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Carl Jacobi (1805-1840)            .        ,                      ,              ,                   ,        !              ,                      4 , .. 5 .                          .   , ,     (      )        (D Alembert 1717-1783)      -                 .   ,             ,    ,   5 ,  E. Galois (1811-1832)       21  (               ),              (      4       )             4 . , ,                        5        Z%T   |  ) X h  e  n7J < ez2p     (1) 9,  9v,             , ,           Kelvin (1824-1907)                        .    (  )          (287  212 ..)    Newton (1643 - 1727)   Euler (1707-1781)    Gauss (1777- 1855)     Kepler (1571-1630)         Tycho Brache (1546-1601),              ,    ,           .                       ,         ,   .,         ()   , ,     (-)     .     .  ,          ,         ,         ,          .  ,             ,         (Euler), ,                 .     .        10    : .                 ,     Cramer    ,    (  )    : (1)  i=1,2,& 10  Ai, i=1,2,& ,10       ,    i-       bi, i=1,2,& ,10 (  ). ,        11  ,    ,    10! = 3 628 800 ,         10 . ,       ,            36 .        ,            (2.5*1018).             (1). ^%PP h }  Y  4  =  15 !F N#&osl8p     (2) 9,  9      ;     (..),              .           (..)       ..,     . ,     ,      ..       (17 )            ,             . ,       ..       ..,          ( ..   )             .              ..                      ! (             , ..                  20     ,       Cramer              1.000.000 /sec,    1.000      2.5*1018   !  ..             ).             ,      ..,      ..       1947,      (U.C.L.A.)     .  ,      ,     ..         (1959),    ,             ..,                 ..           .. ~z %K    * ` ! qe 44`XT+@36=1K(OXfb c)rj9p     (3) 9,  9   ..    .             (S. Parter, Comm ACM, 1969, 12, 661-663).            (P. Henrici),   Hartree      ..    ,                      .      ..        ... , ,               . ,     , (       )     ( ). ,    ..,             ,   : ) , )  , )  .      .                (-)      (..      ,          ). ,    (     ),    (..                ).          ,        ,  ..,          ...,         ,           (      ).       ,          .     ,  , ..           : ,  ,   : `z %5 R E 7 '  c 0  -   B B * V  2  8  ?    ~l62 pljm#:p     (4) 9,  9 ,            ,           ,     , ,       ,      ..,      , ,  ,         :   .          :  .       ..           .         (rate of convergence)  .                      ,     (error bound),    ,    . ,        (stability)       ,     (..  ),         (   round off errors).          ,             .                .               . ,        (well-posed)     ( ),      () ,    : 1.   ||||    : ||||<    (+). 2.   : ||(+)-()||0  ||||0. (           ) .   t   '  O V ^ G     $ -  :sbu4<?   k  _ =n;p     (5) 9,  9                  .      (well-conditioned)               .       (ill-conditioned).                           ,        (..      ).            (..        ).      ,     ():      =1, =1. ,             (       ). ,          (    )     ,          ,     ,       .      . ,     ill- conditioned ,       ,            Hilbert           (     Hilbert  3).      : (1, 2, 3)=(9,-36,30).  ,        2        (1) : (1, 2, 3)=(7,-23,17).            Hilbert.       2        5 ..         (1, 2, 3)=(9.00613,-36.03206,30.0303).     g  1 F   V F       +U    Y\3.       (..): 1.1: 1.2:    ..:0l&lED4. 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̙33___PPT10i.