C:/大宮司先生原稿/数値積分数値微分.dvi
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1 6 u(x) 6. ::: x ; x x x ::: ::: u ; u u u ::: (dscrete) u = u(x ) ( = :::) x ;x = h =cnst: x <x<x x u(x) u(x) = x ;x h u x;x h u (x;x )(x;x ) x x u (x) = u ;u h Z x u(x)dx = x;x x (x;x ;x ) ; u u(x) ; (x;x ) x x = x x u = u = u ;u O(h) h Z x u(x)dx = x h(u u )O(h ) [ ]=[ ][ ] (trunctn errr) 6.:
2 x Tylr u(x) =u (x;x )u! (x;x )! (x;x ) u (x) =u (x;x )! (x;x ) Z x u(x)dx =(x;x )u x! (x;x ) u! (x;x ) 4! (x;x ) 4 u = u hu! (h)! (h) ( = :::) u u Tylr x;x x;x h O(h ), O(h ) u ; u u x ; <x<x x u(x) u(x) = h n (x;x )(x;x )u ; ;(x;x )(x;x ; )u (x;x ;)(x;x )u! (x;x ;)(x;x )(x;x ) u (x) = h (x;x ;x )u ; ;(x;x ; ;x )u (x;x ; ;x )u Z x! (x;x )(x;x )(x;x )(x;x ; )(x;x ; )(x;x ) u(x)dx = (x;x ;) (x;x ;x x; h )(u ; u );4(x;x ;h)u ; (x;x ;)(x;x ) h 4! (x;x ;) (x;x ) x u = h (;u ;u )O(h ) u = h (u ;;4u u )O(h ) Z ; u(x)dx = h(u ;4u u )O(h 4 ) 4 4 u
3 6. (numercl ntegrtn, numercl qudrture) u(x) dx u(x) u ( = ::: n) (dscrete) [ b] u(x) Newtn-Ctes Newtn-Ctes Guss-Legendre Guss Legendre u 6.. u(x) u u ::: u n u = u(x ) x = <x < <x n = b n n p n (x) n p n (x) = x x n x n 6. n u u ::: u n A A ::: A n p n (x) dx = A p n (x )A p n (x ) A n p n (x n ) (6.) ::: n n ::: n 6.: (dscntnutes) : : :
4 4 n dx =A A A n xdx=a x A x A n x n (6.) x n dx =A x n A x n A n x n n dx = b; xdx= b ; x n dx = bn ; n n A A A n (6.) x x x n :::::::::::::::::: x n x n x n n C A A A. A n C A = b; (b ; )=. (b n ; n )=(n) C A (6.) Vndermnde x x ::: x n A A ::: A n (6.) A A ::: A n u(x) dx = A u A u A n u n (6.4) n p n (x) b 6. u u ::: u n n (6.4) u = p n (x ) u = p n (x ) u n = p n (x n ) p n (x )= (x ) n (x )( = ::: n) (6.) u(x) dx = A p n (x )A p n (x )A n p n (x n ) = (A A A n ) (A x A x A n x n ) n (A x n A x n A n x n n ) = dx xdx n x n dx = p n (x)dx n = 4 A
5 5 SUBROUTINE NUMINT(x,u,f,s) DIMENSION x(:f),u(:f),a(:4),x(:4) REAL(8) v(5,6),w(6) w=x(f)-x() FORALL(=:f)x()=(x()-x())/w DO =,5 v(,6)=./float() IF(==)THEN FORALL(j=:5)v(,j)=. ELSE FORALL(j=:5)v(,j)=x(j-)*v(-,j) ENDIF ENDDO det=. e=. CALL GAUSS(v,w,5,6,det,e) FORALL(=:f)A()=w*v(,6) s=. DO =,f s=sa()*u() ENDDO ENDSUBROUTINE! Slutn