1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1.

Transcription

1 Section Title Pages Id z Id URL 1

2 1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1.3) (1.3) 4 x 1 + x 1 = x = = (1.1) 1 + x =

3 1. 2 cos θ > 0 1 cos θ 1 cos θ = sin2 θ θ 0 = 35 θ = cos θ x x = 0 x 1, x 2 6 = = x 1, x 2 = ± ± = = , x 1 x 2 s 2 2 = 1 N x 2 i x 2 (1.6) N x 2 = x 1 = x 2 y 2 x = y = x 2 y 2 = = sin θ = sin θ 0 = θ θ 0 (1.4) N x i (1 i N) x s 2 x = 1 N N x i (1.5) (x y)(x + y) = = N s 2 = ( ) 2 x y 3 = = 31 x, y x 2, y 2 10 x s 2 4 x i x sin θ sin θ 0 x = 1 N (x i x) 2 + x (1.7) N i=1 sin θ sin θ 0 = 2 cos θ + θ 0 sin θ θ 0 s 2 = 1 N (x (1.4) i x) 2 ( x x) 2 (1.8) N 2 2 i=1 i=1 i=1 x i i x i i x i i x i x = = x 2 i =

4 1. 3 (1.6) x = 0 sin 8 ϕ x = x x x γ µgal ϕ 355 ϕ 0 x = 345 γ 0 γ γ 0 = (sin 2 ϕ sin 2 ϕ 0 ) i x i x i x i x i x i x [( sin 2 ϕ 0 ) (sin 2 ϕ sin 2 ϕ 0 )] sin 2 ϕ sin 2 ϕ 0 = sin(ϕ + ϕ 0 ) sin(ϕ ϕ 0 ) ϕ ϕ (1.7) (1.8) 0 = ±10 x x = 1 γ γ 0 (xi x) = N 5.16 sin 70 sin 10 = 0.84 [gal] s 2 = = = γ γ 0 µgal 6 ϕ 0 = 35 (1/N) x i sin 2 35 = x (1/N) x 2 i x2 x γ γ x i x 0 = sin(ϕ + 35 ) sin(ϕ 35 ) x [ sin(ϕ + 35 ) sin(ϕ 35 )] e x e x = 1 + x 1! + x2 2! + x3 3! x x = 10 n a n S n ϕ ( ) e e+2 γ = ( sin e e+2 ϕ e e sin 4 ϕ) [gal] e e+1 4 µgal e e e e+1

5 2. 4 a n n S n n e x = +10 n a n S n e e e e e e e e e e e e+4 ( e ) e 10 e /e a b c, d, e, f, a b c d e f n 2 31 = n = x x = ±(d 1 β 1 + d 2 β d t β t )β e 32 = ±mβ e (2.1)

6 d i < β 0 m < β m e (1bit) (8bits) (23bits) 0/1 e d 1 d 2 d 23 IBM HITACHI 1 β = UNIX 127 = x = e β = 2 x = e = 1 d 1 = e e 128 m = ( ) 2 β = 16 x = e = 0 d 1 = 4 m = ( ) = = ( ) = (1.0) ( ) 2 = ( ) 16 x = = (1.1) 2 ( ) 2 β = 2 e = 3 m = ( ) 2 = (3fc0 0000) = (1.0) ( ) 2 = (3f ) 16 Fortran x=0 1 x=x if( x.ne.1 ) goto 1... UNIX x = ±(1.d 1 d 2 d 23 ) 2 2 e 0.1 = ( ) ( ) 2 = (3dcc cccd) β 16 (1bit) (7bit) (24bit) UNIX 0/1 e + 64 d 1 d 2 d 24 IEEE 24 UNIX UNIX x

7 c/(a b) a b = = (machin epsilon) ε M 1 36 ε M = Min{ε : 1 + ε > 1} ε > ε 6 Fortran x 2 + y 2 eps = 1 x y x 2 y 2 do i=1, 64 eps1 = 1 + eps ( ) ( ) x 2 + y 2 if( eps1.eq.1 ) goto 1 eps = eps/2 enddo ( ) stop 1 eps = eps*2 print eps NaN(not a number) eps x y x2 + y 2 = x 1 + (y/x) 2 eps = c/a/b

8 2. 7 a + (b + c) = (a + b) + c A/D y = x + x x 0 y = x 1 1 deltax =... ( 16 )... if( y.ne.x+deltax ) goto A/D x x eps1=1+eps ( ) if(x.eq.y)

9 x = u + v (3.5) u 3 + v 3 + q + 3(u + v)(uv + p) = 0 u 3 + v 3 = q uv = p x 2 + 2bx + c = 0 (3.1) u 3 = 1 2 [ q ± q 2 + 4p 3 ] x = b ± b 2 c (3.2) v 3 = p3 u 3 (3.6) u 3 1 b 2 c D = q 2 + 4p 3 (3.7) x 1 = b b 2 c x 2 = c b > 0 x 1 x 1 = b + b 2 c x 2 = c b < 0 (3.3) D u 3 v 3 x 1 u v 1 2 b (3.5) (3.4) b u 3 v 3 1 y 3 + a 2 y 2 + a 1 x + a 0 = 0 y = x a 2 3 x 3 + 3px + q = 0 (3.4) u 3 = u 1 e 2πi/3 p = 1 3 a 1 [ 2 q = a 2 3 ( a2 ) 2 3 ( a2 3 ) 2 a1 ] + a 0 u 3 = p 3 e iϕ v 3 = p 3 e iϕ (3.8) D tan ϕ = D < 0 q D < 0 p u 1 = pe iϕ/3 v 1 = pe iϕ/3 u 2 = u 1 e 2πi/3 v 2 = v 1 e 2πi/3 (3.9) v 3 = v 1 e 2πi/3 u v (3.5)

10 3. 2 y 4 + a 3 y 3 + a 2 y 2 + a 1 y + a 0 = 0 y = x a 3 4 n p n (x) = a n x n + a n 1 x n a 1 x + a 0 (3.13) p n (x) x k p n (µ) = ( ((a n µ + a n 1 )µ + a n 2 )µ a k x 4 + px a 1 )µ + a 0 + qx + r = 0 (3.10) ( a3 ) 2 b n+1 = 0 p = a b k = a k + µb k+1 (3.14) ( a3 ) ( a3 ) 3 q = a 1 2a k = n, n 1,, ( a3 ) ( a3 ) 2 ( a3 ) 4 b r = a 0 a 1 + a p n (µ) (3.10) (x 2 + α) 2 (βx q 2β )2 = 0 p n (µ) = b 0 b k (3.10) p n (x) x µ p n (x) = (x µ)(b n x n 1 + b n 1 x n 2 + 2α β 2 = p α 2 q2 4β 2 = r (3.11) + b 2 x + b 1 ) + b 0 (3.15) β (3.13) (3.15) b k (3.14) 4(α 2 r)(2α p) = q 2 (3.14) b k p α n (x) x µ α β (3.11) α, β b n x n b 2 x + b 1 x 2 + βx + α q 2β = 0 x 2 βx + α + q 2β = 0 (3.12) (3.10) p n (x) = d n (x µ) n + d n 1 (x µ) n d 0 x µ p n (x) p n (x) = (x µ)[(x µ)(b n x n 2 + n 1 d k x = w + µ w

11 3. 3 x k x ε k p n (x) = (x 2 ux v)(b n x n 2 + b n 1 x n b 3 x + b 2 ) + b 1 (x u) + b 0 (3.16) y 1 = f(x 1 ) = f(x + ε 1 ) = f (x)ε f (x)ε 2 1 b n+1 = b n+2 = 0 b k = a k + ub k+1 + vb k+2 (3.17) k = n, n 1,, 0 u = 2µ v = µ 2 p n (x) = (x µ) 2 (b n x n 2 + b n 1 x n b 3 x + b 2 ) + b 1 (x 2µ) + b 0 p n (µ) = b 0 µb 1 p n(µ) = b 1 (3.18) x k = x + ε k (3.21) f(x) = 0 (3.20) x 3 ε 3 = f (x) 2f (x) ε 1ε 2 (3.22) x 2 ε 2 ε 1 f (x) 2f (x) ε 2 y = f(x) = 0 (3.19) x 1 x 3 x 2 x 1 x 3 x 2 f(x) (x 1, x 2 ) f(x) f(x) y 1 + y 2 y 1 x 2 x 1 (x x 1 ) f(x) = 0 x 3 = x 1 x 2 x 1 y 1 (3.20) y 2 y 1 (x 1 y 2 x 2 y 1 )/(y 2 y 1 ) y 3 = f(x 3 ) y 1 y 2 f(x) (robust ) x 4 = x 2 + x 3 2 y 3

12 3. 4 (3.22) ( ) f 2 ( f f(x) x f(x) ε 4 = 2f f f ) 3f (ε 1 ε 2 ) 2 (3.28) x = f 1 (y) y (x 1, x 2, x 3 ) 4 x 4 x [1234] (y 1, y 2, y 3 ) (3.26) x (y y 2)(y y 3 ) (y 1 y 2 )(y 1 y 3 ) x 1 + (y y 3)(y y 1 ) (y 2 y 3 )(y 2 y 1 ) x 2 3 (3.26) + (y y 1)(y y 2 ) (y 3 y 1 )(y 3 y 2 ) x 3 (3.23) y = y i x = x i y = f(x) = 0 x y = 0 k x k y 2 y 3 x 4 = (y 1 y 2 )(y 1 y 3 ) x y 3 y (y 2 y 3 )(y 2 y 1 ) x x 2 y 1 y 2 + (y 3 y 1 )(y 3 y 2 ) x 3 (3.24) f(x k + x) = 0 f(x k ) + f (x k ) x + = 0 x 1, x j x [1j] x 1 x j x 1 x y 1 j = 2, 3 (3.25) k+1 = x k + x = x k f(x k) f y j y (3.29) (x k ) 1 (3.20) x 3 x [12] x 3 x 4 x 4 = x [123] x [12] x [13] x [12] y 3 y 2 y 2 (3.26) k ε k ε k+1 = f (x) 2f (x) ε2 k (3.30) (3.24) (3.26) (3.26) (3.25) x 1, x 2, x 3 (3.24) x 4 0 ε 4 = f ( f 2f f f ) ε k+1 ε k 3f ε 1 ε 2 ε 3 (3.27) f(x) x x 1, x 2, x 3 x 3 x 1 x 2 x 3 = x f (x)f(x) < 1 [12] [f (x)] 2

