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1
??????S????? ????O? ????? ??T??????O? ?????S
?????G?S???O? ??T??????O? ??? ?????F?????S
????µ?t??? ?????s? ?etapt???a?? ?????aµµa
??ad?µa???? ?t??? 2008-09
???t? 21 ??t?ß???? 2008 ( ?a?ad?se??
F???t?st???a 9 11 , ???t? ?et??t? , ?/?
235 )
1? ?ßd?µ?da
2
F???s?f?a St???? t?? ????µ?t????
?????s??(?.?.?????????, Ge???? Seµ??????
?a??µat????, ??µ?? 1, Se?.39-56)

??S?GOG???
????µ?t??? ?p???s? ???s?se??
G?aµµ??? S?st?µata
?? G?aµµ??? S?st?µata
?d??t?µ?? ?d??d?a??sµata Tet??? ???sµ????
???a?e?
Te???a ???s????s?? ( Lagrange Hermite Splines
)
Te???a ?e?t?st?p???s?? Optimazation ( ?as???)
????µ?t??? ?a?a????s? ????????s? ????????s?
?at? Gauss
????µ?t??? ?p???s? S?????? ??af?????? ???s?se??
(?as??? sta p??ß??µata Initial Boundary
Value Problems )
????µ?t??? ?p???s? ??af?????? ???s?se?? µe
?e????? ?a?a?????? (?as??? p??? st?? st?at??????
t?? Finite Differences -- Finite Elements)
S?µp????µata

3
? ? ? T ? ? ? ? ?

K. I. I??da??d? ?fa?µ?sµ??? ????µ?t???
?????s?, ????? 2006.
. Conte, S. D. Elementary
Numerical Analysis,
-DeBoor
McGraw Hill.
. Powell M. J. D. Approximation
theory and methods, Cambridge
University Press,1981 .
Varga R. S. Matrix Iterative
Analysis, Prentice ,1962.
Young D. M. A Survey of Numerical
Mathematics, - R. T. Gregory
Addison-Wesley,73.
Schendel U. Introduction to Numerical
Methods for Parallel Computers,Ellis Horwood
Lmt.1984.

???T?????? ?????S? ???SOS?ON
(a) ??t?p?sµ?? ?????.
(ß) ?p?????st???? a??????µ?? e??es?? µ?a?
???a?.
(?) Ge?µet????? e????e? ?p?????st????
a??????µ??.
(d) ?p?????st???? a??????µ?? e??es?? ????
t?? ?????

a??eß????? e??s?s??(deflation)
???ad???? ???e?.
(e) ??a?s??s?a ????? se ????ß? (
aßeßa??t?ta ) t??
s??te?est?? t?? a??eß????? e??s?se??.
(st) ???es? ???a? µe??st?? µ?t??? (
Bernoulli-Rutishauser ).
(?) ?a??t?ta s?????s?? a??????µ??.
(?) ?p?t????s? S?????s?? ß?ad???
s????????sa? epa?a??pt????
µe??d?? (?p?????st???? ???????µ??).

5
( a ) ??????S??S ???O?

1. G??F??OS (Ge?µet?????) ??a ???e e??s?s?, µe
t?? ß???e?a d?a??aµµ?t?? ???st?? s??a?t?se??
.
2. ?e ????????? t??p?(???eß????? ???s?se?? )
(?) ????????e? Sturm,
(??) ????p???s? ???st?? ?e???µ?t??
( Bolzano , Descarte, ?.?.p.
),
(???) ???s? ?ata?????? e?d???? µe??d??
( ????d?? Bernoulli - ??a t??
ap???t?? µe?a??te?? ???a ,
??af???? / ??????? ??a ??e? t??
???e?).

6
1. G??F??OS

?e ap?? ?at?????? ??af??? pa??stas?
(G.?.) t?? eµp?e??µ???? ???st?? s??a?t?se??.
?.?. st?? a??????e? e??s?se??
1.1
1.2
?a ????µe µe ????sµ? ( d??spas? ) st??
eµp?e??µe?e?
d?? ???st?? s??a?t?se?? (a?t? ?a e??a? ? s??????
p?a?t??? )

7
( Ge?µet????? s????e?a )

?a? a?a??t?s? t?? ?????? s?µe??? t?? d??
s??a?t?s?a??? d?a??aµ-
µ?t?? t?? , p?? st?? ??s?a e??a? ?? ???e? t??
a?????? s????t?s??.
?ts?, ??a t?? 1.1 ????µe t?? d??spas?
?a? ta d?a???µµata p?? pa??sta?ta? st?? e????a a.
?? ????? s?µe?? t??? e??a?
? µ??ad??? p?a?µat??? ???a p?? ??e? ? s????t?s?.
?a??µ??a ??a t? 1.2 ????µe t?? d??spas?
?a? ta d?a???µµata t?? e????a? ß , µe µ?a
p?a?µat??? ???a, st?? t?µ? t???.

a
ß
a
8
2. ???eß????? e??s?se?? - ( ? )
????????e? Sturm

O? ???st??,se ???e p??????µ??? e??s?s?
a?t?st????? µ?? a???????a Sturm µe a????? µ??? t?
p??????µ?
?a? t?? pa?????? t??, e?? ? ta?t?t?ta t?? ate????
d?a??es?? t??? d?µ?????e? ??a p??????µ? ?p????p?
p??
µe a??a?µ??? p??s?µ? ap?te?e? t?? ep?µe?? ???.
??e? d?a???se?? µeta?? t?? 2 te?e?ta???
p??????µ??
d?µ???????? ????? ????? ?at? ??a ßa?µ? ?at?te??,
µ???? t?? sta?e??? p??????µ??, p?? e??a? t?
t????.
?a??de??µa 1? St?? e??s?s?
, p??fa??? ?a
????µe
?p?te ??????µe ?at? ta p??a?afe????ta
?a? µet? t?? ?µa??p???s?
St?? s????e?a, ap? t?? ta?t?t?ta t?? ate????
d?a??es?? ????µe
p?? ed? d?µ?????e? t? ?p????p?
?a? d?de? t?? ep?µe?? ?µa??p???µ??? ???
.
??a d?a??es? µeta?? t??
?a? efa?µ??? t?? ta?t?t?ta?
d?de?
?p?te t? ??? ?p????p? ?µa??p???µ??? d?µ?????e?
t?? te???? ??? t?? a?????-
??a? Sturm