\`<+D=' = @B +e K0 `(    c $!Y<$@  0 + "  B'Y }  Y `  c $A j??/0 j`  c $A k??@ kH  0޽h ? ̙33%___PPT10.[ڬ+xwD' = @B D' = @BA?%,( < +O%,( < +D ' =%(%(DK ' =%(D' =4@BBBB%()))D' =+4 ,0B 0.9*3>'Bstyle.fontSize= `B<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*1D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1g%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*1gD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*1gD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*gD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*g+/ K0 h`(    c $DYY<$@  0 Y "  BDY }  Y `  c $A o??@ oH  0޽h ? ̙33WO___PPT10/.[ڬ+xwD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%()))D' =+4 ,0B 0.9*3>'Bstyle.fontSize= `B<*HD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ip%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*IpD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*IpD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*pD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+= K0 <4 (    c $YY Y "xd  B]Y }  Y `  c $A p??@ pH  0޽h ? ̙33___PPT10i.[ڬ+D=' = @B + K0 P((    HfY 'g,  Y   s *x+ Y "xd 0 TA ? ?{ ]  Y`  c $A ??@  0 TA ? ?    +H  0޽h ? ̙33___PPT10i.[ڬ+D=' = @B + K0 ` (    HwY 'g,  Y   s *}Y Y "xd 0 TA  ? ?t   Y`  c $A  ??@  H  0޽h ? ̙33___PPT10i.[ڬ+D=' = @B +m   | p  (  l   BY ?"6@ NNN?N      : (1) x1=-3.60014, x2=1.22859, x3=3.97207, x4=7.39948+ l4(l0        :4h    40e0eA     @  5% 8c8c     ?A)BCD|E||# "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab YH  0޽h ? ff[[ffRR|___PPT10i.+D=' = @B + K0  >(    HLY g,  Y   s *Y Y "xd`  c $A ??@    NGBH ?"6@ NNN?NY9<   <?"6@`NNN?NYS<H  0޽h ? ̙33___PPT10i.[ڬ+D=' = @B +I K0 H@ !(    6dY"  J 0 `  c $A ??@   NGBH ?"6@ NNN?NY9<r  S Yw'  Y <| qev ! #"  eqv Y7   BY ?"6@ NNN?N ev function f(x) implicit none real(8), intent(in) :: x real(8) :: f f = x**4 - 9*x**3 - 2*x**2 + 120*x - 130 end function f function df(x) implicit none real(8), intent(in) :: x real(8) :: df df = 4*x**3 - 27*x**2 - 4*x + 120 end function df !/  ) <2  @`  Bl ?"6@ NNN?Nqe v cQprogram 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) <= eps) exit x = xnew enddo 5 format (2x,"x(0)= ",F12.8) 10 format (2x,"x(",i1,")= ",F12.8) end program newton_raphson R  1 8 @     @`B   <o ?"0@NNN?NqeeB   <o ?"0@NNN?NqvvB   <o ?"0@NNN?NqeqvB   61 ?"0@NNN?N e vB  <o ?"0@NNN?NevH  0޽h ? ̙33___PPT10i.[ڬ+D=' = @B +E \ T  (    9  #"z94 -  B\+ ?"6@ NNN?N l9 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 @`[  B1 ?"6@ NNN?Nl  W4.0 x(0)= 4.00000000 x(1)= 3.97222222 x(2)= 3.97206842 x(3)= 3.97206841XX @`p  B+ ?"6@ NNN?N 9l l1.0 x(0)= 1.00000000 x(1)= 1.21505376 x(2)= 1.22853161 x(3)= 1.22858939 x(4)= 1.22858939mm @`  B1 ?"6@ NNN?N l -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 @`B   <o ?"0@NNN?N9B   61 ?"0@NNN?Nl9lB   <o ?"0@NNN?N9B   <o ?"0@NNN?NB   61 ?"0@NNN?N  B  <o ?"0@NNN?N998  61 k'g b   Newton-Raphson  "21$$*   TA$ $? ?"6@ NNN?N  $H  0޽h ? ff[[ffRR|___PPT10i.%b-+D=' = @B +I K0 H@0,(  , , H\l   l  , s *Yj< l "xd` , c $A <??@ <H , 0޽h ? ̙33___PPT10i.[ڬ+D=' = @B + K0 ` 8(  8 8 H0l ,  l  8 s *6lZ l "xdz 80  :0e0eA     ? A@  A5% 8c8c     ?A)BCD|E||# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab )q   l` 8 c $A ???@ ? 8 B ?"6@ NNN?N  80  @0e0eA     ? @  5% 8c8c     ?A)BCD|E||# "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab (   lH 8 0޽h ? ̙33___PPT10i.[ڬ+D=' = @B + K0  T(  T T HPNl   l  T s *Sl l "xd` T c $A ??@  T B ?"6@ NNN?NH T 0޽h ? ̙33___PPT10i.[ڬ+D=' = @B +E  K0 D <  ` (  ` ` Hhil   l  ` s *(gl l "xd` ` c $A ??@  ` B ?"6@ NNN?Nz `0  :0e0eA     ? A@  A5% 8c8c     ?A)BCD|E||# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abqQ  l  ` HA ? ?"6@ NNN?Nv , H ` 0޽h ? ̙33___PPT10i.[ڬ+D=' = @B + K0 @  pz(  p p HS-   -  p s *Еl - "xd` p c $A ??@  p B ?"6@ NNN?N  p HA! ? ?"6@ NNN?Ng   p HA" ? ?"6@ NNN?NZ    p HA# ? ?"6@ NNN?N T    p HA #? ?"6@ NNN?N  #  p HA% ? ?"6@ NNN?N &f H p 0޽h ? ̙33___PPT10i.[ڬ+D=' = @B +! K0  P  t(  t t Hȁ-   l  t s *l - "xd` t c $A ??@  t B ?"6@ NNN?N  t HA+ a? ?"6@ NNN?NH  a  t HA' ? ?"6@ NNN?NF     t HA( ? ?"6@ NNN?N VH   t HA) !? ?"6@ NNN?N v !H t 0޽h ? ̙33___PPT10i.[ڬ+D=' = @B +o K0 nfp <(  < < H@-  `}  -  < s *l Y~  - "xd < bA eq1"`S  l` < c $A i??@ i` < c $A j??0 H j` < c $A k??}  u k  < VA eq3S   lH < 0޽h ? ̙33___PPT10i.[ڬ+D=' = @B +V K0    @(  @ @ Hl  `}  l  @ s *lG l "xd @ VA eq2  " l` @ c $A l??@ l` @ c $A m??  m` @ c $A* ??k* ` @ c $A o??  o  @ <0n+ ?"6@ NNN?NF @  Z((l2   @ VA eq4 ` " l  @ <`l ?"6@ NNN?NV P  Z((l2   @ <l ?"6@ NNN?N &6  Z((l2 .  @ 6l wG,$D  0     pU 0 H @ 0޽h ? ̙33___PPT10.[ڬ+f D|' = @B D7' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* @ %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* @ D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* @ + K0  D(  D D Hl 'g,  l  D s *l l "xd D0 TA w? ? q  w l` D c $A x??@ x` D c $A y??<2 L y` D c $A z??Mu z` D c $A {?? {  D0 TA |? ?pt c  | lf  D s *A }? ?9 R4  }f  D s *A ~? ? Sh  ~f  D s *A ? ?X  H D 0޽h ? ̙33___PPT10i.[ڬ+D=' = @B +WW K0 QQRRHP(  H H Hl  `  l  H s *x l "xd H0 TA ? ?  ` H c $A ??@ ` H c $A ?? f H s *A ? ? f H s *A ? ?d?  K *  H# #"2& rrrrrrrr*,   H H  ?"6@`NNN?N  _1  @`  H H ?"6@`NNN?N   _1  @`  H HL- ?"6@`NNN?N    _2  @`  H H8& ?"6@`NNN?N 4   _2  @` H H( ?"6@`NNN?N a 4  _2  @` H Hd8 ?"6@`NNN?N a  _2  @` H HhA ?"6@`NNN?N   _2  @`^ H BJ3?"6@`NNN?N  .  & @` H H; ?"6@ NNN?N * X @`( H HN ?"6@ NNN?N *  j 1   @` H Hp^ ?"6@ NNN?N4 *  X @` H Hn ?"6@ NNN?Na *4  X @` H Hp ?"6@ NNN?N *a  X @` H Hp ?"6@ NNN?N*  X @` H Bx3?"6@ NNN?N* X @` H HX ?"6@ NNN?N>   `   @` H H` ?"6@ NNN?Nn > _+  @` H H ?"6@ NNN?N n _+  @` H H\ ?"6@ NNN?N  _+  @`( H Bd3?"6@ NNN?N  p+"&"  @` H HD ?"6@ NNN?N>   `   @` H HȌ ?"6@ NNN?Nn >  _+  @`  H HҌ ?"6@ NNN?N n  _+  @` !H Hڌ ?"6@ NNN?N   _+  @` "H B3?"6@ NNN?N   _2  @` #H H ?"6@ NNN?N>   `   @` $H H ?"6@ NNN?Nn >  _+  @` %H HD ?"6@ NNN?N n  _+  @` &H H  ?"6@ NNN?N   _+  @` 'H B3?"6@ NNN?N   _1  @` (H H ?"6@ NNN?N>4    `   @` )H Hh$ ?"6@ NNN?Nn4 >  _+  @` *H H- ?"6@ NNN?N4 n  _+  @` +H H5 ?"6@ NNN?N4   `   @` ,H B(?3?"6@ NNN?N4   _0  @` -H H,H ?"6@ NNN?N>a  4  `   @` .H HLQ ?"6@ NNN?Nna >4  _+  @` /H HPZ ?"6@ NNN?Na n4  `   @` 0H Hpc ?"6@ NNN?Na 4  `   @`" 1H Btl3?"6@ NNN?Na 4  j  0.25  @` 2H Hu ?"6@ NNN?N>  a  `   @` 3H H~ ?"6@ NNN?Nn >a  `   @` 4H H ?"6@ NNN?N na  _+  @` 5H H ?"6@ NNN?N a  `   @`  6H Bܙ3?"6@ NNN?N a  h  1.6  @` 7H Hࢍ ?"6@ NNN?N>  `   @` 8H H ?"6@ NNN?Nn>  `   @` 9H H ?"6@ NNN?Nn  _+  @` :H H ?"6@ NNN?N  `   @`* ;H BǍ3?"6@ NNN?N  r "&"  @`J    3(x).   @`J =H Bݍ3?"6@ NNN?Nn>  2(x).   @`J >H B3?"6@ NNN?Nn  1(x).   @`J ?H B$3?"6@ NNN?N  0(x).   @` @H B3?"6@ NNN?N gx @`B AH <o ?"0@NNN?N*B BH 61 ?"0@NNN?N*B CH 61 ?"0@NNN?N * B DH 61 ?"0@NNN?Na *a B EH 61 ?"0@NNN?N4 *4 B FH 61 ?"0@NNN?N * B GH 61 ?"0@NNN?N  B HH 61 ?"0@NNN?N * B IH <o ?"0@NNN?N*B JH <o ?"0@NNN?NB KH 61 ?"0@NNN?NB LH 61 ?"0@NNN?NB MH 61 ?"0@NNN?NnnB NH 61 ?"0@NNN?N>>B OH 61 ?"0@NNN?N  B PH <o ?"0@NNN?N**B QH 61 ?"0@NNN?N. 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