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1 M.Shimura Ver 0.97

3 / LF LU LDU QR SVD / /

.................................... 37 1.12................................ 40 2 43 2.1 LU LDU.............................. 43 2.2................................ 45 2.3 QR.................................... 47 2.

4 Markov chain/ The input output model

4................................ 120 6.5................................... 125 6.6.......................... 128 6.

5 5 19 E E Einheistsmatrix 1 I I XA 1 X 1 A X AB BA A t A,A t,a T A t J π,e 1p1.1x1 1r3 1 3 PC 8GB CPU

7 x y = ]D110= ,: cr D x y = ]D111= ,: cr D a 11 x 1 +a 12 x 2 + a 1n x n = b 1 a 21 x 1 +a 22 x 2 + a 2n x n = b a m1 x 1 +a m2 x 2 + a 2mn x n = b m a 11 a 12 a 1n b 1 x 1 a 21. a m1 + x 2 a 22. a m2 a 2n + + x n = a mn b 2. b m

1 1 14,: 50 80 1000 cr D111 1 0 4 1 1 14 0 1 10 50 80 1000 1.1.1 a 11 x 1 +a 12 x 2 + a 1n x n = b 1 a 21 x 1 +a 22 x 2 + a 2n x n = b 2.

8 8 1 a 11 a 12 a 1n x 1 b 1 A = a 21 a 22 a 2n... a m1 a m2 a mn Ax = b. x 2 b 2, x =, b =... x = b ( x = A 1 b) A x n b m Ax = b (A, b) Gabriel Cramer( ) Genova ax +by +cz = j dx +ey + f z = k gx +hy +iz = l a b c x j d e f y = k g h i z i = J a b c j d e f k x g h i l y = a b c z d e f g h i cr=: %.}:"1 cr D

9 1.1 9 x, y, x * Grammar / J ( J =. local) =:(global) =,:= <- J = raminate,:,: 1 1 1, ,: matrix-divide %. J primitive) }:"1 cr=: %. }:"1 J (tacit de f inition) (y) (Explicit de f inition) J 2 cr0=: 3 : y %. }:"1 y B J matrix diviode(%.) A Ax = B x = A 1 B, (or B A ) = = *1 0 e.g e 9 J

10 BC (1642?- 1708) (1683) (1685) ( ) ]D120=: 1 1,: det a c b d = a c b d = ad bc 2 -/. * a NB. determinant a b c a b c det d e f = d e f g h i g h i e f = a h i b d f g i + c d g e h a d g b e h c f i a d g b e h = aei a f h + b f g bdi + cdh ceg = aei + b f g + cdh a f h bdi ceg A A 1 deta 0, 0 (non-singular matrix) 2 2

11 a A = c A 1 = b d 1 a d c b d c b a = Grammar %. D120 2 _0.5 _1 0.5 *2 J * 3 determinant J ) - /. * permanent +/. * +/. * D120 NB. permanent 8 NB. 1*4*2*1 *2 2 2 *3 3 4

12 Carl Frierich Gauss( ) Duke o f Braunshuweig Gauss 1809 * y = 14 y = 7 x + 7 = 10 x = 3 Example 3 3 2x 1 +3x 2 x 3 = 5 4x 1 +4x 2 3x 3 = 3 2x 1 +3x 2 x 3 = x 3 = 3 2x 2 = 4 x 2 = 2 2x 1 = 2 x 1 = 1 *4

13 x 1 +3x 2 x 3 = 5 4x 1 +4x 2 3x 3 = 3 2x 1 +3x 2 x 3 = , x 3 = 3 10, x 2 = 20 x 2 = 2 4x 1 = 4, x 1 = 1 * Gauss Jordan Jordan William Jordan( ) Deutch aˆ 11 0 aˆ aˆ 12 x 1 x 2. x n = ˆ b 1 ˆ b 2. ˆ b n *5

= 2 4x 1 = 4, x 1 = 1 *5 1.3.3 Gauss Jordan Jordan William Jordan(1842 1899) Deutch aˆ 11 0 aˆ 12.

14 14 1 Example gauss jordan D130=. 2 3 _1,4 4 _3,: _2 3 _1 D _1 4 4 _3 _2 3 _1 D130,.=/ i _ _ _2 3 _ Gauss jordan require system/packages /math/linear.ijs gauss_jordan D130,.=/ i _ _ _0.6 _0.2 J %. %.D _ _ _0.6 _0.2

2 3 _1,4 4 _3,: _2 3 _1 D130 2 3 _1 4 4 _3 _2 3 _1 D130,.=/ i.3 2 3 _1 1 0 0 4 4 _3 0 1 0 _2 3 _1 0 0 1 Gauss jordan require system/packages /math/linear.

15 David Salsburg [

16 / rank X (rank) [ = 0 ] [ = 15 ] 0 Example D _1 _ _ _1 0 _5 small_matrix D _ _ _5 _ _1 0 _5 11 _1 0 _ e_15 _

1 rank X (rank) 0 2 1 1 15 2 1 1 15 2 1 1 15 2 1 1 15 4 6 3 41 4 6 3 41 0 4 1 11 0 4 1 11 2 9 3 37 2 9 3 37 0 8

17 1.4 / 17 D non-singular matrix A 1 A = AA 1 = I =/ i J Charles Hermite( ) France A = A A A t A D141=. 1 3j1,: 3j_1 2 D141 : D j_1 1 3j1 3j1 2 3j_1 2 + D j_1 3j1 2

Hermite(1822 1901) France A = A A A t A D141=.

18 1 viewmat require viewmat viewmat [z=:. j. / i:2j16 1.

18 18 1 viewmat require viewmat viewmat [z=:. j. / i:2j viewmat A > 0 detd i > 0, i = 1,, n A Grammar 0 a i j 0, i, j = 1,, n A 0 j 3j2 conjugate + 3j2 detd i 0, i = 1,, n + 3j2 3j_2

1 viewmat A > 0 detd i > 0, i = 1,, n A 0 1.4.

19 1.4 / Chatelin C.Moler D a 22 D143 _149 _50 _ _27 _9 _ λ 6λ 2 + λ 3 = 0 Eigenvalue: D144 _149 _50 _ _27 _9 _ λ 6.01λ 2 + λ 3 = 0 Eigenvalue: D145 _149 _50 _ _27 _9 _ λ λ 2 + λ 3 = 0 Eigenvalue: ±j G 0 1

Eigenvalue: 3 2 1 D144 _149 _50 _154 537 180.01 546 _27 _9 _25 1.67 + 9.26λ 6.