9
( ? ) ????????e? Sturm (S????e?a)

St?? s????e?a , ?atas?e???eta? ? s?et???? p??a?a?
p?? pe????e? t??? ?????
t?? a???????a? Sturm, e?? st?? ?ata????f?
pa??stata? ? e??e?a t?? p?a?µat????
µe ta d?ast?µata p?? µa? e?d?af?????
?ata???f??ta? s???????? ta p??s?µa t??
???? t?? a???????a? se ???e ?p?d??st?µa.
?ts?, p????pte? ? pa?a??t? p??a?a? ?p?? se ???e
d??st?µa ? d?af??? st? p?????
t?? a??a??? p??s?µ?? ?a?????e? epa???ß?? t??
a???µ? t?? ????? st? d??st?µa a?t?
?at? s???pe?a, ?p???e? µ?a µ??? ???a p??
?p??????eta? ?t? e??a? ? x1.3247.

x f0(x) f1(x) f2(x) f3(x) ?????? ???a??? ???s?µ??
8 2
1.6 2
0.25 2
0 2
1 _ 2 1 ???a
2 1 1 ???a
8 1
10
?a??de??µa 2?

St?? e??s?s?
? a???????a Sturm p?? d?µ?????e?ta? e??a? ?
?p?te d?µ???????µe t?? pa?a??t? p??a?a ???
?et???? ?a? a???t???? ???e?
?p??p?? e??a? fa?e?? ?t? ? e??s?s? ??e?
??a a???t??? ???a, ?a?
??e?? ?et???? ???e?.
?S??S? ?a µe?et??e? t? e?d?? t?? ????? t??
e??s?se??

11
( ?? ) ????p???s? ??af???? Te???µ?t??

G?a µe? t? Te???µa Bolzano, e??a? e????? ?
efa?µ??? t??, p.?.
st?? pa?a??t? e??s?s? t?? pa?ade??µat?? 1
?a??de??µa 1? St?? e??s?s? f(?) ?3 -
? 1 0 ,
????µe t?? t?µ??
.
??a, ?p???e? ?p?sd?p?te µ?a ???a st? d??st?µa 1,
2,
? pe??tt? p????? ?????, p???µa ??a t? ?p??? ed?
de? µp????µe
?a ap?fa????µe ( µp????µe ?µ?? µe t?? a???????a
Sturm ).
???????, µe t?? ß???e?a t?? ?a???a t??
?a?t?s???, ??a t??
?d?a e??s?s? epe?d? ?p???e? µ?a (1) a??a??
p??s?µ??, ??a
?p???e? µ?a p?a?µat??? ?et??? ???a.
??? ap? t??
????µe d?? (2) a??a??? p??s?µ??,
??a ????µe e?te 2 e?te 0
a???t???? ???e?,
?a? a?t?? e??a? ?? µ??e? p????f???e? p?? µa?
pa???e? ? ?a???a?.

12
????p???s? ??af???? Te???µ?t?? ( s????e?a )

?a??de??µa 2?
??a de?te?? pa??de??µa, ?st? t?? e??s?s??
p?? e??a? ? e??s?s? a??t?t?? ßa?µ?? p?? µp??e? ?a
ep????e?
a?a??t???, e????a ????µe 1 a???t??? ?a? 3 ?1
?et???? ???e?.
??? de ,t??a, ?????µe ???s? t??MATHEMATICA, ??a
epa???e?s?, t?te µe t?? ep?????
solvex4-9x3-2x2120x-1300,x
µp????µe ?a p????µe t?? a?a??t???? e?f??se?? t??
4
????? , p?? µe 5 de?ad??? ??f?a e??a?
-3.60014, 1.22859, 3.97207 ?a? 7.39948.

13
????????S?

?a pa?ap??? af????? t? ?e???t??? p?e??? t??
a??eß?????
e??s?se??,p?? ??a e??s?se?? a??t??? ßa?µ??,p.?.
5??, 6??,
ad??at??? ?a d?s???, ?e?????, t?p???.
?d? ??e? e?d?af???? ? efa?µ?sµ??? p?e??? t??
??µat?? p??
µe ???s? t?? ?a???a t?? ?a?t?s??? ?a? t?? t????
a??a???
s?µe??? t?? s??te?est??,s?µpe?a??eta? ? ?pa??? 3
?1 ?et????
?????, e?? ap? ta p??s?µa t??
f(-?) ?49?3-2?2 -120? -130
0,
s?µpe?a??eta? ? pa???s?a µ??? a???t???? ???a?.
Ta µp????saµe t???? ,µe ??p??a a???µ?t??? µ???d?,
ap? a?t??
p?? a?????????,?a ß???µe ???e? t??(2) µe
ap???ste?? t??p?
ap? t??? pa?ap??? ?e???t????? t?p???( t??
MATHEMATICA).
?.?. µe t?? Newton-Raphson ?a? t? p????aµµa
p?? a??????e?
?a? µe a?????? t?µ?? x0-3., x01.0, x04., ?a?
x08.0, ????µe ???
t?? a???ße?a t?? 5 d.?. ta e???
ap?te??sµata(???? a???????a?)

