ÌÔÈfiÌÔÚÊË Î·Ù·ÓÔÌ‹ ..............................................................................289 17.3.2 ∆ÚÈÁˆÓÈ΋ ηٷÓÔÌ‹ ..................................................................................291 17.3.3 ∂ÎıÂÙÈ΋ ηٷÓÔÌ‹ ....................................................................................292 17.3.4 ∫·ÓÔÓÈ΋ ηٷÓÔÌ‹ ....................................................................................293 17.3.5 °ÂÓÈ΋ ηٷÓÔÌ‹ ........................................................................................299

ª∂ƒ√™ ∑' – ™Ù·ÙÈÛÙÈ΋ ......................................................................................307 18. ™ÎÔfi˜ Ù˘ ™Ù·ÙÈÛÙÈ΋˜ ..................................................................................309 18.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................309 18.2 ™˘ÏÏÔÁ‹ ÏËÚÔÊÔÚÈÒÓ ..............................................................................309 18.3 ¢ÂÈÁÌ·ÙÔÏË„›· ..........................................................................................310 18.4 ∂ÌÊ¿ÓÈÛË ÙˆÓ ‰Â‰ÔÌ¤ÓˆÓ ........................................................................311 18.5 ¶ÔÈ· Â›Ó·È Ù· ‚·ÛÈο ÚÔ‚Ï‹Ì·Ù· Ù˘ ÛÙ·ÙÈÛÙÈ΋˜ ..............................311 18.6 ª¤ıÔ‰ÔÈ ·ÚÔ˘Û›·Û˘ (ÂÌÊ¿ÓÈÛ˘) ÙˆÓ ÛÙ·ÙÈÛÙÈÎÒÓ ‰Â‰ÔÌ¤ÓˆÓ ......312 18.6.1 ¶·ÚÔ˘Û›·ÛË ÙˆÓ ‰Â‰ÔÌ¤ÓˆÓ Ì ÂÎı¤ÛÂȘ Î·È ·Ó·ÊÔÚ¤˜ ......................312 18.6.2 ¶·ÚÔ˘Û›·ÛË ÙˆÓ ‰Â‰ÔÌ¤ÓˆÓ Û ÛÙ·ÙÈÛÙÈÎÔ‡˜ ›Ó·Î˜ ..........................312

¶∂ƒπ∂Ã√ª∂¡∞

17

19. ª¤ÛÔÈ ..............................................................................................................319 19.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................319 19.2 ª¤ÛÔÈ ÎÂÓÙÚÈ΋˜ Ù¿Û˘ ............................................................................319 19.2.1 ª¤ÛÔ˜ ·ÚÈıÌËÙÈÎfi˜ ‹ ̤ÛË ÙÈÌ‹ ‹ ÛÙ·ÙÈÛÙÈ΋ ÙÈÌ‹ ................................319

20. ∫·ÓÔÓÈ΋ ∫·Ù·ÓÔÌ‹ ......................................................................................325 21. ÃÚÔÓÔÏÔÁÈΤ˜ ™ÂÈÚ¤˜ ∂˘ı›·˜ ∆¿Û˘ ..............................................................327 21.1 ∂ÈÛ·ÁˆÁ‹ ....................................................................................................327 21.2 °Ú·ÊÈÎfi˜ ÚÔÛ‰ÈÔÚÈÛÌfi˜ Ù˘ ÁÂÓÈ΋˜ Ù¿Û˘..........................................327 21.3 ∞Ó·Ï˘ÙÈÎfi˜ ÙÚfiÔ˜ ÚÔÛ‰ÈÔÚÈÛÌÔ‡ ÁÂÓÈ΋˜ Î·È Â˘ı‡ÁÚ·ÌÌ˘ Ù¿Û˘....................................................................................328 21.4 ∞Û΋ÛÂȘ ....................................................................................................331 ª∂ƒ√™ ∏' – Matlab ............................................................................................345 22. µ·ÛÈο ™ÙÔȯ›· Matlab..................................................................................347 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9 22.10 22.11 22.12 22.13