20 20 1 D146 3 _2 _1 _2 4 _1 1 _1 3,.{ : D _2 1 NB. X _2 4 _1 NB. X _1 _1 3 NB X X 1 X 1 X 1 X 2 X 1 X 3 X 2 X 1 X 2 X 2 X 2 X 3 X 3 X 1 X 3 X 2 X 3 X 3 gram_rank D _ _15 21 _5 2 _ Script small_matrix=: 3 : 0 NB. calc determinant of small matrix NB. Usage: u matrix SIZE=..{(;(<: # y), 2)$ 2#}. >: - i. # y NB. make size 2*2 3*3 4*4.. MAT=. SIZE {. L:0 y NB. take matrix 2*2 3*3 4*4... MAT,:(-/. * L:0 MAT) NB. calc determinant of each small matrix ) Reference F. / Springer-Verlag Tokyo 1993/2003( )

_15 21 _5 2 _5 11 +----------+---+ 2 0 2 1.4.7 Script small_matrix=: 3 : 0 NB. calc determinant of small matrix NB. Usage: u matrix SIZE=.

21 J.P.Gramm( ), E.S chmidt( ) 1 (orthogonal matrix) T t = T 1 A t A = AA t = I A t = A 1 U U = UU = I U = U v 1 = v 1 u 1 = v 1 v 1 v 2 = v 2 (v 2, u 1 )u 1 u 2 = v 2 v 2 v 3 = v 3 (v 3, u 2 )u 2 (v 3, u 1 )u 1 u 3 = v 3 v 3 v 4 = v 4 (v 4, u 3 )u 3 (v 4, u 2 )u 2 (v 4, u 1 )u 1 u 4 = v 4 v 4.. Example v 1 v 2 v v 1 = v 1 u 1 = v 1 v 1 = v 2 = v 2 (v 2, u 1 )u 1 = 3 3, = u 2 = v 2 v 2 = v 3 = v 3 (v 3, u 2 )u 2 (v 3, u 1 )u = 1 1, , = u 3 = v 3 v 3 = QR Q,R

22 22 1 D _ !:0 D _ _ _ NB. Q (shumit orthogonal) R a0 a1 =. 128!:0 D150 NB. Q and R a0 +/. * a1 NB. Q +/. * R _ Grammar 128!:0 QR *6 A = Q A 1 = Q 1 a _ _ %. a _ e_16 _ _ A t = Q t A0 A0 t A0 = I Q t Q = I : a _ _ ( : a0) +/. * a0 NB e_ e_ e_16 1 _ e_ e_16 _ e_17 1 *6 J 128!:

23 1.5 23

24 = = /transpose transpose 2 D160= ,: _1 3 1 D161= ,: D _1 3 1 D D160 +/. * D D161 +/. * D NB. AB NB. BA Grammar NB. /comment transpose : : D /. * AB BA AB BA {@> <"0 D

25 Hadamard products = = D162=: 1 2,: 3 1 D163=: 0 3,: 2 1 D162d D D162 * D Kronecher products = = ({@> D162) * L: D163

26 *7* A = 2 4 NB. D120 LF tr trace 1 1 A = 2 4, B = 2 4 P 1 = tr[b 1 ] = = B 2 = = = P 2 = 1 2 tr[b 2] = 2 P[λ] = λ 2 2 i=1 P 1 λ n 1 P[λ] = 2 5λ + λ 2 ( ) J p. p. 2 _ plot (i:10); 2 _5 1 &p. i:10 *7 NJ Joha Randoll *8 Mathematica

27 charactristic equation A 1 = 1 (B 1 P 2 I) P 2 A 1 = = = Grammar LF char_lf 2 2 $ _ LF det(a λi n ) = 0 λ p(λ) = det(a λi n ) Leverrir-Faddeev p(λ) c k p(λ) = λ n + c 1 λ n 1 + c 2 λ n c n t λ 2 + c n 1 λ + c n Tr[A] Tr[A] = a 1,1 + a 1,t + + a n,n

28 28 1 (Bk) n k=1 B 1 = A p 1 = Tr[B 1 ] B 2 = A(B 1 p 1 I) p 2 = 1 2 Tr[B 1] B k = A(B k 1 p k 1 I) p k = 1 k Tr[B k] B n = A(B n 1 p n 1 I) p n = 1 n Tr[B n] p(λ) = λ n p1λ n 1 p 2 λ n 2 p n t λ 2 p n 1 λ p n A A 1 = 1 p n (B n 1 p n 1 I) Newton x f (x) f (x) J D. APL "0 ˆ:_ (ˆ:100) new_1=: 1 : ] - x % x D.1 (ˆ:_)("0) 2 _5 1&p. new_1 i Grammar polinominal p. J 1 p. 2 _5 1 &p.i:10 i. i: i. n 0 n i: n n =/ i.3 tr trace,tra = n i=1 a ii,

29 1.8 LF LF , A = , B 1 = p 1 = Tr[B 1 ] = 2 B 2 = = p 2 = 1 2 Tr[B 2] = = 5 B 3 = = p 3 = 1 3 Tr[B 3] = 1 18 = 6 3 P[λ] = λ 3 3 i=1 p iλ n 1 p[λ] = 6 + 5λ 2λ 2 + λ 3 λ 1, ± p[λ] = 6 5λ 2λ 2 + λ 3 J p. p. 6 _5 _ _

30 *9 A 1 = 1 (B 3 1 p 3 1 I) p 3 A 1 = = = 4/6 7/6 5/6 2/6 5/6 1/6 2/6 8/6 4/6 %. D _ _ _ _ _ Script John Randoll LF tr=: (<0 1)& : NB. diag NB. umatrix=: (=/ )@i.@# char_lf=: 3 : 0 ANS=. TR_SUM=. +/ tr MAT=. y NB. sum of trace UMAT=. =/ i. # y for_lf. i.<: # y do. MAT=. y +/. * MAT - UMAT * TR_SUM TR_SUM=. (% 2+LF)* +/ tr MAT ANS=. ANS,TR_SUM end. (p. POL), (<POL=.(-ANS),1) ) n 1 LF 1 2 p k 2+LF p. λ p n 1 char_lf D _2 1 6 _5 _ f (λ) = 6 5λ 2λ 2 + λ 3 Eigenvalues (1) J p. *9 1 p.

