数値天文学入門 - pholus.mtk.nao.ac.jp

1 2006.. 1. 10.. 2. 11.. 3. 12.. I. 1 . 4.. 13.. 5. II.. 6. 14.. 7. III.. 8. 15.. 9. 16.. (2).. Ralston and Rabinowitz . Kernighan . Numerical Recipes . 1.. IEEE754 .. (integer) (real) (complex). (fixed) . (floating) .. x = s m be (sign) s= 1 (base) b=2 or 16. (mantissa) m (exponent) e . Fortran C . (bit ). Real*4 float 24+8 2-24 ~ Real*8 double 53+11 2-53 ~ Real*10 - 64+16 2-64 ~ 4 Real*16 long 113+15 2-113 ~ double & .. m .. (nominal) . =x/x0, u=v/v0 .. e := exp( ), := *atan ( ) , 2 := sqrt ( ). 1 n . , 2 , n !, n !!, . n n m .. = e = . 1999 . Mathematica, Maple: . MATLAB: . MAXIMA: Mathematica . scilab: MATLAB . awk, bc, perl: . GnuMP: . GNUPLOT: .. IMSL, NAG, LAPACK. NUMPAC. Numerical Recipe . NetLIB: . GSL(=GNU Sci. Lib.): C, C++. SLATEC: Fortran, FFTW: FFT .. Cnm = Ank Bkm k (Strassen) .. : x~0 1-cos(x).. (Taylor) . f ( x) = .. f ( n). ( x0 ).. ( x x0 ). n n =0 n!

2 Cn n .. ( x ). 2 n 2 4 cosh x = c0 ( x 2 ). c0 ( x 2 ). x x cos x = = 1 + . n =0 ( 2n ) ! 2 24. sinh x = xc1 ( x 2 ).. sin x = x . ( x ) 2 n x3 x5. = x + xc1 ( x 2 ). n = 0 ( 2n + 1) ! 6 120 e x = cosh x + sinh x (Horner) .. N. P ( x ) an x n =N(N+3) /2 n=0. Flops =2N-1 . P ( x ) = a0 + x ( a1 + x ( a2 + x ( a3 + ) ) ).. P:=a N ;do ( k=N-1,0,-1){P:=a k +x*P}.. = +1. P ( x ) = ( y c0 ) ( b1 + ( y c1 ) ( b2 + ( y c2 )( b3 + ) ) ) y ( x a). 2.. Ralston and Rabinowitz . (Stumpff) . Stumpff (Himmelsmechanik 1-3, 1959-74).. ( z ). k . 1 z z2 1. cn ( z ) = + = zcn + 2 ( z ). k = 0 ( n + 2k ) ! n ! ( n + 2 )! ( n + 4 )! n! z=x >0. 2 c0 ( x 2 ) = cos x, c1 ( x 2 ) = ( sin x ) / x, c2 ( x 2 ) = (1 cos x ) / x 2. z=-x <0. 2 c0 ( x 2. ) = cosh x , c1 ( x 2. ) = ( sinh x ) / x , c2 ( x 2. ) = ( cosh x 1) / x 2. d2 x dt 2. + x = 0 x = x c 0 0 ( t 2. ) + v tc 0 1 ( t 2. ).. (2). Cn ( z ) n !

3 Cn ( z ). Cn = 1 n ( n + 1) zCn + 2 Cn ( 0 ) = 1.. 1. N C N = C N +1 = 1. N=8(|z|< ), 10(|z|< ), 19(|z|<1). 2.. 3. Cn cn =. c0 = C0 , c1 = C1 n! (2).. x3 2 x5 17 x 7 62 x9 1382 x11 21844 x13 929569 x15. tan x = x + + + + + + + + xt ( x 2 ). 3 15 315 2835 155925 6081075 638512875. x3 3x5 5 x 7 35 x9 63 x11 231x13 143 x15. 1. sin x = x + + + + + + + + xs 1 ( x 2 ). 6 40 112 1152 2816 13312 10240. tanh x = xt ( x 2 ).. xn x 2 x3. log (1 x ) = = x + + +. n =1 n 2 3. sinh 1 x = xs 1 ( x 2 ).. tan 1 x = x . ( x )2 n x3 x5. = x + xt 1 ( x 2 ). n =0 2n + 1 3 5 tanh 1 x = xt 1 ( x 2 ).. (hypergeometric) .. ( a )n ( b )n x n F ( a, b, c; x ) . ( c )n n ! n =0 . (Pochhammer) . F . (a + n). (a) = a ( a + 1) ( a + n 1). a n ( ). a b . a or b .. (Kummer). (congruent) .. ( a )n x n F ( a, c; x ) . ( c )n n ! n =0 . a .. (Pochhammer). (generalized) .. (a ) ( a p )n x n p Fq ( a1 , , a p ; b1 , , bq ; x ).