¶ÂÚÈ‚¿ÏÏÔÓ ............................................................................................347 ∏ ·ÚÈıÌËÙÈ΋ ÙÔ˘ ÚÔÁÚ¿ÌÌ·ÙÔ˜ ........................................................349 ∫·ı·ÚÈÛÌfi˜ ÙˆÓ ÔıÔÓÒÓ ......................................................................351 ∏ ÂÓÙÔÏ‹ format Î·È ÔÈ ·Ú¿ÌÂÙÚÔÈ Ù˘..............................................353 √È Û˘Ó·ÚÙ‹ÛÂȘ ÙÔ˘ Matlab ..................................................................355 ¶ÚÔÙÂÚ·ÈfiÙËÙ· Ú¿ÍÂˆÓ ......................................................................358 √È ÂÓÙÔϤ˜ save Î·È load ......................................................................360 ∏ ÂÓÙÔÏ‹ diary ........................................................................................363 √È ÂÓÙÔϤ˜ who Î·È whos ......................................................................364 √È ÂÓÙÔϤ˜ ÂÌÊ¿ÓÈÛË ·Ú¯Â›ˆÓ dir, ls Î·È what ..................................366 √È ÂÓÙÔϤ˜ ¯ÚfiÓÔ˘ clock, date, tic, toc ................................................367 ∞ÏÊ·ÚÈıÌËÙÈο - Ë ÂÓÙÔÏ‹ input ..........................................................368 ∂ȉÈΤ˜ ÌÂÙ·‚ÏËÙ¤˜ Î·È ÛÙ·ıÂÚ¤˜ ........................................................371

·. º·ÓÙ·ÛÙÈ΋ ÌÔÓ¿‰· ..........................................................................................371 ‚. ∆Ô ¿ÂÈÚÔ............................................................................................................372 Á. ∞ÚÔÛ‰ÈÔÚÈÛÙ›· - ÌË ·ÚÈıÌËÙÈο ·ÔÙÂϤÛÌ·Ù· ..........................................373 ‰. ∞ÎÚ›‚ÂÈ· ÎÈÓËÙ‹˜ ˘ԉȷÛÙÔÏ‹˜ eps................................................................373

22.14 ÕÏϘ ÂÓÙÔϤ˜ ÙÔ˘ ML ..........................................................................374 22.14.1 √È ÂÓÙÔϤ˜ exit Î·È quit ............................................................................374 22.14.2 √È ÂÓÙÔϤ˜ clear Î·È clc ............................................................................374

18

ª ∞£∏ª∞∆π∫√™ § √°π™ª√™ ππ

23. ∆· ª·ıËÌ·ÙÈο ÙÔ˘ Matlab............................................................................377 23.1

™Ù·ıÂÚ¤˜ Î·È ÌÂÙ·‚ÏËÙ¤˜ ......................................................................377 23.1.1 ∂ίÒÚÈÛË ÙÈÌ‹˜............................................................................................377 23.1.2 ¶Ú¿ÍÂȘ Ì ÌÂÙ·‚ÏËÙ¤˜ ..............................................................................380

23.2

™˘Ó·ÚÙ‹ÛÂȘ ‚È‚ÏÈÔı‹Î˘......................................................................381 1. ∞Ϥ˜ ÙÚÈÁˆÓÔÌÂÙÚÈΤ˜ Û˘Ó·ÚÙ‹ÛÂȘ ..............................................................382 2. ∞ÓÒÙÂÚ˜ ÙÚÈÁˆÓÔÌÂÙÚÈΤ˜ Û˘Ó·ÚÙ‹ÛÂȘ ........................................................383 3. §ÔÁ·ÚÈıÌÈΤ˜ - ∂ÎıÂÙÈΤ˜ Û˘Ó·ÚÙ‹ÛÂȘ..........................................................384