31 ( ) ( ) 14 φ(λ) = A λ φ(λ) = 0 φ(λ) λ A φ(a) = O n Example λ 1 1 A = = 2 λ λ λi A = 1 5λ 3λ 2 + λ 3 = 0 I 5A 3A 2 + A 3 = O 3 λ A A n 1. LF 2. ch A n D190 char_lf D _1 _ _1 _5 _ = 1 5λ 3λ 2 + λ 3 ch D190 NB. 0 I A Aˆ2 Aˆ = I 5A 3A 2 + A 3 0

32 I 5A 3A 2 + A 3 = O 3 A 1 A 1 5I 3A + A 2 = O 3 A 1 = 5I 3A + A 2 D ch_inv D190 _1 _1 1 0 _ _2 %. D190 _1 _1 1 0 _ _ Script ch=: 3 : 0 NB. Cayley Hamilton theorem B EV CR =: char_lf y NB. CR is lambda KH0=:(<=/ i. # y), mp_matrix y NB. I A Aˆ2 Aˆ3.. (<+/ > KH0 * L:0 {@> CR),KH0 NB. ans is 0//correct ) mp_matrix=: 3 : 0 NB. matrix multiplication TMP=. y +/. * y ANS=. y ; TMP for_ctr. i.<:<: # y do. TMP=. y +/. * TMP ANS=.ANS,<TMP end. ) ch_inv=:3 : 0 NB. modified 05/06/2008 NB. Cayley Hamilton inverse matrix NB. e.g. Aˆ(-1)= -5I-3A+Aˆ2 EV CR =: }. char_lf y NB. CR is lambda CH0=:(<=/ i. # y), mp_matrix y NB. I A Aˆ2 Aˆ3.. CH1=:(+/ > (}: CH0) * L:0 {@> }. CR)

33 CH1 % {.CR NB. ans is 0//correct )

34 (C m ) (A λi)l = 0 char_lf D _2 1 6 _5 _ f (λ) = 6 5λ 2λ 2 + λ 3 Eigenvalues λ 2 3 B = A λi = 1 1 λ λ λ = l (A λ 1 I)l = m = n D180; (3 *=i.3);a1=.d180-3*=i.3 NB.D180 I D180-lamda I _ _1 _ _ _ _ A λi λ

35 Randoll 1990 B 11 = ( 1) 2 1 λ 1 = 3 1 λ λ 2 4 B 12 = ( 1) = 3 1 λ 2λ + 7 B 13 = ( 1) λ 1 3λ 5 = B 21 = ( 1) λ λ + 2 = B 22 = ( 1) 4 2 λ 3 = 1 1 λ λ 2 λ 5 B 23 = ( 1) 5 2 λ λ + 1 = B 31 = ( 1) 4 1 1λ = λ+ B 32 = ( 1) 5 2 λ λ 8 = B 33 = ( 1) 6 2 λ 2 = 1 1 λ λ 2 3λ + 4 λ 2 4 2λ + 7 3λ 5 ad j(b) = λ + 2 λ 2 λ 5 λ + 1 λ + 2 3λ 8 λ 2 3λ = λ λ char l f 2 3 LF λ

36 36 1 ( ) char_lf_evec_sub=: 3 : 0 TR_SUM=. +/ tr MAT=. y NB. sum of trace ANS=. <UMAT=. =/ i. # y for_lf. i.<: # y do. MAT=. y +/. * TMP=. MAT - UMAT * TR_SUM TR_SUM=. (% 2+LF)* +/ tr MAT ANS=. ANS,<TMP end. ) B k = A(B k 1 p k 1 I) (B k 1 p k 1 I) char_lf_evec_sub D _2 3 _4 7 _ _1 1 2 _ _3 2 _ EIGEN2 3 _ _ char_evec D _ _11 _3 5 _ _1 3 _ _ _ λ 2, λ, 1 char lf evec sub0 Script char_evec=: 3 : 0 EIGEN=: {@> ; 1{ char_lf y EIGEN2=:{@>L:0 EIGEN ˆ/L:0.i.# EIGEN ADJMAT=: char_lf_evec_sub y ANS=. < for_lf. i. # y do. TMP=. +/> ( > LF{ EIGEN2) * L:0 ADJMAT ANS=. ANS,<TMP end. EIGEN,:}. ANS )

37 Example 1 x = 1 2, y = a1 _1 1 2 a2 1 0 _1 Euclidean vecter norm x x x = n i=1 x 2 i x = x x %: a1 +/. * a = 6 distance x and y between x y = (x y) (x y) (a1 - a2) +/. * a1 - a2 14 %: (a1 - a2) +/. * a1 - a cosine o f angle between x and y cosθ = x 7 y x y _3 (a1 +/. * a2) %: a1 +/. * a %: a2 +/. * a = _ = 150 o

38 38 1 unit length x = y = 1 u x = x x a1 % %: a1 +/.* a1 _ KX cm KY (%) KX KY ( ) x x 1 (x x) 2 n stand KD _ _ _ _ _ _ _

39 x x dev=:-(+/%#) dev2=:-"1(+/%#)nb. rank 1 *10 Script stand=: dev % "1 sd dev2=: -"1(+/ % #) var=: # % ([:+/[: *: dev) NB. sd=. %:@var NB Reference 2000

40 x 1 x y 1 ȳ t XX C = N = 1 x 1 x x 2 x... x n x x 2 x y 2 ȳ N y 1 ȳ y 2 ȳ... y n ȳ x 3 x y 3 ȳ x n x y n ȳ 1 (x n x)(x n x) (x n x)(y n ȳ) N (y n ȳ)(x n x) (y n ȳ)(y n ȳ) = x / 1 ni=1 X i n am=: +/ % # Example am STYLE am2=: # % +/ dev2=. -"1 +/%# ( : dev2 STYLE) +/. * dev2 STYLE vartable STYLE _ _

41 cortable STYLE _ _ Grammar *: square 2 (ˆ2 ) %: square root 2 ( ) -"1(+/%#) x x [: Cap

43 LU LDU A A = LU A = LDU LU A = r T 1 = a 21 /a 11, a 31 /a T 2 = a (1) 32 /a(1) A (1) = T 1 A = r A (2) = T 2 A (1) = r T = T 2 T 1 = T 2, T 1 = L = T 1 =

44 A = r = LU P30 0 require system/packages/math/linear.ijs require system/packages/math/matfacto.ijs lud D _ _1r r2 5r r _7r pivot LDU U DU D U U U D A = r r2 1r = r r r2 1r2 A = = LDU r a3 % diag a3 NB. a3 is U0 1 _1r2 _1r2 0 1 _

45 And re Lous Cholesky( ) A L A = LL t A = LDL t LU A = A = T 1 UU(T 1 ) t V = T 1 U A = VV t T 1 = 1r r a 21 /a 11, a 31 /a T 2 = r3 1 a (1) 32 /a(1) A (1) = T 1 AT1 t = A (2) = T 2 A (1) T2 t = Σ = = U T = T 2 T 1 = r2 1 0 = 1r r3 1 3r r6 2r T 1 = 1r r2 2r (T 2 T 1 ) 1 U = 1r r2 2r V(choleski decomposition) = =

46 46 2 A = VV t = Ly = b LDL D220 4 _2 _6 _ _ v % _1r3 1 0 _3 2 1 diag v d diag v *=i.3 v _1 3 0 _3 2 1 c=. :( : v % 2 3 1) +/. * d _1 3 0 _3 2 1 choleski D220 c +/. * : c 4 _2 _6 _ _ c=v D220 DL t x = y J require system/packages/math/matfacto.ijs choleski D220 NB. A _1 3 0 _ Reference 2003

47 2.3 QR QR (R) Gram-shumit (Q) 1961 Francis Kublanovskaya * 1 QR Q,R D _ !:0 D _ _ _ NB. Q (shumit orthogonal) R a0 a1 =. 128!:0 D150 NB. Q and R a0 +/. * a _ NB. Q +/. * R *1 householder matrix

48 48 2 b [ ] [ ] 11 b 12 b 13 e1 e 2 e 3 = v1 v 2 v 3 0 b 22 b b 33 D150 %. a0 NB. A %. Q = R e_ _ QR QR Script 1,2 pwr=: ˆ:100 NB. repeat 100 times (change you like) eval_t=: (>&{: +/. * >&{.)&(128!:0) eval_t pwr D180 NB _ _ e_324 _2 _ e_324 1 char_lf D _2 1 6 _5 _ Grammar power ˆ: :_ 128!:0 QR

49 2.4 SVD SVD singular value decomposition J package/math svd.i js * 2 Jordan SVD SVD SVD * 3 P = U V T σ 1 σ 2 P = U... σ r V r, σ i = λ i, i = 1,, r P M = PP T N = P T P D D260 +/. * : D260 ( : D260) +/. * D M N 0 char_lf D260 +/. * : D _ char_lf ( : D260)+/. * D _ %: pick_evec D260 +/. * : D _ pick_evec ( :D260) +/. * D _ _ *2 LAPACK *3 m 2 m 2, m = SVD

50 50 2 svd D _ e_ _ U V T P P

51 A A A = T AT 1 * Example D180 2 _ _1 6 5λ 2λ 2 + λ 3 char_lf D _2 1 6 _5 _ av=. pick_evec D _ _ (%.av)+/. * D180 +/. * av e_ e_ e_16 _ e_ e_16 1 ( *4

52 52 2

53 (Jordan canonical f orm) Camille Jordan Example 3 3 D261 _1 2 0 _1 1 2 _1 _2 _1 _ _2 _ _1 _1 _1 _ J -/. * D261 %. D261 _1 _2 0 1

54 54 2 _1 _ _2 _1 _2 _6 2 3 small_matrix D _1 2 0 _1 2 _1 _ _1 _2 _1 0 1 _2 _5 _1 _ _2 _5 2 6 _2 _ _ gausss elimination/ J gauss elimination gauss_elimination D _2 _ _1 _ _0.6 _ _ LAPACK LAPACK require addons/math/lapack/lapack.ijs require addons/math/lapack/dgeev.ijs,. }. dgeev_jlapack_ D _1 _1 _1j e_8 _1j_ e_ _ e_ j_ e_ j e_10 _ _ j e_ j_ e_ _ j_ e_9 _ j e_9 _ _

55 *5 (%. ev ) +/. * D261 +/. * ev domain error ( %.ev)+/.*d261+/.*ev gausscp_sub D _ _ _ _ _ _0.4 _ _ Jordan 4 EXample -1 4 λ 1 1 λ 1 λ 1 1, λ 1 1 λ 1 1 λ 1 λ 1, λ 1 1 λ 1 λ 1 λ 1 2 λ 1 small_matrix a _1 3 _1 _2 1 1 *5 }. dgeev_jlapack_ D _ _ _ a1 is eigenvector (%. a1) +/. * D260 +/. * a e_9 _3.8002e_ _

56 _1 _2 _1 1 1 _2 _4 _1 _ _2 _4 2 6 _2 _ _ λ A ( 1)I, (A ( 1)I) 2 A ( 1)I= _1 1 3 _1 _2 _1 _ _2 _4 A ( 1)I) 2 = a1=._1 small_pol D261 a1 +/. * a1 2 *6 2 A ( 1)I a1 +/. * a1 +/. * P ev= , , , ev *6 128!:0 a domain error 128!:0 a

57 P 1 AP (%. ev) +/. * D261 +/. * ev _ _ _ _1 P 1

59 /2 1/ blue red A0 1 1r2 1r2 1 A0 plot_vfield 5 pd eps /temp/vfield2_0.eps EXample

60 60 3 ({@> ;("1),. 5# <i. 5), L:0 {@>. i.5 a1 +/. * L:0 a r r r2 3 5r2 7r2 7r2 4 9r2 9r2 11r r r r2 1 3r2 3r2 5r2 2 7r2 5r2 9r r r

61 , A P 1,.P r P k = (A λ 1 I) k }{{} (A λ ri) (λ k λ 1) k }{{} (λ k λ r ) P 1 = (A λ 2I)(A λ 3 I) (λ 1 λ 2 )(λ 1 λ 3 ) P 2 = (A λ 1I)(A λ 3 I) (λ 2 λ 1 )(λ 2 λ 3 ) P 3 = (A λ 1I)(A λ 2 I) (λ 3 λ 1 )(λ 3 λ 2 ) Example a=.2 _2 3,1 1 1,:1 3 _1 2 _ _1 1 1 P 1 = (A λ 2I)(A λ 3 I) (λ 1 λ 2 )(λ 1 λ 3 ) = 1 (3 ( 2))(3 1) P 2 = (A λ 1 I)(A λ 3 I) (λ 2 λ 1 )(λ 2 λ 3 ) P 3 = (A λ 1I)(A λ 2 I) (λ 3 λ 1 )(λ 3 λ 2 ) = 1 (1 3)(1 ( 2)) A λi MAT _1 _2 3 4 _2 3 1 _2 3 = char_lf a _2 1 6 _5 _ f x = 6 5λ 2 lambda 2 + λ 3 Eigenvalue: = ( 2 3)( 2 1) = 1 6 =

62 _ _ _ λ LAMBDA _2 1 _ _ spectol a r2 1r10 2r5 0 11r15 _11r15 1r2 _5r6 1r3 1r2 1r10 2r5 0 1r15 _1r15 _1r2 5r6 _1r3 1r2 1r10 2r5 0 _14r15 14r15 _1r2 5r6 _1r A = λ 1 P 1 + λ 2 P 2 + λ 3 P 31 LF LF char_evec a _ _11 _3 5 _ _1 3 _ _ _ Φ(λ) = g 1 (λ) (λ λ 1 ) + + g r (λ) m1 (λ λ r ) mr Example issue:

63 D _ _1 2 1 (λ 2) 2 (λ 3) 1 A = (λ 2) 2 (λ 3) λ 2 + B (λ 2) + C 2 λ 3 1 = A(λ 2)(λ 3) + B(λ 3) + C(λ 2) 2 1 = (A + C)λ 2 + ( 5A + B 4C)λ + 6A 3B + 4C A +C = 0 5A +B 4C = 0 6A 3B +4C = a0=.1 0 1,_5 1 _4,: 6 _3 4 a _5 1 _4 6 _ %. a0 _1 _1 1 1 (λ 2) 2 (λ 3) = 1 λ 2 + = λ + 1 (λ 2) λ 3 1 (λ 2) λ 3

64 64 3 char_lf D _12 16 _ A 2I 2 small_pol D _ _1 0 P 1 = ( A + I)(A 3I) P 1 = P 2 = (A 2I) , P 2 = S = 2P 1 + 3P 2 2P 1 + 3P 2 = N = (A 2I)P 1 A = S + N (A 2I)P 1 = A = S + N = Reference 1994

65 a A = b b c, X = x 1 x 2 Example x x2 2 + x x 1x 2 + 4x 2 x 3 + 4x 1 x A = b LAPACK require addons/math/lapack/lapack.ijs require addons/math/lapack/dgeev.ijs ] ev=. ;{: dgeev_jlapack_ b _ _ _ _ _ _ char_lf b _1 _1 _5 _9 _ f (x) = 5 9λ 3λ 2 + λ 3 = 0 5, -1( ( : ev) +/. * b1 +/. * ev _ e_ e_ e_ e_ e_ e_16 _ x t Ax = y y2 2 1y2 3

66 A f f in trans f orm ), 3.2 fern /2 ( ( ) x1 w(x) = w = x 2 θ n ( ) a b = c d ( a b ) ( x1 c d x 2 ) + ( e f ( ) r1 cosθ 1 r 2 sinθ 2 r 1 sinθ 1 r 2 cosθ 2 ) = Ax + t Burnslay IFS Burnslay s f ern parameter of fern by Burnsray w a b c d e f p p

67 parameter calc FERN _ _ _ calc_ifs0 FERN _ _ _ view_pixel calc_ifs0 FERN Script NB section calc_ifs0=:4 : 0 NB. Usage: 100 calc_ifs0 FERN X0=. x Y0=. (\:({:"1 y)){ y NB. sort by prob AD=. 2 2 $ L:0 { 4{."1 Y0 NB. a b c d/box EF=. {4 5{"1 Y0 NB. e,f/box RND=. (? 100)% 100 NB. rand P0= / \{:"1 Y0 NB. probability/0.01 adjustment origin ANS=.< TMP=.(;{.EF) +(>{. AD) +/. *"(1) 0 0 NB. first for_ctr. i. X0 do. IND=.{.(((?100)%100) < P0) # i. # y TMP=. (;IND{EF)+ (>IND{AD) +/. * "1 TMP ANS=. ANS,<TMP end. ;("1),. ANS ) view_pixel=:3 : 0 gopen

68 68 3 gclear glpixel adj_pixel_sub y ) adj_pixel_sub=:3 : 0 TMP=. y MINMAX=. ;("1),.(<./y); >./y TMP=. TMP + -<./ {. MINMAX MINMAX=. ;("1),.(<./TMP); >./TMP TMP * TIMES=. <.<./1000 % {: MINMAX ) Reference Michael F. Barnsley [Fractals Everywhere 2nd Edition] 1993 Morgan-Kaufmann

69 / Ax = b (A t A)x = A t b A t x = (A t A) 1 A t b A = (A t A) 1 A t A A t A J %. y A = (A t A) 1 A t * < SVD (SVD /OLS OLS(ordinaly reast aquare equation) y X X 1 y 1 X *1 K.E.Iverson IBM IBM Iverson v (E.Mcdanell

70 x 1 y 1 1 x 2 α.. β = y 2. 1 x n y n y i = α + βx i Y = α + βx i + u i Y = α + β 1 X 1 + β 2 X 2 u y X 1 y 1, X regx=:3 : ({:"1 y ) %. 1,.}:"1 y NB. y %. 1,X x,y 1 2 x 1, x 2,, x n, y y Example D410 ( issue : shirasago) t Q P I T 1000 yen man en h cortable D410

71 4.1 / _ _ {"1 D410 P I Q=Y {"1 D410 P T Q=y reg {"1 D410 _ _ y = P I reg {"1 D410 _ y = P Y native reg mreg

72 72 4 X 1 1,. 1 3{"1 D410 X1 X X Q estimˆq ({."1 D410),. estim_reg {"1 D410 AIC AIC AIC / AIC *2 AIC MLL = 1 2 log Q n AIC = nlog Q + 2 (k + 1) n Q k + 1 *2 e- ai ci-

73 4.1 / 73 reg0 reg_exam_ad {"1 D410 reg0 reg_exam_ad {"1 D f= _ _ f= _ corr=: corr=: AIC: _ AIC: DW= DW= t=: _ _ t=: _ reg0 reg_exam_ad {"1 D f= _ corr=: AIC: _ DW= t=: _ AIC 3 P t 1