13 3. 5 x f 2 x f 2 x = a 3 a 0 0 f 2 f(x) = x 2 a = 0 (3.29) x k+1 = x k x2 k a = 1 ) (x k + axk 2x k 2 Fortran C x = ϕ(x) (3.31) x k k x k+1 = ϕ(x k ) (3.32) ϕ(x) ϕ(y) q x y 0 < q < 1 (3.33) a f 1 (x) = x 3 a = 0 f 2 (x) = x 2 a x u = nt + e sin u f 1 : x k+1 = 1 ( 2x k + a ) 3 x 2 k f 2 : x k+1 = x k(1 + 2a/x 3 k ) n t e 2 + a/x 3 k u 3 u t 10 ϕ(u) = nt + e sin u k f 1 f 2 e ϕ(u) (3.33) f 2 (x) = 2(x3 a) x 3 f 1 x k+1 x k q x k x k 1 (e = ) nt = π/6

14 3. 6 k u k b u u + b 0 v v = b 0(u, v) b u u + b 1 v v = b 1(u, v) b 0 / u c k = b k d k = b k 1 (3.35) u v (3.17) u c n+1 = c n+2 = 0 c k = b k + uc k+1 + vc k+2 (3.36) k = n, n 1,, 0 v d n+1 = d n+2 = 0 d k = b k + ud k+1 + vd k+2 n p n (x) x 2 ux v k = n, n 1,, 0 p n (x) = (x 2 ux v)(b n x n 2 + b n 1 x n b 2 ) + b 1 (x u) + b 0 (3.34) b k (3.17) c 1 u + c 2 v = b 1 (3.37) b 1, b 0 0 x 2 ux v p n (x) u v p n (x) u, v 6 b 0 (u, v) = 0 b 1 (u, v) = 0 (u, v) +4x 3 + 3x 2 + 2x + 1 (3.38) (u, v) ( u, v) b 0 b 1 b 0 (u + u, v + v) = 0 b 1 (u + u, v + v) = 0 d k = c k c 0 u + c 1 v = b 0 p 6 (x) = 7x 6 + 6x 5 + 5x 4

15 3. 7 k u v b 0 b e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 8 n 2 p n (z i ) z i = a n j i (z (3.40) i z j ) i = 1, 2,, n (u, v) (3.29) (u, v) z i = p n(z i ) p n(z i ) (3.39) p n(z) z = z i (3.40) (3.40) z DKA n i p n (z) z i (i = 1, 2,, n) z i (3.39) p n (z) = a n (z z 1 z 1 )(z z 2 z 2 ) (z z n z n ) (3.39) p n (z) = z z j j z j p n (z) = a n a n n i=1 n i=1 (z z i ) z i (z z j ) j i p n (z i ) p n(z i ) z = z i (3.18) p n (z i ) 0 (3.40) p n (z i ) = z i a n (z i z j ) j i p n(z) z = z i 1 z i = j i 1 1 p n(z i ) z i z j p n (z i ) (3.41) z i

16 3. 8 P (re iϕ ) = r n e inϕ n 1 0 P (w) = 1 ( p n w a ) +c n 2 r n 2 e (n 2)iϕ + + c 0 = 0 n 1 a n na n e inϕ = w n + c n 2 w n c 0 (3.42) r n + c n 2 r n 2 e 2iϕ + + c 0 e niϕ = 0 c k n 1 r > r 0 S(w) = w n c n 2 w n 2 c n 3 w n 3 c 1 w c 0 (3.43) 0 +( c n 3 + c n 3 e 3iϕ )r n 3 S(w) + + ( c 0 + c 0 e niϕ ) = δ w > 0 S(0) < 0 S( ) > 0 P (w) w > r 0 S(w) w = r 1, r 2 r n 2 S(r 2 ) r n 1 S(r 1 ) = c n 2 (r 2 1 r 2 2 ) + c 0 (r n 1 r n 2 ) = 0 r 1 r 2 S(r) > 0 r > r 0 0 r r S(r) w > 0 1 r 0 r 0 S(r) > 0 r S(r 0 ) = 0 r 0 > 0 (3.44) p n (z) r z j = a n 1 na n S(r) > 0 r > r 0 { [ 2(j 1)π +r exp i + π ]} (3.46) P (w) n 2n j = 1, 2,, n r 0 w r 0 (3.45) S(r) = r n c n 2 r n 2 c 0 δ > 0 ( c n 2 + c n 2 e 2iϕ )r n 2 (3.40) r 0 w = re iϕ r > r 0 (3.40) zi

17 3. 9 δ k (3.40) Durand Kerner (3.46) n Aberth x j M DKA 6 (3.38) (3.46) (3.41) p n (x) = a n (x x j ) j=1 4 x k n M M p n (x) M n M M 3 3 x 1 < x 3 < x 2 f(x) x 1 < x < x 2 n p n (x) a k δ k f(x 3 ) < f(x 1 ) f(x 3 ) < f(x 2 ) x 0 x 4 x 3 (a n + δ n )x n + (a n 1 + δ n 1 )x n 1 + x 4 + (a 0 + δ 0 ) = 0 ( ) x = x 0 + ε ε, δ k f(x) p n (x 0 + ε) + n δ k x k 0 = 0 k=1 p n (x 0 ) + p n(x 0 )ε + k k ε = δ kx k 0 p n(x 0 ) δ k x k 0 = 0 (3.47) x 1 x 3 x 2

18 3. 10 x 3 x 3 x 4 x 3 x 1 x 2 x 1 = u x 4 x 3 x 2 x 1 = v x 1 x 4 x 3 x 2 (x 1, x 3, x 2 ) x : x 4 x 3 u + v = 1 u f(x 4 ) f(x 4 ) > f(x 3 ) x 3 x 4 (x 1, x 3, x 4 ) u = v 1 u 0.62 u 2 3u + 1 = 0 u = 3 5 = (3.48) 2 v = 5 2 = u v x 1 x 3 x 4 x 2 f(x 4 ) < f(x 3 ) (x 3, x 4, x 2 ) x x 3 A B : = 1 : x x 3 v x 4

19 a (3) 33 x 3 = b (3) 3 (f) a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 (a) a (2) 22 x 2 + a (2) 23 x 3 = b (2) 2 (d) a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3 (a) (b) (c) (a) (b) x 1 (a) a 21 /a 11 (b) a (2) 22 x 2 + a (2) 23 x 3 = b (2) 2 (d) l 21 = a 21 a 11 a (2) 22 = a 22 l 21 a 12 a (2) 23 = a 23 l 21 a 13 b (2) 2 = b 2 l 21 b 1 (c) x 1 x 1, x 2 x k 1 (b) (c) (a) (a) a 31 /a 11 a (1) 11 x 1 + a (1) 12 x 2 + a (1) 13 x a (1) 1n x n = b (1) 1 (c) l 31 = a 31 a 11 a (2) 32 = a 32 l 31 a 12 a (2) 33 = a 33 l 31 a 13 b (2) 3 = b 3 l 31 b 1 a (2) 32 x 2 + a (2) 33 x 3 = b (2) 3 (e) x 1 x k (d) (e) x 2 l 32 = a(2) 23 a (2) 22 a (3) 33 = a(2) 33 l(2) 32 a(2) 23 b (3) 3 = b (2) 3 l (2) 32 b(2) 2 a (3) 33 x 3 = b (3) 3 (f) (a) (d) (f) x 3 = b(3) 3 a (3) 33 x 2 = 1 a (2) 22 ( ) b (2) 2 a (2) 23 x 3 x 1 = 1 a 11 ( b 1 a 12 x 2 a 13 x 3 ) n (g) a i1 x 1 + a i2 x a in x n = b i (4.1) i = 1, 2,, n a (2) 22 x 2 + a (2) 23 x a (2) 2n x n = b (2) (4.2) a (k) kk x k + + a (k) kn x n = b (k) k a (k) k+1k x k + + a (k) k+1n x n = b (k) k a (k) nk x k + + a (k) nn x n = b (k) n a (1) ij = a ij b (1) i = b i k k + 1 n l ik = a(k) ik a (k) kk a (k+1) ij b (k+1) i = a (k) ij = b (k) i k < i j n (4.3) l ik a (k) kj (4.4) l ik b (k) k (4.5) 0 k < n

20 4. 2 Fortran do k=1, n-1 do i=k+1, n lik=a(i,k)/a(k,k) do j=k+1, n a(i,j)=a(i,j)-lik*a(k,j) enddo b(i)=b(i)-lik*b(k) enddo enddo (4.4) k (n k) 2 a (1) 11 x 1 + a (1) 12 x 2 + a (1) 13 x a (1) 1n x n = b (1) 1 ( ) a (2) 22 x 2 + a (2) 23 x a (2) 2n x n = b (2) 2 x n = b(n) n a (n) nn x i = 1 a (i) ii a (3) 33 x a (3) 3n x n = b (3) 3 (4.6)..... ( b (i) i n j=i+1 a (n) nn x n = b (n) n ) a (i) ij x j (4.7) i = n 1, n 2,, 1 (4.1) (4.4) a (k) ij 1 3 n3 + O(n 2 ) (4.8) a (k+1) ij a (k) ij a (k) n ij a ij n = 100 n = 1000 (4.5) b (k) i b i (4.3) a (k) ik (4.8) l ik 1/3 2/3 a ij b i (4.5) (4.7) 1 0 n (4.4) (4.5) (4.7) n Fortran (4.4) k = 1 n 1 n (4.4) n 1 (n k) n3 (4.3) kk ( ) k 0 k=1 (4.3) (4.5) (4.7) n (4.3), (4.5), (4.7) 1 2 n2 a (k)

21 x 3 x 3 x 1 x 2 x 1 x 2 x 1 x 2 (4.10) k k a (k) ik k i n (4.10) i i k x i 0 LU a (k) k ij n n A (k) A (k) A (k+1) a (k) ij k i, j n A (k+1) = M k A (k) i, j i k j k M k n n k = 1 M l M 1 = det[a ij ] = a (1) 11 a(2) 22 a(k) kk... a(n) nn (4.9) l n1 0 1 M k (k, k) [ l k+1,k, l k+2,k,, l n,k ] T ( T ) n 1 0 A U n = 3 x 1 a (2) 22 = a(2) 32 = 0 M n 1 M 2 M 1 A = U (4.11) U (4.6) a (2) 23 x 3 = b (2) 2 a (2) 33 x 3 = b (2) 3 b (2) 2 a (2) 23 = b(2) 3 a (2) 33 (4.10) L = M 1 1 M 1 2 M 1 n 1 (4.12) (4.11) A A = LU (4.13)