4 1 n . n = 0 ( b1 ) n ( bq )n n ! . a1, a2, , ap . pFq .. F ( a, b, c; x ) 2 F1 ( a, b; c; x ) F ( a, c; x ) 1 F1 ( a; c; x ).. e = 0 F0 ( x ) , (1 x ) = 1 F0 ( a; x ) , log (1 x ) = xF (1,1, 2; x ) , x a 3 x2 3 x2 . sin x = x 0 F1 ; , sinh x = x 0 F1 ; , 2 2 2 2 . 1 x2 1 x2 . cos x = 0 F1 ; , cosh x = 0 F1 ; , 2 2 2 2 . 1 1 1 3 2 1 1 1 3 . sin x = xF , , ; x , sinh x = xF , , ; x 2 , 2 2 2 2 2 2 . 1 3 1 1+ x 1 3 2 . tan 1 x = xF ,1, ; x 2 , tanh 1 x = log = xF ,1, ; x . 2 2 2 1 x 2 2 .. (Pad ) (continued fraction). x 3 x 5 97 x 7. x+ + + +. ex = 1 +. x = 1 +. x log (1 + x ) = 12 30 5040.. Bk x k x x2 x4 x x 3 7 x 5 181x 7.. k =0 k ! 1 + . 2 12 720. + 1+ + + +. 2 12 180 7560. +.. Bn (Bernouli) B2n+1=0 (n>1). {( 1) n } 1 1 1 1 5 691 7 3617 43867 174611 854513 236364091 . B2 n = , , , , , , , , 6 30 42 30 66 2730 6 510 798. , 330. , 138. , 2730. , .. (m,n) . m n m+n m k p x k Rmn ( x ) = k =0.

5 N = f ( x ) + O ( x m + n +1 ). 1+ q j x j j =1. m=n-1 or n m+n= . m+n .. (2). m+n f ( x ) fi xi i =0. m n . p(x) p ( x ) = m p x k q ( x ) = 1 + n q x j k j q(x). k =0 j =1. p ( x ) = f ( x ) q ( x ) + O ( x m + n +1. ). q p n k f j =1. m +i j q j = f m+i ( i = 1, , n) pk = f k + f k j q j j =1. (3).. (n,n) .. x x x2 x x 2 x3. 1+ 3 1+ + 5 1+ + + 7. x 2+ + 2 12 + x 2 10 120 x exp ( x ) = = 2. + = 2 3. +. x 12 x x 720 x x x 100800. 1 1 + 1 + . 2 2 12 2 10 120.. an | a1. b0 + b0 +. n =1 | b a2. n b1 +. b2 +.. tan(x). tan x =. x =. x .. x 2. x2. 1+ 1 . n =1 2k + 1 x2. 3 . 5 . (2).. n ak | An =. k =1 | b Bn k A0 = 0, B0 = 1, A1 = a1 , B1 = b1. A = b A. n n n 1 + an An 2. Bn = bn Bn 1 + an Bn 2. (4). tan(x) . 3. x x . tan x =. x x 2. + O ( x 5. ) =. 2x 2 (. 15 + O x 7. ). 1 1 . 3 5. 3 3 5. 2x x x x x +. =. 3x 2. 21. x 4. + O ( x 9. ) = 9 945 + O x11. 4x 2. x 4 ( ). 1 + 1 +. 7 105 9 63.