23.3 23.4

ªÈÁ·‰ÈÎÔ› ·ÚÈıÌÔ› ................................................................................385 ¶›Ó·Î˜ ..................................................................................................388 23.4.1 °ÂÓÈο ..........................................................................................................388 23.4.2 ∞ÓÙ›ÛÙÚÔÊÔ˜ ›Ó·Î·˜ ................................................................................400 23.4.3 ∞Ó¿ÛÙÚÔÊÔ˜ ›Ó·Î·˜..................................................................................400 23.4.4 ∞ÚÈÛÙÂÚ‹-‰ÂÍÈ¿ ‰È·›ÚÂÛË ..........................................................................400 23.4.5 ∞fiÛ·ÛÌ· ̤ÚÔ˘˜ ›Ó·Î· ......................................................................401 23.4.6 ™˘Ó¤ÓˆÛË-Â·‡ÍËÛË ÈӿΈÓ. ∏ Û˘Ó¿ÚÙËÛË cat ....................................404 23.4.7 ∂ÓÙÔϤ˜ ÂȉÈÎÒÓ ÈÓ¿ÎˆÓ ..........................................................................409 23.4.8 ¶›Ó·Î·˜ Pascal ............................................................................................416 23.4.9 ¶›Ó·Î·˜ Pascal (·Ú¿ÁÔÓÙ·˜ Cholesky) ..................................................417 23.4.10 ¢˘Ó¿ÌÂȘ ÈÓ¿ÎˆÓ ....................................................................................418 23.4.11 ¶›Ó·Î˜ Ì ‚‹Ì· ......................................................................................421

23.5

¢È·Ó‡ÛÌ·Ù· ............................................................................................423 23.5.1 °ÂÓÈο ..........................................................................................................423 23.5.2 ™˘Ó·ÚÙ‹ÛÂȘ ‰È·Ó˘ÛÌ¿ÙˆÓ ........................................................................424 23.5.3 ∆˘È΋ ·fiÎÏÈÛË (Standard Deviation) ..................................................434

23.6 23.7 23.8 23.9 23.10

∞ÚÈÛÙÂÚ‹ ¢È·›ÚÂÛË. ∂›Ï˘ÛË ÁÚ·ÌÌÈÎÒÓ Û˘ÛÙËÌ¿ÙˆÓ ....................435 ¢ÂÍÈ¿ ‰È·›ÚÂÛË. ∂›Ï˘ÛË ÂÍÈÛÒÛÂˆÓ ÈÓ¿ÎˆÓ ..................................437 ∂ÛˆÙÂÚÈÎfi ÁÈÓfiÌÂÓÔ 2 ‰È·Ó˘ÛÌ¿ÙˆÓ. ∏ ÂÓÙÔÏ‹ dot ..........................439 ∂͈ÙÂÚÈÎfi ÁÈÓfiÌÂÓÔ 2 ‰È·Ó˘ÛÌ¿ÙˆÓ. ∏ ÂÓÙÔÏ‹ cross........................439 ∂ȉÈΤ˜ ÂÓÙÔϤ˜ ‰ËÌÈÔ˘ÚÁ›·˜ Û˘ÓfiÏˆÓ ÙÈÌÒÓ ÈÓ¿ÎˆÓ ‹ ‰È·Ó˘ÛÌ¿ÙˆÓ. √È ÂÓÙÔϤ˜ linspace Î·È logspace..............................440

24. ∆· °Ú·ÊÈο ÙÔ˘ Matlab ................................................................................443 24.1 ∏ ÂÓÙÔÏ‹ plot ..........................................................................................443 24.1.1 ∏ Û˘Ó¿ÚÙËÛË linspace ................................................................................445 24.1.2 ∏ Û˘Ó¿ÚÙËÛË logspace ................................................................................445

24.2 24.3

™˘Ó·ÚÙ‹ÛÂȘ ÁÚ·ÊÈÎÒÓ..........................................................................446 ¢È·ÌfiÚʈÛË ÁÚ·ÊËÌ¿ÙˆÓ ....................................................................449

