40 4 4.3 I (1) I f (x) C [x 0 x 1 ] f (x) f 1 (x) (3.18) f (x) ; f 1 (x) = 1! f 00 ((x))(x ; x 0 )(x ; x 1 ) (x 0 (x) x 1 ) (4.8) (3.18) x (x) x 0 x 1 Z x1 f (x) dx ; Z x1 f 1 (x) dx = Z x1 = 1 1 Z x1 Z x1 ff (x) ; f 1 (x)g dx max x1 f 00 ((x))(x ; x 0 )(x ; x 1 ) dx jf 00 ()j Z x1 (x ; x 0 )(x ; x 1 ) dx = 1 max jf 00 ()j h 3 (4.9) 1 x1 (x ; x 0 )(x ; x 1 ) dx = ; 1 6 h3 (4.10) ji ; I (1) j = 1 1 max jf 00 ()j h 3 n ab = 1 1 max jf 00 ()j (b ; a) h (4.11) ab h O(h ) f (x) C 3 [x 0 x ] Z x f (x) dx ; Z x f (x) dx 1 180 max jf (3) ()j h 4 (4.1) x f (x) x 0 x 1 x f (x) f (x) C 4 Z x f (x) dx ; Z x f (x) dx 1 90 max jf (4) ()j h 5 (4.13) x
4.3. 41 h = (x ; x 0 )= (4.13) n= ji ; I () j 1 90 max jf (4) ()j (b ; a) h 5 n ab 1 180 max jf (4) ()j (b ; a) h 4 (4.14) ab h O(h 4 ) Z sin x dx = 1 (4.15) 0 Z 1 e x dx = e x ; 1 (4.16) 0 Z dx 0 5 + 4cosx = (4.17) 3 Z +1 e ; p 1 x dx = (4.18) ;1 (4.18) x [;10 10] h O(h ) O(h 4 ) h (a) (b) (;1 1) (4.17) (4.18) (a) (b)
4 4 4.4 R b I = f (x)dx a f (x) I [;1 1] I = Z 1 ;1 x = tanh sinh t f (x) dx (4.19) (4.0) sinh x = ex ; e ;x cosh x = ex + e ;x tanh x = sinh x cosh x (4.1) (;1 1) (;1 1) dx = cosh t I = Z +1 ;1 f ; dt cosh (4.) sinh t tanh sinh t cosh t ; dt cosh (4.3) sinh t jtj t! 1 sinh t! 1 ejtj y! 1 cosh y! 1 ejyj cosh sinh t! 1 4 exp 4 exp jtj (4.4) 1= cosh ( sinh t) 0 x! a x! b f (x) (4.3) h [N ; h N + h] (4.3) N ; N + XN + I (D) = f n=n; = XN + tanh sinh(nh) cosh(nh) cosh ; sinh(nh) h a n f (b n ) (4.5) n=n;
4.4. 43 a n = b n = tanh ; cosh(nh) h cosh (4.6) sinh(nh) sinh(nh) (4.7) x ; cosh sinh(nh) f (x) R 1 p dx ;1 1;x [a 1] (;1 1)
44 4 [x 0 x ] f (x) C 4 " h = (x ; x 0 )= " 1 90 h5 max x 0 x jf (4) ()j (4.8) x 1 = x 0 + h x = x 0 x 1 x 3 f (x) f (x) f (x) f (x) 3 x = x 1 1 ^f (x) = f (x) + 1 h [f 0 (x 1) ; f 0 (x 1 )] (x ; x 0 )(x ; x 1 )(x ; x ) (4.9) ^f (x) x = x 1 ^f (x) f (x) [x 0 x ] z x = z ^f (x) x = x 0 x 1 x 0 z 6= x 0 z 6= x 1 z 6= x x = z f (x) ^f (x) ~f (x) = ^f (x) + c(x ; x 0 )(x ; x 1 ) (x ; x ) (4.30) x = x 0 x 1 x 0 x = x 1 0 f ~ (x) ^f (x) f (x) x = x 0 x 1 x x = x 1 f 0 (x) c x = z f (x) f ~ (x) C 4 g(x) = f (x) ; f ~ (x) (4.31) g(x) [x 0 x ] x 0 x 1 x z 4 4 3 g 0 (x) 1 f (x) f ~ (x) 1 x = x 1 x = x 1 g 0 (x) g 0 (x) [x 0 x ] 4 g (4) (x) [x 0 x ] 1
