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1 ( ) ( ) R n R n n Euclid R n R n f =(f 1 ;f 2 ;:::;f n ) T : f(x) =0 (1) x = (x 1 ;x 2 ;:::;x n ) T 2 R n 1 \T " f i (1 i n) R n R (1) f i (x) =0 (1 i n) n 2 n x =cosx f(x) :=x 0 cos x (1) 1 Ax = b f(x) :=b 0 Ax (1) (1) 1.2 f 3 f(x) 1 x 3 watanabe@cc.kyushu-u.ac.jp 1 2 Jacobi ( Carl Gustav Jacob Jacobi (1804{1851), Jacobian ) 3 x; y 2 R n, ; 2 R f(x + y) =f(x) +f(y) f 1

2 8 >< >: f 1 (x) = 9 P x 1 4 f 2 (x) = 81 P x x 2 x 3 8 p 2 + PRx 7 p x 1 x 3 8 p 2 + PRx 8 p 2 f 3 (x) = 09 x 1 x 2 4 p +9P x 3 + p 2 PRx 9 2 f 4 (x) =36P x 4 + p 2 PRx 10 f 5 (x) =02 x 5 + x 2 x 7 2 p 2 + x 1 x 8 2 p 2 0 x 4 x 9 p2 + p 2 x 4 x 9 0 x 3 x 10 p2 + p 2 x 3 x 10 f 6 (x) =08 x 6 0 x 1 x 7 p2 0 p 2 x 3 x 9 f 7 (x) =0 x 1 p 0 x 2 x p 2 + x 1 x 6 p2 0 3 x x 3 x 8 4 p x 2 x 9 4 p 2 f 8 (x) =0 x 2 p 0 x 1 x p x 3 x 7 4 p x x 1 x 9 4 p 2 f 9 (x) =0 p 2 x 3 0 x 4 x 5 p2 + p 2 x 3 x x 2 x 7 4 p x 1 x 8 4 p x 9 f 10 (x) =0 p 2 x 4 0 x 3 x 5 p2 0 6 x 10 (= n =10 x P, R 1: f(x) 4 f n \ " 5 : 1 ; x (0) ( ) ; x (0)! x (1) ; x (0)! x (1)!111!x (k)!111 (successive approximation method) 1 ( ) 1.3 Newton-Raphson f Newton-Raphson (Newton-Raphson's method) 6 : x (k+1) = x (k) 0 f 0 (x (k) ) 01 f(x (k) ) k =0; 1; 2;::: 7 1 f(x) :=x 0 cos x =0 Newton-Raphson x (k+1) = x (k) 0 x(k) 0 cos x (k) 1 + sin x (k) k =0; 1; 2;::: n f 0 (x (k) ) f x (k) n 2 n Jacobi [4], [9] 6 Newton (Newton's method) Joseph Raphson (1648{1715) 7 regula-falsi (bisection method) Traub Muller [14], [3] 8 \f 0 (x (k) ) 01 f(x (k) )" Jacobi f 0 (x (k) ) f(x (k) ) 1 f 0 (x (k) )x = f(x (k) ) 2

3 Newton-Raphson 1 f f 0 (x (k) ) 9 1 n 1 Newton- Raphson [3], [10], [12], [16] ( [1], [2], [7] ) 10 ( [3], [5], [9] ) R R n n (norm) 11 : R n kk 3 : 1. x 2 R n kxk 0 kxk =0, x = 0 2. x 2 R n, 2 R kxk = jjkxk 3. x; y 2 R n kx + yk kxk + kyk Minkowski : \jj" n =1 \," ( [11] ) 9 Ax = b x A b ( [11] ) 10 1 ( ) [15] 11 ( norma ) 3