f smultneus lner equtns by Gussn elmntn SUBROUTINE GAUSS(,w,n,n,det,e) REAL(8) (n,n),w(n) det=. e=max(e,.e-5) cycle_: DO k=,n l=k w=(k,k) IF(ABS(w)>e) GOTO l=k l=l IF(l>n)STOP 5555 w=(l,k) IF(ABS(w)<=e) GOTO FORALL(j=k:n) w(j)=(k,j) (k,j)=(l,j) (l,j)=w(j) ENDFORALL det=-det det=w*det FORALL(j=k:n)(k,j)=(k,j)/w IF(k==n)CYCLE cycle_ FORALL(=k:n,j=k:n)(,j)=(,j)-(,k)*(k,j) ENDDO cycle_ DO k=n,,- FORALL(=:k-)(,n)=(,n)-(,k)*(k,n) ENDDO END 6. = x! b = x 4! x s GAUSS Guss,v! n=n w det e
6 6 6.. Newtn-Ctes A A ::: A n A = h ( = ::: n) h n / / 4 Smpsn / /8 Smpsn /8 4 / / / (6/) ( ) (Weddle ) Newtn-Ctes [ b] n N h =(b ; )=N x = x h ( = ::: N) x = x N = b u = u(x ) n = ( trpezdl lw) Z xn ; u(x) dx = h x u u u u N; u N n = Smpsn / N =7 Smpsn /8 Z xn x u(x) dx = h(u 4u u 4u u 4 ) 8 h(u 4u 5 u 6 u 7 ) Newtn-Ctes (trunctn errr) Smpsn / u(x) Tylr u(x) =u (x;x )u! (x;x )! (x;x ) 4! (x;x ) 4 u (4) [x x ] Z x u(x) dx =hu h u 4 x h h4 4 5 h5 u (4) [x x ] Smpsn / Tylr h(u 4u u )=hu h u 4 h h4 5 8 h5 u (4) Z x u(x) dx = x h(u 4u u ) ; 9 h5 u (4) Smpsn / Smpsn / N =(b;)=h e t = ; b; 8 h4 u (4)
7 7 Smpsn / 4 O(h 4 ) Newtn-Ctes n 4 5 ;e t b; h b; 8 h4 u (4) b; 8 h4 u (4) (b;) 945 h6 u (6) 55(b;) 96 h6 u (6) Tylr Tylr n n Newtn-Ctes A n u(x) Newtn-Ctes N Smpsn / Z xn x u(x) dx = h(u 4u u 4u u 4 u N; 4u N; ) Smpsn / Z xn x u(x) dx = h(u 4u u 4u 4 u 5 u N; 4u N ) Z xn x u(x) dx = h(u u u u u N; ) n = n n 6.. Guss-Legendre [x x ] u u u(x = ) h [x x ] Smpsn / u u u Guss-Legendre Guss n Legendre n n (smth) (sucently smth)
8 8 Legendre [; ] [ b] ~x = b b; x [; ] ~u(~x) d~x = b; Z ; u(x) dx n u(x ) ( = ::: n) n n p n (x) p n (x) =p n (x)(x;x )(x;x ) (x;x n ) q n (x) (6.5) q n (x) n [; ] Z Z Z p n (x) dx = p n (x) dx (x;x )(x;x ) (x;x n ) q n (x) dx (6.6) ; ; ; x x ::: x n n Legendre P n (x) (x;x )(x;x ) (x;x n )=P n (x) (6.6) (6.4) (6.5) p n (x k )=p n (x k ) (k = ::: n) Z ; p n (x) dx = nx = A p n (x ) (6.6) Legendre Z ; x r P m (x) dx = (r = ::: m;) Guss-Legendre Z ; p n (x) dx = nx = A p n (x ) (6.7) x n Legendre A (6.) x A n x A Guss-Legendre [ b] e t = (b ; )n n (n )! (n )(n )(n ) u (n) (n )! (6.8)
