0 3 10 71 < < 3 1 7 ; (y k )
0
LU n n
M (2; 4; 1; 2) 2 n 2 = 2 2 n 2 n 2 = 2y 2 n n ' y = x [a; b] [a; b] x n = '(x n 1 ) (x n ) x 0 = 0 S p R 2 ; S p := fx 2 R 2 : kxk p = 1g; p = 1; 2; 1 K i a ii r i ; n = 4 f n ; f n (x) := 1 nx; x 2 [0; 1/n]; f n (x) := 0; x 2 (1/n; 1] p n ; n = 20; f (x) := (1 + 25x 2 ) 1 :
T n ; n = 0; : : : ; 6 p n ; n = 20; f (x) := (1 + 25x 2 ) 1 ; x y s(x); n = 20 f (x) := (1 + 25x 2 ) 1 [x j 2 ; x j +2 ]: H) y y H) U; U = I 2uu T ; x x i x n = 2 3 i x i
x 0 = 3 x 0 = 2 x 0 = 0
LU LU
0; 1; 2; 3; : : : ; 9: 3:14159 3 10 0 + 1 10 1 + 4 10 2 + 1 10 3 + 5 10 4 + 9 10 5 N ; N 1 ; : : : ; 0; 1 ; 2 ; : : : 0 9 N N 1 : : : 0: 1 2 : : : ( N N 1 : : : 0: 1 2 : : : ) 10 N 10 N + N 1 10 N 1 + + 0 10 0 + 1 10 1 + 2 10 2 + : N N 1 : : : 0 p; p(x) = N x N + N 1 x N 1 + + 0;
x = 10; : 1 2 : : : P 1 k=1 kx k ; x = 1/10 4:130 4:12 9 = 4:12999 : : : : 9 P 1 i=1 10 i = 1: k 0 k k 0 k 9; k 9; ( N N 1 : : : 0: 1 2 : : : ) 10 N 0 x x ˇ 2 ˇ = 2 ˇ = 8 ˇ = 16 ˇ 0; 1; : : : ; ˇ 1 ˇ > 10; 10; : : : ; ˇ 1 k ˇ; ( N : : : 0: 1 2 : : : )ˇ Pk=N kˇk: 1 (100110:11) 2 12 5 +12 2 +12 1 +12 1 +12 2 = (38:75) 10 : ˇ ˇ; i ˇ ˇ; (53473) 8 = 5 8 4 + 3 8 3 + 4 8 2 + 7 8 1 + 3 8 0 = (22331) 10 : 3 + 8 (7 + 8
(4 + 8 (3 + 8 5))); x p; p(x) = N x N + N 1 x N 1 + + 0; i; 0 i N; p(x) p(x) = 0 + x( 1 + x( 2 + + x( N 1 + x N )) ) y N i = N 1; : : : ; 0: y i + x y; y p(x): y i + x y; i; x y; y; x p N N x (0 < x < 1); ˇ; (:11) 2 = 1 2 1 + 1 2 2 = 1 2 + 1 4 = (:75) 10: ii ˇ: (369) 10 0; 1; 2; : : : (369) 10 = ( : : : 2 1 0) 8 = 0 + 8 ( 1 + 8( 2 + ) ): 0 369 : 8; 0 = 1: (46) 10 ; 1 + 8( 2 + ): 1 46 : 8; 1 = 6:
2 + 8( 3 + ): 2 = 5 3 + = 0 (369) 10 = (561) 8 : x ˇ x = (: 1 2 : : : )ˇ = 1ˇ 1 + 2ˇ 2 + : ˇ ˇx = ( 1 : 2 : : : )ˇ: 1 ˇx (: 2 : : : )ˇ (: 2 : : : )ˇ ˇ; 2 x = (:372) 10 1 ; 2 ; : : : (:372) 10 = (: 1 2 : : : ) 2 : 2x = 0:744 1 = 0; 1 := 0:744 2 1 = 1:488 2 = 1; 2 := 0:488 2 2 = 0:976 3 = 0; 3 := 0:976 2 3 = 1:952 4 = 1; 4 := 0:952 2 4 = 1:904 5 = 1; 5 := 0:904 : (:372) 10 = (:01011 : : : ) 2 : x x i i i +j j 1; i+j = i : i + 1 x j; (i+1) ; (i+2) ; : : : ; (i+j ) `; (i++`j ) = (i+) ; = 1; 2; : : : ; j: i = 0; x j = 0 j > i: x ˇ1 ˇ2: x (0 < x < 1)