4 \k k" x x i (1 i n) kxk 1 = kxk 2 = nx i=1 jx i j nx i=1! 1 jx i j 2 2 kxk 1 = max 1in jx ij 1: R n 1 1- l Euclid, l l 1 a(> 0) x 2 R 2 kxk 1 = a \ ", kxk 2 = a \ ", kxk 1 = a \ " ( [11], [13] ) 2 13 n n 1 2 n x = (sin(1); sin(2);:::;sin(n)) T n 2 Fortran (IEEE ; 64 ) 2: x =(sin(1); sin(2);:::;sin(n)) T n kxk 1 kxk 2 kxk 1 1 0:84 0:84 0: :48 2:23 0: : : : : : A kxk A := kaxk kk a, kk b c 1;c 2 x c 1kxk a kxk b c 2kxk a 1=c 2kxk b kxk a 1=c 1kxk b 4

5 1 2 n 3 : kxk 2 kxk 1 p nkxk 2 ; kxk 1 kxk 2 p nkxk 1 ; kxk 1 kxk 1 nkxk 1 : 2 ( [1] ) Fortran 4.5 x; y 2 R n x 0 y 15 n 1 x, y x; y i s 16 1 s=0.0d0 do i=1,n s=s+abs(x(i)-y(i)) end do! 0! 1 n!! do Fortran 90 SUM s=sum(abs(x-y))! s 1 SUM ( MAXVAL ) x, y n 0 2 s=0.0d0 do i=1,n s=s+(x(i)-y(i))**2 end do s=sqrt(s)! 0! 1 n!! do! Fortran 90 SUM : s=sqrt(sum((x-y)**2))! s (Euclid) 15 s \s=0.0d0" \D0" \s=0" \s=0.0" abs, sqrt max dabs, dsqrt, dmax1 16 Fortran / 5

6 s=0.0d0 do i=1,n s=max(abs(x(i)-y(i)),s) end do! 0! 1 n! x(i)-y(i) s s! do Fortran 90 MAXVAL : s=maxval(abs(x-y))! s 3 VPP700/56 ( -Wv,-sc ) 1 i n x(i)=sin(i), y(i)=cos(i) n = Fortran CPU 3: (VPP700/56 ) VPP700/56 Fortran (Fujitsu Fortran90/VP V10L10 L98061) n 2 n A = (a ij ) kak 1 kak 1 = max nx 1in j=1 [11] ja ij j JIS Fortran 4 DOUBLE PRECISION 4 Fortran 6

7 3 3.1 ( ) = + = 0 21 x ^x ( ) kx 0 ^xk (= k^x 0 xk) (absolute error) 22 x 6= 0 ^x kx 0 ^xk kxk (relative error) ( ) x kxk kx 0 ^xk (3) k^xk (3) ^x x (2) (3) ^x 0 x k^x 0 xk \ " (2) 7

8 3.2 x 1 x ^x 0:1 x = 100, ^x =99:9 x =0:2, ^x =0:1 2 0:1 (0:1) x 100 x 0:2 ^x 99:9 ^x 0:1 2: jx 0 ^xj =0:1 x ^x ( ) x =100,^x =99:9 0:001 x = 0:2, ^x = 0:1 0:5 2 \log 10 ( )" x = 1000, ^x = x = 100, ^x = 99: x ^x 5 6 ( 3) 1 0: ( 4) 1 1=99 0:0101 ( ) 3: 6 5 ( ) 4: ( ) (2) (3) [4] 23 8

9 x ^x ( [3], [5], [9] ) (machine epsilon) 24 1+">1 (4) [5] (4) Fortran 90 VPP700/ program machine_epsilon! ( ) implicit none! integer(kind=4) :: i! 4 integer(kind=8) :: j! 8 real(kind=4) :: x! real(kind=8) :: y! real(kind=16) :: z! 4 intrinsic epsilon,huge,tiny!! ( ) write(6,*) epsilon(x)! write(6,*) epsilon(y)! write(6,*) epsilon(z)! 4 write(6,*)! ( ) write(6,*) huge(i)! 4 write(6,*) huge(j)! 8 write(6,*) huge(x)! write(6,*) huge(y)! write(6,*) huge(z)! 4 write(6,*)! ( ) write(6,*) tiny(x)! write(6,*) tiny(y)! write(6,*) tiny(z)! 4 end program machine_epsilon! 5 24 IEEE