22 4. 4 A LU L x 2 x 3 M 1 k M k l i,k l L = l 31 l l n1 l n2 l n3 1 (4.14) L A A (4.13) LU l ij n (4.3) Fortran A det A = det L det U det L = 1 A (4.9) a(i,j) det=1 do k=1, n akk=a(k,k) det=det*akk a(k,k)=1 n = 3 do j=1, n+1 a 11 a(k,j)=a(k,j)/akk x 1 + a 12 x 2 + a 13 x 3 = b enddo 1 do i=1, n if( i.ne.k ) then aik=a(i,k) a 21 a 31 a(i,k)=0 x 1 do j=1, n+1 a 22 a(i,j)=a(i,j)-aik*a(k,j) enddo x 1 + a 12 x 2 + a 13 x 3 = b 1 x 2 + a 23 x 3 = b 2 a 32 x 2 + a 33 x 3 = b 3 a 12 x 1 = b 1 x 2 = b 2 x 3 = b 3 n + 1 b i a(i,n+1) a 32 endif enddo enddo x 2

23 4. 5 LU (4.6) 0 A A = L U u 11 u 12 u 13 u 1n 0 u 22 u 23 u 2n U = 0 0 u 33 u 3n u nn (4.15) L (4.14) L U L U = A (1,1) γ i x i 1 + α i x i + β i x i+1 = b i (4.18) i = 1, 2, 3,, n γ 1 = β n = 0 α i, β i, γ i a ij a i x i + β i x i+1 = c i i = 1, 2,, n (4.19) u 11 = a 11 a i, c i a 1 = α 1 c 1 = β 1 l i1 = a a i = α i γ i β i 1 (4.20) i1 a i 1 i = 2, 3,, n u 11 c i = b i γ i c i 1 a L U i 1 i = 2, 3,, n (1,2) (2,2) u 12 = a 12 u 22 = a 22 l 21 u 12 l 21 x i = 1 (c i β i x i+1 ) (4.21) u 12 u 22 a i i = n, n 1,, 1 l i2 = 1 ( ) a i2 l i1 u 12 i = 3, 4,, n u 22 k k + 1 j = 1, 2,, n i 1 l ij = a ij l ik u kj i = 1, 2,, j (4.16) k=1 l ij = 1 ( a ij u jj j 1 ) l ik u kj k=1 i = j+1, j+2,, n (4.17) a i, c i (4.12) x i α i > β i + γ i (4.22) LU (4.1) n n A = [a ij ] b i b = [b 1, b 2,, b n ] T ( T ) x = [x 1, x 2,, x n ] T (4.1) ( ) Ax = b (4.23)

24 4. 6 (4.31) (4.29) A 1 x = A 1 b (4.24) x = 0 (4.32) x x x 0 x = 0 x = 0 (4.25) αx = α x x + y x + y A λ i α ρ(a) = max λ i (4.34) L 2 ( n x 2 = x i 2) 1/2 i=1 x 1 = L 2 (4.26) n x i L 1 (4.27) i=1 x = max 1 i n x i L (4.28) A b Ax A = sup x 0 x (4.29) x Ax = A x b (4.29) L 1 L A 1 = max a ij j A = max i A(x + δx) = b + δb δx = A 1 δb (4.25) i a ij (4.33) j L 2 L 1 L L 2 i A A L 2 A 2 = ρ(a) (4.35) Ax = b b δb x + δx AB A B (4.30) Ax A x (4.31) b = Ax A x δx = A 1 δb A 1 δb

25 4. 7 δx x κ(a) δb b (4.36) κ(a) = A A 1 (4.37) κ(a) κ(a) A κ (4.1) A 1 1 δa x + δx (A + δa)(x + δx) = b δx = A 1 δa(x + δx) U T z = e L T v = z (4.40) δx x + δx κ(a) δa A (4.38) e v δa / A A L 2 e 1 A 1 A 1 (4.35) κ 2 (A) = max i λ i min i λ i (4.39) (3.29) A 1 1 = A T = sup A T e e A 1 A T A T e = v e = max e i = 1 (4.40) A A z 1 = e 1 = 1 u 11 u 11 ( ) z k = 1 k 1 e k u ik z i (4.41) L 2 k = 2, 3,, n A A 1 e k = ±1 z k L 1 A 1 (4.33) sign(x) A 1 ( ) 1 k 1 A 1 e k = sign u ik v i z (4.40) (4.33) A A 1 = A T v n = z n i v = max i v i A LU v U T z z v e T = [1, ±1, ±1, ] 1 u kk i=1 i=1

26 4. 8 v k = z k n i=k+1 l ik z i (4.42) k = n 1, n 2,, 1 v A 1 1. = max v i (4.43) i κ 1 (A) = A 1 A 1 1 Ax = b x δx x + δx v A(x + δx) = b A 1 1 δx A = Aδx = b Ax = r (4.44) x r x + δx L 1 (4.43) n n 3 /3 κ x LU (4.43) L 2 n 2 κ 2 = 103 A n < 10 n A

27 x 0, x 1,, x n f[x] = f(x) f(x) f n (x) f[x 0, x 1,, x k, x] = f(x) f[x 0, x 1,, x k 1, x] f[x 0, x 1,, x k 1, x k ] f n (x k ) = f(x k ) k = 0, 1,, n (5.1) x x k (5.3) f n (x) x a k = f[x 0, x 1,, x k ] 3 Π n (x) = (x x 0 )(x x 1 ) (x x n ) n = (x x i ) i=0 n Π n (x) f n (x) = (x x k )Π k=0 n(x k ) f(x k) (5.2) (5.1) (5.3) (5.4) 1 f(x 0 ) x 0 x (5.2) I 0i (x) = x i x 0 f(x i ) x i x (5.2) i = 1, 2,, n f n (x) x n 1 I 01 (x) x 1 x I 01i (x) = f n (x) = a 0 + a 1 (x x 0 ) x i x 1 I 0i (x) x i x +a 2 (x x 0 )(x x 1 ) + i = 2, 3,, n f n [x 0, x] f n(x) f n (x 0 ) x x 0 f n [x 0, x] = a 1 + a 2 (x x 1 ) + f n [x 0, x 1 ] = a 1 +a n (x x 1 ) (x x n ) f n (x) = f[x 0 ] + f[x 0, x 1 ](x x 0 ) +f[x 0, x 1, x 2 ](x x 0 )(x x 1 ) + +f[x 0, x 1,, x n ](x x 0 )(x x 1 ) (x x n ) +a n (x x 0 )(x x 1 ) (x x n ) 1 I 012 k 1i (x) = x i x k 1 I 012 k 1 (x) x k 1 x f n (x 0 ) = a 0 I 012 k 2i (x) x i x i = k, k+1,, n (5.4) (5.5) x(i) x i y(i) f(x i ) x = x c y c real do x(0:n), y(0:n), f(0:n) i=0, n f(i)=y(i) enddo do k=0, n-1

28 5. 2 do i=k+1, n f(i)=f(k)+(f(k)-f(i)) * *((x(k)-xc)/(x(i)-x(k))) enddo enddo yc=f(n) f(i) (5.2) + x j x f j 1 + x x j 1 f j 1 j n (5.6) (5.4) (5.5) h j h j x n x j 1 < x < x j h j = x j x j 1 f j = f(x j ) x 0, x 1,, x n S j (x j ) = f j S j (x j 1 ) = f j 1 [ 1 f(x) = S j(x) = h ( ) 2 j x 2 6 M xj x j ] h j [ x = 1.0, 0.8,, 0.8, h ( ) 2 j x 6 M xj 1 j 3 1] + f j f j 1 S x j x x x j 1 j (x) = M j 1 + M j h j h j 10 M j x j 1.0 x j 1 x x j [ (xj ) 3 ( ) ] S j (x) = h2 j 6 M x xj x j 1 h j h j [ (x ) 3 ( ) ] + h2 j 6 M xj 1 x xj 1 j h j h j x j S j(x j ) = S j+1(x j ) h j h j h j 6 M j 1 + h j + h j+1 M j + h j M j+1 = f j+1 f j f j f j 1 (5.7) h j+1 h j M 0 = M n = 0 j = 1, 2,, n 1 M 1 M n 1 4 x 0 < x 1 < (5.7) < x n

29 5. 3 a i n a i (5.7) M j S n = f 2 (5.6) f(x) [a, b] ϕ i (x) u(x) v(x) L 2 (5.11) ϕ i (x) (u, v) = b a u(x)v(x)dx (5.8) u 2 a 2 = (u, u) (5.9) i n u 2 (4.25) n = 10 n = 11 L 2 (5.11) 2 { 1 i = j (ϕ i, ϕ j ) = δ ij = (5.12) 0 i j ϕ i (x) f(x) [a, b] δ ij ϕ i (x) n f(x) n f n (x) = a i ϕ i (x) (5.10) i=0 a i = (f, ϕ i ) (5.13) b n S n = f(x) a i ϕ i (x) 2 dx n a i=0 n S n = f 2 a 2 i a k i=0 b n [f(x) a i ϕ i (x)]ϕ k (x)dx = 0 n S n ( a i=1 ) n a i n a i (f, ϕ i ) = f 2 f n 2 lim f f n = 0 (5.14) n n a i (ϕ i, ϕ k ) = (f, ϕ k ) (5.11) i=0 f(x) (f, ϕ i )ϕ i (x) (5.15) k = 0, 1,, n i=0 x i (5.11) (5.10) (5.11) i = k i=0

30 5. 4 f(x) (f, ϕ i ) (5.15) x f(x) n = 5 1 ϕ i cos kx, sin kx [ π, π] π f(x) 0 π f(x) 1 2 a 0 + a k = 1 π b k = 1 π (a k cos kx + b k sin kx) (5.16) k=1 π π π π f(x) cos kxdx f(x) sin kxdx 1/π cos kx, sin kx f(x) x π f n (x) b k f n (x + π) = f n (x) f n (x) = n [ 1 π ] f(y) sin kydy sin kx π k=1 0 f n (x) x f(x) = π n [cos k(x y) 2π 0 k=1 cos k(x + y)]dy 1 0 < x π x f(x) = 1/2 x = 0 (5.17) x y 0 π x < 0 df n (x) = 1 π n d [cos k(x y) dx 2π 0 dy k=1 + cos k(x + y)]dy 2 a 0 = 1 b 2k 1 = (2k 1)π df n (x) = 1 n [cos kx(1 cos kπ)] dx π a 0 a k b k 0 k=1 df 2n 1 (x)/dx f 2n 1 (x) = π n k=1 sin(2k 1)x 2k 1 n = 10 sin 2nx n = 5, 10 = π sin x π df 2n 1 (x) dx = 2 π 1 0 n cos(2k 1)x k=1 π (5.18)

