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05.13.18 - ,

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29 2012 . 14.00 212.142.03 ͻ :

127055, , , . 3.

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10 2012 .

212.142.03 .

[ ] Ai Bi Yi+1 = WiYi, i = 0, 1,...,, N, Wi = Ci Di, 2n 2n [ ] 0 I WiT JWi = J, J =, i = 0,..., N.

-I 2n, n N, -(p(t)x(t)) = x(t), x(0) = x(l) = 0.

[ ] -C(t) A(t)T JT y = H(t)y, H(t) =, H(t)T = H(t) A(t) B(t) H(t), (1) (0) -(ri xi) + ri xi+1 = (i, i + 1] (1) xi+1 = 0, xi/(ri xi+1) xi = 2n n Y (t) = [X(t)T U(t)T ]T rangY (t) = n, XT (t)U(t) = UT (t)X(t), det X(t) = 0 m(t) = defX(t) = n rangX(t).

Yi = [XiT UiT ]T (i, i + 1] KerXi+1 KerXi, XiXi+1Bi 0, KerA A, A A A A = AT m(Yi) (i, i + 1] N F(x) = f(i, x(i + 1), x(i)) inf x(i) Rn i= i = 0,..., N + N F2(z) = {zT (i + 1)Q(i)z(i + 1) + 2zT (i + 1)R(i)z(i) + i=+ (z(i))T P (i)z(i)}.

F2(z) (1) (0) -(ri xi) + ri xi+1 = xi+1, x0 = = xN+1 = 0, = b, b R (0, N +1] [ ] (1) 1 1/ri yi+1 = yi, y0 = [0 1]T, (0) (0) (1) ri - 1 + (ri - )/ri = b.

, [a, b] R,, [i, i + 1) (0, N + 1] [0, N + 1).

(a, b] (-, a] (-, b] a, b R.

(a, b] 27 5 1 400 14 5 H(t), (t), H(t) (t), B(t) 0, t [t0, t1] Y (t), (t) l0(, t0, t1) - l0(Y, t0, t1) ind(Q - Q)(t0+) - ind(Q - Q)(t1-) l0(, t0, t1), l0(Y, t0, t1) Y (t), (t) (t0, t1) t0+, t1- indA A (t) H(t), (M, N + 1], 2n n Y = [XT UT ]T, = [XT T ]T, 2n n Y = [XT UT ]T, = [XT T ]T T w = w(Y, ) = Y J.

(Y, ) = 1(Y, ) + 2(Y, ) ( ) 1(Y, ) = rangM, M = I - XX w, 2(Y, ) = indP, ( ) T P = T wT XX T, T = I - MM.

P = PT M ( ) M = I - XX X.

1(Y, ) = 0 Im(X) Im(X) ImA A T = I, (Y, ) = 2(Y, ) = ind(XT [Q - Q]X) Q, Q XT QX = XT U, XT QX = XT .

P ( ) T P = T XT [Q - Q]X T Q, Q (Y, ) = 0 Im(X) Im(X), XT (Q - Q)X 0.

(Y, ), (Y, ) = (DY, D ), D = diag(I, -I).

Z, Y, Y = Z[0 I]T, = [0 I]T, k(Y C1, C2) = k(Y, ), k = 1, 2, C1, C k(LY, L ) = k(Y, ), k = 1, 2, L k(Z[0 I]T, [0 I]T ) = (Z-1[0 I]T, Z-1[0 I]T ), k = 1, 2.

k 2(Z[0 I]T, [0 I]T ) = ([0 I]T, Z[0 I]T ), (Y, ) + (, Y ) = rang(w(Y, )), 1(Y, ) = rangX -rangX +1(, Y ) (Z[0 I]T, [0 I]T ) = ([0 I]T, [0 I]T ) - ([0 I]T, Z[0 I]T )+ +([0 I]T, Z[0 I]T );

0 (Y, ) min(rangw, rangX), w Z, , W (W Z[0 I]T, W [0 I]T ) = (Z[0 I]T, [0 I]T ) + (W Z[0 I]T, W [0 I]T )-(W [0 I]T, W [0 I]T ).

2n.

(Y, ), (Y, ) (Y, ) = i-(), (Y,] ) = i+[ ] (), [ [ ] 0 (I - XX)X X X =, Y =, Y =, U XT (I - XX) XT (Q - Q)X i-() = ind() i+() .