¶∂ƒπ∂Ã√ª∂¡∞

19

24.4 24.5 24.6 24.7 24.8 24.9 24.10 24.11

°Ú·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ê·ÓÙ·ÛÙÈÎÒÓ Î·È ÌÈÁ·‰ÈÎÒÓ ·ÚÈıÌÒÓ ............450 √È ÂÓÙÔϤ˜ ezplot Î·È fplot....................................................................453 ∏ ÂÓÙÔÏ‹ fplot ........................................................................................459 ∫›ÓËÛË ÛÙÔ ÁÚ¿ÊËÌ· - ∏ ÂÓÙÔÏ‹ comet ..............................................461 ¶ÔÏÏ·Ϥ˜ ÁÚ·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ ÛÙÔ ›‰ÈÔ Û‡ÛÙËÌ· ......................461 ™˘ÛÙ‹Ì·Ù· Û˘ÓÙÂÙ·ÁÌ¤ÓˆÓ ..................................................................470 °Ú·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ Û ÔÏÈΤ˜ Û˘ÓÙÂÙ·Á̤Ó˜............................472 °Ú·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ Û ڷ‚‰ÔÁÚ¿ÌÌ·Ù·, ÈÛÙÔÁÚ¿ÌÌ·Ù· Î·È Î˘ÎÏÈÎÔ‡˜ ÙÔÌ›˜. √È ÂÓÙÔϤ˜ bar Î·È hist ....................................475 24.12 ™˘Ó·ÚÙ‹ÛÂȘ ÛÙÔÓ ∆ÚÈۉȿÛÙ·ÙÔ ¯ÒÚÔ – °Ú·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ ÛÙȘ 3 ‰È·ÛÙ¿ÛÂȘ ............................................................479 24.12.1 ∏ Û˘Ó¿ÚÙËÛË meshgrid ............................................................................479 24.12.2 √È ÂÓÙÔϤ˜ mesh(z) Î·È surf(x,y) ............................................................479 24.12.3 ∏ Û˘Ó¿ÚÙËÛË Sphere ................................................................................482

24.13 ∞Ôı‹Î¢ÛË °Ú·ÊÈÎÒÓ ¶·Ú·ÛÙ¿ÛÂˆÓ ..............................................484 25. ∆· ¶ÔÏ˘ÒÓ˘Ì· ÛÙÔ MATLAB ......................................................................485 25.1 °ÂÓÈο..........................................................................................................485 25.2 ¶ÚfiÛıÂÛË – ∞Ê·›ÚÂÛË ÔÏ˘ˆÓ‡ÌˆÓ ....................................................485 25.3 ¶ÔÏÏ·Ï·ÛÈ·ÛÌfi˜-¢È·›ÚÂÛË ÔÏ˘ˆÓ‡ÌÔ˘ Ì Ú·ÁÌ·ÙÈÎfi ·ÚÈıÌfi ....487 25.4 ∞ÓÒÙÂÚ˜ Û˘Ó·ÚÙ‹ÛÂȘ ÙˆÓ ÔÏ˘ˆÓ‡ÌˆÓ ..............................................488 26. ∞گ›· ∂ÓÙÔÏÒÓ Î·È ™˘Ó·ÚÙ‹ÛÂˆÓ M-Files..................................................493 26.1 °ÂÓÈο..........................................................................................................493 26.2 ¢ËÌÈÔ˘ÚÁ›· ·Ú¯Â›ˆÓ Script ......................................................................497 26.3 ¢ËÌÈÔ˘ÚÁ›· ·Ú¯Â›ˆÓ Function - ™˘Ó·ÚÙ‹ÛÂˆÓ ......................................500 26.4 ¢ËÌÈÔ˘ÚÁ›· ·Ú¯Â›ˆÓ Û˘Ó·ÚÙ‹ÛÂˆÓ ÙÔ˘ ¯Ú‹ÛÙË ....................................507 26.5 ∞ÓÒÓ˘Ì˜ Û˘Ó·ÚÙ‹ÛÂȘ ............................................................................510 27. ¶ÚÔÁÚ·ÌÌ·ÙÈÛÌfi˜ MATLAB ........................................................................513 27.1 ™ÙÔȯ›· Ì·ıËÌ·ÙÈ΋˜ ÏÔÁÈ΋˜ (¿ÏÁ‚ڷ Boole) ..................................513 27.2 ™¯ÂÛÈ·ÎÔ› ÏÔÁÈÎÔ› ÙÂÏÂÛÙ¤˜ – Relational operators..............................516 27.3 §ÔÁÈΤ˜ Û˘Ó·ÚÙ‹ÛÂȘ - ™¯ÂÛȷΤ˜ Û˘Ó·ÚÙ‹ÛÂȘ ....................................518 27.4 §ÔÁÈΤ˜ Û˘Ó·ÚÙ‹ÛÂȘ is ............................................................................519 27.4.1 ∏ Û˘Ó¿ÚÙËÛË ischar(a) ..............................................................................519 ¢È·Ù¿ÍÂȘ ÎÂÏÈÒÓ ........................................................................................520 27.4.2 ∏ Û˘Ó¿ÚÙËÛË isequal – ∂›Ó·È ›ÛÔ ..............................................................521 27.4.3 ∏ Û˘Ó¿ÚÙËÛË isletter – ∂›Ó·È ·ÏÊ·‚ËÙÈÎfi˜ ¯·Ú·ÎÙ‹Ú·˜ ......................521