31 5. 5 df 2n 1 (x)/dx 0 x = mπ 2n m = ±1, ±2,, 2n 1 f 2n 1 (x) x = 0 n sin x x = π/2n Sinc(x) =. 1 x2 n 2 3! + x4 5! f 2n 1 (x) (5.18) x 6 /7! f 2n 1 (π/2n) = π = nπ π 0 π/2n sin 2ny 0 sin y dy sin y sin(y/2n) dy n 1 π sin y 2nπ sin(y/2n) dy 1 π sin y π y dy 0 0 (Lanczos, Si(x) = x 0 sin y y dy x = π n 1 Sinc(x) = x 2 + x4 (5.21) 120 (5.20) (5.21) x 2 Sinc(x) = sin x x (5.19) x = 0 0/0 sin x x = 0 sin x sin x/x x = 0 sin x (5.20) x = 0 (5.20) x = π/2n x x 2 < 1/2 1 1 = ! 23 0 ) x 2 1/2 x 2 = 1 2 sin θ Si(π) = f 2n 1 (π/2n) x 6 7! = 1 7!2 3 sin3 θ = 1 ( 3 7!2 3 4 sin θ 1 ) sin 3θ 4 f 2n 1 (π/2n) n sin θ sin 3θ !2 3 4 sin θ = 1 3 7!2 3 2 x2 x (5.20)

32 e-5 2.0e-5 1.0e c k (5.24) x cos nθ cos nθ x n T n (x) = cos(n cos 1 x) (5.26) T 3 (x) = 4x 3 3x T 4 (x) = 8x 4 8x sin nθ x n T n+1 (x) 2xT n (x) + T n 1 (x) = 0 (5.27) x 1 cos f(x) (5.24) x x = cos θ (5.22) cos f(cos θ) θ (5.16) (5.23) c k cos kθ f(cos θ) 1 x 1 f(x) 2 c 0 + c k cos kθ (5.23) k=1 f(x) = a n x n π c k = 1 f(cos θ) cos kθdθ π π a n x cos kθ 1 c k n x n = 1 [n/2] ( ) n 2 n 1 T n 2k (x) (5.28) k k=0 f n (x) = 1 n 2 c [n/2] n/2 0 + c k cos kθ (5.24) n T 0 (x) k=1 1/2 ( n + 1 p.89 ) c n+1 (5.24) f(x) = c k T k (x) x 8 c k c k V n = V n+1 = 0 V k 1 = c k + 2xV k V k 1 (5.25) k = n, n 1,, 1 f n (x) = 1 2 c 0 + xv 0 V 1 x n T 0 (x) = 1 T 1 (x) = x T 2 (x) = 2x 2 1 n=0 k=0 (5.27)

33 5. 7 Sinc Sinc (5.19) θ k n n cos mθ k = sin mθ k = 0 m 0 (5.30) (5.19) x π θ (5.20) k x T n (x) 1 x 1 n Sinc(π x) = a n x n n=0 a n = ( 1)n π 2n (2n + 1)! a n c n n 0 i j T i (x k )T j (x k ) = n/2 i = j 0 (5.32) n a n c n k=1 n i = j = e e+0 i, j < n e e+0 T i (x) T j (x) x k e e e e 2 1 x 1 f(x) e e e d 4 c k = 2 n f(x j )T k (x j ) (5.33) e e 6 n e e e e e e 11 f n (x) f n (x k ) = f(x k ) k=1 Sinc (5.33) (5.34) (5.32) ( ) [a, b] f(x) n Sinc p n (x) = a 0 + a 1 x + + a n x n c k cos nθ 0 θ π ε = max p n(x) f(x) (5.35) a x b n θ k = (k 1/2) π (a 0, a 1,, a n ) k = 1, 2,, n (5.29) n k=1 T n (x k ) = 0 x k = cos θ k (5.31) (5.30) j=1 f n (x) = 1 n 1 2 c 0 + c k T k (x) (5.34) k=1

34 5. 8 tan x/x n + 2 ( ) x = π y 1 y 1 4 y 4 tan x/x (5.21) ( y tan tanh T 5 (y) ) 4 (5.36) y = 1 n T n+1 (x) T n+1 (x) x 1.0e-4 x 1 f(x) n T n+1 (x) y T n+1 (x) (5.29) n n + 1 n e-4 a i x j a 0 + a 1 x j + a 2 x 2 j + + a n x n = f(x j ) (5.36) tan x/x j = 1, 2,, n + 1 tan x = x 3 x x x6 n + 1 a i x8 + (5.38) x 8 e(x) = p n (x) f(x) y 0.3 e(x) x k y 4 e(x k ) x k n + 2 e(x k ) e(x k ) (5.8) ε w(x) 0 p n (x k ) = f(x k ) ± ε (5.37) k = 1, 2,, n + 2 p k (x) a i ± e(x k ) (p j, p k ) = w(x)p j (x)p k (x)dx = λ k δ jk a (5.39) p k (x) x k µ k 0 µ 0 = 1 b

35 5. 9 [a, b] w(x) λ n α k β k γ k P n (x) [ 1, 1] 1 2/(2n + 1) 2 1/k 0 1 1/k T n (x) [ 1, 1] 1/ 1 x 2 π/ L n (x) [0, ) exp( x) 1 1/k 2 1/k 1 1/k H n (x) (, ) exp( x 2 ) π2 n n! 1 0 k 1 p k (x) (5.41) x = x j y x j y = x l p 0 (x) = µ 0 p 1 (x) = 0 (5.32) p k (x) = µ k x k + p k (x) = (α k x β k )p k 1 (x) γ k p k 2 (x) α k = µ k β k = α k (xp k 1, p k 1 ) (5.40) µ k 1 λ k 1 γ k = α c k = 1 n w j p k (x j )f(x j ) kλ k 1 λ k j=1 α k 1 λ k 2 (5.43) α k β k n 1 p k (x) p k 1 (x) f n (x) = c k p k (x) (5.44) k=0 p k 2 (x) n 1 p k (x)p k (y) = µ n 1 λ k µ n λ n 1 (5.43) k=0 p n(x)p n 1 (y) p n 1 (x)p n (y) [ ] (5.41) n n 1 p k (x j )p k (x) x y f n (x) = w j f(x j ) x = x l (5.42) x j = l f n (x l ) = f(x l ) x n (5.44) p n (x), p n 1 (x), p k (x) (5.40) (5.41) n p n (x) f n (x) = p k (x) p n (x) (x x j=1 j )p n(x j ) f(x j) (5.45) x 1, x 2,, x n (5.41) n 1 1 [p k (x j )] 2 w j λ k k=0 n 1 k=0 = µ n 1 µ n λ n 1 p n 1 (x j )p n(x j ) (5.42) p k (x j )p k (x l ) λ k = 0 j l j=1 k=0 λ k (5.2) x j p n (x) ( )

36 5. 10 f(x) f(x) = b 0 + b 1 + a 1 a 2 b 2 + a 3 b (5.46) N+1 = b N+1 a i, b i x n = N, N 1,, 0 0 f(x) f(x) = b 0 + a 1 a 2 a 3 a 4 b 1 + b 2 + b 3 + b 4 + (5.47) N (5.47) tan x = x x 2 x e x = 1 + x x x x x 2n + 1 log(1 + x) = x x x 2x 2x nx nx 2+ 2n + 1+ tan 1 x = x x 2 4x n 2 x 2 2n + 1+ x 2+ f n (5.47) a n, b n j = 1, 2 Z ν 1 (x) + Z ν+1 (x) = 2ν x Z ν(x) f j f j 1 Z ν 1 (x) Z ν (x) Z ν+1 (x) Z ν (x) = 2ν x 1 Z ν (x)/z ν+1 (x) = 2ν x 1 1 2(ν + 1)/x 2(ν + 2)/x 1 2(ν + 3)/x = x 2(ν + 1) x 2 2(ν + 2) n+1 n = b n + a n+1 n+1 (5.48) N A 1 = 1 B 1 = 0 A 0 = b 0 B 0 = 1 A j = b j A j 1 + a j A j 2 (5.49) B j = b j B j 1 + a j B j 2 j = 1, 2, f n = A n B n (5.49) A j, B j C j = x 2 2(ν + 3) f j = f j 1 C j D j C j D j A j A j 1 D j = B j 1 B j (5.47) f 0 = C 0 = b 0 D 0 = 0 n C j = b j + a j (5.50) C j 1 n = b n + a n+1 1 D j = j = 1, 2, b n+1 + b j + a j D j 1

37 5. 11 C j D j 1 f(x) m n C j D j 0 m f(x) A a i x i m(x) B n (x) = i=0 (5.51) n 1 + b j x j j=1 (5.49) a i b j (5.50) a i b j a i b j b 0 = 1 (5.50) m+n+1 (5.49) x i a 0 + a 1 x i + a 2 x 2 i + + a m x m i tan x 3 tan x x. = 1 1 x 2 3 x 2 5 = 15 x2 15 6x 2 a 0 a m, b 1 b n x i tan x/x x tan x/x m + n e e e e e e e e e e e e e e-4 e(x) = A m(x) B n (x) f(x) e e e e-3 m + n e e-2 x j tan x (5.38) x > 0.5 A m (x j ) = [f(x j ) ± ε] B n (x j ) (5.53) j = 1, 2,, m + n ε ± e(x) m x j ε n a i b j m + n + 2 e(x) = f(x i )(1 + b 1 x i + b 2 x 2 i + + b n x n i ) i = 1, 2,, m + n + 1 (5.52) x i

38 5. 12 (5.53) m+n+1 a i b j m + n + 2 (5.50) ε a i (5.52) tan x/x m = n = 2 A 2 (x) = x e-3x 4 B 2 (x) = x e-2x 4 A m (x), B n (x) f(x) 0 x x 8 f(x) = c k x k x = k=0 (5.51) tan x B n (x) c k x k A m (x) k=0 x = 2 x i a i, b j 10 4 n i b j c k x j+k = b j c i j x i tan x j=0 k=0 i=0 j=0 x m A x x4 B i x x4 b j c i j = a i i = 0, 1,, m (5.54) j=0 A m (x) B n (x) x m A m (x) B n (x) 0 A m (x)/b n (x) i c k b j c i j = 0 (5.55) j=0 c k (5.55) b 1 b n π/2 8 x > 1 tan x x = π/2 (5.49) j = 5 i = m + 1, m + 2,, m + n i = m + n b j b 0 = 1 b 1 b n n n i b j e b n min(i, n) x 1 + x/2 + x2 /10 + x 3 /120 1 x/2 + x 2 /10 x 3 /120

