untitled

Size: px

Start display at page:

Download "untitled"

あつのあきます
5 years ago
Views:

1 2 (numerical algorithm) 2. = ratio of the circumference to its diameter, number, Ludolph s number... sin, cos = e x2 dx = /2 etc. : 999 : 99 : 974 P.: 973, J.-P.: 200 J. Arndt, Ch. Haenel: Unleashed, Springer, Berlin Heidelberg, mathematics a, b. b = Pythagoras ( B.C.) b =2/ 3 3 <<3 2 =2 3=

2 a b 2.. half hexagon 2.2. n 2n O, /n LMN; LON : n /n n = LN =2a; LOT : n /(2n) n /2 = LT = b; LOM : 2n /(2n); 2n = LM =2a ; POQ : 2n /(2n) 2n = PQ =2b ; n OT LON LT LT H OTL LH : LT = OL : OT. OT 2 =+b 2 a = b +b 2 (2.) 2n a = b +b 2 (2.2) n 2n TPM TLH bb = a(b b ) (2.3) 6

3 T P b M Q L 2a a H N O 2.2. n-polygons 7

4 (2.) (2.3) a b = b + +b 2 (2.4) n =2A, n =2B, 2n =2A, 2n =2B A = na = nb +b 2, B = nb, A = 2na = 2nb +b 2, B =2nb = C = +b 2,C = +b 2 A = B/C, B = 2B +C, 2nb + +b 2 ( = ( B 2 B + )), A (2.5) (2.6) A = B /C n =4 2 m+ (m =, 2,...) 2l m, 2L m 2l m 2A 2L m 2B L =4, (2.7) { l n = L n / +(L n /2 n+ ) 2, (n =, 2,...) (2.8) L n+ = 2/ (/l n +/L n ), (recurrence formula) L,l, }{{} L 2,l 2, }{{} 8 L 3,l 3,... }{{} 6 8

5 2.2.2 () 0 <l n <<L n (2) l n <l n+ <L n+ <L n {l n } {L n } (3) 2 2=l <l 2 < <l n <l n+ < <L n+ <L n < <L 2 <L =4 (4) l = lim n l n, L = lim n L n (5) l = L = (2.8) L n L n+ = L n = ( ) Ln + L n+ 2 l n ( Ln 2 n+ ) n O(2 4n ) (n ) L n = O(2 2n )=O(4 n ) (n ) (2.9) log 0 4= n L n Ludolph van Ceulen(540-60, ) m+ (m =, 2,...) 2 n+ n ln Ln

6 real numbers R integers Z rational numbers Q algebraic numbers transcendental numbers 0

7 J.H. Lambert (728-77, ): (76) C.L.F. Lindemann ( , ): (882) 2.3 tan x tan 6 =, tan 4 = etc. Taylor (Maclaurin ) f(x) =f(0) + f (0)x + f (0) x Gregory Leibniz J. Gregory ( , ) G.W.F. Leibniz (646-76, ) tan x arctan x 4 = arctan d (arctan x) = dx +x 2 0 dx +x 2 = 4 +x 2 = x 2 + x 4 x 6 + +( ) n x 2n +, ( x < ) dx +x 2 = x 3 x3 + 5 x5 7 x7 + + ( )n 2n + x2n+ + arctan x Maclaurin x = 4 = ( )n (2.0) 7 2n + ( ) n = 4 2n + = (4m + )(4m +2), 4 n=0 M m=0 m=0 (4m + )(4m +2) = O ( ) (4M + 3)(4M +4) ( ) = O 6M 2

8 M =0 (6M 2 ) /( + x 2 ) Maclaurin Sharp / 3 0 dx +x 2 = arctan 3 = 6 6 = ( ( )n (2n +)3 + n / 3 3 Gregory Leibniz Sharp 699 ) n 2n LON = 2 n, LOH = n, LH =sin n, LT =tan n 2 n+ 2l n 2 n+ 2L n sin, tan Maclaurin l n =2 n+ sin 2, L n+ n =2 n+ tan (2.) 2 n+ sin x = x 3! x3 + 5! x5 7! x7 +, ( x < ) tan x = x + 3 x x x7 + ( x < 2 ) (2.) N =2 n+ ( l n = ( ) 2 ( ) ( 4 + ), L n = + ( ) 2 2 ( ) ) 6 N 20 N 3 N 5 N ( 3 (2l n + L n )= + ( ) 4 + ) 20 N (/N) 2 {l n }, {L n } { 3 (2l n + L n )} ( l n+ = (n+) + 60 ) (n+) 2

