????

1
????µ?t??? ?????s?

• ?. ??f?d??

2
?? e??a? ? ????µ?t??? ?????s?

• ???a? ?p?st?µ?
• ?s???e?ta? µe µe??d??? ep???s?? µa??µat????
  p??ß??µ?t?? µe ???s? a???µ?t???? p???e?? (µe ?/?)
  ?a??? ?a? µe t?? a????s? t?? sfa?µ?t?? st??
  p??s????s? t?? ??se??.
• ???a? ?????
• ?f??? st?? ep????? e?e???? t?? µe??d?? p?? e??a?
  p?? ?at?????? ??a t?? ep???s? e???
  s???e???µ???? p??ß??µat??.
• ?pa?te? a??pt??? de???t?t?? ?a? d?a?s??s??.

3
?a??de??µa ?p???s? G?aµµ???? S?st?µat??
???s?se?? (1)
?
4
?a??de??µa ?p???s? G?aµµ???? S?st?µat??
???s?se?? (2)

• ????d??
• ??p?? t?? Cramer (?)
• xA-1b (?2)
• ?µese? µ???d?? Gauss (?)
• ?p??
• ?e µe???? ?d???s?
• ?e ????? ?d???s?
• ?pa?a??pt???? µ???d??
• Jacobi
• Gauss-Seidel
• ?.?.

5
?a??de??µa ?p???s? G?aµµ???? S?st?µat??
???s?se?? (3)

• ???t???a ep?????? µe??d??
• Ge????
• ????p????t?ta (?p?????st???, µ??µ??, ???p???s??)
• ?a??t?ta s?????s?? (??a epa?a??pt???? µe??d???)
• ????ße?a
• ???e?t???t?ta se a???µ?t??? sf??µata
  (a?apa??stas?? ded?µ???? ?a? p???e??)
• ??a?t?µe?a ap? t? s???e???µ??? p??ß??µa
• ?at?stas? t?? p??a?a t?? s?st?µat??
• ?????? µ?de????? st???e??? t?? p??a?a
• ??ast?se?? t?? p??ß??µat??
• ????? s?µµet??e? st?? p??a?a, ?.?.

6
?a??de??µa ?p???s? G?aµµ???? S?st?µat??
???s?se?? (4)

• ?.?. ?p???s? t?? s?st?µat??
• µe 4 s?µa?t??? ??f?a
• ????ß?? ??s? (µe 4 ??f?a)
• ?p?? µ???d?? Gauss
• ????d?? Gauss µe µe???? ?d???s?

7
?e??e??µe?a

• ????µ?t??? t?? ?p?????st?
• ???s????s? ?a? sf??µata
• S?st?µata ??aµµ???? e??s?se??
• ?d??t?µ?? ?a? ?d??d?a??sµata
• ?a?eµß???
• ?p???s? µ? ??aµµ???? e??s?se??
• ????µ?t??? pa?a????s? ?a? ????????s?
• ?fa?µ???? se C / Mathematica

8
S?????µµata

• G. S. ?apa?e?????? ?a? ?. G. ?s?t???a?,
  ????µ?t??? ?????s? (µe efa?µ???? se Matlab ?a?
  Mathematica), 3? ??d?s?, ??d?se?? S?µe??, ????a
  2004.
• ?. ?. F?a??????, ????d?? ????µ?t???? ?????s??,
  ??µ?? ? Te???a ?a? ?fa?µ???? ?a? ? Mathematica
  ?a? ?? efa?µ???? t??, ??d?se?? ?f?? ????a??d?,
  Tessa?????? 2005.
• ?. ??µ?t???d?? ?a? ?. ?????a?, ?fa?µ?sµ???
  ????µ?t??? ?????s?, 2? ??d?s?, ??d?se?? ????
  ?e?????????, ????a 2008.
• G. S. S?f?a??? ?a? ?. T. ????p?????, ????µ?t???
  ?????s?, ??d?se?? ?. Staµ?????, ????a 2005.
• ?. ?. ??a??t??, ????µ?t??? ?????s?, ??d?se??
  ???????? G??µµata, ????a 2002.
• S?µe??se?? Mathematica.