6 A + bx y=. c + dx . ( x). n x 2 x3. log (1 + x ) = = x + . n =0 n + 1 2 3. x>0 x y=. 2+ x 1+ y . y 2n log (1 + x ) = log = 2 y . 1 y n = 0 2n + 1. (2). (Euler) a = 1 . k k .. n . k =0 2. k +1 j . a n =0 j =0 j .. a0 b0 1 k 1 k 1 . an + an +1 = bn an = + + k +1 bj . n =0 2 4 k = 2 2 j =0 j . ( 1). n 1. an =. bn 3. n . f ( x ) sin ( kx ) dx .. sin cos . 2t 1 t2 2t t tan sin = , cos = , tan =. 2 1+ t 2. 1+ t 2. 1 t 2.. x 2t 1+ t2 2t t tanh sinh x = , cosh x = , tanh x =. 2 1 t 2. 1 t 2. 1+ t2. sin cos . sin cos . 0< < /4. t:=tan ( * ) ; t2:=t*t; d:= ( +t2). s:=(t* )*d; c:=( )*d . sin cos sincos . sin cos .. (2). atan2 ( y, x ).. ( ). sin 1 s = atan2 s, 1 s 2 , cos 1 c = atan2 ( ). 1 c 2 , c , tan 1 t = atan2 ( t ,1).. tanh-1 . 1 1 s 1 1 c2 1 x 1 . 1. sinh s = tanh , cosh c = tanh , log x = 2 tanh . 1+ s 2 c x + 1 . log . ( ) (. sinh 1 s = sign ( s ) log s + 1 + s 2 , cosh 1 c = log c + c 2 1 , ) tanh 1 t =.

7 1. 2. 1+ t . log . 1 t .. 1: . 2: do ( k=0,K ) {c k :=F ( k )}. f = Ak F ( k ) do ( n=1,N ) {f n :=A n0 *c0. k do ( k=1,K ){f n +=A nk *c k }}.. F(xk)=xk . c0 := ; c1:=x; do ( k=2,K ){ck :=ck-1*x}. 2.. (Chebyshev) . (Legendre) . (associated) . (Bessel) . (Hermite) . (Laguerre) .. 1 1 x . Tn ( x ) = F n, n, ; . Tchebysheff 2 2 . 1 x . Pn ( x ) = F n, n + 1,1; . 2 . ( n + m )! ( ) 1 x . m Pn ( x ) = m m 1 x 2. F m n, n + m + 1, m + 1; . 2 m !( n m ) ! 2 . (2). or . 1 x . n x2 . Jn ( x) = 0 F1 ; n + 1; . n! 2 4 .. 1 x2 . H 2 n ( x ) = ( 1) ( 2n 1) !! F n; ; . n . 2 2 . 3 x2 . H 2 n +1 ( x ) = ( 1) ( 2n + 1) !! xF n; ; . n 2 2 .. Ln ( x ) = F ( n;1; x ).. k !! k ( k 2 )( k 4 ). (recurrence) .. bFa ,b +1,c = ( b a ) Fa ,b ,c + aFa +1,b ,c ( c b ) Fa ,b 1,c = ( a b )(1 x ) Fa ,b,c + ( c a ) Fa 1,b,c ( c a ) Fa 1,b,c = c 2a + ( a b ) x Fa ,b,c + a (1 x ) Fa +1,b,c . aFa +1,c = ( x + 2a c ) Fa ,c + ( c a ) Fa 1,c.

8 X cos .. T U T x cos n ,U x sin n . n( ) n( ).. sin . 1. ( 2k 1) . Tn ( xk ) = 0 xk = cos . ( n) ( n). ( k = 1,.., n ). 2n . 2. k .. dx ( ). dTn ( n ) (n). yk = 0 yk = cos . n .. ( k = 0,1,.., n ). 3. Tn ( yk (n). ) = ( 1) k (2). 1. T1. T2. 0. T3. T4. T5. -1. -1 0 1. (3).. T0 = 1,U 0 = 0, T1 = x,U1 = 1.. Tn +1 = xTn (1 x ) U n , U n +1 = xU n + Tn 2.. Tn +1 = 2 xTn Tn 1 , U n +1 = 2 xU n U n 1. (4). : Wn, Vn y~0 Tn . Un Vn . Wn 1 Tn , Vn yU n = sin n y sin .. t:=tan ( * ) ; y:=t* ( +t*t ) ; w:=t*y; w2:=w* ; z:=r2+ ;. W0 := ; T0 := ; V0 := ; W1:=w; T1:= ; V1:=y;. do ( n=1,N ){Wn+1:=z*Wn +w2-Wn-1 ; Tn+1:= ; Vn+1:=z*Vn -Vn-1}. (5). T0 ( x ) = 1, T1 ( x ) = x T2 ( x ) = 2 x 2 1. T3 ( x ) = 4 x 3 3 x T4 ( x ) = 8 x 4 8 x 2 + 1. T5 ( x ) = 16 x5 20 x3 + 5 x T6 ( x ) = 32 x 6 48 x 4 + 18 x 2 1. T7 ( x ) = 64 x 7 112 x5 + 56 x3 7 x T8 ( x ) = 128 x8 256 x 6 + 160 x 4 32 x 2 + 1.

9 T9 ( x ) = 256 x9 576 x 7 + 432 x5 120 x3 + 9 x (6).. [ n / 2] n 1 . x = n 1 Tn 2 k ( x ). n 2 k =0 k . 1 = T0 16 x 5 = T5 + 5T3 + 10T1. x = T1 32 x 6 = T6 + 6T4 + 15T2 + 20T0. 2 x 2 = T2 + T0 64 x 7 = T7 + 7T5 + 21T3 + 35T1. 4 x 3 = T3 + 3T1 128 x8 = T8 + 8T6 + 28T4 + 56T2 + 70T0. 8 x 4 = T4 + 4T2 + 6T0 256 x 9 = T9 + 9T7 + 36T5 + 84T3 + 126T1.. Tn ( x ) cos ( ni ) , Vn ( x ) sin ( ni ) ( x )k = xk cos k .. n 1 d . ( x 1). Pn ( x ) 2 n 2 n ! dx . n . 1. Pn ( x ) = ( 1) Pn ( x ). n . 2.. ( 2n 1) !! P2 n ( 0 ) = ( 1) pn 0 , P2 n +1 ( 0 ) = 0 pn 0 . n ( 2n )!! Pn (1) = 1, Pn ( 1) = ( 1) N !! N ( N 2 )( N 4 ). n (2). p ( x ). n P2 n ( x ) = ( 1) 2 k n k =0. nk 2n . pnk ( x ). n P2 n +1 ( x ) = ( 1) x 1 + 2 k n k =0 2k + 1 .. 2n + 1 . p0,0 = 1 pn +1,0 = pn ,0. 2n + 2 . pn ,k +1 =. ( n k )( 2n + 2k 3). pn ,k ( k + 1)( 2k + 1). (3). P0 ( x ) = 1, P1 ( x ) = x P ( x ) = ( 3 x 1) / 2.

10 2. 2. P ( x ) = ( 5 x 3x ) / 2. 3. 3. P ( x ) = ( 35 x 30 x + 3) / 8. 4. 4 2. P ( x ) = ( 63 x 70 x + 15 x ) / 8. 5. 5 3. P ( x ) = ( 231x 315 x + 105 x 5 ) /16. 6. 6 4 2. P ( x ) = ( 429 x 693 x + 315 x 35 x ) /16. 7. 7 5 3. P ( x ) = ( 6435 x 12012 x + 6930 x 1260 x + 35 ) /128. 8. 8 6 4 2. (4). 1. P1. P2. 0 P3. P4. P5. -1. -1 0 1. (5). Pn . ( n + 1) Pn +1 = ( 2n + 1) xPn nPn 1.. P0 := ;P1:=x;do ( n=1,N ){Pn+1:=A n *x*Pn -Bn *Pn-1}. An = 1 + Bn , Bn = 1 I n +1. In 1. In . n (6). Pn .. P6 . P3 : 3 / 5, P4 : (15 ). 120 / 35, P5 : (35 ). 280 / 63. P6 : , , P7 : , , P8 : , , , (7). Pn P : , 9 , , P10 : , , , , P11 : , , , , P12 : , , , , , P13 : , , , , , (8). dPn 1 dPn . Qn = x sin . dx cos d .. 2 x kpnk ( x ). n 2 k 1. Q2 n ( x ) = ( 1). n +1. k =1. ( 2n + 2k + 1) p ( x ). n Q2 n +1 ( x ) = ( 1) 2 k n nk k =0. (9).. ( n + 1) Qn+1 = ( 2n + 1)( Pn + xQn ) nQn 1. Pn . Q0 = 0, Q1 = 1.