4.4. 45 g (4) () = f (4) (4) d4 (x) ; ^f (x) ; c (x ; )(x ; x dx 4 1 ) (x ; x ) = 0 x= (4.3) ^f (x) 3 4 0 c = 1 f (4) () (4.33) 4! (4.30) x = z ~ f (z) = f (z) f (z) ; ^f (z) = 1 f (4) ()(z ; x 0 )(z ; x 1 ) (z ; x ) (4.34) 4! x = z z [x 0 x ] z (4.9) z x f (x) ; f (x) = 1 f (4) ((x))(x ; x 0 )(x ; x 1 ) (x ; x ) 4! + 1 h [f 0 (x 1) ; f 0 (x 1 )] (x ; x 0 )(x ; x 1 )(x ; x )(4.35) x 0 x 1 I I 1 Z x (x ; x 0 )(x ; x 1 ) (x ; x ) dx = ; 4 x 0 15 h5 (4.36) Z x x 0 (x ; x 0 )(x ; x 1 )(x ; x ) dx = 0 (4.37) [x 0 x 1 ] [x 1 x ] ji ; I j = 1 4! Z x x 0 1 max jf (4) ()j 4! x 0 x Z x x 0 f (4) ((x))(x ; x 0 )(x ; x 1 ) (x ; x ) dx (x ; x 0 )(x ; x 1 ) (x ; x ) dx = 1 90 h5 max x 0 x jf (4) ()j (4.38) Rainer Kress: Numerical Analysis, Springer-Verlag, 1998.
46 4 4 1 (1) R = 0 sin xdx n = 4 8 16 h = (=)=n () R b I = a f (x) dx h I h I h n = (b ; a)=h 3 f (x) C m 1X f (x) = ;1 c k e ikx (4.39) f (x) 1 Z I = 0 f (x) dx (4.40) (1) I = c 0 () [0 ] n I I n I n ; I = 1X l=1 c ln (4.41) (3) f (x) C m c k = o(k ;m ) o(k ;m ) k ;m 0 ji n ; Ij h = =n f (x) C 1 (3) ji n ; Ij h 0 (4.17)
4.4. 47 4 (1) f (x) x! 1 f (x) jxj R +1 I = ;1 f (x) dx x = sinh( sinh t) h [;Nh Nh] R +1 () I = ;1 1 1+x dx h = 1 0:5 0:5 N Nh = 4 I = 5 R 1 p (1) I = 1 ; x dx ;1 h () h (3) () (4.11) (4.14) (4)
49 5 f (x) f 0 (x) f (x) 5.1 f (x) x f 0 (x) h f (x + h) x f (x + h) = f (x) + hf 0 (x) + 1 h f 00 (x) + 1 3! h3 f (3) (x) + = f (x) + hf 0 (x) + O(h ) (5.1) f 0 (x) = f (x + h) ; f (x) h + O(h) (5.) f 0 (x) h! 0 h O(h) 1hf 00 (x)+o(h ) f (x ; h) f (x ; h) = f (x) ; hf 0 (x) + 1 h f 00 (x) ; 1 3! h3 f (3) (x) + = f (x) ; hf 0 (x) + O(h ) (5.3) f 0 (x) f 0 (x) = f (x) ; f (x ; h) h + O(h) (5.4)
50 5 (5.1) (5.3) f 0 (x) = f (x + h) ; f (x ; h) h + O(h ) (5.5) O(h ) (5.1) (5.3) f (x + h) ; f (x) + f (x ; h) = h f 00 (x) + 4! f (4) (x) + (5.6) f 00 (x) f 00 (x) = f (x + h) ; f (x) + f (x ; h) h + O(h ) (5.7) 5. h h 0 h h f (x) f (x + h) f (x) 53 54 " = ;54 f (x) jf (x)j" f (x + h) jf (x)j" jf (x)j " + jf (x)j " = jf (x)j " (5.8) h h 1 f 00 (x) h + O(h ) (5.9) jf (x)j " h + 1 f 00 (x) h (5.10) h = s jf (x)j jf 00 (x)j " (5.11)
5.. h 51 f 00 (x) jf 00 (x)j jf (x)j h p = " (5.1) h ;7 1 jf (x)j h jf (x)j p " (5.13) f (x) jf (x)j" h