9 9 I = Z dx x =:5 5 7 Smpsn / 4 Smpsn /8 n = Guss-Legendre trpezdl (n=) 7.55 :5 Smpsn / (n=).546 : : :9 Smpsn /8 (n=) 4.59 : ;:5 Newtn-Ctes (n=4) 5.54 :4 Newtn-Ctes (n=6) ;:444 Guss-Legendre (n=) ;: Guss-Legendre (n=) ;: Guss-Legendre I = Z ; dx (:5:5x) = :55555 (:;:7746) :88888 :55555 = :49988 : (::7746) x Smpsn /8 7 Newtn-Ctes (n=6) Guss-Legendre 6. u = u(x ) u(x) u(x) Lgrnge 6.. Lgrnge Lgrnge Lgrnge
10 Lgrnge u(x) = X k= C k() u ;k C () =; (h ;)(h h ;) h (h h )(h h h ) C () = (h )(h ;)(h h ;) h h (h h ) C () = (h )(h h ;) C (h )(h ;) () =; (h h )h h (h h h )(h h )h = x;x h = x ;x ; h = x ;x h = x ;x u(x) =(; )u u (; ; )( h ; ) h ( h )( h h ) u (; ; )(; h h ; ) h ( h ) (; )( h ; h ) (; )( h ( h u ; h ) ) h ( h )( h h ) u = =h h = h =h h = h =h [x x ] Z x x u(x) dx = h (u u ) h ; h h h ( h ) u h ; h h ( h ) u ; h ; h h ( h )( h h ) u ; h h ( h )( h h ) u u (6.9) (6.) [x x ] Z x u(x) dx = x h (u u ) h h 4h ; h ( h )( h h ) u em 4 h h h ( h h ) u ; h h h h ( h ) u h h ( h h )( h h ) u (6.b) = =h h = h =h h = h =h h = h = h = h Z x u(x) dx = h x 4 (;u ;u u ;u ) (6.) Z x u(x) dx = h x 4 (9u 9u ;5u u ) (6.b) Z xn n 8 u(x) dx = h x 4 u 4 u 4 u 5 4 u u 4 u 5 u N;4 5 4 u N; 4 u N; 4 u N; 8 4 u N (6.c)
11 [x x ] S(x) = (;) (;) (6.) S(x ) S(x ) S (x ) S (x ) S(x) = x (;) (;) u ; x 6 6 u u ; x 6 u (6.) x = x ;x =(x;x )=x u [x x ] Z x Z u(x)dx x S()d x = x 4 (u ) x nu ; x 6 u u ; x 6 u = x (u u ); x 4 (u ) (6.4) 4.. x 5 u(4) =5! S(x) = snh ; (;)x p snh(x p ) (;) u(x) = p ; snh snh (;)x x p u p (;) u ; p u ; p snh(x p ) (6.5) p x Z x u(x)dx csh(x x x p ); (u u ) p snh(x p ) ; x ( p ) (6.6) 4.. (6.) u(x) = 6 x u (;) 6 x u (;) (6.7)
12 [x x ] j 4.4. Z x u(x)dx x ) x ( ) (6.8) x 4 (u u(x) (;)u u R; R ; R; ; u (6.9) =(x;x )=h u = = u ;u u = u = ;u ;= R = u = u [x x ] Z x u(x) dx h ; ; u x u R = lg R ; ; u R; R; = h hu u R; lg R ; R u (6.) R; [x ; x ] [x x ] Z x dx h hu ;u x ;u(x) R; R lg R R; u R; Z x h u(x) dx h ; u u R x R; lg R ; R; u R; (6.b) (6.c) Rmberg u(x) T () b; = h T () = h fu(x )u(x h )g = I (b;)h u (x )O(h ) (6.) I O(h ) T () ( h =(b;)=) T () = h fu(x )u(x h )u(x h )g = I (b;)h u (x )O(h ) (6.) T () T () h h h! T () = () (4T ;T () )=I O(h ) (6.)