Y, [ ] [ ] [ ] 0 I X 0 w(Y, ) T [V ] = V V - = [V ]T, V = [Y ], Y =, 0 0 U 0 w(Y, ) (Y, ) = ind([V ]) - ind(XT U).

2n 2n W 4n 2n I [ ] A B A B W =.

0 -I, W = C D C D W , n 2n, 4n 2n W , W, W , (W , ) rang(W - ), -1 - (W , ) = (W , ), (W , ) = rang(B) - rang(B) + ( , W ) - (W , ) = (W [0 I]T, [0 I]T ) + ( W , I) = -1 -= (W [0 I]T, W [0 I]T ) + (W , I).

W, Z, , (W Z[0 I]T, [0 I]T ) = (Z[0 I]T, [0 I]T ) + (W Z[0 I]T, W [0 I]T )-( [0 I]T, [0 I]T )+ -+(W , ) - (-1 W Z, -1Z), -(W , ) - (-1 W Z, -1Z) = -= (-1Z, -1 W Z) - ( , W ).

-|(W , ) - (-1 W Z, -1Z)| rang(W - ), W = m(Yi) (i, i + 1] Zi Yi = Zi[0 I]T, -mk(Yi) = k(Zi+1[0 I]T, Wi[0 I]T ) = (Zi+1[0 I]T, Zi-1[0 I]T ), k = 1, 2, k m (Yi) = m1 (Yi) + m2 (Yi) (i, i + 1] Yi, = 1 + 2 .

Yi, i m(i) (i, i + 1] m(Yi) m(i) - m(Yi) + (i) = 0, (i) = (Yi, i) N l(, M, N) - l(Y, M, N) = (M) - (N + 1), l(Y, M, N) = m(Yi), i=M l(Y, M, N) Yi (M, N + 1], l(, M, N) |l(Y, M, N) - l(, M, N)| rangw(Yi, i) n.

H(t), (t).

Zi, i i+1 = ii Yi = Zi[0, I]T, i = i[0, I]T.

#i(, Z) = (Gi, Gi+1) - (i, Wi), Gi = i-1Zi, W m(i) (i, i + 1] m(Yi) m(i) - m(Yi) + (i) = #i(, Z), l(, M, N) - l(Y, M, N) = (M) - (N + 1) + #(, Z, M, N), N #(, Z, M, N) = #i(, Z), i=M |l(, M, N) - l(Y, M, N) - #(, Z, M, N)| n.

#(, Z, M, N) (Wi, i) 0, l(, M, N) - l(Y, M, N) (M) - (N + 1) Yi(M) (M) M, YM = [0 I]T :

i(N+1), Zi(M) (M) (N+1) ZM [0 I]T = ZN+1 [0 I]T = [0 I]T.

(M) (M) l(, M, N), l(Y, M, N) (M, N + 1] (M) (M) l(, M, N) - l(Y, M, N) = #((N+1), Z(M), M, N), #((N+1), Z(M), M, N) i(N+1), Zi(M).

(i, Wi) 0 (Wi, i) l(Y, M, N) = 0, (Wi, i), i = M,..., N.

m(Yi) [i, i + 1) (i, i + 1) [ ] T T Di -Bi Yi = Wi-1Yi+1, Wi-1 =.

-CiT AT i Yi Yi i i + Yi m(Yi) [i, i+1), m (Yi) = m (Yi) + m (Yi), 1 ( ) T m(Yi) = rangMi, Mi = I - XiXi Bi, Ti = I - MiMi, T m(Yi) = ind(TiT Xi+1XiBi Ti), Bi Wi.

Zi Yi = Zi[0 I]T, -m(Yi) = (Zi[0 I]T, Wi-1[0 I]T ) = k(Zi-1[0 I]T, Zi+1[0 I]T ), k = 1, 2, k k m (i) = m (i) + m (i) [i, i + 1) Yi, 1 = 1 + 2 .

m(Yi), m(Yi) :

m(Yi), m(Yi) (i, i + 1] [i, i + 1) m(Yi) - m(Yi) = rang(Xi+1) - rang(Xi), l(Y, M, N) - l(Y, M, N) = rang(XN+1) - rang(XM), N l(Y, M, N) = m(Yi).

i=M (M) l(Y, M, N) (N+1) Yi(M) (M, N + 1], l(Y M, N) Yi(N+1) [M, N + 1), (M) (N+1) l = l(Y, M, N) = l(Y, M, N).