10 5: Fortran ( ) EPSILON TINY HUGE VPP700/56 : E E-16 (= =2 023 (= = E-0034 (= 4 = (= 4 = (= 8 = E+38 (= =( ) (= =( ) E+4932 (= 4 =( ) E (= = (= = E-4932 (= 4 = R n x (k) =(x (k) 1 ;:::;x(k) n ) T a =(a 1 ;:::;a n ) T fx (k) g a (converge) x (k) i! a i (k!1) (1 i n) (5) (5) \lim k!1 x(k) = a" 25 \x (k) i! a i (k!1)" \kx (k) 0 ak!0 (k!1)" ( [11] ) 0 (5) x (k) 25 : ">0 K k>k kx (k) 0 ak <" 10

11 4 x (k) (5) a f(x) = 0 x x (k) (1) 27 ">0 kf (x (k) )k <" (6) f(x (k) ) (1) (residual) 1 Ax = b \b0ax (k) " \Ax (k) 0 b" x (k) kf (x (k) )k kx (k) k 1 28 [1] <" (6) n 1 (1 ) f i (x (k) )(1 i n) " 28 1 kb 0 k Ax(k) <" kbk 29 11

12 f(x (k) ) f(x (k) ) ">0 (1) kx (k) 0 x (k01) k <" (7) x (k) (7) (2) ">0 kx (k) 0 x (k01) k kx (k01) k <" (8) x (k) (8) (8) 3.2 x (k), x (k01) (7), (8) 4.3 (6), (7), (8) x (k) (1) a ( ) : kx (k) kx (k) 0 ak 0 ak <"; <" kak

13 (f ) 33 " x (k) a a [16] 1 5 y y = f (x) " 0 a x (k+1) x (k) x 5: 5 " a f(x) =0 x (k) x (k+1) jx (k+1) 0 x (k) j <" ja 0 x (k+1) j < " kf(x (k) )k < " Newton-Rapson f (x) ( 4.6 ) ( [14] ) 4.3 x (k) a x (k) f C>0 kx (k) 0 ak Ckx (k) 0 x (k01) k C [10] 13

14 4.5 Fortran 1 f f(x) = 0 Newton-Raphson Fortran 36 \Newton_Raphson" f \Generate_Function" Jacobi \Generate_Jacobian" 1 Gauss SSL II \DVLAX" 37 1 R = 10, P= epsz (6) (8) program Newton_Raphson! implicit none! integer(kind=4),parameter :: n=10! real(kind=8),dimension(n) :: v,f,u,vw! real(kind=8),dimension(n,n) :: G! Jacobi real(kind=8),dimension(2) :: error!, real(kind=8) :: P,R,epsz!, integer(kind=4) :: i,k,imax,icon,isw,is! integer(kind=4),dimension(n):: ip! dvlax intrinsic abs,maxval! external dvlax,generate_function,generate_jacobian! v=1.0d0! 1 P=1.0D0! P R=10.0D0! R epsz=2*epsilon(epsz)! imax=100! isw=1! LU (dvlax) do k=1,imax! u=v! u call Generate_Function(u,v,P,R)! f(u) call Generate_Jacobian(u,G,P,R)! Jacobian call dvlax(g,n,n,v,epsz,isw,is,vw,ip,icon)! 1 (dvlax) if(icon /= 0) then!-+ write(6,*) 'Error occurred in DVLAX:',icon! +-dvlax stop! end if!-+ v=u-v! v call Generate_Function(v,f,P,R)! error(1)=maxval(abs(f))! error(2)=maxval(abs(v-u))/maxval(abs(u))! write(6,'(i3,2e15.7)') k,error(1),error(2)! if( error(1)<epsz.or. error(2)<epsz ) then!-+ write(6,*) 'approximate solution: '! do i=1,n! +- write(6,'(i4,e25.13)') i,v(i)! end do!, exit! end if!-+ end do! end program Newton_Raphson! 1 \&" \intent" ( ) DVLAX kyu-vpp man dvlax SSL II \.f90" 14