9 l () n 3 (4l n+ l n )= ( 3 (l n+ l n )+l n+ = 3 ) (n+) L () n 3 (4L n+ L n )= ( 3 (L n+ L n )+L n+ = 3 ) (n+) extrapolation , 2 0 (n =9) 4 2 n+ n ln Ln accelerated

10 Euler Machin L. Euler ( , ) key idea tan(a B) = a =tana, b =tanb ( ) a b A B = arctan +ab tan A tan B +tana tan B (2.2) Euler (748) 4 = arctan = arctan ( ) ( ) + arctan 2 3 (2.3) John Machin (706) 4 = arctan = 4 arctan ( ) 5 ( ) arctan 239 (2.4) 4 arctan 20 = A tan A = 5 9 (2.4) Machin arctan ( 4 = ) ( ) O(5 2n ) 4

11 n Gregory-Leibniz Sharp Machin Gregory Leibniz O(n 2 ) n 5

12 Machin O(c an ) (c>) E. Salamin R.P. Brent (976). 2. Landen (Landen transformation) 3. Legendre (Legendre formulae) 206, 58, 430, 000 (999 0 ) ftp://pi.super-computing.org/readme.our latest record Pascal A 0 =,B 0 =,T 0 = 2 4,X 0 = n := 0 while abs(a n B n ) >εdo begin A n+ := (A n + B n )/2; B n+ := A n B n ; T n+ := T n X n (A n A n+ ) 2 ; X n+ := 2X n ; n := n +; end; := (A n+ + B n+ ) 2 /(4T n+ ) 2.4. agm arithmetic-geometric mean (agm) a 0,b 0,c 0 :, a 2 0 = b c 2 0 a n = 2 (a n + b n ), b n = a n b n (c 2 n = a2 n b2 n ) (2.5) lim a n = lim b n agm(a 0,b 0 ) 6

13 . a n >b n 2. b n <a n <a n 3. b n <b n 4. b n <b n <a n <a n 5. {a n }, {b n } 6. ā = lim a n, b = lim b n n n 7. 2a n = a n + b n ā = b agm c n = 2 (a n b n ) 0 < <c n+ <c n < <c 2 <c c 2 n =4a n+c n+ c n+ = c2 n = 4a n+ 4a n+ ( ) c 2 2 n = = 4a n c < 0 c 2(n+2) ( (n+) )(a n+ a n a ) I(a, b) = /2 0 dt /2 a2 cos 2 t + b 2 sin 2 t, J(a, b) = 0 a 2 cos 2 t + b 2 sin 2 tdt agm Landen I(a n,b n )=I(a n+,b n+ ), J(a n,b n )=2J(a n+,b n+ ) a n b n I(a n+,b n+ ) ( agm(a 0,b 0 ) I(a 0,b 0 )= 2, J(a 0,b 0 )= a ) 2 j c 2 j I(a 0,b 0 ) (2.6) j=0 Legendre (b/a) 2 +(b /a ) 2 = a 2 I(a, b)j(a,b )+a 2 I(a,b )J(a, b) a 2 a 2 I(a, b)i(a,b )= 2 aa (2.7) a 0 = a 0 =,b 0 = k, b 0 = k (k 2 + k 2 =) (2.6), (2.7) = 4agm(,k)agm(,k ) 2 j (c 2 j + c 2 j ) j= 7

14 k = k =/ 2 = 4 ( agm(, / 2) ) 2 2 j+ c 2 j j= k, k <k,k <, k 2 + k 2 = a 0 =,b 0 = k {a n,b n,c n } agm = agm(,k) a 0 =,b 0 = k {a n,b n,c n} agm = agm(,k ) NN 4a N+ a N + N N 2 j c 2 j 2 j c 2 j j= j= NN < [ ( N exp agm ) )] agm agm agm 2N+ +2 N exp ( agm agm 2N + (2.8) 2 NN = N N < 2 2 N+4 agm 2 exp( 2 N+ ) (2.9) 0 ( ) ( ) log 0 N > 2 N+ N log log log 0 agm N O(c an ); O(c an ) 8

2 8 BASIC (4) WWW Taylor BASIC 1 2 ( ) 2 ( ) ( ) ( ) (A.2.1 ) 1

2 8 BASIC (4) WWW Taylor BASIC 1 2 ( ) 2 ( ) ( ) ( ) (A.2.1 ) 1 2 8 BASIC (4) 203 6 5 WWW http://www.math.meiji.ac.jp/~mk/syori2-203/ Taylor BASIC 2 ( ) 2 ( ) ( ) ( ) 2 (A.2. ) 6B 2 2. f a f(x) = f (n) (a) n! (x a) n = f(a)+f (a)(x a)+ f (a) 2 (x a) 2 + + f (n) (a)