# БЫСТРЫЙ АЛГОРИТМ ПРОВЕРКИ ВЫРОЖДЕННОСТИ ГАНКЕЛЕВЫХ МАТРИЦ

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• 12 2 (2011)

VIII : ,

190- 120-

510.52

.., .. (. ) ykuz@niisi.ras.ru,petrunin@niisi.ras.ru

2009-2010 .. (

- ..) , -

-

. -

O(n log2 n), -

468.

- -. [1]-[3] 2009-2010 .. ( - ..) - - , . , ( ) .

.

-

, 09-01-00287- 11-01-12002---2011

• ... 61

C R - . - : . [4].

- ( O(n log2 n)), - , - [6].

C O(n log2 n). -. C 468. , (. [8]).

-.

. a b, deg a > deg b.

u0 = a, u1 = b,

v0 = 0, v1 = 1,

w0 = 1, w1 = 0,

ui+1 = ui1 qiui,vi+1 = vi1 qivi,

wi+1 = wi1 qiwi,

qi , ui1 ui, ui+1 .

, i 0

wiu0 + viu1 = ui,

(vi, wi) = 1,

i = k, uk+1 = 0, ..uk uk1. uk (a, b), wk vk , wku0 + vku1 = (a, b).

2 ui ui+1 {uj}, ui+1, ui+2 (

ui+1ui+2

)=

(0 11 qi+1

)(uiui+1

)

• 62 . . , . .

(uiui+1

)=

(qi+1 11 0

)(ui+1ui+2

)

(q 11 0

), deg q > 0 , -

. ,

(ab

)= Q

(uiui+1

), i, Q

Q1 =

(wi viwi+1 vi+1

).

, , ,

(ab

)=

Q

(a

b

), Q deg(a) > deg(b), a b

, .. i a = ui b = ui+1 [7]. - [5]. -

a b, deg(a) > deg(b), Q

,

(ab

)= Q

(uiui+1

), deg(ui+1) < deg(a)2 deg(ui).

- - , - .

- - , , . 1973 . , - , - . - , [5] 8- , , , . 1980- , [6], , - , . , , . - . [5] . , 10 - . . . [7], , . -, .

- [7], HGCD. , EMGCD [6]

• ... 63

. HGCD, ui, ui+1, .

HGCD

: a, b, deg b < deg a = n.: Q , deg(c) n/2 > deg(d), (

c

d

)= Q1

(ab

).

HGCD(a, b):

1. m := deg a2

; deg(b) < m, E;

2. a0 := a div xm; b0 := b div xm;

R := HGCD(a0, b0);(a

b

):= R1

(ab

);

3. deg(b) < m, R;

4. q := a div b;

(cd

):=

(b

amodb

);

5. l := deg(c); k := 2m l;6. c0 := c div xk; d0 := d div xk;

S := HGCD(c0, d0);

7. Q := R (q 11 0

)S;

Q.

, -- O(n log2 n). - .

1. - HGCD -

n 117n log22(n) + O(n log(n)) n = 2k 234n log22(n) +O(n log(n)) n.

. T (n) HGCD - a, b, , n = deg a > deg b, M(n) n.

n = 2k.

• 64 . . , . .

T (n) HGCD. 2 a0 b0, , -

. a0, b0 T (n

2), .. deg(b0) < deg(a0) = nm = n2 .

, R , -, Q, . 7 :

Q := S (0 11 q

)R.

, R. , R ( ), R1. 1 [7]. -, R deg(a0) n2 .

, 2 4M(n) +O(n).

4 a div b amodb. D(n) f -

g, deg(g) = n deg f 2n1. , m deg(b) < deg(a) deg(uj).

, .

1. - n n 234n log22(n) + O(n log2(n)) n = 2k 468n log22(n) +O(n log2(n))

. a b, HGCD - n- uj . , , HGCD - 2n+1 468n log22(n)+O(n log2(n)). n = 2k 1 234n log22(n) +O(n log2(n)).

[1] . . , . . -, S- - , . ., 200, 11, 2009, 15-44

[2] . . , , , 430, 3, 2010, 318-320

[3] . . , . . -, - - , ,433, 2, 2010, 154-157

[4] Tyrtyshnikov E.E. Fast algorithms for toeplitz and quasi-toeplitz systems. J.Numer. Anal. Math. Modelling, 4, 5, 1989, 419-430.

[5] Aho A. V., Hopcroft J. E., Ullman J. D., The design and analysis of computeralgorithms. Addison-Wesley, Reading, Mass., 1976.

[6] Richard P. Brent, Fred G. Gustavson, David Y. Y. Yun, Fast solutionof Toeplitz systems of equations and computation of Pade approximants,Journal of Algorithms 1, 1980, 259-295. ISSN 0196-6774. MR 82d:65033. URL:http://web.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub059.html.

[7] K. Thull and C. Yap. A unied approach to HGCD algorithms for po-lynomials and integers. Manuscript, 1990, http://cs.nyu.edu/cs/faculty/yap/allpapers.html/.

• ... 67

[8] . . , - , , 22, 1, 2010, 17-49

[9] .., , -, , 2008, 277-289, http://parallel.ru/info/VVV/18.pdf

[10] Bernstein D. J., Fast multiplication and its applications. In: AlgorithmicNumber Theory. Lattices, Number Fields, Curves and Cryptography (Mathe-matical Sciences Research Institute Publications), 44 (Buhler J. P. et al., eds.).Cambridge, 2008, pp. 325-384.

17.10.2011