13 (6.) (6.) O(h ) T () =4 ( h =(b;)=4 =h =) T () = h fu(x )u(x h )u(x h )u(x h )u(x 4h )g (6.4) = I (b;)h u (x )O(h ) T () T () h! T () T () = () (4T ;T () )=I O(h ) (6.5) T () T () h! T () T () = T () ;T () ; = I O(h 4 ) (6.6) k k T (k) k; ( h k =(b;)= k; = h k; =) T (k) = h kfu(x )u(x h k )u(x h k )u(x k; h k )g (6.7) = I (b;)h k u (x )O(h ) T (k;) T (k) h! T (k) T (k) = T (k) ;T (k;) ; = I O(h ) (6.8) T (k;) T (k) h! T (k) T (k) k T (k) m = m T (k) (k;) m; ;T m; m = I O(h m ) ( <mk) (6.9) ; m= k; k T () T () T () T () T () T ()... k; T (k;) T (k;) T (k;) T (k;) k; k. T (k) T (k) T (k) T (k) k; T (k) k jt (k) k ;T (k) (k) j < T k; k
14 4 6. (nte-derence frmuls) Tylr n k n;k Lgrnge u(x) =(;)u u (;)h h = x ;x =(x;x )=h d=dx =(=h)d=d u (x) = h (;u u ) ; ; hu (6.) O(h) == O(h ) = x = x u = u = h u = O(h) u = h u ;= O(h) (6.) h = x ;x ; 6.. u(x) =u e u=! (;) e u! (m; )(;)h eu ;= = u ;u ; e u = m u= e = u ; ;u u= e = u ;u m m ; ( u= e ; u;= e ) e u = m ( u= e ; u= e ) h = x = m ; = x ;= =h m = x = =h u (x) = h n eu=! (;) e u! (m ; )(;)(;) h (6.) (x) = h e u (m; ;)h (6.b)
15 5 O(h ) O(h) = u = h eu=; e u O(h ) (6.) = h e u O(h) (6.b) e u e u u = h eu= ; e u O(h ) (6.c) = h e u O(h) (6.d) - ; m ; - - m u(x) =u e u= (;) n e u (m; ) e u = 4! (m; )(;)(;;m )h 4 u (4) e u = = m ; m ( e u ; e u u=@x = ;m ; (;)(m; ) e u = u(( m )h) =u u (m )m (m ; m ) m ; m m m = == Lgrnge 4 u ; u u u x ; x x x 4 (m ; )(;)(;;m ) Tylr h 4 u (4) 4 (=4!)h 4 u (4) 4
16 6 u (x) = hn eu=! (;) e u! C() e u = 4! (m ; )(;)C()(;;m ) h u (4) (6.4) (x) = h n e u (m; ;) e u = 4! C()(m ; ;)(;;m ) h u (4) (6.4b) (x) = h e u = 4 (4m; ;;m )hu (4) (6.4c) C() =(;)(m ; )(;) O(h ) O(h ) = u = hn eu= ;! e u ;! m; e u = O(h ) (6.5) = h n e u (m; ;) e u = O(h ) (6.5b) - ; m ; - - m - m - m - Lgrnge u(x) =u e u= (;) e u ;;m e u = O(h 4 ) eu 5= = m (u ;u ) e u = e u = = m ( e u ; e u ) m m ( e u5= ; e u= ) m =(x ;x )=h m =m m u (x) = hn eu=! (;) e u! C () e u = O(h ) (6.6) (x) = h n e u (;;m ) e u = O(h ) (6.6b) (x) = h e u = O(h) (6.6c) C () =(;)(;)(;;m ) = u = hn eu= ;! e u! (m ) e u = O(h ) (6.7) = h n e u ; (m ) e u = O(h ) (6.7b)