Ri i -i = Ri Yi Yi i - i+1 = Wii, Wi = Ri+1WiRi.

- Yi Ri Yi.

Ri Wi i + - Ri J.

B(t) 0, C(t) 0, Y (t), JY (t) t +.

Yi Ri :

- Yi i = Ri Yi.

-1 -1 -m(Ri Yi) - m(Yi) - (Ri Yi, Ri [0 I]T ) = ui, - ui = (Ri+1[0 I]T, Wi[0 I]T ) - (Ri[0 I]T, Wi-1[0 I]T ), - Wi = Ri+1WiRi, - m(Yi) m(Ri Yi) (i, i + 1] Yi i (M, N + 1] N -1 -l(R-1Y, M, N) - l(Y, M, N) - (Ri Yi, Ri [0 I]T )|N+1 = ui, M i=M fi|K = fK - fM.

M i - (Ri+1[0 I]T, Wi[0 I]T ) = (Ri[0 I]T, Wi-1[0 I]T ).

i +.

- Ri J, [ ] Ai Bi Wi = Ci Di T ind(-AT Ci) = ind(AiBi ) i, Yi, JYi i i +.

R yi+1() = Wi()yi, i = 0,..., N, yi = [xi ui]T, x0 = xN+1 = 0, i+1() = i()i, i = 0,..., N, i = [xi i]T, x0 = xN+1 = 0, [ ] I T Wi() = Si, Si JSi = J, Wi = WiT, Wi 0, -Wi I i().

1, #{ l| J}, l = 1, J.

Wi() i() :

(a, b] N #{ 1|a < b} = (Gi[0 I]T, Gi+1[0 I]T ), ( )-1 i=Gi = Zi(N+1)(b) Zi(0)(a), b a, Zi(N+1)(), Zi(0)() (N+1) (0) ZN+1 (b)[0 I]T = Z0 (a)[0 I]T = [0 I]T.

Wi(b), Wi(a), (0) (0) a, b (Zi+1(a))-1 Wi(b)Wi-1(a) Zi+1(a).

[a, b).

Zi(0)(), i(N+1)() Yi(0)() = Zi(0)()[0 I]T, i(N+1)() = i(N+1)()[0 I]T Gi = i(N+1)(b)-1Zi(0)(a) a, b, P, a, b R #{ 2| b} - #{ 1| a} = #(Z(0)(a), (N+1)(b), 0, N) - P, N #(Z(0)(a), (N+1)(b), 0, N) = {(Gi, Gi+1) - (i(b), Wi(a))}, i= P = #(Z(0)(0), (N+1)(0), 0, N) = p - p, (N+1) (0) p = l( (0), 0, N + 1), p = l(Y (0), 0, N + 1), 0 < min{ 1 2}.

yi+1 = Wi()yi, i = 0,..., N, R0x0 - R0u0 = 0, RN+1xN+1 + RN+1uN+1 = 0, yi+1 = Wi()yi, i = 0,..., N, N+R0x0 - R0u0 = 0, RN+1xN+1 + RN+1uN+1 = 0, T T R0R0 = R0R0 T, RN+1RN+1 = RN+1RNT, rang[R0 -R0] = rang[RN+1 RN+1] + N+= n R0, R0, RN+1, RN+1.

1 = 2 = .

0 k 2n k = rangwL + rangwR, wL = [R0 R0]J[R0 R0]T, wR = [-RN+1 RN+1]J[-RN+1 RN+1]T, i+k 2, i 1, i 1, i i+k. i+k 1, i 1, i 2, i i+k.

[-1, N +2] i = 0,..., N i = -1, i = N + N+ p = m(i(0)) i=- (-1, N +1] -1(0) = [0 I]T.

p = 0, ([R0 R0]T, [R0 R0]T ) = ([-RN+1 RN+1]T, [-RN+1 RN+1]T ) = i , i 1, i , i 1, i i.