x x 1 1 10 = X 1 2 + 1 = 2 4 + 2 5 + 2 8 + 2 9 + 4n 2 4n+1 n=1 (:1) 10 = (:0001100110011 : : : ) 2 : x; ˇ; x = (:d 1 d 2 : : : )ˇe; d 1 0; d i ˇ e ( :d 1 d 2 : : : ); ˇ: e: (:00598) 10 = :59810 2 ; (111:001) 2 = :1110012 3 e ˇ: ˇ; t;
L U e ˇ L; U L U x 0 x = :d 1 d 2 : : : d t ˇe; :d 1 d 2 : : : d t t ˇ d 1 0 e L e U: M = M (ˇ; t; L; U ) ˇ = 2; t = 4; L = 1 U = 2: M = M (2; 4; 1; 2) :1000 2 1 = 1/4 :1111 2 2 = 3:75: [ 1 ; 1 ]; [ 1 ; 1] [1; 2] 4 2 2 M: [ 1 ; 1 ] 4 2 :10002 1 ; :10012 1 ; :10102 1 ; : : : ; :11112 1 ; :10002 0 ; 1 + i ; i = 0; 1; : : : ; 8; [ 1 ; 1] :1000 4 32 2 20 ; :1001 2 0 ; :1010 2 0 ; : : : ; :1111 2 0 ; :1000 2 1 ; 1 + i ; i = 0; 1; : : : ; 8; 2 16 [1; 2] :10002 1 ; :10012 1 ; :10102 1 ; : : : ; :11112 1 ; :10002 2 ; 1 + i ; i = 0; 1; : : : ; 8: [2; 3:75] 8 M; :10002 2 ; :10012 2 ; :10102 2 ; : : : ; :11112 2 ; 2 + i ; i = 0; 1; : : : ; 7: M 4 4 2 1 1 2 1 4 0 1 4 1 2 1 2 4 M (2; 4; 1; 2): ˇ; t; L; U; M d i = ˇ 1; 1 i t; e = U :10 : : : 0ˇL M e: M :1 ˇL M:
ˇ = 10; t = 5; 1 (= :110 1 ) 10 5 (= :110 4 ) M; 1 + 10 5 = 1:00001 ˇ; t; L; U M; M t [L; U ]: M M x 0 < jxj < :1 ˇL: x x :1 ˇL :d 1 d 2 : : : d t ˇU ; d i = ˇ 1; i = 1; : : : ; t; (x): (x) x M jx (x)j jx yj y 2 M: x 0 x = 0; (x) = x
ˇ t L U ˇ1 t 16 6 64 63 9:5 10 7 16 14 64 63 2:2 10 16 2 24 125 128 1:2 10 7 2 53 1021 1024 2:2 10 16 2 64 16381 16384 1:1 10 19 10 10 98 100 1 10 9 ˇ ˇ (x) x ˇ 1 ˇ1 t : x 2 (x) = x; x x 0 ; x 00 x 0 < x < x 00 : j (x) xj 1 2 jx0 x 00 j j (x) xj jxj jx0 x 00 j 2jxj : x x = qˇk; q = :d 1 d 2 : : : d t d t+1 : : : x; x 0 = :d 1 d 2 : : : d t ˇk x 00 = (:d 1 d 2 : : : d t + ˇ t )ˇk; jx 0 x 00 j = ˇk t : 1ˇ :d 1 q < 1; jxj = qˇk ˇk 1 : j (x) xj jxj jx0 x 00 j 2jxj 1 2 ˇ1 t :
(x) x t x ˇ = 10; t = 5; x = (: 1 : : : 5 6 : : : ) 10 k : 6 5; 5 (x) = x 00 := (: 1 2 : : : 5 + 10 5 ) 10 k : 6 < 5; (x) = x 0 := (: 1 2 : : : 5)10 k : 6 = 5 i = 0; i 7; x 00 x 0 : x (x) x = (:d 1 d 2 : : : d t d t+1 : : : )ˇk; (x) = (:d 1 d 2 : : : d t )ˇk; d t : 1 2 x 0 :1 ˇL ˇU (x) j (x) xj jxj u; u; 1 ˇ1 t ; u = 2 ˇ1 t : u 1:
? +; ; ; : x; y M; x? y z := (x)? (y) : x; y (x); (y); (x)? (y) 2t (x)? (y) M (): ˇ = 10; t = 5; U = L = 10; () x = 5891:26 y = :0773414 z (x) = :58913 10 4 ; (y) = :77341 10 1 ; (y) +4; (x) + (y) = :5891377341 10 4 : z = (x) + (y) = :58914 10 4 : z; x + y x + y (= 5891:3373414); (x + y) (= :58913 10 4 ) (x) + (y) R: M = 1; ˇ = 3 10 5 ; = 3 10 5 : ( + ˇ) + 1; + (ˇ + ) 1:0001: 1 + x = 1 x = 0: x = 4 10 5 (x 2 M ); 1 + x (1 + x) =
(1:00004) = 1: x 2 R 0 < x < 5 10 5 x 0 < x < ˇ1 t /2 1 + x = 1 x 10 11 : 1 ˇ1 t /2; 1: ˇ; t " 1 1 + " > 1: " "/2; "? : x; y; x? y (x)? (y) x? y x? y (x) = x(1 + ") " = "(x); j"j u:
" = (x) x x " i ; 1 i m; j" i j u < 1; "; j"j u; my (1 + " i ) = (1 + ") m : i=1 := Q m i=1 (1 + " i); (1 u) m (1+u) m : (1 + x) m [ u; u]; " 2 [ u; u] = (1 + ") m ; x y (x) = x(1+" 1 ) (y) = y(1+" 2 ); j" i j u: z := (x) (y) = xy(1 + " 1 )(1 + " 2 )(1 + " 3 ) = xy(1 + ") 3 j"j u; ˇ ˇz xy ˇ = ˇˇ(1 + ") 3 1ˇˇ = ˇˇ3" + 3" 2 + " 3ˇˇ 3u + 4u 2 : xy u 1; u 2 u: u; (x) x(1 + "1 ) z := = = x (y) y(1 + " 2 ) y 1 + " 1 1 + " 2 (1 + " 3 )
j" i j u: 1/(1 + " 2 ) = 1 + ı ı = " 2 /(1 + " 2 ); jıj u/(1 u): j"j u ˇ ˇz x/y ˇ = ˇˇ(1 + ") 2 (1 + ı) 1ˇˇ = ˇˇ2" + ı + " 2 + 2"ı + ı" 2ˇˇ 3u + ; x/y u 2 ; z := (x) + (y) = x(1 + " 1 ) + y(1 + " 2 ) = x(1 + " 1 )(1 + " 3 ) + y(1 + " 2 )(1 + " 3 ) = x(1 + ") 2 + y(1 + ı) 2 ; j"j; jıj u: z = (x + y) + 2("x + ıy) + " 2 x + ı 2 y u 2 ; z (x + y) + 2("x + ıy): ˇ ˇz (x + y) x + y ˇ + ıy ˇ 2ˇ"x ˇ 2u jxj + jyj : x + y jx + yj x; y jxj + jyj = jx + yj 2u: x; y x y " u; ı u; 2uˇˇ(x y)/(x + y)ˇˇ; 2u x y: x = :45142708 y = :45115944; x + y = :26764 10 3 : ˇ = 10; t = 5; U = L = 10; z := (x) + (y) = (:45143 :45116) = :00027 = :27000 10 3 :
ˇ (x + y) ˇz ˇ 88 10 4 ; x + y 88 2u = 10 4 z = :27000 10 3 3; 4 5 :451 x; y; (x); (y) (x) + (y) z: x = 451852000; y = 451851000 x+y = 1000 z = (x)+ (y) = 0 t = 10 x y z x y z = (x + y) = (x + y)(1 + "); ˇ ˇz (x + y) x + y ˇ = j"j u: ˇ = 10; t = 5; x = :12346; y = :12345: z = (x) + (y) = (x + y) = x + y = 10 5
ˇ = 10; t = 10 p 7892 p 7891 : p p 7892 = :8883692926 10 2 ; p 7891 = :888313007910 2 : :5628470000 10 2 ; p x p y = x y p x + p y ; p 7892 p 7891 = 1 p 7892 + p 7891 = :5628468294 10 2 f (x) = x x jxj x!0 ( x)/x = 1; x = x (x 3 /6) + "(x); j"(x)j jxj 5 /120; f (x) x 3 /6 jxj 5 /120: x 3 /6