14
?a? p?? ?p????????ta? ?t? e??a? d?ad????? ??
(1) x1-3.60014, x21.22859, x33.97207,
x47.39948.
15
Newton Raphson in action
program newton_raphson implicit none real(8), external f,df real(8) eps, x,xnew integer maxiter,iter,i eps .5e-5 read,x print 5, x do i 1, 10 xnew x - f(x)/df(x) print 10, i,xnew if (dabs(xnew - x) lt eps) exit x xnew enddo 5 format (2x,"x(0) ",F12.8) 10 format (2x,"x(",i1,") ",F12.8) end program newton_raphson ??????? S???? , pa???s???eta? ? a????? ?p?????sµ?? t?? ???a? µe??st?? µ?t??? . ?p??e?te ?a t? ???ete a?t? ??a t?? e??s?s? (2) function f(x) implicit none real(8), intent(in) x real(8) f f x4 - 9x3 - 2x2 120x - 130 end function f function df(x) implicit none real(8), intent(in) x real(8) df df 4x3 - 27x2 - 4x 120 end function df
16
?p?te??sµata efa?µ???? t??Newton-Raphson ??a t??
-3.0 x(0) -3.00000000 x(1) -3.84018265 x(2) -3.62184706 x(3) -3.60033544 x(4) -3.60013528 x(5) -3.60013527 1.0 x(0) 1.00000000 x(1) 1.21505376 x(2) 1.22853161 x(3) 1.22858939 x(4) 1.22858939
4.0 x(0) 4.00000000 x(1) 3.97222222 x(2) 3.97206842 x(3) 3.97206841 8.0 x(0) 8.00000000 x(1) 7.53431373 x(2) 7.40845667 x(3) 7.39952108 x(4) 7.39947746 x(5) 7.39947746
17
( ß ) St?at?????? ???es?? ?????
?p?????st???? ???????µ??

?. ????d?? ????t?µ?s?? ( Bisection Method )
??. ?sfa?µ???? T?s?? ( Regula Falsi )
???. ????d?? t?? ???d?? ? t?µ???sa? (Secant)
?V. ????d?? Newton Raphson
V. Whittakers method
VI. Mullers method
VII. Bernoullis method
VIII. Q D method

18
St?at?????? ???es?? ????? (S????e?a)

1. G?af???? e????e? ?a? ?d?a?te?a ?a?a?t???st???
t?? ?p?????st???? a??????µ??
2. ???s? ep?st?µ?????? pa??t??, ?p?? ta IMSL /
MATLAB /MATHEMATIKA ?.?.p.
3. ???p???se?? t?? ?p?????st???? a??????µ??
4. ?a??t?te? s?????s?? t?? ?p?????st????
a??????µ??

19
?. Ge???? ?pa?a??pt???

???ß??µa ?a ß?e?e? µ?a ???a t?? e??s?s??
(Leonardo)
?p??t?s? ??ad?at?ss??µe t??? ????? t?? e??s?s??
?a? d?µ???????µe t?? a???????a
?p?te ????µe t?? t?µ??
p?? ? te?e?ta?a d?de? t?? ???a µe a???ße?a 9
de?ad???? ??f???
(d.?.),p???µa p?? ?d?se ? Leonardo t? 1225 ?????
p?t? ?a?e??
?a ??????e? p??
???a ??a t?? a??? s?????s? t?? a???????a?, a?
?p?????s??µe
t?? pa?????? t?? epa?a??pt???? s????t?s?? F(?)
ß??s???µe
p?? s?µa??e? ?t? 2 µe 3 epa?a???e?? apa?t???ta?
??a 1 ???? d.?.

20
Ge???? epa?a??pt??? (s????e?a)

?a?at???se??
1. Sta ep?µe?a ?a d??µe p?? µe t?? d?ad??as?a
Aitken µp????µe ?a ep?ta?????µe, a????
s????????se? a???????e?.
2. ???a? p??fa??? ?t? t?? a?ad??ta?? t?? e??s?s??
de? t?? ?????µe t??a?a, a??? µe st??? t??
pa???s?a µ??? epa?a??pt???? s????t?s?? µe ?s? t?
d??at? µ????te?? ap???t? t?µ?, st?? pa?????? t??.
3. ?? sf??µa t?? ? epa??????? ?a e??a?

21
??. ? µ???d?? t?? d???t?µ?s??

?p?te?e? t?? sta?e?? a??? a??? d?ad??as?a
p??s????s?? µ??? ???a? ,p?? ??e? e?t?p?s?e?
e?t?? s???e???µ???? d?ast?µat??.
??ad?????? d???t?µ?se?? t?? d?ast?µat?? ?a?
?at?????e? ep?????? ?p?d?ast?µ?t?? µp????? ?a
d?s??? d?ad?????? p??se???se?? µ??? ???a? µe
d?a???? a??a??µe?? a???ße?a.
. ? a???ße?a p?? ep?t???????µe µet? ?
d???t?-µ?se??, ??a t?? ?p?????sµ? t?? ???a? ? ,
e??a?

22
???. ????d?? t?? esfa?µ???? ??s?? - ?a?eµß?????
µ???d??

? µ???d?? apa?te? t?? e???e?sµ? t?? ???a? se
??p??? d??st?µa, st? ?p??? st?? s????e?a
e??µ????eta? ? d??e?sa s????t?s? µe t?? e??e?a ?
?p??a d????eta? ap? ta a??a?a s?µe?a t??.
?ts? , ? t?µ? t?? e??e?a? a?t?? µe t?? ????a ??
d?de? t?? ??t??µ??? p??s????s? t?? ???a?.
. ???
t?te ? ep?µe??? ???? t?? a???????a? e??a? ?
. ? µ???d?? e??a? ta??te?? t?? d???t?µ?se?? a???
µp??e? ?a pa???s??se? ep?ß??d??s?, ?p?te ?? d??
pa?a??a??? t?? ( de? e??a? ?? µ??e? )
Ilinois ?a?
Pegasus
ap??a??st??? t?? ta??t?ta µe ?at??????
?p?d?p?as?sµ? ( ? µe??s?) t?? t?µ?? t??
s????t?s?? st? ???? p?? µ??e? sta?e??, ?ts? ?ste
?a ?p???e? s??e??? e?a??a?? t?? ?????,p???µa p??
p??sd?de? t?? µe??st? d??at? ta??t?ta.

23
IV. ????d?? t?? ???d?? (Secant - ?a?eµß????? )

? µ???d?? de? apa?te? e???e?sµ? t?? ???a? ,e??a?
ta??te?? t?? esfa?µ???? ??s??, ???e ???? ???? t??
a???????a? pa???eta? ap? t??? d?? p??????-µ?????.
?ts?,e??
.
?p???e? ??a p??ß??µa st? µ???d?, p?? e??a?
?t? de? ??e? e?asfa??sµ??? t?? s?????s?.
?a???? t? p??ß??µa, ? µ???d?? e??a?
d?µ?f???? ??at? ??e? ????? ta??-t?ta, ??e? ap??
a??????µ? ?a? de? apa?te? t?? ???s? t??
pa?a?????.