17 u(x) =u u= e h (;) e u n (m; ) e u = 4 (;;m ) e 4 u 5! (m;; m ; )(m ; )(;)(;;m )h 5 u (5) eu ;= = m ;; (u ;;u ; ) e u ; = m ;; m ; ( e u;= ; e u;= ) e u ;= = m ; ( e u ; e u ; ) e 4 u = 4 m ( e u = ; e u ;= ) m ;; =(x ; ;x ; )=h m ; = m ;; m ; m = m ; m u (x) = hn eu=! (;) e u! C() e u = ; (m ; )(;)C()(;;m ) e 4 u O(h 4 ) 4! (x) = n e u h (m; ;) e u = ; C()(m ; ;)(;;m ) e 4 u O(h ) (6.8) (6.8b) (x) = h n e u = 4 (4m; ;;m ) e 4 u O(h ) (6.8c) u (4) (x) = h 4 e 4 u O(h) (6.8d) = u = h n eu=;! e u ;! m; e u = 4! m; (m ) e 4 u O(h 4 ) = h n e u (m; ;) e u = ; ; m ; (m ; ;)(m ) e 4 u O(h ) (6.9) (6.9b) = h e u = 4 (m; ;;m ) e 4 u O(h ) (6.9c) 4 e 4 u e 4 u u (x) = hn eu=! (;) e u! C() e u = ; (m ; )(;)C()(;;m ) e 4 u O(h 4 ) 4! (x) = n e u h (m; ;) e u = ; C()(m ; ;)(;;m ) e 4 u O(h ) (6.4) (6.4b) (x) = h n e u = 4 (4m; ;;m ) e 4 u O(h ) (6.4c) u (4) (x) = h 4 e 4 u O(h) (6.4d)
18 8 = u = hn eu= ;! e u ;! m; e u = 4! m; (m ) e 4 u = h n e u (m; ;) e u = ; O(h 4 ) ; m ; (m ; ;)(m ) e 4 u O(h ) (6.4) (6.4b) = h e u = 4 (m; ;;m ) e 4 u O(h ) (6.4c) eu 5= = m (u ;u ) e u = m m ( e u5= ; e u= ) e u = = m ( e u ; e u ) e 4 u = 4^m ( e u = ; e u = ) m =(x ;x )=h ^m = m ; m - ; ; 4 m ;; - m ; - - m - m -m - m; - m; - m - m - m - ^m - m - 4 Lgrnge u(x) =u u= e n (;) e u ;;m e u ;m = 4 eu 7= = m (u 4;u ) e u = m m ( e u7= ; e u5= ) e u 5= = m ( e u ; e u ) e 4 u = 4 m ( e u 5= ; e u = ) e 4 u O(h 5 ) m =(x 4 ;x )=h m =m u (x) = hn eu=! (;) e u! C () e u = ; (;)(;;m )C ()(;m ) e 4 u O(h 4 ) 4! (x) = n e u h (;;m ) e u = ; C ()(;;m )(;m ) e 4 u O(h ) (6.4) (6.4b) (x) = h n e u = 4 (4;;m ;m ) e 4 u O(h ) (6.4c) u (4) (x) = h 4 e 4 u O(h) (6.4d)
19 9 = O(h 4 ) u = hn eu= ; e u!! (m ) e u = ; 4! (m )m e 4 u = n e u h ; (m ) e u = ; m (m )m e 4 u O(h ) (6.4) (6.4b) = h n e u = ; 4 (m m ) e 4 u O(h ) (6.4c) Lgrnge (6.) (6.b) u j u = = n h; h ; h h (u ;u ) h (u ;u ; ) h ; n (u ;u ); (u ;u ; ) h ; h h h ; O(h ) O(h) h = x ;x u = n h (h h ) (u ;u ); h h h h (u ;u ) h (h h )h h h h h h h h h h h (u ;u ) h = h h h O(h ) 6..5 m ;; = m ; = m = = u = u ;u ; h O(h) u = u ;u h O(h) u = h (u ;;4u ; u )O(h ) u = h (;u ;u )O(h ) u = h (;u 4u ;u )O(h ) u = 6h (;u ;;u 6u ;u )O(h ) u = 6h (;u 8u ;9u u )O(h ) u = h (u ;;8u ; 8u ;u )O(h 4 ) u = h (;u ;;u 8u ;6u u )O(h 4 ) u = h (;5u 48u ;6u 6u ;u 4 )O(h 4 )
20 = h (u ;;u ; u )O(h) = h (u ;;u u )O(h ) = h (u ;u u )O(h) = h (u ; 5u 4u ;u )O(h ) = h (;u ;6u ; ;u 6u ;u )O(h 4 ) = h (u ;;u 6u 4u ; u )O(h ) = h (5u ;4u 4u ;56u u 4 )O(h ) = h (;u ;u ;u u )O(h) = h (;u u ;u u )O(h) = h (;u ;u ; ;u u )O(h ) = h (;u ;u ;u 6u ;u )O(h ) = h (;5u 8u ;4u 4u ;u 4 )O(h ) u (4) = h 4 (u ;;4u ; 6u ;4u u )O(h ) u (4) u (4) = h 4 (u ;;4u 6u ;4u u )O(h) = h 4 (u ;4u 6u ;4u u 4 )O(h) j n O(h j;n ) (6.b) = 6.4 Newtn