S n = 1 + nx k=1 1 k 2 + k : 1 k 2 + k = 1 k 1 k + 1 ; S n = 1 + 1 1 1 + 2 2 1 1 + + 3 n 1 = 2 1 n + 1 n + 1 : S 9 = 1:9 S 99 = 1:99 S 999 = 1:999 S 9999 = 1:9999: S n S 0 = 1; S k = S k 1 + S 1 1 ; k = 1; : : : ; n; k(k + 1) k = 1; 2; : : : ; n: 1 S S + k(k + 1) :
n zs n 9 1:900000000 99 1:990000003 999 1:999000003 9999 1:999899972 ˇ = 10; t = 10 zs n S n = T n T 0 = 1 n(n + 1) ; T k = T k 1 + T n = T n 1 + 1; 1 ; k = 1; : : : ; n 1; (n k)(n k + 1) zt n n zt n 9 1:900000000 99 1:990000000 999 1:999000000 9999 1:999900000 n N i; 1 i N; i
s N = P N i=1 i; s 1 = 1; s k = s k 1 + k; k = 2; 3; : : : ; N; s k ; k = 1; 2; : : : ; N: i s k s 1 = 1; s k = ( s k 1 + k); k = 2; 3; : : : ; N: " i ; j" i j u; s 1 = 1 s 2 = ( s 1 + 2) = ( s 1 + 2)(1 + " 1 ) = 1(1 + " 1 ) + 2(1 + " 1 ) s 3 = ( s 2 + 3) = ( s 2 + 3)(1 + " 2 ) = 1(1 + " 1 )(1 + " 2 ) + 2(1 + " 1 )(1 + " 2 ) + 3(1 + " 2 ) s 4 = ( s 3 + 4) = ( s 3 + 4)(1 + " 3 ) = 1(1 + " 1 )(1 + " 2 )(1 + " 3 ) + 2(1 + " 1 )(1 + " 2 )(1 + " 3 ) : + 3(1 + " 2 )(1 + " 3 ) + 4(1 + " 3 ) s 4 ; jı i j u s 4 = 1(1 + ı 1 ) 3 + 2(1 + ı 1 ) 3 + 3(1 + ı 2 ) 2 + 4(1 + ı 3 ): j"j u (1+") k = 1+k"+ O(k 2 u 2 ); k = 2; 3: u 1 s 4 1(1 + 3ı 1 ) + 2(1 + 3ı 1 ) + 3(1 + 2ı 2 ) + 4(1 + ı 3 ) = s 4 + (3ı 1 ) 1 + (3ı 1 ) 2 + (2ı 2 ) 3 + ı 3 4; i 1 > 2 > 3 > ;
1 < 2 < 3 < ; ı i : s N ; [ u; u] ; " 1 ; " 2 2 [ u; u]; " 3 2 [ u; u] " 1 + " 2 = jj + jj " 3 : x = " 1 + " 2 ; jxj jj + jj u: " 3 := x/ jj + jj ; j" 3 j u: s 1 = 1: jıj; j" 2 j u; s 2 = ( s 1 + 2)(1 + ı) = ( 1 + 2)(1 + ı) = s 2 + s 2 ı = s 2 + js 2 j" 2 : jı 0 j u; s 3 = ( s 2 + 3)(1 + ı 0 ) = s 2 + js 2 j" 2 + 3 (1 + ı 0 ) = s 3 + js 2 j" 2 (1 + ı 0 ) s 3 + js 2 j" 2 + s 3 ı 0 ; js 2 j " 2 ı 0 = O(u 2 ): " 3 2 [ u; u] s 3 s 3 + js 2 j + js 3 j " 3 O(u 2 ): s N s N + js 2 j + + js N j " N j" N j u; O(u 2 ): N = js 2 j + js 3 j + + js N j;
ˇ ˇ s N s N N ˇˇˇ / s N js N j u =: N u; s N u N = N /js N j; k N = (N 1) 1 + (N 1) 2 + (N 2) 3 + + N s N i zs n zt n N = n + 1; 1 = 1; k = s N = S n = 2 1/(n + 1); N = 2n n: Η προεπισκόπηση των επόμενων σελίδων δεν είναι διαθέσιμη m = 1 ; k = 2; : : : ; N; k(k 1) 1 2 + + 1 n + 1 Z m 1 2n n 1 dx m > 0; x m 2; 1 2 + + 1 m < m < 1 + 1 2 + + 1 m 1 ;