24
v. ????d?? Newton - Raphson

? ta??te?? ap??e? p?? p??a?af????sa?.
??e? ?µ?? t?? a????? t?? ???s?? t?? pa?a?????.
?p??e? ?a ?e????e? ???a?? pe??pt?s? t??
µe??d?? t?? ???d?? ?ta? ta d?? s?µe?a s?µp?s???.
. ?????e? ap? ??a t??a?? s?µe?? t?? ped???
???sµ?? (p?? ?a?? e??a? ???a? p??s??? t?? ???a?)
. ? a??????µ?? t?? µe??d?? d?deta? ap? t?? t?p?
. ? ta??t?ta t?? e??a? tet?a??????, ?ste ???t?
st? ???a ?a d?p?as???e? µe ???e ??a epa?????? t?
p????? t?? a???ß?? ??f???, p?? ????? ?? t?te
?p?????s?e?.

25
VI. ????d?? Whittaker

.? µ???d?? e??a? pa??µ??a t?? Newton-Raphson
de? apa?te? ???s? pa?a?????, a??? ???s?-
µ?p??e? µ?? sta?e?? ???s? ?a? a??????µ?
pa??-µ??? t?? N.-R., p?? e??a? ? a????????
. S?????? ?? t?? sta?e?? t?µ? t?? a ?aµß??e-
ta? ? t?µ? t?? pa?a????? t?? s????t?s?? ?a-
t? t?? e?????s?.
. ???fa???, ? ta??t?ta t?? ?p??e?peta? t??
?.- R.,p??? ,?µ??, ? ap??? a??????µ?? t?? t??
???e? e???st???.

26
VII. ????d?? Muller (?a?eµß?????)

? ????d?? e??a? ?e???e?s? t?? µe??d?? t?? ???d??
p??, a?t? e??e?a?, pa?eµß???e? pa?aß??? (
de?te??-ß??µ?a s????t?s?).
?? p?e????t?µa t?? e??a? ?t? µp??e? ?a
p??sd????se? ?a? µ??ad???? ???e?.
?????e? µe t??a s?µe?a p?? de? ?e??ta? se
e??e?a,a??? ß??s???ta? e?at????e? t?? a???? t??
a?????.
? ta??t?ta t?? e??a? s?ed?? tet?a?????? (1.84 ),
de? apa?te? ???s? pa?a????? ?a? de? e??a?
e?a?s??t? se p???ap??? ???e? .
???a? ? µ???d?? p?? a???p??e?ta? sta pa??tta
?t??µ?? ?p?-p????aµµ?t?? ??a t?? e??es? t?? ?????
p??????µ??.

27
VIII. ????d?? Bernoulli (????a??? ???a)

?e??pt?s? 1?.
????a??? ???a µ??ad??? - ??a?µat???
St?? a??eß????? e??s?se?? ? µ??ad???
?at?ap???t? t?µ? µe?a-
??te?? ???a µp??e? ?a ?p?????s?e? µe ??a
pa????p?? t??p?, µe
ß?s? t?? pa?at???s? ?t? se a?ad??µ???? s??se?? (?
e??s?se??
d?af????), ? ?e???? t??? ??s? e?f???eta? µe t??
ß???e?a t??
????? t?? ?a?a?t???st???? t??? p??????µ??.
?ts?, a????????ta? a?t?st??f? p??e?a, ??a ???e
e??s?s?
a?t?st?????µe t?? a?ad??µ??? s??s? ( e??s?s?
d?af???? )
µe t?? a?????? t?p???? s?????e? ,p??ta t?? ?d?e?
?e ß?s? t?? (3) ? a???????a p?? d?µ?????e?ta? ap?
t?? (2)

ap?de????eta? ?t? ??e? t?? ?d??t?ta
?e??pt?s? 2?
? ????a??? ???a e??a? µ??ad??? ( 2 ????a??e? )
St?? pe??pt?s? a?t?,? ???a p??pe? ?a ??e? t??
??f?as?
?p?te,t?te ap?de????eta? ?t? ?? ???? t?
(4)p?????? t??
?p??p?? e????a ?p????????ta? ?? d?? µ??ad????
???e?

29
?p?de??? t?? (5)

St?? e??s?se?? d?af???? (2), ? ?e???? t??? ??s?
d?deta? ap? t??
???? t?? (7), ? (5) ???feta?
p?? µe e?a???? ?????? pa?????ta , ap? t??
a???µ?t? t??
?a? t?? ap? t?? pa?a??µast?, ?a???
?a? t?? e?t??es?
t?? s?et???? ap??p???s??, e????a ?aµß????µe t??
s??s? (5), af?? p??fa??? st??? ????? t??
???sµat?? ?a ????µe
?p?te t? te???? ap?t??esµa ?a e??a?

30
?p?de??? t?? (6)

St?? pe??pt?s? a?t? ?? 2 µ??ad???? ???e? ?a
p??????
??? ? ?e???? ??s? t?? (2) ?a d?deta? ap? t??
?a? s????e?? µ??ad????
?p?te µe e?a???? ?????? pa?????ta ?a? ???e??
?????
?????a d?ap?st??eta? ?t? ???? t?? (12), ta ?
p?????? t?? e??s?se??

? ep???s? t?? ??aµµ???? s?st?µat?? (13) ?? p???
t???
?p?te ?? p?????? e?f??se?? t?? ????? e??a?
p??fa?e??.
?a??de??µa (?? µ??ad???? ???e? t?? e??s?se?? t??
Leonardo)
? a?ad??µ??? s??s? ???feta?
p?? ??a t?? a?????? s?????e? (3),d?de? t?? t?µ??