39 x + 5x 2 cos x x + x x + 14x 2 2T j (x)t k (x) = T j+k (x) + T j k (x) x x + 5x 2 tan 1 x x x 4 x x x 4 x 3 x x = 0 T j (x) c k T k (x) x ε 0.8 < x < 1.2 ε = < x < 2.0 ε = x 1 n b j c (j) k = 0 k = m+1, m+2,, m+n f(x) = c k T k (x) j=0 k=0 n b j c (j) k = a k k = 0, 1, 2,, m (5.56) (5.51) A m (x), B n (x) b j m n A m (x) = a i T i (x) B n (x) = b j T j (x) a i i=0 j=0 B n (x)f(x) 1963 : k=0 = 1 2 = k=0 c (j) k c k [T j+k (x) + T j k (x)] k=0 c (j) k T k(x) c (0) k = c k n B n (x)f(x) = b j c (j) k T k(x) j=0 j=0 k=0 m a i T i (x) i=0

40 A = QR Q = [q 1, q 2,, q m ] (6.3) 4 R = [r ij ] A LU QR Q n m R m m r 11 r 12 r 1m r 22 r 2m QR R = r mm q k Q n m (n m) A n a j A = [a 1, a 2,, a m ] (6.1) A n m QQ T a j (6.4) q 1 = a 1 r 11 = a 1 r 11 L 2 a 2 q 1 q 1 a 2 q j a 2 q 1 q k u 2 = a 2 q 1 r 12 r 12 = q T 1 a 2 a j T u 2 q 1 u 2 1 q 2 = u 2 r 22 j=1 r 22 = u 2 q k = u k r kk r kk = u k (6.2) u k a k r k 1,j = q T k 1a (k 1) j j = k, k+1,, m (6.5) q 1, q 2,, q k 1 q k = a(k) k r kk = a (k) k j=1 j=1 Q T Q = I m (6.4) I m m m A = [a (1) 1, a(1) 2,, a(1) m ] q 1 q 1 q 2 a (1) 2, a(1) 3,, a(1) m k q k q 1 a (2) 2 k 1 u k = a k q j r jk r jk = q T q k 1 j a k q k a (k) j = a (k 1) j r kk q k 1 r k 1,j u k q m k 1 k a k = q k r kk + q j r jk = q j r Q R (6.3) jk (6.4)

41 6. 2 (6.5) r kk a w a (k) P a = a w(w j (j k) T a) = [s, 0,, 0] T (6.9) s 2 = a 2 a (k) j a (k) k P a 0 a T Q A R a 2 (w T a) 2 = sa 1 Q a n = m Ax = b 1 a w T a Q T (6.9) w w Q T QRx = Q T b Rx = Q T b w T a = s(s a 1 ) s = a sign ( a 1 ) r jk w = 1 w T a [a 1 s, a 2,, a n ] T (6.10) A b a (1) n+1 (6.5) s s a 1 n (n + 1) R r i,n+1 n P = I + uut (6.11) n s(a 1 s) a (n+1) n+1 0 u T = [a 1 s, a 2,, a n ] b P P b = b + βu β = ut b s(a 1 s) QR 2 n m (n m) A w a A a 1 P P = I ww T w 2 = 2 (6.6) P P T = P P T P = I (6.7) P 1 P 1 A P 1 A n 1 a P 2 P 1 A I P 2 a (1, 1) 1 0 P k (k, k) P k P a 2 = a T P T P a = a 2 (6.8) I k 1 0 P (6.6) P k = 0 I n k+1 ww T QR

42 6. 3 I k k k A = QL P k Q Q 0 A m(m n) [ ] R P m P m 1 P 1 A = 0 (6.12) R m m 0 a m 1 q m (n m) m m = n u m 1 = a m 1 l m,m 1 q m P m l m,m 1 = q T ma m 1 q m 1 = u m 1 l m 1,m 1 = u m 1 Q = P 1 P 2 P m (6.13) l m 1,l 1 Q (6.3) Q n n Q (6.12) m [ ] u k = a k q j l jk l jk = q T j a k (6.14) Q T R j=k+1 A = 0 q k = u k l kk = u k QQ T = I [ ] (6.14) A = QL R A = Q 0 n > m Q m QL (6.3) (6.12) m 1 R Q n 1 0 R A 0 (6.12) [ ] 0 Ax = b (n = m) A P k P m P m 1 P 1 A = L b P k P n 1 P n 2 P 1 Ax = Rx = P n 1 P n 2 P 1 b Q n n QL A Q R A = QR Q l 11 0 l 21 l 22 L = l m1 l m2 l mm q m = a m l mm r kk l mm = a m A Ax = λx (6.15) λ A x λ det(λi A) = 0

43 6. 4 P a T 1 B = P 1 AP (6.16) A B a 1 2 (w T a 1 ) 2 = a 2 (6.15) P sa 21 a 1 2 = a s 2 P 1 AP P 1 x = λp 1 x By = λy x = P y y = P 1 x w 0 w = [0,,,, ] T a 11 a 21 a 1 s = sign( a 21 ) n B w T a 1 = s(s a 21 ) (6.17) P A B w A B y w = 1 w T [0, a 21 s, a 31,, a n1 ] T (6.18) a 1 B = P 1 AP 1 B P (1, 1) 1 b 11 = a 11 b 12 = b 21 = s 0 P 0 A 0 B = P 1 AP = P AP A n n n A B = 0. B = P T AP α 1 β β 1 α 2 β 2 0 A a 1. 0 β 2 α 2 β.. 3 = P P 1 a αn 1 β n 1 P 1 a 1 = a 1 w(w T a 1 ) = [a 11, s, 0,, 0] T 0 0 β n 2 α n i=2 a 2 i1 w s a 21 P 1 (6.11) P 1 = I + uut s(a 21 s) P = I ww T w 2 = 2 u T = [0, a 21 s, a 31,, a n1 ] QR P 1

44 6. 5 A P = P 1 P 2 P n 2 (6.19) P T AQ = (6.21) P 1 A P = P 1 P 2 P n Q = Q P Q 2 Q n (6.19) B = 0 (6.20) A P QR..... c s P = A 1 0 s c. P 1 (6.10) B = P 1 A (6.22) B b T 1 Q 1 = [b 11, s, 0,, 0] b T 1 Q 1 (6.17) (6.18) P T P = I B P 1 AQ 1 = p kk = p ll = c = cos ϕ p kl = p lk = s = sin ϕ B = P 1 AP = P T AP B = P T A B = B P

45 6. 6 B k, l B k, l ϕ B B cos ϕ sin ϕ { b kj = a kj cos ϕ a lj sin ϕ { b lj = a kj sin ϕ + a lj cos ϕ b ik = b ik cos ϕ b il sin ϕ b il = b ik sin ϕ + b il cos ϕ 1 j n 1 i n (6.23) b ik b il i = k, l A 1 B (k, l) c = s = ct 1 + t 2 b kl = b lk = (a kk a ll ) sin ϕ cos ϕ +a kl (cos 2 ϕ sin 2 ϕ) (6.24) a kl 0 0 ϕ (6.25) (k, l) (l, k) B 0 τ = a ll a kk A D ± t 0 s 0 0 τ > 0 A B = P T 1 AP t = τ + τ > 0 τ b 2 ij = τ < 0 a 2 ij sign( τ) i,j i,j t = τ ( /τ 2 ) (6.23) sign b 2 kk + 2b 2 kl + b 2 ll = a 2 kk + 2a 2 kl + a 2 ll b kl 0 ϕ b 2 kk + b 2 ll > a 2 kk + a 2 ll a kl 0 b ll a kk ϕ (6.24) b kl = 0 2a kl tan 2ϕ = a ll a kk b kl = 0 t 2 + a ll a kk a kl t 1 = 0 t = tan ϕ t = a kk a ll 2a kl ± (akk ) 2 a ll + 1 (6.25) 2a kl P (6.23) (6.25) 2a kl (6.23) b kk = c 2 a kk + s 2 a ll 2csa kl c s a ll a kk = c2 s 2 cs a kl a ll b kk = a kk ta kl b ll = a ll + ta kl (6.26)

46 6. 7 (6.23) D λ j b kj = a kj s(a lj + γa kj ) P j p j b lj = a lj + s(a kj γa lj ) (6.27) γ = s j k, l Ap j = p j λ j 1 + c 0 a kl (k, l) n 2 /2 (k, l) = (6.19) (1, 2), (1, 3) (1, n), (2, 3), (2, 4) p n (λ) det(λi B) 0 λ n A k k (6.23) P 1, P 2, p k (λ) A p 1 (λ) = λ α 1 P = P 1 P 2 λ α 1 β 1 p 2 (λ) = β 1 λ α 2 P T AP = D AP = P D p k (λ) = λ α 1 β 1 0 β 1 λ α 2 β β 2 λ α 3 β λ αk 1 β k β k 1 λ α k (6.28) p k (λ) = (λ α k )p k 1 (λ) β 2 k 1p k 2 (λ) (6.29) k = 2 n n A p 0 (λ) = 1 p k(λ) = (λ α k )p k 1(λ) β 2 k 1p k 2(λ) +p k 1 (λ) (6.30) n p n (λ) p k (λ)

47 p n (λ) (6.29) (6.30) p n (λ) n n A λ 1 > λ 2 > > λ n λ p n (λ), p n 1 (λ),, p 0 (λ) u k N(λ) N(λ) λ Au k = λ k u k (6.34) a < λ < b p n (a) 0 p n (b) (a, b) p n (λ) x 0 = c 1 u 1 + c 2 u c n u n N(a) N(b) (6.31) r < λ < r lim ( ) (a, b) (6.33) c 1 0 x 0 u 1 x j A( ) λ 2 u 2 x 0 u 1 x j = Ax j 1 (6.32) j x j A λ 1 x 0 λ 1 lim j x T j x j x T j x = λ 1 (6.33) j 1 c 1 0 (6.32) x j = A j x 0 = c 1 λ j 1 u 1 + c 2 λ j 2 u c n λ j nu n [ = c 1 λ j 1 u 1 + c ( ) j 2 λ2 u 2 + c 1 λ 1 (6.19) + c ( ) ] j n λn u n c 1 λ 1 r = max ( β i 1 + α i + β i ) 1 i n j x j x j = u 1 u 1