15 subroutine Generate_Function(v,f,P,R) implicit none real(kind=8),dimension(10),intent(in) :: v real(kind=8),dimension(10),intent(out) :: f real(kind=8),intent(in) :: P,R real(kind=8) :: S intrinsic sqrt S=sqrt(2.0D0) f(1)= (9*P*v(1))/4 + (9*v(2)*v(3))/(8*S) + (P*R*v(7))/S f(2)= (81*P*v(2))/4 + (9*v(1)*v(3))/(8*S) + (P*R*v(8))/S f(3)= (-9*v(1)*v(2))/(4*S) + 9*P*v(3) + S*P*R*v(9) f(4)= 36*P*v(4) + S*P*R*v(10) f(5)= -2*v(5) + (v(2)*v(7))/(2*s) + (v(1)*v(8))/(2*s) - (v(4)*v(9))/s + & S*v(4)*v(9) - (v(3)*v(10))/s + S*v(3)*v(10) f(6)= -8*v(6) - (v(1)*v(7))/s - S*v(3)*v(9) f(7)= -(v(1)/s) - (v(2)*v(5))/(2*s) + (v(1)*v(6))/s - (3*v(7))/2.0D0 + & (3*v(3)*v(8))/(4*S) + (3*v(2)*v(9))/(4*S) f(8)= -(v(2)/s) - (v(1)*v(5))/(2*s) - (3*v(3)*v(7))/(4*S) - (9*v(8))/2.0D0 - & (3*v(1)*v(9))/(4*S) f(9)= -(S*v(3)) - (v(4)*v(5))/s + S*v(3)*v(6) - (3*v(2)*v(7))/(4*S) + & (3*v(1)*v(8))/(4*S) - 3*v(9) f(10)= -(S*v(4)) - (v(3)*v(5))/s - 6*v(10) end subroutine Generate_Function \;" Fortran 90 subroutine Generate_Jacobian(v,G,P,R) implicit none real(kind=8),dimension(10),intent(in) :: v real(kind=8),dimension(10,10),intent(out) :: G real(kind=8),intent(in) :: P,R real(kind=8) :: S intrinsic sqrt S=sqrt(2.0D0) G=0.0D0 G(1,1) = (9*P)/4.0D0 ; G(7,1) = -(1/S)+v(6)/S G(1,2) = (9*v(3))/(8*S) ; G(7,2) = -v(5)/(2*s) + (3*v(9))/(4*S) G(1,3) = (9*v(2))/(8*S) ; G(7,3) = (3*v(8))/(4*S) G(1,7) = (P*R)/S ; G(7,5) = -v(2)/(2*s) G(2,1) = (9*v(3))/(8*S) ; G(7,6) = v(1)/s G(2,2) = (81*P)/4.0D0 ; G(7,7) = -1.5D0 G(2,3) = (9*v(1))/(8*S) ; G(7,8) = (3*v(3))/(4*S) G(2,8) = (P*R)/S ; G(7,9) = (3*v(2))/(4*S) G(3,1) = (-9*v(2))/(4*S) ; G(8,1) = -v(5)/(2*s) - (3*v(9))/(4*S) G(3,2) = (-9*v(1))/(4*S) ; G(8,2) = -(1/S) G(3,3) = 9*P ; G(8,3) = (-3*v(7))/(4*S) G(3,9) = S*P*R ; G(8,5) = -v(1)/(2*s) G(4,4) = 36*P ; G(8,7) = (-3*v(3))/(4*S) G(4,10)= S*P*R ; G(8,8) = -4.5D0 G(5,1) = v(8)/(2*s) ; G(8,9) = (-3*v(1))/(4*S) G(5,2) = v(7)/(2*s) ; G(9,1) = (3*v(8))/(4*S) G(5,3) = -(v(10)/s) + S*v(10) ; G(9,2) = (-3*v(7))/(4*S) G(5,4) = -(v(9)/s) + S*v(9) ; G(9,3) = -S + S*v(6) G(5,5) = -2.0D0 ; G(9,4) = -(v(5)/s) G(5,7) = v(2)/(2*s) ; G(9,5) = -(v(4)/s) G(5,8) = v(1)/(2*s) ; G(9,6) = S*v(3) G(5,9) = -(v(4)/s) + S*v(4) ; G(9,7) = (-3*v(2))/(4*S) G(5,10)= -(v(3)/s) + S*v(3) ; G(9,8) = (3*v(1))/(4*S) G(6,1) = -(v(7)/s) ; G(9,9) = -3.0D0 G(6,3) = -(S*v(9)) ; G(10,3) = -(v(5)/s) G(6,6) = -8.0D0 ; G(10,4) = -S G(6,7) = -(v(1)/s) ; G(10,5) = -(v(3)/s) G(6,9) = -(S*v(3)) ; G(10,10)= -6.0D0 end subroutine Generate_Jacobian 15