32
?a??de??µa (?? µ??ad???? ???e? t?? e??s?se?? t??
leonardo-s????e?a)

??e? e?d?af???? ? e?a??a?? t?? p??s?µ?? t?? ?,
p???µa p?? ?p?d????e? t?? pa???s?a t??
µ??ad???? ?????.
?e???? , ?a ????µe

33
IX. ????d?? ??af???? ??????? (Quotients
Differences)

? µ???d?? e??a? ?e???e?s? t?? µe??d??
Bernoulli, d?µ?????e? ?s????µe? p??? t?? ???e?
t?? a??eß????? e??s?s?? a???????e?
(p??????, q-sequences ) ,?? ?p??e? ?e????
s????????? p??? t?? p?a?µat???? ?a? d?a?e???µ??e?
???e? t?? e??s?s??.? p??t? a???????a t?? µe??d??
e??a? ? ???st? µa? Bernoulli. ?????, µeta?? t??
pa?ap??? a????????? pa?eµß?????ta? ?? a???????e?
d?af???? ( d-sequences), p?? st?? ??s?a e??a?
µ?de?????. ?ts?, ??a t?? e??s?s? (1),
?ept?µe??a?? ?a ????µe ??a t??
??? d?ad??as?a p?? ap?d?deta? ap? t??
pa?a??t? p??a?a ?a? t??? d?? ?a???e? s??µat?sµ??
t?? ???? t?? a?????????, t?? q- a???????a? ?a?
t?? d- a???????a?,p?? e??a? ? p??t? ??aµµ?, ?
p??t? ?a? ? te?e?ta?a st??? t?? p??a?a e??a?
sta?e??? ?a? ? µe? ??aµµ? s??µat??eta? ap? t???
s??te?est?? t?? e??s?s?? ,?? de st??e? e??a? d??
a???????e? µe µ?de????.?a
st???e?a t?? p??t?? ??aµµ?? e??a? ta
e?? ? p??a?a? ?a ??e? t?? a??????? a?????
µ??f?

34
???a?a? ??af???? ??????? (??a??? t?? p??a?a)

S?µp?????eta? de µe d?ad?????? ??aµµ?? µe
t?? ß???e?a t?? e??? d?? t????????? ?a?????
1?? ?a???a? ??a ta q st???e?a ???e q st???e??
?s??ta? µe t? p??????µe?? t??, s?? () t? de???
t??, µe??? () t? a??ste?? t??, t?? p??????µe???
??aµµ??.
2?? ?a???a? ??a ta d st???e?a ???e d st???e??
?s??ta? µe t? p??????µe?? t??, p???ap?as?asµ???
ep? t? de??? t?? ?a? d????µ??? d?? t?? a??ste???
t?? st???e??? t?? d???? t?? ??aµµ??.
?a pa?ap??? ap?d?d??ta? ap? t??? pa?a??t? t?p???
(1)
???fa???, se ???e ??a ??aµµ? s?µp???????ta? p??ta
ta q st???e?a ?a? µet? ta d st???e?a.

35
?a??de??µa ?? ???e? t?? e??s?s?? p?? a??????e?
e??a? 4,3,2?a? 1.

? e??s?s?
? p??a?a? d?af???? p?????? ?a e??a?
?p? t?? ?p??? e??a? saf?? ?? s?????se?? t??
q-a????????? st?? ???e?, e??
?? d-a???????e? e??a? ap??? µ?de?????.

36
?e??pt?s? ???ad???? ?????

St?? µ??ad???? ???e? ?? a???????e? q de?
pe????e???ta? ap? µ?-
de????? d a???????e?,a??? d?µ????????ta?
d?? ???e? a?????-
??e? ap? t??? ????? t?? µ? s????????s?? q
a?????????
p?? s????????? st? ?????sµa ?a? st? ????µe??
t?? e??s?s?? p?? ?? ???e? t?? p??sd???????? t?
??t??µe?? µ??ad??? ?e????.
?a??de??µa St?? e??s?s? t?? Leonardo
O? ???st? ????µe t?? p?a?µat??? ???a 1.369 ?a?
t?? d?? µ??ad????
,?ts? e?? ?atas?e??s??µe t??
s?et??? p??a?a ?a ????µe

37
???a?a? d?af???? p?????? (e??s?s? Leonardo )
38
?a???t?µa. F???s?f?a ????µ?t???? ??a??se??

Se pa?a??te?e? ep???? ?ta? s?ed?? ap?de?t? ?
?p??? t?? Carl Jacobi (1805-1840) ?t?
?a µa??µat??? ?p??et???
t?p?te ???? ap? t?? t?µ? t?? a????p????
p?e?µat??.
??t? ? ??? µ??? af???µ??? ?a? ?e???t???
t?p???t?s?, a??? ?a? ? ??p?? ?pe??fa?? ??s?, p??
?ta? f?s??? ap?t??e-
sµa ??? µ??? t?? ?e???t???? e?as????s?? t??
p?e????t?ta? t?? µa??µat???? t?? ep???? t??, a???
?a? t?? ?pa?s??
p?? e??e d?µ??????se? ? a???a? t?? ???????
st??? t?te d?a???t??, e??e ?? s???pe?a t?
?a??t?µ??? e?e??? µa ?a?????
t?µ?t??? ??f?as? ?t?
?? µa??µat???? ????????? p?? ?a ??s???
??a p??ß??µa, a??? de? µp????? ?a ß???? t? ??s?
t??! .
??a pa??de??µa, ?a ?a??st??se saf?ste?? t?
sp??da??t?ta t?? pa?ap??? ??f?as??. G?? t? s??p?
a?t?, a? ?p???-
s??µe p?? ????µe ??a ???sµ??? p??ß??µa p?? µet?
µ?? s?et??? µa??µat??? a????s? ?ata???e? st??
ep???s? µ???
a??eß????? e??s?s?? a??t???? t?? 4?? ßa?µ??, p.?.
5?? ßa?µ??. F?s??? ? µa??µat???? µa? µp??e? ?a
?s????ste? ?t?
t? p??ß??µa ??e? ???e? ?a? ?t? ? ??s? t?? d?deta?
ap? t?? ???e? t?? a??eß????? e??s?s?? p??
p??????e.
?p? t?? s?et??? ??e??a, de, p?? st? s????e?a
???e? ( a?af????? µe t?? ???e? t?? a??eß?????
e??s?se?? ) ß??s?e? ?t?
t? Teµe???de? Te???µa t?? ????ß?a? ( DAlembert
1717-1783 ) e?asfa???e? se ???e a??eß???? e??s?s?
?-?st?? ßa?-
µ??, µe s??te?est?? p?a?µat????? ? µ??ad?????,
t?? ?pa??? ? a???ß?? ????? µ?sa st? s?µa t??
µ??ad???? a???-
µ??.
?? ?e???µa d?st????,??? µ??? de? p??sf??e?
?aµ?a ?p?de??? ??a t?? t??p? e??es?? t?? ?????
a?t??, a??? st??
p???e?µ??? pe??pt?s?, t?? e??s?se?? 5?? ßa?µ??, ?
E. Galois ( 1811-1832 ) se e??as?a p?? s?????a?e
se ?????a
21 et?? (pe?????? t?? ?p??a? ap?ste??e se f???
t?? t?? pa?aµ??? t?? µ???µa??a? p?? t?? ?d???se
st? ???at?),
ap?de??e µe t? ß???e?a t?? ?e???a? t?? ?µ?d??
t? ad??at? t?? e??es? t?p?? ( p?? ?a pe????e?
µ??? t?? 4