48 6. 9 A λ k µ µi A B = (µi A) 1 (6.35) λ k A u k B 1 u k = (µi A)u k = (µ λ k )u k 1 Bu k = u k µ λ k B (µ λ k ) 1 µ λ k λ i µ λ k < µ λ i i k B A QR (µ λ k ) 1 x 0 B A 1 = A = Q 1 R 1 A 2 = R 1 Q 1 lim j lim j x j x j = u k u k x T j x j x T j x = j 1 1 µ λ k (6.36) (µi A) 1 (µi A)x j = x j 1 (6.37) ( ) 2 2 x j 1 x j (6.36) j RQ A Q A (6.37) λ k. = µ x T j x j 1 x T j x j λ k µ QR (6.37) µ QR (6.37) LU QR µ µi A A λ v (6.37) A vi λ v A vi QR λ v λ v x j 1 = 1 µ λ k QR A QR A Q Q 1 AQ = Q 1 (QR)Q = RQ A RQ A 2 QR A k = Q k R k A k+1 = R k Q k (6.38) A k A A

49 6. 10 (6.40) A k v k I = Q k R k B [q 1, q 2, ] A k+1 = R k Q k + v k I (6.39) h 11 h 12 A k QR A k v k I QR = [q 1, q 2, ] h 21 h 22 0 h 32 (6.41) R k Q k = Q T k A k Q k v k I A k+1 = Q T k A k Q k Bq 1 = q 1 h 11 + q 2 h 21 A k+1 A k k A k q j QR v k q T 1 Bq 1 = h 11 v k A k q 1 (n, n) h 11 h 11 v k q 2 h 21 = Bq 1 q 1 h 11 n (n, n) λ n λ n n n h 2 21 = Bq (n 1) (n 1) 1 q 1 h 11 2 h 21 = Bq 1 q 1 h 11 v k h 21 q 2 = 1 (Bq h 1 q 1 h 11 ) 21 q 2 (6.41) B Bq 2 = q 1 h 12 + q 2 h 22 + q 3 h 32 BQ = QH (6.40) Q H h 12 = q T 1 Bq 2 h 22 = q T 2 Bq 2 Q Q H H h 12, h 22 Q H h 32 = Bq 2 q 1 h 12 q 2 h 22 Q q 3 = 1 (Bq h 2 q 1 h 12 q 2 h 22 ) 32 H Q h 32 q 3 Q q j H Q = [q 1, q 2,, q n ] q 1, q 2

50 6. 11 QR (6.9) (6.10) P k Q = P n 1 P n 2 P 1 A A Q QR P 1 A k v k I = Q T k R k A k+1 = R k Q k + v k I (6.42) A k+1 v k+1 I = Q T k+1r k+1 A k+2 = R k+1 Q k+1 + v k+1 I (6.43) QR Q T m 11 = p 1 = a 2 11 (v k + v k+1 )a 11 + a 12 a 21 R +v k v k+1 (6.42) (6.43) m 21 = q 1 = a 21 (a 11 + a 22 v k v k+1 ) (6.45) P 1 P 1 M 0 A k M A k a ij m 31 = r 1 = a 21 a 32 A k+1 = Q k A k Q T k A k+2 = Q k+1 A k+1 Q T k+1 A k+1 (6.43) s = sign( p 1 ) p q2 1 + r2 1 (6.46) A k v k+1 I = Q T k Q T k+1r k+1 R k (6.42) w 0 M (A k v k+1 I)(A k v k I) P 1 = Q T R (6.44) Q = Q k+1 Q k R = R k+1 R k Q A k+2 (6.10) w = 1 s(s p1 ) [p 1 s, q 1, r 1, 0,, 0] T A k+1 P 1 A k+1 A k P 1 A k A k+2 = QA k Q T A k+2 A k P 1 A k = 0 0 A k Q T = Q T A k P 1 Q T 0 Q (6.44) P 1 M 0 Q M P 1

51 6. 12 p 2 q 2 P 1 A k P 1 = r q 2, r 2 0 P n l a l,l 1 P 2 = 0 I n 1 ww T l = n a nn n n ww T (6.46) p 1, q 1, r 1 p 2, q 2, r 2 0 l n n 1 A k+2 (6.45) Q = P n 1 P n 2 P 1 Q a 11 a ll a 12 a l,l+1 a 32 a l+2,l+1 QR P k v k, v k+1 A k 2 2 a l,l 1 0 l λ a n 1,n 1 a n 1,n a n,n 1 λ a n,n = 0 (6.45) QR v k + v k+1 = a n 1,n 1 + a n,n v k v k+1 = a n 1,n 1 a n,n a n 1,n a n,n 1 A A k a n,n 1 a n,n A k (n 1) (n 1) n m (n m) A a n 1,n 2 A k l a l,l 1 0 l n l l 1 n l + 1 l = n (1,1) (l, l) 2 2 A = UΛV T (6.47)

52 6. 13 U n m (6.47) U V V m m Λ m m U V U T U = I m V T V = I m B S T V V V T = I m U UU T U (6.47) U V (6.47) A Λ A U = P S V = QT 0 0 p A ( ) n m B p m p = m p < m A κ(a) = max(λ i) min(λ i ) (6.48) U T AV = Λ S T BT = Λ T 1 = 0 B A 0 0 r 3 0 B = BT = n m(n m) A (6.9) P k 0 (6.17) 0 (2,1) 0 B Q k T 1 B(6.21) ϕ 1 θ 1 S 1 B = P T AQ (6.49) 0 P = P 1 P 2 P m B = S T B = 0 Q = Q 1 Q 2 Q m 1 m = 4 B w 1 r B (2,1) 0 θ 1 0 w 2 r 3 0 (1,3) w 3 r 3 B = (6.50) T w 4... T 2 ϕ (1,3) 0 (3,2) 0 0 S T 2 0 cos ϕ 1 sin ϕ 1 0 sin ϕ 1 cos ϕ I

53 6. 14 l w l 1 0 r l 0 B T k S k I k 1 c s s c 7 I m k 1 l l = m S T m 1S T m 2 S T 1 BT 1 T 2 T m 1 T 1 ϕ A = T [cos ϕ 1, sin ϕ 1, 0,, 0] T B T B si e+0 s e e 3 [w1 2 s, w 1 r 2, 0,, 0] T e 5 s B T B 2 2 [ w 2 m 1 + r 2 m 1 w m 1 r m ] w m 1 r m wm 2 + rm 2 6 (ill-conditioned) QR B U V QR S T U S B T V T r m, w m 1, r m 1, w m 2, U V r l 0 w l 1 = 0 l l r l l 1 l r l (l 1, l) 0 (l 1, l + 1) l 1 l + 1 (l 1, l + 1) 0 (l 1, l + 2) l 1 k (l 1, k) l (6.48) κ(a) = w m

54 f(x) I = b a f(x)dx (7.1) h = b a ( a + b ) I C = hf 2 I I C = h2 4! [f (b) f (a)] (7.2) 7h ! [f (3) (b) f (3) (a)] f(x)dx = h + 31h6 x 0 2 (f 0 + f 1 ) ! [f (5) (b) f (5) (a)] + h3 12 f (ξ) (7.5) x2 f(x)dx = h x 0 3 (f 0 + 4f 1 + f 2 ) h5 90 f (4) (ξ) (7.6) x3 I T = h f(x)dx = 3h x [f(a) + f(b)] (7.3) 0 8 (f 0 + 3f 1 + 3f 2 + f 3 ) 2 I I T = h2 3h5 2 3! [f (b) f (a)] 80 f (4) 3 (ξ) 8 (7.7) x4 + h4 6! [f (3) (b) f (3) (a)] h6 3! 7! [f (5) (b) f (5) (a)] + f(x)dx = 4h x (989f f 1 928f f f f 5 928f f f 8 ) 2368h f (10) (ξ) (7.9) h 2 f (b) f (a) hf (ξ) I S = h [ ( a + b ) ] f(a) + 4f + f(b) (7.4) 6 2 (7.4) (7.4) = 1 3 (2I b a = h (7.6) C + I T ) x 2 x 0 = 2h h4 I I S = 2 2 6! [f (3) (b) f (3) (a)] + 5h6 2 3! 8! [f (5) (b) f (5) (a)] + f(x) x f j = f(x j ) h = x j+1 x j ξ x1 x 0 x8 f(x)dx = 2h 45 (7f f f 2 +32f 3 + 7f 4 ) 8h7 945 f (6) (ξ) (7.8) 2

55 7. 2 h x 0 x n x 0 x n f(x)dx = 3h f(x) 2 (f 1 + f 2 ) + h3 4 f (ξ) f(x) = e iωx 4 (7.8) h(iωh) eiωξ f(x) 8 (7.9) (7.3) (7.4) h(iωh) eiωξ ωh > 1 (7.9) cos 2 θdθ = π 0 2 f(x) h = π/2 θ = 0, π/2, π 10 h ωh < 2π/10 < 1 I T = π ( = 2) π 2 x2n f(x)dx = h [ 1 x 0 2 f 0 + f 1 + f 2 + +f 2n f ] 2n (7.10) h2 [ f (x 2n ) f (x 0 ) ] + 12 (7.4) h 4 h 6 x2n f(x)dx = h x 0 3 [f h 0 + 4f 1 + 2f f 2n 2 + 4f 2n 1 + f 2n ] (7.11) h4 180 [ f (3) (x 2n ) f (3) (x 0 ) ] + x3 x 0 f (2k 1) (x) 0 π (7.9) h (7.8) h h x 2n x 2 dx 1 f (x) = 24x(1 x2 ) (1 + x 2 ) 4 [a, b] n h = b a n x j = a + jh j = 0, 1, 2,, n