74 / 18 ( George King s Law,is An estimate of by how much a deficiency in the supply of corn will raise the price of corn. 10% 30% 20% 80% 30% 160% 40% 280% 50% 450% 450 KING line,marker plot { : KING pd eps temp/king0.eps 4.2 kings rule OLS y i = c 0 + c 1 x 1 + c 2 x *3 1 x 1 x 2 1 c 0 y 1 y 2 1 x 2 x c 1 =.. 1 x n xn 2 y n *3 Vandermonde Matrix

75 4.2 / KIngs rule 2 estim_poly KING estim_poly KING f = x x x speed and breaki KING ˆ/ % ˆ/ _

76 poly2 KING 28 _ f = x + 0.2x 2 f = x x x 3 3 poly2 KING e_ * 4 3 poly2 a e_ y = x x 3 ({."1 a1),. 3 estim_poly a1 _50 _450 _ _40 _280 _ _30 _160 _ _20 _80 _ _10 _30 _ e_ actual estim kings law extend to minus 3 ({:"1 a1)%. 1,.{."1 a e_ y = x AIC AIC MLL = n 2 log Q n AIC = 2 MLL + 2 (k + 1) Script NB. polynominal *4

77 4.2 / 77 ({:"1 a1),. +/"1 (({:"1 a1)%. 1,.{."1 a1)&p.1,.{."1 a1 _50 _450 _ _40 _280 _ _30 _160 _ _20 _80 _ _10 _30 _ ({:"1 a1)%. 1,.{."1 a e_ exact estim kings law extend to minus NB. Usage: e.g. 3 reg_poly y reg_poly=: 4 : 0 if. 1= +/ * $ y. do. x. poly1 y. else. x. poly2 y. end. ) NB polynominal regression poly1=: 4 : y. %. (>:i. # y.) ˆ/ i. >: x. poly2=: 4 : ({:"1 y.) %. ({."1 y.)ˆ/("1)i.>: x. estim_poly=: 4 : 0 NB. single type/ double type is same NB. 2 estim_poly KING if. 1=+/*$ y do.y0=. y.,. ((>:i. # y.)ˆ/ i.>: x.) +/. * x. poly1 y. else. Y0=. ({:"1 y.),. ((;}:"1 y.)ˆ/ i.>: x.) +/. * x. poly2 y. end. Y0 ) NB. modified estim_poly 5/Dec./ Reference Meyberk/Vachenauer

78 D430=: , y n k y n k y n (autocovariance function) y n k (autocorrelation), ACF(k) = Σn t=k 1 (Y t Ȳ)(Y t k Ȳ) Σ n t=1 (Y t Ȳ) 2,X(t), Cov[X(t 1 ), X(t 2 )] t 1, t 2, τ = t 1 t 2 pre f ix, su f f ix D430 J pre f ix, su f f ix (<\. D430),..(<\D430 <\D430 NB. prefix <\.D430 NB. suffix prefix t Y t, Y t 1, Y t 2, Y t 3...,

79 ACF Ȳ tmp=: ((<\.D430),..@(<\ ) D430)- L:0 (+/%#) D _ _1.5 _ _ _1.5 _ _1.5 _ _ _1.5 _ _1.5 _ _ _ _1.5 _ _ _ _ _ */ L:0,. ({."1 tmp),: L:0 {:"1 tmp _ _ _ _6.25 _0.75 _3.75 _ _

80 _ = ) ;+/ L:0 TMP _5.25 _ _6.25 = TMP1 % (+/)@:( ˆ&2)@ dev y NB. 1 _ _ _ ACF acf D430 NB. Cov % Var 1 _ _ _ A t A t A t A t 1 A t A t 2 A t A t 3 A t A t 4 A t A t ACF Script, ACF(k) = Σn t=k 1 (Y t Ȳ)(Y t k Ȳ) Σ n t=1 (Y t Ȳ) 2 SCRIPT acf=:3 : 0 NB. refine ACF2 code NB. ACF autocorellation coefficients (many times at once) C1=. ( <\. y ),..@(<\)y C3=. C1 - L:0 C2=. mean y NB. dev C4=.> +/ L:0 */ L:0 <"2 >C3 NB. change box style

81 NB. sum (Y_t-Y )*(Y_(t-k) - Y ) C4 % +/ (y - mean y )ˆ2 NB. C4 % VAR y ) Autoregression (autoregressive model,ar) AR Yule-Walker,, Yule Walker Yule-Walker ACF Yull-Walker C M a M = c M ACF c(k) = 1 N N k s=1 x(s)x(s + k), (k = 0, 1, 2,..., M) Yull-Walker c(0) c(1)... c(m 2) c(m 1) C M = c(1) c(0) c(1)... c(m 2) c(m 2)... c(1) c(0) c(1) c(m 1) c(m 2)... c(1) c(0) c M = (c(1), c(2),...c(m)) T ACF(k) C M, Yull-Walker yull-walker AR a 1 a 2.. a p = C 1 C 2.. C p ACF ACF acf a=: _ _ _ Yull-Walker Yull-Walker Yull-walker acf 3 C M c M c(0) c(1) c(2) c(1) c(1) c(0) c(1) c(2) c(2) c(1) c(0) c(3)

82 yw a NB. YW 1 _ _ _ _ _ _ _ _ cr=: %.}:"1 YW NB. Cramer Method 3 yw a 1 _ e_ e_16 _ _ e_ e_16 _ _ e_16 _ e_ y(t) = x t x t x t 3 Script yw_sub0=:4 : 0 NB. Yull- Waker sub NB. make C(m) Matrix NB. for univariate AR T1=: (.}. M), M=: y. NB. reverce and connect T2=:> x. # < T1 NB. copy x. vertical T3=: ({. $ T2), <: {: $ T2 NB. decrease $ to (0 _1) T4=:(-x.) {. ("1) T3 $, T2 NB. take backward (_x.) column ) yw=:4 : 0 NB. Yule Walker Method(Univariate) NB. Data type is Yoko list NB. X(t)=X1(t-1)+ X2(t-2) + X3(t-3)... NB. Usage: x. jisuu // y. Data Y1=: acf y. Y3=:}. Y2=: (>: x.) {. Y1 Y4=: (x. yw_sub0 Y2),. Y3 Y5=: cr Y4 NB. cramer decomposition )

83 NB. y4=: 4 : (x. yw_sub0 Y2),. }. Y2=:(>: x.) {. acf y. Yull-Walker,,, Reference Stephen A. DeLurgio [Forcasting Principles and Applications] McGRAW-HILL 1998