9
??ß?????af?a

• S. ??a?a???, Mathematica ?a? ?fa?µ????, ?a?/???
  ??d?se?? ???t??, 2001.
• G. S. ?apa?e??????, ?. G. ?s?t???a?, ?a? ?. T.
  Faµ????, S??????? ?a??µat??? ????sµ??? Matlab
  Mathematica, ??d?se?? S?µe??, ????a 2004.
• E. Don, Mathematica, ??d?se?? ??e?d????µ??, 2005.
  
• R. L. Burden and J. Douglas Faires, Numerical
  Analysis, 6th ed., Brooks/Cole, 1997 (d?a??s?µ?
  st? ß?ß???????).
• A. Ralston and P. Rabinowitz, A First Course in
  Numerical Analysis, 2nd ed., Dover, 1978.

10
???s?µe? ??e????se??

• ?st?se??da µa??µat?? http//www.unipi.gr/faculty/
  kofidis/teaching.htm
• ??sa???? st? Mathematica (S?µe??se??)
• http//www.unipi.gr/faculty/mbouts/sim/Intro_to_M
  athematica08.pdf

11
???tas? - ?a?µ?????s?

• ???as?a (at?µ???) ??a t? sp?t?
• G?apt? e??tas?
• ?e????? ßa?µ??
• max(ßa?µ?? ??apt??,
• 0.3ßa?µ?? e??as?a?0.7ßa?µ?? ??apt??)

12
???s????s? ?a? Sf??µata (1)

• ???? pepe?asµ???? µe?????? µ??µ??, ? a?apa??stas?
  a???µ?? st?? ?/? ?a? ?? p???e?? µ a?t??? (µp??e?
  ?a) e?????? a?a???ße?e?.
• ??? a?t?? ep??e????? t?? a???ße?a µe t?? ?p??a
  ?p??????eta? ? ??s? e??? p??ß??µat?? µe ??p??a
  a???µ?t??? µ???d?

13
???s????s? ?a? Sf??µata (2)

• ?a??de??µa
• ?? e??a? ?? p??se???se?? t?? a???µ??
  , µe s?et??? sf??µata
• p??? e??a? t? s?et??? sf??µa st?? p??s????s? t??
  ????µ???? t???,
• (?p??t?s? a?
  a??et? µ????.)

14
S?st?µata G?aµµ???? ???s?se?? (1)

• ?p?????sµ?? t?? t?µ?? t?? n a???st?? x st?
  s?st?µa t?? n e??s?se?? Axb, µe d?sµ??a t?? nxn
  p??a?a A ?a? t? nx1 d????sµa b.

15
S?st?µata G?aµµ???? ???s?se?? (2)

• ????d?? apa???f?? Gauss
• ???????p???s? t?? p??a?a t?? s?st?µat??
• ?p???s? t?? ?s?d??aµ?? t????????? s?st?µat?? µe
  p?s? a?t??at?stas?

?
x
b
16
S?st?µata G?aµµ???? ???s?se?? (3)

• ?fa?µ????
• ?p???s? p????? s?st?µ?t?? µe t?? ?d?? p??a?a
• ?p?????sµ?? a?t?st??f?? p??a?a
• ?p?????sµ?? ??????sa? p??a?a
• ?a?a???t?p???s? p??a?a se ??? ?a? ??t?
  t?????????? pa?????te?