56 7. 3 h I T (h) = h [ 1 2 f 0 + f f n f ] n (7.12) h h/2 h/4 h = 0 m = 5 I I T (h) = αh 2 + βh 4 + α, β h I I T (h/2) = 1 4 αh βh4 + h 2 (h = 0.5) I = 1 3 [4I T (h/2) I T (h)] 1 4 βh4 + k S (k) 0 S (k) 1 S (k) 2 S (k) 3 (7.9) (7.10) h S (k) S (0) 0 = I T (h/2 k 3 ) k = 0, 1, 2, 32 S (k) m S (k) m = S (k+1) m 1 + S(k+1) m 1 S(k) m 1 4 m 1 m = 1, 2, (7.13) m k e x2 dx S (0) 0 S (0) 1 S (0) 2 S (1) 0 S (1) 1 S (2) 0 m 2 S (0) 0 h S (1) 0 S (0) h = 1 1 h S (2) 0 S (1) 0 S (2) 0 S (1) x = 0, ±1, ±2, ±3, ±4 1 S (0) 1 S (0) π = h = S m (k) S (1) 0 S (0) 0 S (0) 1 S (0) 0 S (0) 1 1 I = f(x)dx S (1) 1 S (1) 0 1 S (0) 2 S(1) 1 S(2) 0 x h x = tanh y dx = dy f(x) cosh 2 (7.14) y 0 x 4 log(x x 2 )dx (7.3) 0 f(x) = e x2 x = ± 0

57 7. 4 y I = dy f(tanh y) cosh 2 y y f(tanh y)/ cosh 2 y y = ± e 2 y 0 π dx = 2 cosh y ( cosh y 2 π )dy 2 sinh y h I SE = h w k f(x k ) (7.15) y ± k= 1 y k = kh x k = tanh y k w k = cosh 2 y k ± w k 0 (DE) h 1/2 1/4 h I DE (h) = h w k f(x k ) (7.18) x k w k k= ( h = 1/2 M π ) y k = kh x k = tanh 2 sinh y k π h = 1/2, 1/4, 1/8, x k w k w k = 2 cosh y ( k (7.15) h cosh 2 π ) 2 sinh y k w k (7.16) h I SE (h) ( I I SE (h) exp c ) (7.16) h c f(t) I I SE (h/2) (I I SE (h)) 2 I SE (h/2) I I I SE (h/2) I ISE (h/2) I SE (h/2) I SE (h) h t f(x) = 1 2 (7.14) ( π ) x = tanh 2 sinh y (7.17) ( 1975) y exp( c exp y ) F (p) = 0 f(t)e pt dt p > 0 p e pt t p pt = e y dt = 1 p ey dy (7.19) h k w k = 2 F (p) = 1 f(t)w(y)dy (7.20) p

58 7. 5 w(y) = exp(y e y ) (7.21) n 2 ±x k n 3 ±x k F (p) = h f(t k )w k (7.22) p k= y k = kh pt k = e y k w k = w(y k (7.16) (7.16) y = 0 (7.19) 9 0 w(y) y = 20.7 pt = y = 2.0 pt = 20.7 e pt pt > 20.7 t = 0 1 [ 1, 1] 1 n I = f(x)dx I G = w k f(x k ) (7.24) 1 I Ch = 2 n f(x k ) (7.23) n k=1 f(x) n x 2n 1 I Ch 0 x k n = 2 n = 3 k=1 2n x k, w k f(x) x 1 + x 2 = 0 x x 2 2 = 2 3 w 1 + w 2 + w 3 = 2 x 2 = x 1 = 1 w 1 x 1 + w 2 x 2 + w 3 x 3 = 0 3 w 1 x w 2 x w 3 x 2 3 = 2 3 w 1 x w 2 x w 3 x 3 3 = 0 n = 8 n > 9 w 1 x w 2 x w 3 x 4 3 = 2 5 w 1 x w 2 x w 3 x 5 3 = 0

59 7. 6 x 2 = 0 x 1 = x 1 x 2 1 = 3 5 n ±x k w k w 1 = w 3 = 5 w 2 = w(x) [a, b] p j (x) 5 p n (x) x k (k = 1, 2,, n) n 1 [ n 1 n ] 1 f n 1 (x) = w k p j (x k )f(x k ) p j (x) λ j=0 j k=1 f(x) f n 1 (x) I G = b a n 1 = λ j=0 j w(x)f n 1 (x)dx [ n ] 1 b w k p j (x k )f(x k ) w(x)p j (x)dx k=1 b a w(x)p j (x)dx = 1 µ 0 b a w(x)p 0 (x)p j (x)dx = λ 0 µ 0 δ 0j j j = 0 I = b n a w(x)f(x)dx. = I G = k=1 a w k f(x k ) (7.25) w(x) = 1 x = y 2 p n (x) P n (x) P n (x k ) = 0 k = 1, 2,, n w k = 2(1 x2 k ) [np n 1 (x k )] 2 (7.26) n 1 x = 0 2n f(x)e x dx. = k w k f(x k ) 1 g(x) dx x 0 g(x) x = 0 I = 2g(y 2 )dy 0

60 7. 7 I = g(x) xdx 0 x = 0 x x = 0 x = 0 (7.5) (7.6) (7.8) (7.9) I = 2g(y 2 )y 2 dy 0 h I(h) h/2 I(h/2) I = g(x)x ±α dx 0 < α < 1 h 0 I(h) 4 (7.8) x = 0 I I(h) = 8h7 945 f (6) (ξ) (7.16) y = ± h/2 I(h/2) x x = ±1 I I(h/2) = 8h [f (6) (ξ L ) + f (6) (ξ R )] ξ L ξ R x = 1 f (6) (ξ) (7.16) exp ( π f ) (6) (ξ) 1 x = 2 sinh y ( π ) I I(h/2) = 1 [I(h/2) I(h)] (7.27) cosh 2 sinh y 63 (7.17) f(x) ε x 1 x 1 I(h/2) I(h) < ε (7.28) x = 1 f(x) 63 x = 1 I(h/2) I(h/2) (7.23) I =. I(h/2) + 1 [I(h/2) I(h)] (7.29) 63 I(h/2) I(h/4)

61 7. 8 ε ε/

62 y n = Az1 n + Bz2 n (8.4) A B n = 0 y 0 y 1 y 0 = A + B y 1 = Az 1 + Bz 2 (8.5) d 2 y(t) dt 2 + ω 2 y(t) = 0 (8.1) t = n t y n = y(n t) z 1 d 2 y(t) dt 2. = y n+1 2y n + y n 1 t 2 (8.3) A B z 1 < z 2 B 0 n z 1 A = 0 N y N y N+1 A B (8.3) n = N, N 1,, y n+1 [2 (ω t) 2 ]y n + y n 1 = 0 (8.2) y n 1 = 1 β (αy n + y n+1 ) n y n+1 + αy n + βy n 1 = 0 (8.3) z 2 (8.3) α β n z 1 < z 2 (8.3) y 0 y 1 z 1 n = 0, 1, y n z 2 z 1 = z 2 α β n y n z 1 < 1 (8.3) n y n y n z n (8.3) n + z n z n (8.3) z 1 > 1 z 2 > 1 (8.6) (8.3) z 2 + αz 1 + β = 0 z z z1 1, z 1 2 = 1 [ α ± ] ω 2 > 0 (8.2) α 2 2 4β 1 z 1

63 8. 2 (8.2) ω t 1.. z 1, z 2 = 1 ± iω t = e ±iω t d 2 Z n (x) dx (8.2) (8.1) ) dz n (x) + (1 n2 x dx x 2 Z n (x) = 0 (8.8) y(t) = Ae iωt + Be iωt (8.3) α = 2 cos θ β = 1 z 1 1, z 1 2 = cos θ ± i sin θ = e ±iθ 1 (8.3) x J n (x) y 0 = 0 x A + B = 0 J 0 (x) x = 10 y 1 = A(e iθ e iθ ) = 2iA sin θ k a k S k A = e e+0 2i e e+1 (8.3) e e+2 y n = sin nθ y 0 = 1 y n = cos nθ e e e e+0 (8.3) e e+0 sin(n + 1)θ + sin(n 1)θ = 2 cos θ sin nθ cos(n + 1)θ + cos(n 1)θ = 2 cos θ cos nθ (8.7) sin θ J 0 (10) cos θ cos nθ sin nθ J 0 (x) Z n (x) n 0 x 0 ( x ) n ( 1) k ( x ) 2k J n (x) = (8.9) 2 k!(k + n)! 2 k= e e e e e e e e e e+1 a k (8.9) S k k k = x = 5

64 8. 3 J n (x) Z n (x) Z n+1 (x) 2n x Z n(x) + Z n 1 (x) = 0 (8.10) Z 0 (x) Z 1 (x) y Z 2 (x), Z 2 (x), N = ε y N+1 = 0 (8.13) ε Z n (x) J n (x) (8.10) J 0 (1) J 1 (1) n J n (1) y n J n (x) n = 5 3 (8.10) (8.3) α n J k (x) J k (x) =. y k (8.15) z1 1, z 1 2 = n S n x ± 2 x 2 1 ε n x n 1 (8.14) (8.10) n y k y k S n x (8.10) 1 (8.10) N n (x) n (8.10) Z n (x) z1 n z 1 < 1 N 0 (x) N 1 (x) J n (x) J 0 (x) + 2 J 2 0 (x) + 2 J 2n (x) = 1 (8.11) n=1 Jn(x) 2 = 1 (8.12) n=1 J n (x) N n N y k 1 = 2k x y k y k+1 (8.14) S = y k = N, N 1,, 1 N 1 (x) n J n (x) 0 (8.10) N k=1 y 2k (8.11) S J 0 (x) J 1 (x) (8.13, 14) J n (x)n n+1 (x) J n+1 (x)n n (x) = 2 πx (8.16) J 0 (x) J 1 (x) N 0 (x)

65 8. 4 x < 5 N 0 (x) K n (x) π 2 N 0(x) = (ln x ) 2 + γ K 0 (x) K 1 (x) J 0 (x) K n (x) + 1 ( x ) 2 1 ( x ) 4 (1 + 1/2) (1!) 2 2 (2!) ( x ) 6 +(1 + 1/2 + 1/3) I n (x)k n+1 (x) + I n+1 (x)k n (x) = 1 (8.22) (3!) 2 2 x K 0 (x) γ K n (x) x γ = K 0 (x) N 0 (x) d 2 Z n (x) dx dz n (x) x dx (1 + n2 x 2 ) Z n (x) = 0 (8.17) j 0 (x) = x 1 sin x I n (x) j 1 (x) = x 2 (sin x x cos x) (8.23) I n (x) = i n J n (ix) (8.18) I n (x) (8.9) I n (x) I n+1 (x) + 2n x I n(x) I n 1 (x) = 0 (8.19) j n (x) j n 1 (x) + j n+1 (x) = 2n + 1 j n (x) x (8.24) (8.23) z1 1, z 1 2 = n n n x ± 2 x J n (x) 1 J n (x) (8.24) n I n (x) (8.19) j 0 (x) (8.23) I 0 (x) + 2 I n (x) = e x (8.20) n=1 j n (x) x j 2 = x 3 [(3 x 2 ) sin x 3x cos x] j 0 (x) x = kπ 0 j 1 (x) n n (x) j n (x) (8.24) N n (x) (8.17) K n (x) K n+1 (x) 2n x K n 0 (x) = x 1 cos x n(x) K n 1 (x) = 0 (8.21) n 1 (x) = x 2 (cos x + x sin x) (8.25) I n (x) n 2 (x) = x 3 [(3 x 2 ) cos x + 3x sin x] n n