84 Working Example Q t = α 0 + α 1 P t + α 2 Y t + u 1t Q t = β 0 + β 1 P t + β 2 T t + u 2t Q t = P t = Y t = T t = Example D410 : 1998/2007 Haavelmo *5 α 0 + α 1 P t + α 2 Y t + u 1t = β 0 + β 1 P t + β 2 T t + u 2t α 1 P t β 1 P t = α 0 α 2 Y t + u 1t + β 0 + β 2 T t + u 2t P t (α 1 β 1 ) = α 0 + β 0 α 2 Y t + β 2 T t u t1 + u 2t P t = α 0 + β 0 α 1 β 1 + α 2Y t + β 2T t + u t1 + u 2t α 1 β 1 α 1 β 1 α 1 β 1 P t Q t = α 0 + α 1 P t + α 2 Y t + u 1t ( α0 + β 0 = α 0 + α 1 α 1 β 1 + α 2Y t α 1 β 1 + β 2T t + u 1t + u 2t ) + α2 Y t + u 1t α 1 β 1 α 1 β 1 = α 0 + α 1( α 0 + β 0 ) + α 1( α 2 Y t ) + α 1(β 2 T t ) + α 1( u 1t + u 2t ) + α 2 Y t + u 1t α 1 β 1 α 1 β 1 α 1 β 1 α 1 β 1 *5 Trigve M.Haavelmo Norway 1989

85 = α 0(α 1 β 1 ) + α 1 ( α 0 + β 0 ) + α 1( α 2 Y t ) + α 2 Y t (α 1 β 1 ) α 1 β 1 α 1 β 1 + α 1(β 2 T t ) + ( α 1u 1t + α 1 u 2t ) + (α 1 u 1t β 1 u 1t ) α 1 β 1 α 1 β 1 = α 1β 0 + α 0 β 1 α 1 β 1 + α 2β 1 Y t α 1 β 1 + α 1β 2 T t α 1 β 1 + α 1u 2t β 1 u 1t α 1 β 1 π 10 = α 1β 0 α 0 β 1 α 1 β 1 π 11 = α 2β 1 α 1 β 1 π 12 = α 1β 2 α 1 β 1 π 20 = α 0 + β 0 α 1 β 1 π 21 = α 2 α 1 β 1 π 22 = u 1t = u t1 + u 2t α 1 β 1 u 2t = α 1u 2t β 1 u 1t α 1 β 1 β 2 α 1 β 1 Q t = α 0 + α 1 P t + α 2 Y t + u 1t Q t = β 0 + β 1 P t + β 2 T t + u 2t Q t = π 10 + π 11 Y t + π 12 T t + u 1t P t = π 20 + π 21 Y t + π 22 T t + u 2t * Q t = α 0 + α 1 P t + α 2 Y t + u 1t Q t = β 0 + β 1 P t + β 2 T t + u 2t Q t α 1 P t = α 0 + α 2 Y t + u 1t 1 Q t β 1 P t = β 0 + β 2 T t + u 2t 2 Q t = P t = Y t = T t = (X t 1 ) * 7 *6 W.H.Green II(2000)p835 *7

86 [ Qt P t ] [ 1 2 α 1 [ Qt P t ] [ 1 β 1 [ Qt P t ] [ 1 1 ] = [ 1 Y t T t ] [ α 0 α 2 0 ] = [ 1 Y t T t ] [ β 0 α 1 β 1 Q 1 P 1 Q 2 P [ α 1 β 1 Q n P n 0 β 2 ] + u 1t ] + u 2t ] = [ 1 Y t T t ] [ α 0 β 0 ] = 1 Y 1 T 1 1 Y 2 T Y n T n α β 2 [ α 0 β 0 α β 2 ] + (u 1 + u 2 ) t ] + (u 1 + u 2 ) t YB = ZΓ + E Y = ZΓB 1 + EB 1 Y = ZΓB 1 Y = ZΓ B J B B Γ Γ OLS [ Qt P t ] [ 1 1 α 1 β 1 ] [ 1 Yt T t ] [ α 0 β 0 α β 2 ] b=.shira_sub20 D _ g=.shira_sub30 D _

87 B Γ B Γ A/B B Γ 1 α 1 α 0 α β 1 β 0 0 β 2 OLS OLS ( ) Q = α 0 + α 1 P + α 2 Y Q = β 0 + β 1 P + β 2 T shira_sub0 D410 _ _ _ OLS Q = P Y Q = P T B [ 1 α1 1 β 1 ] b=.shira_sub2 D _ B 1 B 1 β 1 α 1 α 1 β %. shira_sub2 D _

88 88 4 Γ [ α0 α 2 0 β 0 0 β 2 ] g=.shira_sub3 D410 shira_sub3 D410 _ _ Z Z B 1 Γ ( ) (%. b) +/. * g _ _ Y T Y T ˆQ (1,. 2 3 {"1 D410) +/. * : (%. b) +/. * g

89 YB = ZΓ Y = ZΓ B. B B 1 OLS OLS B Γ 1 Γ OLS YB = ZΓ Y = ZΓ B shirasago_model D Q P shirasago_model D YB = ZΓ Y = ZΓ B ZΓ B Y (Ŷ), ZΓ%.B 1 %. B Γ

90 90 4 B 1 α 1 1 β 1 B 1 1 β 1 α 1 α 1 β b _ ( ) ( ) %. b _ [ ] 1 β 1 α 1 α0 α 2 0 α 1 β β 0 0 β 2 Γ ( β 1, +α 1 ) (α 0, β 0 ) = α 0 β 1 + α 1 beta 0 = π 10 ( β 1, +α 1 ) (α 2, 0) = α 2 β 1 = π 11 ( β 1, +α 1 ) (0, β 2 ) = α 1 β 2 = π 12 ( 1, +1) (α 0, β 0 ) = α 0 + β 0 = π 20 ( 1, +1) (α 2, 0) = α 2 = π 21 ( 1, +1) (0, β 2 ) = β 2 = π 22 1 α 1 β 1 ) Klein Model Klein(1950) *8 Lawrence Robert Klein, (1920-) USA 1920 UC Berkley MIT Oxford ( ) ( ) ( ) ( ) ( ) C t = α 0 + α 1 P t + α 2 P t 1 + α 3 (WP t + WG t ) + e 1t I t = β 0 + β 1 P t + β 2 P t 1 + β 3 K t 1 + e 2t WP t = γ 0 + γ 1 X t + γ 2 X t 1 + γ 3 A t + e 3t X t = C t + I t + G t P t = X t T t WP t *8 June4 1992