21 6.4. Newtn Lgrnge Strlng 4 u(x) =u u =u ;=! u! ( ;) u = u ;= 4! ( ;) 4 u O(h 5 ) (;= =) u (x) = hn u= u ;= u 6 ( ;) u = u ;= ( ;) 4 u O(h 4 ) (6.44) (x) = n u h u = u ;= (6 ;) 4 u O(h ) (6.44b) (x) = n u = u ;= 4 u h O(h ) (6.44c) u (4) (x) = h 4 4 u O(h) = u = hn u= u ;= ; 6 u = u ;= O(h 4 ) (6.44d) (6.45) = u h ; 4 u O(h ) (6.45b) = u = u ;= O(h ) h x = 4 Bessel (6.45c) u(x) = u u ; ; u=! (;) u u (;); ;! u = 4! ( ;)(;) 4 u 4 u O(h 5 ) ( ) == u = = u =; 4 u = O(h 4 ) = u u = ; 5 4 = = u = O(h ) = 4 u 4 u O(h) = u (4) 4 u 4 u O(h ) (6.46) (6.46b) (6.46c) (6.46d)
22 (6.) u (x) = x ;(;) ;u x 6 u u ; x 6 u (6.47) (x) =(;) (6.47b) x = x ;x =(x ; x )=x j (6.5) u (x) = (x) = p snh x p ;csh ; (;)x p u ; u ; x p u ; x p csh(x p ) (6.48) snh x p snh ; (;)x p u snh(x p ) (6.48b) p 4.. j (6.7) u (x) = x ;(;) (; ) (6.49) x (x) =(;) (6.49b) [x x ] j FORTRAN (6.9) u (x) = hu = lg R x R; R; R ; u (6.5) (x) = (lg R) x (R;) R u (6.5b) R = u = u <! ; = ; =
1 Edward Waring Lagrange n {(x i, y i )} n i=1 x i p i p i (x j ) = δ ij P (x) = p i p i (x) = n y i p i (x) (1) i=1 n j=1 j i x x j x i x j (2) Runge
Edwrd Wring Lgrnge n {(x i, y i )} n x i p i p i (x j ) = δ ij P (x) = p i p i (x) = n y i p i (x) () n j= x x j x i x j (2) Runge [] [2] = ξ 0 < ξ < ξ n = b [, b] [ξ i, ξ i ] y = /( + 25x 2 ) 5 2,, 0,,
1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1.
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HN- 95 HN- 93 HN- 90 HN- 87 HN- 85 HN- 82 HN- 80 HN- 77 HN- 75 HN- 72 HN- 70 HN- 67 HN- 65 HN- 60 HN- 55 HN- 50 HN- 45 HN- 40 HN- 35 HN- 30 HN- 25 HN- 20 HN- 15 HN- 10 H02-80H H02-80L H02-70T H02-60H H05-60F
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C
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1F90/kouhou_hf90.dvi
Fortran90 3 33 1 2 Fortran90 FORTRAN 1956 IBM IBM704 FORTRAN(FORmula TRANslation ) 1965 FORTRAN66 1978 FORTRAN77 1991 Fortran90 Fortran90 Fortran Fortran90 6 Fortran90 77 90 90 Fortran90 [ ] Fortran90
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- II
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構造と連続体の力学基礎
II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton