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1 bit 8, 6, 3 bit byte = 8 bit, char (C )BYTEFORTRAN) 055, -87 word = 6 bit int (C) INTEGER (FORTRAN) -3,7683,767 unsgined int (C) 065,535 double word = 3 bit, longint (C )-,47,483,648,47,483,647 CD-audio 6 bit, 44. kbpsch74 DVD-audio, Hi-Res audio 4 bit, 9 kbps4.7gb /
2 Facebook Byte/ 8 kb Instagram Byte/ 3 MB iphone 7s (5.5 inch) Byte/ 6 MB ipad Pro 9.7 inch Byte/ 9 MB dpi = mm 0.0 mm bit/l7 mm 89 mm 5 MB Byte/ 5 fps 0 Mbps HDTV fps.5 Gbps 4K fps 6 Gbps IEEE bit = 4 byte 4 byte 3 bit /
3 s e e e 3... e 7 e 8 f f 3 f 4... f 3 f (7) 54 e e!e 8 = ± f f 3 4! f 4 e 7 e 6!e 8 7 = ( ) = 0 3 s = 0, f f 3! f 4 s =, f f 3! f 4 3 = ± f f 3 4! f 4 6 ( e e!e 8 ) = ( ) = 55 = ( 00!0) = = ( 00!0) = ( f f 3! f 4 ) 00!0 = NAN (not a number = ! 7 = bit = 8 byte 3 /
4 s e e... e 0 e f f 3... f 5 f e e!e = ( ) ( e e!e ) ( 0) =06 = ( ) s f f 3 4! f 53 e 0 e 9!e 03 ( e e!e ) = ( ) = 0 5 s = 0, f f 3! f 53 s =, f f 3! f 53 5 = ± f f 3 4! f ( e e!e ) = ( ) = 047 = ( 00!0) = = ( 00!0) = ( f f 3! f 53 ) 00!0 = NAN (not a number = ! 03 = /
5 π = elementary functions 6 eponential function : logarithmic function : ln log log e trigonometric functions ep( ) = e sine function : sin cosine function : tangent function : cos tan cosecant function : secant function : cotangent function : inverse trigonometric functions cosec = sin sec = cos cot = tan arcsine function : arcsin sin arccosine function : arccos cos arctangent function : arctan tan csc arccosecant function : arccosec cosec arccsc csc 5 /
6 arcsecant function : arcsec sec arccotangent function : arccot cot hyperbolic functions hyperbolic sine function : sinh = e e hyperbolic cosine function : cosh = e e hyperbolic tangent function : tanh = e e e e hyperbolic cosecant function : cosech = e e csch hyperbolic secant function : sech = e e hyperbolic cotangent function : inverse hyperbolic functions inverse hyperbolic sine function : arcsinh = sinh = ln ± coth = e e e e e y e y = 0 e y = ± inverse hyperbolic cosine function : arccosh = cosh = ln( ± ) e y e y = 0 e y = ± inverse hyperbolic tangent function : arctanh = tanh = ln ( e y ) = e y e y = inverse hyperbolic cosecant function : arccosech = cosech = ln ± arccsch = csch 6 /
7 inverse hyperbolic secant function : arcsech = sech = ln ± inverse hyperbolic cotangent function : y = a y = ep lna Γ t e t Γ dt 0 arccoth = coth = ln ep ep( ) =! 3 3! 4 4!! N N! = ! ep = 7 /
8 ! =.5! 3! =.666!! 3! 4! = !! 3! 4! 5! =.76666!!! 3!!! = ! = f ( 0) f ' ( 0) f! f '' 0 3! f ''' 0 4! f (4) 0 N! f (N ) 0 = a j j j = 0 a j = d j f j! d j = j! f ( j ) ( 0) N =0 3 4! N tan tan = 3 5 7! 8 /
9 tan = ! 3 = =.555! 3 =.55737! 5 7 y = a 0 a a! y = a 0 a ( a a ) ( a 3 a )! = a 0 = a 0 = a 0 a ( a a ) a 3 a a = a 0 ( a a ) ( a 3 a )!! ( a a ) ( a 3 a )! a ( a a ) ( a 3 a ) ( a 4 a )! ( a a ) ( a 3 a )! [ ] a ( a a ) ( a a ) ( a 3 a )! ( a 3 a ) ( a 4 a )! a = a 0 ( a a ) ( a a a a 3 ) ( a 4 a a 3 a )! ( a 3 a ) ( a 4 a )! a = a 0 ( a a ) ( a a a a 3 ) [( a 4 a a 3 a ) ( a 3 a a a )]! ( a 3 a ) ( a 4 a )! { } 9 /
10 = a 0 a ( a a ) ( a 3 a a a ) ( a 3 a ) ( a 4 a )! [( a 4 a a 3 a ) ( a 3 a a a )]! a = a 0 ( a a ) ( a 3 a a a ) [( a a a a 4 3 ) ( a 3 a a a ) a 3 a ]! [( a 4 a a 3 a ) ( a 3 a a a )]! f ' ( ) = lim f ( ) f h h 0 h (4.3.) f f 3 (4.3.) = f ( ) = = ep( ln ) = ep( ln) f ( ) = d ep ln d = ln ep ln f ( 3) = 3 ln = ! f ( 3) f h = = ln = = ! 0 /
11 numerical differential f S = b a f d (4.4.) numerical integral quadrature w f a b a S b a = b a N ( ) w f a ( b a) ( j ) w j f a b a j=! w N f ( a ( b a) N ) (4.4.) { w j } = w,w,!,w N { } { j } = {,,!, N } 0 < <! < N w w! w N = (4.4.3) (4.4.4) mid-point method ( b a) S! b a f a 0.5 b a N N f a.5 b a N 0.5 N! f a N = b a N N j= ( j 0.5) f a b a N (4.4.5) /
12 { j } w j { } (4.4.) j = j 0.5 N w j = N (4.4.6) (4.4.7) (4.4.3) (4.4.4) S = d 0 (4.4.8) Fig π / 4 = y Fig (4.4.8) (4.4.8) /
13 S = d = 0 sinθ d=cosθ dθ : 0 θ : 0 π/ π/ sin θ cosθ dθ = cos θ dθ 0 π/ 0 (4.4.9) cosθ cos θ = S = π/ cos θ dθ = 0 0 ( cosθ ) d = θ sinθ π/ 0 = π sinπ sin0 0 = π 4 (4.4.0) (4.4.8) 5 N = 5 Fig /
14 y Fig (4.4.8) 5 S! = 0.79! S = π / 4 = 0.785! 0 N = 0 S! 0.788! Riemann (4.4.) j = j N (4.4.) 4 /
15 w j = N N [ j =, N ] [ j =,3,!, N ] (4.4.) (4.4.3), (4.4.4) 5 (4.4.8) S! π / 4 = = 0.749! Fig ! y Fig (4.4.8) 5 (4.4.) 5 /
16 j = j N w j = 3 N 4 3 N 3 N [ j =, N ] [ j =,4,!, N ] [ j = 3,5,!, N ] (4.4.3) (4.4.4) 5 (4.4.8) S! 0 π / 4 = ! Fig = 0.770! 0.79! y Fig (4.4.8) 5 6 /
17 Gauss-Legendre M. Abramowitz & I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York (965) ± i [-, ] w i Table 4.4. (4.4.) S b a w N f! w N f a b ( b a) N a b ( b a) N w N f a b b a N = b a j = j w j = w j N j= N w j f a b ( b a) j (4.4.5) (4.4.6) (4.4.7) (4.4.3), (4.4.4) < N < N <! < N < w N w N! w N = (4.4.8) (4.4.9) Table 4.4. ± i w i /
18 Table 4.4. (4.4.8) S! = ! π / 4 = ! Gauss-Chebyshev ± i [-, ] 4.4. ± i (4.4.8) S! = ! 8 /
19 π / 4 = ! Monte Carlo ep ln t = u = t t v = u u y = v u f ( ) = a 0 a a a 3 3 a = a 0 a a ( a 3 a 4 ) f ( ) 9 /
20 f ( ) = a 0 a a a 3 3 a 4 4 3! 4! = a 0 a a 3 a 3 a 4 f Horner 4 b c = 0 = b ± b c b > 0 c > 0 c b b c b = c = =, b b c = b b c c = b b c IEEE 0 /
21 /
0104.pages
bit 8, 6, 3 bit byte = 8 bit, char CBYTEFORTRAN) 055, -87 word = 6 bit int (C) INTEGER (FORTRAN) -37683767 unsgined int (C) 065535 double word = 3 bit, longint C -4748364847483647 CD-audio 6 bit, 44. kbpsch74
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