16 6 VPP700/56(IEEE 64 ) (6), (8) : VPP700/56 1 0: E +01 0: E : E +02 0: E : E +01 0: E : E +01 0: E : E +00 0: E : E : E : E : E : E : E : E : E 0 07 sin, cos 38 n 2 n A =(a ij ) n b =(b i ) a ij = 8 >< >: s s 2 ij n +1 sin n +1 2 ij n +1 sin n +1 (i 6= j) 2 2n (i = j) ; b i = nx a ij j=1 1 Ax = b SOR 39! 1:5 wisdom(fujitsu S-4/1000E) IEEE 4 6 n =10 ( ) SOR (Successive Over-Relaxation method) A L D U A = L + D + U 1 <!< 2 x (k) := (D +!L) 01 (!b + ((1 0!)D 0!U)x (k01) )! =1 Gauss-Seidel [7] 40 16

17 10-1 maximum norm of residual single precision double precision quadruple precision number of iteration 6: SOR 41 [14] 42 n n ( ) n n (6) (7) 1 n 2 p n 4.7 (1) 41 [10] 42 ( ) 17

18 ( ) [6], [8], [10] 1 f(x) = 0 x ^x ( ) ( ) 43 f

19 ( ) [1] Richard Barrett, Michael Berry, Tony F. Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine, and Henk Van der Vorst: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, SIAM, Philadelphia, PA, URL : Templates,,, ISBN X, URL PostScript FORTRAN, MATLAB 1 4,600 ( ) [2], :,,, ISBN , ,200 [3] :, 13,, ISBN , ,000 [4] :, 8,, ISBN , Fortran 2,330 [5] : FORTRAN 77 ( ),,, ISBN , FORTRAN 77 IEEE Fortran 95 3,200 [6], :, :,, ISBN X, ( ) 3,000 [7], :,, ISBN , Gauss-Seidel SOR H 2,000 [8] :,, Vol.8, No.4 (1998) pp.42{54. IEEE 754 OS [9] : 2, 4,, ISBN X, B5 Fortran 90 2,300 [10], :,, ISBN ,

20 ,600 [11] : { {, M-11,, ISBN X, ,800 [12] :, =15,, ISBN , Fortran 1,600 [13] :, Information & Computiong-37,, ISBN , [14] : UNIX C,, ISBN , C C 3,800 [15] : 1 Gauss,, Vol.28, No.4 (1995), pp.291{349. URL 1 URL PostScript [16] :, =14,, ISBN , ,800 illustration: H 2 Naoko 20

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