20

ª ∞£∏ª∞∆π∫√™ § √°π™ª√™ ππ 27.4.4 √È Û˘Ó·ÚÙ‹ÛÂȘ iskeyword (Â›Ó·È Ï¤ÍË ÎÏÂȉ›) Î·È invarname (Â›Ó·È fiÓÔÌ· ÌÂÙ·‚ÏËÙ‹˜ ÙÔ˘ ML). ......................................................................522 27.4.5 ∏ Û˘Ó¿ÚÙËÛË isvector – ∂›Ó·È ‰È¿Ó˘ÛÌ· ................................................522 27.4.6 √È Û˘Ó·ÚÙ‹ÛÂȘ isfinite, isinf Î·È isnan ....................................................523

27.5 ∏ Û˘Ó¿ÚÙËÛË find ......................................................................................523 27.6 §ÔÁÈÎÔ› ÙÂÏÂÛÙ¤˜ – Logical Operators ..................................................525 27.7 ∂ÓÙÔϤ˜ ·Ó·Î˘ÎÏÒÛÂˆÓ ............................................................................528 27.7.1 √È ÂÓÙÔϤ˜ for...end ....................................................................................528 27.7.2 √È ÂÓÙÔϤ˜ while-end ..................................................................................533 27.7.3 √È ÂÓÙÔϤ˜ if-else-elseif-end ......................................................................537 27.7.4 √È ÂÓÙÔϤ˜ switch-case-otherwise-end ......................................................541 27.7.5 ∏ Û˘Ó¿ÚÙËÛË menu ....................................................................................544

27.8 ∏ ÂÓÙÔÏ‹ feval ............................................................................................548 27.9 ∞fiÏ˘ÙË ÙÈÌ‹ ›Ó·Î· Î·È ‰È·Ó‡ÛÌ·Ù· ....................................................550 28

¶ÚÔÛÂÁÁÈÛÙÈΤ˜ ̤ıÔ‰ÔÈ Ì MATLAB ÌÂ Û˘Ì‚ÔÏÈÎfi ÚÔÁÚ·ÌÌ·ÙÈÛÌfi ....................................................................551 28.1 ÀÔÏÔÁÈÛÌfi˜ ·Ú·ÁÒÁˆÓ Û˘Ó·ÚÙ‹ÛˆÓ. ∏ Û˘Ó¿ÚÙËÛË diff ................551 28.1.1 √ÚÈÛÌfi˜ ÌÂÙ·‚ÏËÙ‹˜ – ∏ ¤ÎÊÚ·ÛË syms..................................................551 28.1.2 ÕÌÂÛË ·Ú·ÁÒÁÈÛË ....................................................................................552 28.1.3 ¶·Ú·ÁÒÁÈÛË Ì ·Ú¿ÌÂÙÚÔ ÛÙË Û˘Ó¿ÚÙËÛË ............................................553 28.1.4 ∂‡ÚÂÛË ÙÈÌ‹˜ ·Ú·ÁÒÁÔ˘ ÛÂ Û˘ÁÎÂÎÚÈ̤ÓË ı¤ÛË ∏ Û˘Ó¿ÚÙËÛË subs ......................................................................................554