#{ 1| } , p N {0}, b R (0) l(Y (b), 0, N) = #{ 1| b} + p, N (0) l(Y (), 0, N) = m(Yi(0)()), i= (0, N + 1] Yi(0)() #{ 1| < } p N {0}, b R (0) l(Y (b), 0, N) = #{ 1| < b} + p, N (0) l(Y (), 0, N) = m(Yi(0)(), i= [0, N + 1) N0 [0, N + 1].

.

[a, b] p, p m(Y ) [i, i + 1) i Yi = [XiT UiT ]T [ ] T -XiT Ui UiT Bi [Vi] =, i = 0,..., N, BiUi BiAT i Hi ki ki -XiT Ui m(Yi) = ind([Vi]/Hi) - ind((-XiT Ui)/Hi), A/H H A.

rang(XiT Ui) = ki, i = 0,..., N, { ind([Vi]/Hi), ki > 0, m(Yi) = ind([Vi]), ki = 0.

(P (x)X) + [R(x) - Q(x)]X = 0, X = X(x, ), 0 < x < l, X(0, ) = X(l, ) = 0.

r 1 = 4.1584576910447 n = 2, Q(x) 0, l = 1, P = [ ] 1 I, R(x) = (1 + x)-2 r = 2, 4 h = 10-2, 10-3, 10-4, r = 2 r = 10-10-10- 1.

Nj [ ] Fj Gj Nj = -Gj Fj, Fj + Gj = I, Gj {0, 1} j Gj.

Y = [XT UT ]T j {0, 1,... 2n - 1}, det(Xj) = 0, Xj = FjX - Gj U.

LU Y Nj [ ] [ ] I 0 Xj NT Y =, j Qj I -Uj = GjX + Fj U, Qj = QT, Qj = UjXj, j Qj1 1 + 2(n - 1).

Yi i = 0, 1,..., N + 1. j = j(i), i = 0, 1,..., N + 1 Yi.

j = j(i), i = 0,..., N + - Qj(i) = Uj(i)Xj(i) Ci - Qj(i+1)i + DiQj(i) - Qj(i+1)BiQj(i) = 0, Wi = NT WiNj(i).

j(i+1) ( + 2) grad div u - rot rot u + grad div u + 2 u + + 2 (grad , grad) u + grad rot u = 0, = (t), = (t), = (t) t t .

t .

[ ] 10-3, 1.05 10-3, t [9.72, 9.59], (t) = t4 + 1, (t) = 2t3 + 1, (t) = t.

j (t, ) Qj (t, ).

Yi j = j(i), i = 0, 1,..., N + 1 Qj(i) j = j(i) Yi Qj(i) [ ] Gj(i)Qj(i)Gj(i) Pi [Vi] = PiT Ri, T Pi = -Gj(i)(I - Qj(i)Fj(i))Bi, T Ri = BiAT + BiFj(i)Qj(i)Fj(i)Bi.

i Hi ki ki Gj(i)Qj(i)Gj(i) [i, i + 1) m(Y ) = ind([Vi]/Hi) - ind((Gj(i)Qj(i)Gj(i))/Hi).

i ki = rang(Gj(i)Qj(i)Gj(i)), [Vi], i.

[ ] sin iI - cos iI Ri =, cos iI sin iI det Xi = det(sin(i)Xi + cos(i)Ui) = 0, Qi C(n), i = M,..., N + 1, Qi = (- cos(i)Xi + sin(i)Ui)(sin(i)Xi + cos(i)Ui)-1.

sin(2i) = 0, i = 0,..., N + 1.

(0, N + 1] N l(Y, 0, N) = qi - ind(tan(i)I - Qi)|N+1, i=T T T T qi = ind(Bi Di - Bi Qi+1Bi) - ind(Bi Di - tan i+1Bi Bi)+ T +ind(BiAT -] iBiBi ), tan i [ [ ] Ai Bi T i Bi Wi = Ci Di, Wi = Ri+1WiRi = .

Ci Di i i [ti, ti+1] i i i+1. i = k k Qk k+ Y = Wi()Ri[I Qi]T, i = k, k+1 Y Y = [XT UT ]T, R det(X + U) = 0, rang(UUXU + UU) = rangUU.

Q = (-X + U)(X + U)-1 = (-UUXU + UU)(UUXU + UU)-(I - UU), Q Q C(n), C(n) n X, U, = 0.

i, i = N +1 i = 0, N0 [0, N + 1].

[i, i + 1) (0, N +1] [0, N +1), i (a, b]

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