39
- 2 -

???t??, e?? st??? pe?asµ????? a???e? ?
?a??????e?a t?? ?e???t???? p?e???? t??
µa??µat???? ?ta? ??t???, e?t??t???,
de? ?ta? ????? e?e???? p?? s?µµe?????t? t??
p?st? t?? ???d?? Kelvin ( 1824-1907) sta
a???µ?t??? ap?te??sµata ?a? ? ?p????
s??????e ?a ??e?
de? ??? ??deµ?a ??a??p???s? ap? t???
µa??µat????? t?p??? e?t?? e?? a?s????µa? t?
a???µ?t??? t??? µ??e???.
??t? ? efa?µ?sµ??? ( ?? ??? d?a?s??t??? )
p?e??? t?? µa??µat???? p?? s????pa?e ??? µ??? t??
????µ?d? ( 287 212 p.?.)
a??? ?a? t?? Newton (1643 - 1727) ?a? t?? Euler
(1707-1781) ?a? a?t?? t?? Gauss (1777-1855) ?d?se
t?? p???p???se?? st??
Kepler (1571-1630) ??a t?? ?ea??st???
a???p???s? t?? ?????d?? pa?at???se?? t?? Tycho
Brache (1546-1601), p?? st? s????e?a
t?? ?d???sa? st?? a?a?????? t?? ?µ???µ?? ??µ??
t?? ????s?? t?? p?a??t?? , p?? ap??p???se t??
µ??a???? t??
s?µpa?t?? ?a? daµ?st??e t? ??sµ??? ????.
?a s???????st??? ap?te??sµata ta ?p??a
ap???ea? ap? t? d??at?t?ta ??? µ??? t??
ap?de??e?? t?? ?pa???? µ??? ??s??
a??? ?a? t?? ??a??t?ta? p??sd????sµ?? a?t??,
e??a? ?? s???pe?a a???ße?? p??ß???e?? e??e??e??
????? ?a? se????? ,eµf???s?
??µ?t?? ??p., µ?????t? ?ta? f?s??? ?a t???s???
pe??ss?te?? t?? ????µ?de?a ( efa?µ?sµ??? )
t?s? t?? µa??µat????,
e?t??t???, te???? ?ta? ? ????e?de?a
(?e???t???-??????) p?e??? t?? µa??µat???? p??
e?d??aµ????e.
? ßas???? ????? ?ta? d?tt??. ???t?? µe?,
d??t? ???? t?? p?????? t?? p??ß??µ?t?? ?a? t??
??tas?? t???, a??? ?a? ????
t?? ?e????t?ta? t?? µa??µat???? d?e?e???s?? , de?
?ta? d??at? ? a?e??es? t?p?? se ???e pe??pt?s?,
e?? ? d??at?t?ta
ap?de???? t?? ?pa???? ??se?? ?e???t????
ap?de?????e e??e??ste?a. ?e?te??? de, pa?a??????
µe ta p??ß??µata
e?e??a sta ?p??a ?ta? d??at? ? ??f?as? t??
??se??, a?t? ?d????se e?te se a??????µ??? ?p?
µ??f? se???? ( Euler ),e?te,
?e????te?a se d?ad??as?e? t?? ?p???? t? µ??e???
t?? a???µ?t???? ?p?????sµ?? ?ta? ?pe???? t??
?p?????st????
d???µe??, t?? ep????.

40
- 3 -

?a? t? e??t?µa e??a? t? d??? ?e??s?a?
F?s???, ? ????µ?t??? ?????s? , a?a??te?
ße?t??µ??e? µe??d??? µe t?? ?p??e? ?a µe???eta?
?at? t? d??at? t? p????? t?? p???e??.
?? epa?astat??? ?µ?? µ?s? e??a? ? pa???s?a t??
??e?t??????? ?p?????st? (?.?.) ?a? ? a???p???s?
t?? µe t??
?.?.,t?? ?p??a ??e? ??????? µetaß???e?, d??t?,
de? p??pe? ?a ??sµ????µe ?t?, ? p??t? µe? ?e?t?ta
t?? ?.?. a???eta?
e?? t?? ep??? t?? ?e?t??a (17?? a???a?), ? de
ap?d???µ?s? ?a? ? pa?a?????sµ?? t?? ep???e ???
t?s? ????
ad??aµ?a?, a??? ???? t?? apa???e?t???? p??????
t?? ?p?????sµ?? p?? apa?t???ta? ap? t??
d?ad??as?e? t??.
?ts?, ?a??stata? fa?e?? ??at? ? eµf???s? t??
?.?. ?d?se t?? te??st?a a?t? ???s? st?? ?.?., ?
?p??a a???p????sa ?a?
t?? ??e? af???µ??e? µe??d??? a????s?? ( ?p?? p.?.
t? s??a?t?s?a?? a????s? ) d????eta? t??a µ?a
pe???d? p??t?-
fa???? ????s?? ?a? p??sf????, st?? efa?µ?sµ???
ep?st?µ? ?a? te???????a.
?a p??sfata de ep?te??µata t?? te???????a?,
e??a? ap??? ad?a???ta ????? t?? s?µß??? t?? ?.?.
?? a???s?µe??t?
p??t?? ??a t??? µa??µat????? e??a? ?t? ?ts?
e?asf???sa? t?? a???p?st?a ?t? µp????? ep?t?????
?a ß??s???? ?a? t??
??se?? t?? p??ß??µ?t??!
G?a ?a µ? ?e????e? ?t? ?ta? µ??? t??
?p?????st? ? s?µß??? ? ?p??a ?at?st?se, p.?.
d??at? t?? µe?a?e??d? ??t??
e?e?e???s? t?? d?ast?µat?? µe t??? te???t???
d???f?????, a??? ?a? ? p???d?? st?? a???µ?t????
d?e??as?e? p??
s???ßa?a? ap?fas?st??? st? ??? ep?te??µa,
epa?e???µe??? st? pa??de??µa t?? ep???s?? t?? 20
??aµµ???? e??s?-
se?? µe ?s????µ??? a???st???, s?µe?????µe ?t?
ep??????ta? t?? ???st? µ???d? t?? Cramer ??a t??
?p?????sµ? µ???
?a? µ??? ??????sa? µe t? ß???e?a ?p?????st? p??
e?te?e? 1.000.000 p???e??/sec, ?a apa?t???t?
pe??p?? 1.000
a???e?,??a t?? e?t??es? t?? 2.51018p???e?? p??
p??a?af??aµe, e?? µe t?? apa???f? Gauss µ???
?ept? a?????!