66 8. 5 Z 0 S c n c n c k = a k P 0 (x) = 1 P 1 (x) = x x 1 (n + 1)P n+1 (x) (2n + 1)xP n (x) +np n 1 (x) = 0 (8.26) ( P n+1 (x) 2 1 ) xp n (x) n + 1 ( ) P n 1 (x) = 0 n + 1 n Z k x = cos θ (8.7) (8.23) Z k = U k + V k sin θ (8.29) (c) c n S = N c n e inθ (8.27) n=0 V N = 0 a N z = e iθ V 0 sin θ S S z N U 0 (a) k = 0 S = c 0 + z(c 1 + z(c z(c N ) ) n Z n (8.30) k = 0 S = Z 0 = c 0 + zz 1 Z 1 = c 1 + zz 2 Z k = c k + zz k+1 a k (8.30) V 0 V 1 Z N = c N N N a k cos nθ a n sin nθ Z N+1 = 0 Z k = c k + zz k+1 k = N, N 1,, 0 (8.28) (8.28) Z k+1 = z 1 (Z k a k ) Z k 1 = zz k + a k 1 Z k 1 + Z k+1 = (z + z 1 )Z k + a k 1 z 1 a k V N = 0 V N+1 = 0 (a) (b) (c) V k 1 + V k+1 = 2 cos θv k + a k (8.30) k = N, N 1,, 1 U 0 = V 1 + V 0 cos θ + a 0 (8.31) U 0 = V 1 V 0 cos θ n=0 n=0

67 8. 6 p n+1 (x) = α n p n (x) + β n 1 p n 1 (x) (8.32) p n (x) S = N a n p n (x) (8.33) n=0 V 1, V 2, V 3 (8.32) V 0, V 1 V 2 p n+1 (x) a n+1 sin nθ cos nθ (8.7) Fig a n V 0 sin θ 1 1 a k n S n = a k V k V n = sin nθ (d) k=0 V n V n 1 + V n+1 = 2 cos θv n (e) S n sin cos V n+1 (8.31) S n+1 = S n + a n+1 V n+1 (f) Fig. 8.1 (8.32) (8.33) Fig. 8.2 q n p n (x) α p n 2 cos θ β 1 sin, cos β β q N+1 = q N+2 = 0 1 V 1 V 2 sin θ α V 3 α q n = a n + α n q n+1 + β n q n+2 (8.34) n = N, N 1,, 1 a 1 a 2 a 3 q 1 q 0 = a 0 + β 0 q 2 S 1 S 2 Fig. 8.1 S 3 S N (8.30) Fig. 8.1 (d) sin nθ cos nθ Fig. 8.1 V 1 V 1 S = q 0 p 0 (x) + q 1 p 1 (x) (8.35) 1 V 1, V 2 a n

68 p 1 (x) p 0 (x) q 0 β 0 β 1 q 1 q 2 α 1 β N 2 q N 1 αn 1 q N a 0 a 1 a 2 a N 1 a N S 0 S 1 S 2 S N 1 S N Fig. 8.2 x n (8.3) x n = x 0 δ n x 0 y n + αy n 1 + βy n 2 = x n (8.36) y y n = x 0 h n n = 0 n = k x k x n = 0 y n = 0 n < 0 k (8.36) y n = x k h n k n 0, 1, 2, y n h n x 0, x 1, x 2 { n n 1 n = 0 h n +αh n 1 +βh n 2 = δ n = 0 n 0 (8.37) y n = x k h n k = h k x n k (8.39) k=0 k=0 δ n h n x n h n h n n 0 n y n h k x n k max x k h k k=0 k=0 { 1 n = 0 (8.38) h n = 0 n < 0 n (8.6) (8.6) (8.36) y n = x n αy n 1 βy n 2 (8.40) h n n = 0, 1, 2, h n = αh n 1 βh n 2 n = 1, 2, (8.39) h n < (8.38) (8.39) (8.40) n=0 (8.4) (8.6)

69 t x j x j 2 < (9.1) j= x j ( 14) x j X D (ω) = x j e ijω (9.2) j= x j ω f x j t ω = 2πf t (9.3) X D (ω) X D (ω + 2π) = X D (ω) (9.4) ω π (9.5) f f N = 1 (9.6) 2 t 2 t t 2 t x j (9.1) x j < (9.2) (9.1) π N x j e ijω X D (ω) 2 dω = 0 lim N π j= N X D (ω) (9.2) x j = 1 π X D (ω)e ijω dω (9.7) 2π π (9.2) x j e ijω (9.3) x j = fn f N tx D (ω)e ijω df tx D (ω) (9.2) (9.7) 5 (5.16) x f(x) (a k, b k ) x j x(t) t t x(t) (aliasing) Fig. 9.1 ( ) x(t) X(σ) = x(t)e iσt dt (9.8)

70 9. 2 σ σ = 2πf x(t) = 1 2π X(σ)e iσt dσ (9.9) X D (ω) X(ω) x j = x(j t) (9.2) j= x(j t)e ijω x(t) (9.9) f(t) F (σ) αβ = 2π α, β β f(mα) = F (nβ) (9.10) 2πα m= n= f(t) f(t) = x(t)e iωt/ t j f(j t) f(t) F (σ) = X(σ + ω/ t) α = t X D (ω) = = 1 t m= n= x(m t)e imω ( ) ω + 2πn X t X(σ) X D (ω) ( ) ω + 2πn tx D (ω) = X (9.11) t n= X D (ω) ω X(σ) σ X Fig. 9.2 X(ω) X D (ω) X(ω) ω > π π < ω X(ω) ω < π π < ω X(ω) f X D (ω) ω > π X( ) 0 X D ( ) 0 Fig. 9.2 f f f, f ± 1/ t, f ± 2/ t,, f ± n/ t, f ± 2nf N f Fig. 9.1 f = 13 Hz t = 0.1 s f N = 5 Hz f = = 3 Hz 3 Hz X(σ) 0 X D (ω) X(σ) tx D (ω) = X(ω/ t) (9.12)

71 9. 3 σ > π X(σ) = 0 t X D (ω) D X(ω) x j, y j z j = x j y j (9.13) Z(ω) Z(ω) = 1 2π π π X(σ)Y (ω σ)dσ (9.14) z j x j y j X Y z j = x k y j k = x j k y k (9.15) k= k= z j Z(ω) = X(ω)Y (ω) (9.16) y j = x j (9.15) { } z j = k x k x k j = k x j+k x k x k y j Y (ω) = X(ω) z j Z(ω) = X(ω) 2 z j = 1 2π π π X(ω) 2 e ijω dω j = 0 z 0 = x k 2 = 1 π X(ω) 2 dω (9.17) 2π k= π x j t X(ω) 2 (9.2) (9.7) (9.8) (9.9) N x 0, x 1,, x N 1 c k c k = N 1 j=0 ( ) 2πijk x j exp N k = 0, 1, 2,, N 1 x j = 1 N N 1 k=0 ( c k exp 2πijk ) N (9.18) (9.19) (9.18) N 1 [ ] { 2πik(l j) N l = j exp = N 0 l j k=0 (9.18) (9.2) ω k = 2π N k c k = X(ω k ) (9.20) X(ω) ω = 2π N

72 9. 4 c k (9.7) (9.19) X(ω) (9.7) (0, 2π) N 1 2π 2π 0 X(ω)e ijω dω. = ω 2π = ω 2π N k=0 N 1 k=0 X(ω k )e ijω k X(ω k )e ijω k k = 0, N 1/2 X(ω) X(0) = X(ω N ) X(ω k ) = c k (9.19) (9.7) ω N N N N = 2M + 1 2M + 1 x j ( j M) M X(ω) = x j e ijω j= M c k = X(ω k ) ω k = 2πk 2M x j = 1 2M M k= M k M (9.21) c k e ijω k (9.22) j = M j = M 1/2 (9.21) c k X(ω) (9.7) ( π, π) 2M ω (9.15) x j, y j j = 0 N 1 N z j (9.15) (9.16) z j = 1 2π 2π 0 X(ω)Y (ω)e ijω dω FFT x j, y j N z j ~ z x z Fig. 9.3 x, y N z x y x y N z j j = 0 2N 2 N z j z j N z j+n = z j z j j = N 2N 1 j = 0 N 1 z j = n= z j+nn z j z j 2N y

73 9. 5 x j (9.18) (9.19) N N N x j (9.18) c k c N k = c k N c k k = 0 N/2 c k c k = a k + ib k a k, b k a k = b k = N 1 j=0 N 1 j=0 x j cos jω k ω k = 2π N k x j sin jω k (9.23) k = 0, 1, 2,, N/2 b 0 b N/2 0 a k, b k N N (9.19) k N 1 k=0 c k = N 1 k=n/2 N/2 1 k=0 N/2 c k = = N/2 k=1 k=1 c k exp c k + [ c N k exp N 1 k=n/2 ( 2πijk ) N c k ] 2πij(N k) N x j = 2 [ N/2 1 1 N 2 a 0 + (a k cos jω k + b k sin jω k ) k=0 + 1 ] 2 a N/2 cos 2πj (9.24) N k N/2 a N/2 N (0, 2π) N 2M +1 (9.21) a k = b k = M j= M M j= M x j x j cos jω k ω k = 2πk 2M x j sin jω k (9.25) x j = 1 [ M 1 1 M 2 a 0 + (a k cos jω k + b k sin jω k ) k=1 + 1 ] 2 a M cos jπ (9.26) j = + x j (9.2) X M (ω) = M j= M x j e ijω X(ω) X M (ω) x j (9.7) π X M (ω) = 1 M X(σ) e ij(ω σ) dσ 2π π j= M = 1 π X(σ)W M (ω σ)dσ (9.27) 2π π = 1 π X(ω σ)w M (σ)dσ 2π π W M (ω) = M j= M e ijω = sin(n + 1/2)ω sin ω/2 (9.28) (9.27) W M (ω) Fig. 9.4 X M (ω)