91 ( ) K t = K t 1 + I t G T + WG A 1930 K P X 1 C t α 1 P t α 3 WP t = α 0 + α 2 P t 1 + α 3 WG t + e 1t 2 I t β 1 P t = β 0 + β 2 P t 1 + β 3 K t 1 + e 2t 3 WP t γ 1 X t = γ 0 + γ 2 X t 1 + γ 3 A t + e 3t 4 X t C t I t = G t 5 P t X t + WP t = T t 6 K t I t = K t 1 ( ) B = B C 1 I t WP t X t P t K t C α 3 0 α 1 0 I t β 1 0 WP γ X t P t K t Γ = Γ 1 P t 1 WG t K t 1 X t 1 A t G t T t C 1 α 0 α 2 α I t β 0 β 2 0 β WP t γ γ2 γ X t P t K t U = +(u 1 + u 2 + u 3 )i

92 92 4 OLS ]a=. klein_sub0 KLEIN (1) (2) _ (3) B klein_sub2 KLEIN 1 0 _ _ _ _ _1 _ _ _ Γ klein_sub3 KLEIN _ _ REG : (%. klein_sub2 KLEIN) +/. * klein_sub3 KLEIN (1) (2) (3) (4) (5) (6) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ transpose :

93 RMSE,. rmse klein_model KLEIN L.Klein Reference 1998/2007

95 Markov chain/ Andrei A.Markov( ) (1906) The Markov property means that,given present state,future states are independant of past state P(1, 1) P(1, 2) P(1, 3) P(2, 1) P(2, 2) P(3, 3) P = P(3, 1) P(2, 2) P(3, 3).... Example ] M0=. 1r2 1r4 1r4,1r2 0 1r2,:1r4 1r4 1r2 Rain Nice S now Rain 1r2 1r4 1r4 Nice 1r2 0 1r2 S now 1r4 1r4 1r2 _1 x: M0 ([ +/. * ])ˆ:10 M

96 96 5 issue:kemeny,snell,thompson[introduction to Finete Mathematics 3rd.ed.] 1974 Prentishall P n+1 = P n P n π = πp markov_loop M P n

97 5.1 Markov chain/ char_lf M r4 _1r4 1r16 _1r16 _ cahr_evec M r4 _1r r8 3r16 3r8 _3r16 0 3r16 1r16 _1r8 1r16 3r8 3r16 3r _1r4 1r2 _1r4 3r8 3r16 3r8 3r16 0 _3r16 1r16 _1r8 1r f (λ) = λ λ2 + λ 3 Eigenvalue is a1=. {.;{.{: char_evec D510 3r8 3r16 3r8 _1 x: a1 % +/ a * / (A,B,C,D) M3 *1

98 98 5 Beer A B C D A M3 =: B C D {: 100 markov_loop M3 A B C D (issue: 7.1) A 48% B 25%,C 18% D 9% [ ] M p 1 q q 2 0 q P = q q 3 0 q q 4 q p i + q i = 1 p 5 = 1

99 5.1 Markov chain/ 99 p 1, q i p 1 = 0.03 q 1 p 2 = 0.20 q 1 q 2 p 3 = 0.60 q 1 q 2 q 3 p 4 = 0.15 q 1 q 2 q 3 q 4 p 5 = 0.15 p 1, q i p 1, q i r_markov M ;{: 100 markov_loop r_markov M4 NB Reference Script ] M0=: 1r2 1r4 1r4,1r2 0 1r2,:1r4 1r4 1r2 mmp=: [ +/. * ] NB. M0 +/. * M0 +/. * M0 NB. M0 mmp (ˆ:10) M0 markov_loop=: 4 : 0 NB. markov chain NB. x is time of loop NB. y is markov matrix TMP=. y +/. * y ANS=. y ; TMP for_ctr. i. x do. TMP=. TMP +/. * y

100 100 5 ANS=.ANS,<TMP end. ANS=._1 x: L:0 ANS NB. rational to natural number TMP1=.reform_sub L:0 ANS NB. pick-off dust,.(;$ L:0 {. ANS)$ L:0 TMP1 NB. reform ) reform_sub=: 3 : (0) (I. (,y) < 1e_6)},y

101 ( 1 Q 1 Q n 0(n ) I + Q + Q 2 + = (I Q) 1 *2 issue: Hirashita p49 D524 age age mat sub D524 death age0 age1 age2 age3 death age age age age = 1 0 R Q Q = 0 r r r , Q 2 = 0 0 r 0 r r 1 r , Q 3 = r 0 r 1 r , r 0 + r 0 r 1 + r 0 r 1 r r 1 + r 1 r r *2

102 markov_loop a />2 markov_loop a alive D I Q (I Q) 1 NB. I Q matrix Q (I Q) 1 alive0 D524 N

103 P.H.Leslie( ) Oxford Oxford Bureau o f Animal Population 1940 Bxernardelli(1941),Lewis(1942),Leslie(1945) 1959 Leslie 1966 J.H. Pollard stocastic version time t t+1 n1 f1 n1 n2 n3 n-w-1 n-w f2 f3 fw-1 fw pw-2 p1 p2 p3 n2 n3 n-w-1 n-w n 1 n 2 f 1 f 2 f 3 f k 1 f k p n 1 n 2 n 3 n k 0 p 2 (t + 1) = 0 0 p 3 n P k 1 0 n k (t) RACCOON Leslie 5 RACCOON ; ; Leslie Matrix leslie_mat }. RACCOON

104 a4 * P NB. birth NB. die NB. die a4 +/. * P0 NB leslie_loop RACCOON t old total(n)

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1 Leverrie Faddeev 1.1 2007 John Randell NJ Rutgers Univ. (characteristic equation) AX = λx λ : A X : λ (A λi n )X = 0 f (λ) = A λi n = 0 a 11 a 12 a

1 Leverrie Faddeev 1.1 2007 John Randell NJ Rutgers Univ. (characteristic equation) AX = λx λ : A X : λ (A λi n )X = 0 f (λ) = A λi n = 0 a 11 a 12 a Masato Shimura JCD02773@nifty.ne.jp 20 1 24 1 Leverrie Faddeev 2 1.1.......................... 2 1.2 LF....................... 3 1.2.1........................... 3 1.2.2.................. 6 1.3................................