(I) GotoBALS, kkimur/charpoly.html 2
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- かずゆき こうじょう
- 7 years ago
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1 sdmp Maple - (Ver.2) ( ) September 27,
2 (I) GotoBALS, kkimur/charpoly.html 2
3 (II) Nehalem CPU GotoBLAS Intel CPU Nehalem CPU, GotoBLAS, Hyper-Thread technology off 3
4 Windows BIOS off Mac Hyper-Thread technology off Mac OS root, terminal, nvram SMT=0 OFF 4
5 Maple,, 4, sdmp rpearcea/ 4,? 5
6 4 (1) (2) (3)Cylindrical Algebraic Decomposition Projection (4)Dixon (5), 6
7 Dixon f(x, y) = x 2 + y 2 3, g(x, y) = xy 1 f(x, y) g(x, y) f(α, y) g(α, y) x α = ( α 1 ) y 1 1 y 3 + 3y = 0 x 1 α 7
8 y = λ, v = x 1, λ 1 1 λ 3 + 3λ v = 0 λ 1 1 λ 3 + 3λ = 0, y,, 3 8
9 (1) (2) 1 (2+3x)(14+15x) x 4+5x (6+7x)(14+15x) 11+12x+13x 3 (2+3x)(16+17x) 6 + 7x (2 + 3x)(4 + 5x) 8(2 + 3x)( x) (9 + 10x)( x + 13x 3 ) (3) a b(4 + 5x) 8c ad( x + 13x 3 ) a = x, b = x, c = x, d = x, 9
10
11 1., O(2 n ), 2. -,???, 3.fraction-free Gauss, O(n 3 ), 4.,???, mod 5.Berkowitz, Fadeev, O(n 4 ), 10
12 fraction-free Gauss n n C, ĉ ( 1) 0,0 = 1, ĉ (0) i,j = C i,j, ĉ (k) i,j = (ĉ(k 1) k,k ĉ (k 1) i,j ĉ (k 1) k,j ĉ (k 1) i,k )/ĉ (k 2) k 1,k 1 k + 1 i n, k + 1 j n, k = 1,, n 1, ĉ n,n (n 1), 0,, Jacobi,, 11
13 fraction-free Gauss n n C, τ 0,0 ( 1) = 1, τ i,j (0) = C i,j, τ i,j (k) = (τ k,k (k 1) τ i,j (k 1) τ k,j (k 1) τ i,k (k 1) )/τ k 1,k 1 (k 2) k + 1 i n, k + 1 j n, k = 1,, n 1, τ (n 1) n,n, 0,, Jacobi,, 12
14 Berkowitz (I) A = T A = A r S R a r+1,r a r+1,r RS a r+1,r RA r 2 r S RA r 3 r S 1 RA r 1 r S RA r 2 r S a r+1,r+1 13
15 Berkowitz (II) [procedure Berkowitz(A)] if dim A=1 then return [1, A 1,1 ] else r (dim A) 1 Decompose A = A r S R a r+1,r+1 Compute T A [X r+1,, X 1 ] Berkowitz(A r ) [X r+2,, X 1 ] T A [X r+1,, X 1 ] return [X r+2,, X 1 ] 14
![R a r+1,r+1 Compute T A [X r+1,, X 1 ] Berkowitz(A r )](/docs-images/49/20185101/images/page_15.jpg "[X r+2,, X 1 ] T A [X r+1,, X 1 ] return [X r+2,, X 1 ]")
16 ? 1., , 15
17 Resultant( ) 1.Collins subresultant
18 f = a 2 x 2 + a 1 x + a 0,g = b 1 x + b 0 Sylvester res x (f, g) = a 2 a 1 a 0 b 1 b b 1 b 0 m n (m + n) (m + n) 17
19 f = a 2 x 2 + a 1 x + a 0,g = b 1 x + b 0 Bezout res x (f, g) = b 0 b 1 b 1 a 0 b 0 a 2 b 1 a 1 m n (m > n) m m 18
20 1., 2. - Quantifier Elimination Q.E., Cylindrical Algebraic Decomposition C.A.D., Resultant( ) 19
21 a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 c 1 c 2 c 3 c 4 d 1 d 2 d 3 d 4 = +a 1 (b 2 (c 3 d 4 d 3 c 4 ) c 2 (b 3 d 4 d 3 b 4 ) +d 2 (b 3 c 4 c 3 b 4 )) b 1 (a 2 (c 3 d 4 d 3 c 4 ) c 2 (a 3 d 4 d 3 a 4 ) + d 2 (a 3 c 4 c 3 a 4 )) +c 1 (a 2 (b 3 d 4 d 3 b 4 ) b 2 (a 3 d 4 d 3 a 4 ) +d 2 (a 3 b 4 b 3 a 4 )) d 1 (a 2 (b 3 c 4 c 3 b 4 ) b 2 (a 3 c 4 c 3 a 4 ) + c 2 (a 3 b 4 b 3 a 4 )) (bottom up),! 20
22 - L 1 = a 4,4 a 5,5 a 4,5 a 5,4, L 2 = a 4,3 a 5,5 a 4,5 a 5,3, L 3 = a 4,3 a 5,4 a 4,4 a 5,3, L 4 = a 4,2 a 5,5 a 4,5 a 5,2, L 5 = a 4,2 a 5,4 a 4,4 a 5,2, L 6 = a 4,2 a 5,3 a 4,3 a 5,2, L 7 = a 4,1 a 5,2 a 4,2 a 5,1, L 8 = a 4,1 a 5,5 a 4,5 a 5,1, L 9 = a 4,1 a 5,4 a 4,4 a 5,1, L 10 = a 4,1 a 5,3 a 4,3 a 5,1 det(a) = +(a 1,1 a 2,2 a 1,2 a 2,1 )(a 3,3 L 1 a 3,4 L 2 + a 3,5 L 3 ) (a 1,1 a 2,3 a 1,3 a 2,1 )(a 3,2 L 1 a 3,4 L 4 + a 3,5 L 5 ) +(a 1,1 a 2,4 a 1,4 a 2,1 )(a 3,2 L 2 a 3,3 L 4 + a 3,5 L 6 ) (a 1,1 a 2,5 a 1,5 a 2,1 )(a 3,2 L 3 a 3,3 L 5 + a 3,4 L 6 ) +(a 1,2 a 2,3 a 1,3 a 2,2 )(a 3,1 L 1 a 3,4 L 8 + a 3,5 L 9 ) (a 1,2 a 2,4 a 1,4 a 2,2 )(a 3,1 L 2 a 3,3 L 8 + a 3,5 L 10 ) +(a 1,2 a 2,5 a 1,5 a 2,2 )(a 3,1 L 3 a 3,3 L 9 + a 3,4 L 10 ) +(a 1,3 a 2,4 a 1,4 a 2,3 )(a 3,1 L 4 a 3,2 L 8 + a 3,5 L 7 ) (a 1,3 a 2,5 a 1,5 a 2,3 )(a 3,1 L 5 a 3,2 L 9 + a 3,4 L 7 ) +(a 1,4 a 2,5 a 1,5 a 2,4 )(a 3,1 L 6 a 3,2 L 10 + a 3,3 L 7 ) 21
23 =, Lisp Lisp Hash Hash,? 22
24 index,
25 w[k], index, (2 3 4), c[i][j] = i j 1 n=5; i=3; u=1; b1=n-1; for(k=i;k>0;k--){ for(b2=b1;b2>=w[k];b2--){ u=u+c[b2][k]; } b1=b2-1; } 24
26 geobucket,, 25
27 , 1 heap (sdmp ) (Singular ) (f 1 + f 2 ) (g 1 + g 2 ) = f 1 g 1 + f 1 g 2 + f 2 g 1 + f 2 g 2 26
28 ?, f = 5x 2 y + 6x + 7y + 8 (5, [2, 1]) (6, [1, 0]) (7, [0, 1]) (8, [0, 0]) 27
29 (heap ) X j, Y j,, m n (X 1 + X X m )(Y 1 + Y Y n ) = X 1 (Y 1 + Y Y n ) + X 2 (Y 1 + Y Y n ) + X m (Y 1 + Y Y n ) 28
30 (X 1 Y 1 + X 1 Y X 1 Y n ) + (X 2 Y 1 + X 2 Y X 2 Y n ) + (X m Y 1 + X m Y X m Y n ) Y 1, Y 2,, Y n,, X j,, m 29
31 , =m, 1,, =1, 2, 2 root,, 2, root 2, 2,, 2, 30
32 sdmp, 31
33 heap u = ( a 1 b 1 c 1 ) v = b 2 b 3 c 2 c 3 a 2 a 3 c 2 c 3 a 2 a 3 b 2 b 3 a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 b 2 b 3 c 2 c 3 b 1 a 2 a 3 c 2 c 3 + c 1 a 2 a 3 b 2 b 3 = u v 32
34 u[1] v[1], u[2] v[2], u[3] v[3], u[1] v[1], u[2] v[2], u[3] v[3],, 1 33
35 sdmp Maple, kkimur/susemi etc.html 1.sdmp 2. 1,, 34
36 sdmp sdmp, exponent vector 1 word (pack ) f = 5x 2 y + 6x + 7y + 8 (5, [2, 1]) (6, [1, 0]) (7, [0, 1]) (8, [0, 0]) 35
37 32bit, pack=2 16bit 16bit 2 1,, exponent vector 36
38 sdmp 1 + x 2 + 2x 2 + 2x 4 + 4x = 0,, 37
39 a 1 + a 2 x a 3 + a 4 x a 5 + a 6 x a 7 + a 8 x generic position ( ) 38
40 , O(n 2 ) 39
41 , Linear Assignment Problem 40
42 Hungarian algorithm ,, Hungarian algorithm, O(n 3 ) bound cf. Hungarian algorithm, maple, 41
43 1., kkimur/susemi etc.html 4. kkimur/susemi interpolation.html,, 42
1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac
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Google (20/08/2 ) ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67 Introduction ( ) Random Walk & Google Page Rank Agora on Aug. 20 2 / 67 Introduction Google ( ) Random Walk & Google Page Rank
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White-Box Takayuki Kunihiro Graduate School of Pure and Applied Sciences, University of Tsukuba Hidenao Iwane ( ) / Fujitsu Laboratories Ltd. / National Institute of Informatics. Yumi Wada Graduate School
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Appendix A BASIC BASIC Beginner s All-purpose Symbolic Instruction Code FORTRAN COBOL C JAVA PASCAL (NEC N88-BASIC Windows BASIC (1) (2) ( ) BASIC BAS
Appendix A BASIC BASIC Beginner s All-purpose Symbolic Instruction Code FORTRAN COBOL C JAVA PASCAL (NEC N88-BASIC Windows BASIC (1 (2 ( BASIC BASIC download TUTORIAL.PDF http://hp.vector.co.jp/authors/va008683/
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Section Section Section Table of Contents Section 6 Section 7 Section 8 Section 4 Section 0 Section 5 Section 9 Section Section Section Section 4 Section 5 Section 9 Section 0 Section Section 6 Section
24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
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1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C
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C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 新装版 1 刷発行時のものです.
C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009383 このサンプルページの内容は, 新装版 1 刷発行時のものです. i 2 22 2 13 ( ) 2 (1) ANSI (2) 2 (3) Web http://www.morikita.co.jp/books/mid/009383
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