28.2 ÀÔÏÔÁÈÛÌfi˜ ÌÂÚÈÎÒÓ ·Ú·ÁÒÁˆÓ (M¶) 1˘ Ù¿Í˘ ............................555 28.3 ÀÔÏÔÁÈÛÌfi˜ ÌÂÚÈÎÒÓ ·Ú·ÁÒÁˆÓ (ª¶) ·ÓÒÙÂÚ˘ Ù¿Í˘..................556 28.4 √ÏÈÎfi ‰È·ÊÔÚÈÎfi (√.¢.) 1˘ Ù¿Í˘..........................................................558 28.5 √ÏÈÎfi ‰È·ÊÔÚÈÎfi (√.¢.) ·ÓÒÙÂÚ˘ Ù¿Í˘ ..............................................559 28.6 √ÏÔÎÏ‹ÚˆÛË – ∞fiÚÈÛÙÔ ÔÏÔÎϋڈ̷. ∏ Û˘Ó¿ÚÙËÛË int ....................561 28.7 √ÏÔÎÏ‹ÚˆÛË Û˘Ó¿ÚÙËÛ˘ Ì ·Ú·Ì¤ÙÚÔ (ÛÙ·ıÂÚ¿) ............................562 28.8 √ÚÈṲ̂ÓÔ ÔÏÔÎϋڈ̷ ............................................................................563 28.9 ¢ÈÏ¿ ·fiÚÈÛÙ· ÔÏÔÎÏËÚÒÌ·Ù· ................................................................564 28.10 ¢ÈÏ¿ ÔÚÈṲ̂ӷ ÔÏÔÎÏËÚÒÌ·Ù· ..........................................................566 28.11 °ÂÓÈÎÂ˘Ì¤Ó· ÔÏÔÎÏËÚÒÌ·Ù· (ÛÙÔ ¤Ó· ¿ÎÚÔ) ....................................567 28.12 °ÂÓÈÎÂ˘Ì¤Ó· ÔÏÔÎÏËÚÒÌ·Ù· Î·È ÛÙ· ‰‡Ô ¿ÎÚ· ................................569 28.13 ¶ÔÏÏ·Ï¿ ÔÏÔÎÏËÚÒÌ·Ù· (·ÓÒÙÂÚ· ÙˆÓ ‰ÈÏÒÓ) ............................571 28.14 ∂‡ÚÂÛË ÚÈ˙ÒÓ Û˘Ó¿ÚÙËÛ˘....................................................................572 28.14.1 °ÂÓÈο ........................................................................................................572

¶∂ƒπ∂Ã√ª∂¡∞

21

28.14.2 ª¤ıÔ‰Ô˜ Ù˘ ‰È¯ÔÙfiÌËÛ˘........................................................................573 ™˘Ó¿ÚÙËÛË ‚È‚ÏÈÔı‹Î˘ ..........................................................................573 ™˘Ó¿ÚÙËÛË ÙÔ˘ ¯Ú‹ÛÙË ............................................................................578 28.14.3 ª¤ıÔ‰Ô˜ Newton-Raphson......................................................................580 ™˘Ó¿ÚÙËÛË ‚È‚ÏÈÔı‹Î˘ ..........................................................................586

28.15 °Ú·ÌÌÈο Û˘ÛÙ‹Ì·Ù· ............................................................................589 28.15.1 °ÂÓÈο ........................................................................................................589 28.15.2 ∂›Ï˘ÛË ÁÂÓÈÎÒÓ ÁÚ·ÌÌÈÎÒÓ Û˘ÛÙËÌ¿ÙˆÓ ............................................590 28.15.3 √ÌÔÁÂÓ‹ ÁÚ·ÌÌÈο Û˘ÛÙ‹Ì·Ù· ................................................................592 28.15.4 ∏ ÂÓÙÔÏ‹ rref(A) ......................................................................................595 28.15.5 ∂Ê·ÚÌÔÁ¤˜ ÛÙ· ÔÌÔÁÂÓ‹ ÁÚ·ÌÌÈο Û˘ÛÙ‹Ì·Ù· ....................................600

28.16 ¡fiÚÌ· ‰È·Ó‡ÛÌ·ÙÔ˜ ................................................................................602 28.17 ¡fiÚÌ· ¶›Ó·Î·..........................................................................................603 28.18 ÷ڷÎÙËÚÈÛÙÈο ÔÏ˘ÒÓ˘Ì· – π‰ÈÔÙÈ̤˜ Î·È È‰ÈԉȷӇÛÌ·Ù· ÂÓfi˜ ›Ó·Î·........................................................................................................604 28.19 ¶ÔÏ˘ÒÓ˘ÌÔ ÚÔÛ·ÚÌÔÁ¤˜ ÂÏ·¯›ÛÙˆÓ ÙÂÙÚ·ÁÒÓˆÓ ..............................605 29 ∂ȉÈο £¤Ì·Ù· ÙÔ˘ ML ..................................................................................607 29.1 ™˘Ì‚ÔÏÈΤ˜ ÂÎÊÚ¿ÛÂȘ ..............................................................................607 29.2 ™˘Ì‚ÔÏÈÎÔ› ÙÂÏÂÛÙ¤˜ ..................................................................................607 29.3 ∏ ÂÓÙÔÏ‹ pretty ..........................................................................................608 29.4 ™˘Ì‚ÔÏÈΤ˜ ÌÂÙ·‚ÏËÙ¤˜ ∏ ÂÓÙÔÏ‹ symvar ..............................................609 29.5 ¶·Ú·ÁÒÁÈÛË Û˘Ì‚ÔÏÈÎÒÓ ·Ú·ÛÙ¿ÛÂˆÓ ................................................610 29.6 √ÏÔÎÏ‹ÚˆÛË Û˘Ì‚ÔÏÈÎÒÓ ·Ú·ÛÙ¿ÛÂˆÓ ................................................611 29.7 ∂›Ï˘ÛË Û˘Ì‚ÔÏÈÎÒÓ ÂÍÈÛÒÛˆÓ. ∏ ÂÓÙÔÏ‹ solve ................................612 29.8 ∂›Ï˘ÛË Û˘Ì‚ÔÏÈÎÔ‡ Û˘ÛÙ‹Ì·ÙÔ˜ ÂÍÈÛÒÛÂˆÓ Ì ›Ó·Î˜ ....................615 29.9 ™˘Ì‚ÔÏÈÎÔ› ›Ó·Î˜ ..................................................................................616 29.10 ∂ÓÙÔϤ˜ - ™˘Ó·ÚÙ‹ÛÂȘ Û˘Ì‚ÔÏÈÎÒÓ ·Ú·ÛÙ¿ÛÂˆÓ (™.¶.)..................618 29.10.1 ∏ ÂÓÙÔÏ‹ isstr ............................................................................................618 29.10.2 ∏ ÂÓÙÔÏ‹ numeric......................................................................................618 29.10.3 ∞ÏÔÔ›ËÛË ·Ú¿ÛÙ·Û˘ - Ë ÂÓÙÔÏ‹ simple Î·È sipmplify ..................619 29.10.4 ∂ÎÙ¤ÏÂÛË Ú¿ÍÂˆÓ - ∏ ÂÓÙÔÏ‹ expand ..................................................621 29.10.5 ¶·Ú·ÁÔÓÙÔÔ›ËÛË ·Ú¿ÛÙ·Û˘ . ∏ ÂÓÙÔÏ‹ factor................................624 29.10.6 ÀÔÏÔÁÈÛÌfi˜ Ù˘ ÙÈÌ‹˜ ÌÈ·˜ Û˘Ó¿ÚÙËÛ˘ Û ¤Ó· ÛËÌÂ›Ô ÙÔ˘ ¶.√. ∏ Û˘Ó¿ÚÙËÛË subs ....................................................................................624 29.10.7 ∞ÓÙÈηٿÛÙ·ÛË Û ·Ú¿ÛÙ·ÛË ÌÂÙ·‚ÏËÙ‹˜ Ì ¿ÏÏË ............................626 29.10.8 ¢È·¯ˆÚÈÛÌfi˜ ·ÚÈıÌËÙ‹ Î·È ·ÚÔÓÔÌ·ÛÙ‹ ÚËÙ‹˜ ÔÏ˘ˆÓ˘ÌÈ΋˜ ·Ú¿ÛÙ·Û˘. ∏ Û˘Ó¿ÚÙËÛË numden ......................................................627

22

ª ∞£∏ª∞∆π∫√™ § √°π™ª√™ ππ 29.10.9 ∞ÓÙ›ÛÙÚÔÊË Û˘Ó¿ÚÙËÛË Û˘Ó¿ÚÙËÛ˘. ∏ ÂÓÙÔÏ‹ finverse ......................627

29.11 ™ÂÈÚ¤˜ ........................................................................................................629 ∫ÚÈÙ‹ÚÈÔ Û‡ÁÎÏÈÛ˘ ÛÂÈÚ¿˜ D' Alembert ............................................................631 ∫ÚÈÙ‹ÚÈÔ Û‡ÁÎÏÈÛ˘ ÛÂÈÚ¿˜ Cauchy ....................................................................632

30. ¢È·¯Â›ÚÈÛË ∞ÔÙÂÏÂÛÌ¿ÙˆÓ............................................................................635 30.1 ∏ ÂÓÙÔÏ‹ disp ............................................................................................635 30.2 ∏ ÂÓÙÔÏ‹ num2str ......................................................................................635 30.3 ∏ ÂÓÙÔÏ‹ sprintf ........................................................................................636 30.4 ∂ÎÙ‡ˆÛË Û ÂÎÙ˘ˆÙ‹ - ∂ÓÙÔÏ‹ print....................................................642 30.4.1 ∂ÎÙ‡ˆÛË ÂÚÈÔ¯‹˜ ÂÓÙÔÏÒÓ ....................................................................642 30.4.2 ∂ÎÙ‡ˆÛË m-file (ÚÔÁÚ¿ÌÌ·ÙÔ˜) ............................................................645 30.4.3 ∂ÎÙ‡ˆÛË ÈÛÙÔÚÈÎÔ‡ (history) Î·È ÂÚÈÔ¯‹˜ ÂÚÁ·Û›·˜ (workspace) ................................................................................645

30.5 ∂ÎÙ‡ˆÛË Û¯Â‰›ˆÓ - figure ......................................................................646 30.6 ∞Ôı‹Î¢ÛË Û ·Ú¯Â›Ô – ¢È¿‚·ÛÌ· ·Ú¯Â›Ô˘ ......................................646 30.6.1 ¢ËÌÈÔ˘ÚÁ›· ·Ú¯Â›Ô˘ - ∞Ôı‹Î¢ÛË ÛÙÔȯ›ˆÓ ¢È¿‚·ÛÌ· ÛÙÔȯ›ˆÓ. ∏ ÂÓÙÔÏ‹ fopen ......................................................646 30.6.2 ∞Ôı‹Î¢ÛË ÂÚÁ·ÛÈÒÓ - workspace. √È ÂÓÙÔϤ˜ save-load....................649 30.6.3 AÔı‹Î¢ÛË ÂÓÙÔÏÒÓ. Command window ..............................................650

30.7 ÕÏϘ ÂÓÙÔϤ˜ Î·È Û˘Ó·ÚÙ‹ÛÂȘ ÙÔ˘ ML ................................................650 30.7.1 ∏ ÂÓÙÔÏ‹ pause ............................................................................................650 30.7.2 ∏ ÂÓÙÔÏ‹ ginput ..........................................................................................652

¶·Ú¿ÚÙËÌ· ............................................................................................................655 µÈ‚ÏÈÔÁÚ·Ê›· ..........................................................................................................693 ∂˘ÚÂÙ‹ÚÈÔ................................................................................................................695