17
S?st?µata G?aµµ???? ???s?se?? (4)

• ?p?????sµ?? ?d??t?µ?? (?) ?a? ?d??d?a??sµ?t?? (x)
  e??? p??a?a A ?x?x
• ????d?? t?? d???µe?? (power method) ??a
  ?p?????sµ? t?? µ???st?? (?at ap???t? t?µ?)
  ?d??t?µ?? ?a? a?t?st????? ?d??d?a??sµat??
• ??????p???s? x(0)
• ?pa?????? (k0,1,2,)
• x(k1)A x(k)
• ?a??????p???s? t?? x(k1)

18
S?st?µata G?aµµ???? ???s?se?? (5)

• ?fa?µ??? ?????s? se p??te???se? s???st?se?
  (Principal Component Analysis (PCA))

19
?a?eµß??? (Interpolation) (1)

• ?? ????????µe t?? t?µ?? µ?a? (s??e????)
  s????t?s?? f µ??? sta k1 s?µe?a x0 , x1 , , xk
  , p?? µp????µe ?a p??se???s??µe t?? t?µ?? t?? se
  ???a e?d??µesa s?µe?a
• ???????µ??? p??s????s? ???se??????µe t?? f µ
  ??a p??????µ? k ßa?µ??, Pk(x) f(x), ?a?
  ?p????????µe t?? t?µ?? a?t?? t?? p??????µ?? a?t?
  t?? f.

20
?a?eµß??? (Interpolation) (2)

• ?a??de??µa
• ???????µ? Taylor 2?? ßa?µ?? ???? ap? t? x0
• ???????µ? Newton 2?? ßa?µ??

21
?a?eµß??? (Interpolation) (3)
22
?p???s? ??-G?aµµ???? ???s?se?? (1)

• ??sµ???? µ?a? µ?-??aµµ???? s????t?s?? f, p??e?
  e??a? ?? ???e? t?? e??s?s?? f(x)0
• ?????t??? p????f???a
• ???st?µa a,ß st? ?p??? ?e?ta? ? ??t??µe?? ???a,
  ?.
• ????µ?? ???t? st?? ?p???? p?ste?eta? ?t?
  ß??s?eta? ? ???a.

23
?p???s? ??-G?aµµ???? ???s?se?? (2)

• Ge???? µe??d?????a ?pa?a??pt??? ße?t??s? t??
  p???t?ta? t?? d?a??s?µ?? e?t?µ?s?? ??a t? ???a,
  ?.
• ?????? e?t?µ?s? ?0
• ?pa?????? (k0,1,2,...)
• ?k1F (?k)
• ??? ?t?? t? ?k1 ?a e??a? a??et? ???t? st?
  ?k,
• ?p?? F e??a? ? µ???d?? ße?t??s?? t?? e?t?µ?s??.
  

24
?p???s? ??-G?aµµ???? ???s?se?? (3)

• ????d?? Newton-Raphson

25
?p???s? ??-G?aµµ???? ???s?se?? (4)

• ?a??de??µa

26
????µ?t??? ?a?a????s? (1)

• ?? ??a µ?a s????t?s? f ????????µe µ??? t?? t?µ??
  t?? se k1 ?sap????ta (?at? h) s?µe?a x0, x1 , ,
  xk , p?? µp????µe ?a p??se???s??µe t?? t?µ?? t??
  pa?a????? t?? s a?t? ta s?µe?a ?a? se e?d??µes?
  t???

27
????µ?t??? ?a?a????s? (2)

• Ge???? µe??d?????a ???se??????µe t?? f µ ??a
  p??????µ? ?a? ?p????????µe t?? t?µ?? t??
  pa?a????? a?t?? t?? p??????µ?? a?t? t?? pa?a?????
  t?? f.
• ?.?. ? pa??????? st? ?e?t???? s?µe?? t????
  s?µe??? ?p??????eta? ??

28
????µ?t??? ?a?a????s? (3)
29
????µ?t??? ????????s? (1)

• ?e ta ?d?a ded?µ??a, p?? µp????µe ?a
  p??se???s??µe t? ????????µa t?? f ap? x0 ??? xk
• Ge???? µe??d?????a ????????s? t?? p??????µ?? p??
  pa?eµß???e? t?? f sta s?µe?a a?t?.

30
????µ?t??? ????????s? (2)

• ?a???a? ??ape???? (S???et??)

31
????µ?t??? ????????s? (3)

• ?a???a? Simpson (?p???)

32
?e????? ap? t??? d?µ????????