41
-4-

?? ?e????? e??a? ?t? ? ????µ?t??? ?????s?
ep???e? t??p???, ?p?? p.?. t?? ?pa???f? Gauss ??a
t?? e??es?
t?? ??s?? e??? ??aµµ???? s?st?µat??, µe t???
?p????? µ?sa se ???a ?ept? t?? ??a?
ep???eta? t? p??????µe?? p??ß??µa.
???????,e?? ?p???s??µe ?t? ? p?? ?a? ? ?at?
t?? te?e?ta?? µe???? p??eµ? pe???d??, ?ta? ?
pe???d?? ???s?? t??
?.?., t?te t? ?e?????? t?? s???????? ?.?. ?a
p??pe? ?a t?p??et??e? t? ?t?? 1947, ?ta? st?
?a?ep?st?µ?? t??
?a??f????a? (U.C.L.A.) ?d?????e t? ??st?t??t?
????µ?t???? ?????s??.
???a? e?d?af????, ep?s?? t? ?e????? ?t? st??
?????a, ? ??f??d? e?s??a?e t?? ?.?. st? p????aµµ?
t?? µet? ap? µ?a
???????? de?aet?a (1959), e?? st?? ?at??da µa?,
??? µ??? s ??a ta ?a?ep?st?µ?? t?? ??e? e?sa??e?
t? a?t??e?µe??
t?? ?.?.a??? ?a? se p?e?ste? ???e? pe??pt?se??
?p???e? µ?a e?p???t??? ??t?? ??t?s? ??a
a???µ?t???? µe??d???
?a? p????aµµat?sµ? ?.?., µe t?? ?p???
s??d???eta? ? f?s??? ?a? ef??t? efa?µ??? t??
?.?.
? ???sµ?? t?? ?.?. de? e??a? epa???ß??
?a????sµ????. ????? ???e ?t? e??a? ? µe??t?
µe??d??
?a? d?ad??as??? p??? ???? p??se???st????
apa?t?se?? (S. Parter, Comm ACM, 1969, 12,
661-663).
????? ??????? ?t? e??a? ? ?e???a ?atas?e?ast????
µe??d?? t?? µa??µat???? a????s?? (P.
Henrici), e?? ? Hartree ???e? ?t? t? ??µa t??
?.?. s?et??eta? µe t?? ep?st?µ?, t?? t???? t??
a???µ?t???? ?p?????sµ?? ?a? e?d???te?a µe t??
d?ad??as?e? e?e??e? µe t?? ?p??e? ?aµß?-
???µe ???sµ??a a???µ?t??? ap?te??sµata ap?
???sµ??a a???µ?t??? ded?µ??a.

42
- 5 -

???a? ?s?? ? d??aµ?sµ?? t?? ?.?. p??
a??e?ta? t?? ap?d??? t?? ?a????? t?? ???sµ??...!
?µe??, ed?, ?a t???s??µe t?? ?????a t??
p??se???st???? ??s?? ?a? s??ep?? t?? a??????
???sµ?? t??
sf??µat?? p??s????s??. ?p?s??, t?? a?????
e?t??es?? a???µ?t???? p???e?? - µe t?? ß???e?a
e??? µ??a?????
µ?s?? pe?????sµ???? a???ße?a? - ß?se? µ?a?
p???a????sµ???? d?ad??as?a? - t?? a??????µ??.
?ts?, ????µe ??a t?? ?.?., t?? ?p??a ?a
?e???s??µe ?? t?? t???? t?? d??at?? ?a? t??
ß??t?st?? st?at??????, t?
e??? t??pt???
a) ???????µ??,
ß) ????µ?t???? ?p?????sµ??, ?a?
?) ???sd????sµ?? sf??µat??.
? ????? t?? a??????µ?? e??a? ßas????. St?
pa??? ?a ?e????µe ?t? ? a??????µ?? e??a? ??a
p???e? ?a? ?a???
?a????sµ??? s????? ß?µ?t?? (p???e??-?atas?e???)
??a t?? ep?te??? e??? st???? (p.?. t?? e??es? t??
??s?? e???
p??ß??µat??, ? ?µ??a t?? pa?as?e?? ed?sµat?? ?
pe??pt?s? µa?e?????? a??????µ??).
?????t?t??p??, ????µe a???µ?t?????
a??????µ??? (ta ß?µata e??a? ?? a???µ?t????
p???e??), ?a? µ? a???µ?t?????
(p.?. ?e?µet?????? ?????e?de??? ?e?µet??a e??a?
??a s????? a??????µ?? µe ß?s? t?? ?a???a ?a? t??
d?aß?t?).
?? d??? µa? e?d?af???? ?a s???e?t???e?
ap???e?st??? st??? a???µ?t????? a??????µ???,
pe??pt?se?? t?? ?p????
????????µe ?d? ap? t? ???e??, ?p?? p.?., e??a? ?
a??????µ?? t?? ????e?d? ??a t?? e??es? t??
?.?.?., ? ? a???-
???µ?? t?? tet?a??????? ???a? e??? ?et????
a???µ??, ? ? a??????µ?? t?? apa???f?? ??a t??
ep???s? ??aµµ????
s?st?µ?t?? (t?? ?p??? a??????µ? ?d? a?af??aµe ?a?
?a e?et?s??µe ?ept?µe??? a???te?a).
?? pa?ap??? pe??pt?se?? ap?te????
pa?ade??µata pepe?asµ???? a??????µ??, t?? ?p????
? ??s? ß??s?eta?

43
- 6 -

? p???, ? ?p?????sµ?? t?? µe t?? ß???e?a
t?? se????
G?a t??? at??µ??e? a??????µ???,? ßas??? µa?
apa?t?s? e??a? ?p?? t? sf??µa t?? p??se???st????
t?µ??,???? e?t??es??
pepe?asµ???? µ??? ß?µ?t??, e?att??eta?, ?s?
a????eta? t? p????? t?? e?te???µe??? ß?µ?t??, ?
??a ?a
???s?µ?p???s??µe ???????a ?.?.,?a p??pe? ?
a??????µ?? ?a s??????e?,d??ad?, ?a p??pe?, ??a
µe? t? tet?a?????? ???a ?
a???????a t?? a???µ??
?a e??a? s????????sa
,??a de t?? se??? ?a p??pe? ? a???????a
t?? µe????? a????sµ?t??

?a s??????e?.
??a ap? ta ßas??? ??µata t?? ?.?. e??a? ?
p??sd????sµ?? s??????? µe t?? ?p??e? ??a?
ded?µ???? a??????µ??
s??????e?. S??af?? ??µa e??a? ? ???s? t??
ta??t?ta? s?????s?? (rate of convergence) t??
a??????µ??. ??a ????
sp??da?? ??µa e??a? ? p??sd????sµ?? t?? µe??????
t?? sf??µat??, t? ?p??? µp??e? ?a d?deta? e?te
p??se???st???
µe ??a as?µpt?t??? t?p?, e?te µe ??a f???µa
(error bound), t? ?p??? p??sd?????e? p??fa???,
t?? µ???st? t?µ? t??.
?????,??a ep?s?? µe???? ??µa e??a? ? e?st??e?a
(stability) e??? a??????µ??, a?af????? µe t??
ep?d?as? sfa?µ?t??,
e?te sta a????? ded?µ??a ( p.?. pe??aµat???
ded?µ??a ), e?te ?at? t?? d????e?a t?? e?t??es??
t?? p???e??
(sf??µata st???????p???s?? round off errors). ?
ad???e?a t?? a??????µ?? sta sf??µata
st???????p???se??
?a?e?ta? a???µ?t??? e?st??e?a, e??a? de s????t?s?
?a? t?? p?????? t?? ??f??? ta ?p??a
???s?µ?p?????ta? st???
?p?????sµ???. ? e?a?s??s?a t?? a??????µ?? ?? p???
t?? µetaß???? t?? a?????? ded?µ???? d?a????eta?
se d??
?e????? pe??pt?se??. ? p??t? s?et??eta? µe t?
f?s? t?? p??ß??µat?? ?a? af??? t?? t??p?
t?p???t?s?? t??
p??ß??µat??.

44
- 7 -

? de?te?? pe??pt?s? s?et??eta? µe t??
s?µpe??f??? t?? ??s?? t?? p??ß??µat?? ?ta? ta
ded?µ??a t?? ?f?sta?ta?
µ????? µetaß????. ?ts? ?aµß????µe ta ?a???
s?µpe??fe??µe?a (well-conditioned) p??ß??µata sta
?p??a µ?????
a??a??? ded?µ???? ep?f????? a?t?st????? µ?????
µ??? a??a??? st? ??s? t?? p??ß??µat??. Se
a?t??et? pe??pt?s?
?a????ta? ?a??? s?µpe??fe??µe?a
(ill-conditioned). ? d?af??? µeta?? ?a???
te?e?µ???? ?a? ?a??? s?µpe??fe??µe???
p??ß??µ?t?? e??a? ?t? sta µe? ?a???
s?µpe??fe??µe?a p??ß??µa e?? e??a? d?s???? ?a
ep????e? ????
a???µ???? e?a?s??s?a? t?? ??s?? t??, e?t??t???
µp??e? µe ?at?????? ?e???sµ? ?a a?t?µet?p?s?e?
(p.?. µe
a???s? t?? a???ße?a? t?? ?p?????sµ??). ??t???t??
st? ?a??? te?e?µe?? p??ß??µa ? ??s? t?? e??a?
??e?
???µat?? (p.?.a?e?a?t?t?? t?? ???s?µ?p????µe???
a???ße?a? ? ??s? t?? e??a? ad??at?).
O? pa??de??µa e??? ?a??? te?e?µe???
p??ß??µat??, a? p????µe t? s?st?µa (S)
p?? ??e? p??fa??? t? ??s? ?1, ?1. ?p?p????, a?
?e???s??µe ?t? st? ??s? t??
??asµ?t?? ??t??µe t??? a?t?st?????? de?ad?????
a???µ??? (?p???t??ta? ?t? e??a??µaste
µe µ?a ???sµ??? ?p?????st??? µ??a??). ???fa???,
ta ???sµata ?a e?f?ast??? µe ??a
?a????sµ??? p????? ??f??? (µe ap???p? ??f??? ?
st???????p???s??) ?p?te t?
s?st?µa ?a??stata? ad??at??, af?? ? de?te??
e??s?s? ?a d?af??e? p??t?te e?af??? t?? p??t??,
?? p??? t? de?te??
µ????, e?? ta p??ta µ??? ?a e??a? ?sa. ?????, ??
??a pa??de??µa e??? ill-conditioned p??ß??µat??,
a?af????µe t??
??ass??? pe??pt?s? t?? ??aµµ???? s?st?µat??, t??
?p???? ? p??a?a? t?? s??te?est?? e??a? t?? ???s??
t??
p?????? Hilbert, ???st?? ??a t?? ?a???
s?µpe??f??? t???
µe
St?? p???e?µe?? pe??pt?s? ?aµß????µe t? ??t???
s?st?µa ??aµµ???? e??s?se?? (µe p??a?a
s??te?est?? t?? p??a?a

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