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4. 31 GRADIENT TECHNIQUES u1 (Vf(uA m u , ) V f ( U 0 ) ) uo u = 0. + 01du uo 01, uo + 0 1 ~d u du >0 uo u1 u2 by al, 01, du. {u,} {a,} by un+1 = U n - %fG1(un) W uo n ) modiJied optimal {a,} f du 3). 7. Proof. ) Of Vf(u)) > 0. H. I = u1 32 1. +: f~( u) l l ) ] n y Y f ( u n )+ 0 --f a3 u,, + uo. ) AZL= -ZZF1 Tf(u) Of -E(Tf(M), I / - If. 1 2)(Yf(U), I 1 -'GIl-1 T f ( U ) ) Y) 0. 4. 33 GRADIENT TECHNIQUES H H = =I 01 ) = 1. 11. - - f u* H,(uo) = an 1 on En f {Hn} 0 < A, < 11 H;' /I f2 = {u I f ( U ) < Ao f(u0)).

22 1. ) no of f. u // = 1, -11 Vf(x)il < ( V f ( 4 9 4 d /I +rVf(x) ti = Vf(x)ll]. ) = ( V f ( x ) ,24) = -/I Vfll <0 01. 2). a ( V f ( x ) ,u ) u <0 + nu) < f ( x ) . by x. x, = u) <0 u) Steepest Descent with Respect t o Non-Euclidean N o r m s on En do on u), on En on En. 3. ” u). = I/ u ;1 = E2 I/ u (I2 = E2. ) u :1 - 1 ) = 0, 11 u ;1 + 2XGu = 0 - 1 = 0 2Gu] (u, Gu) u = (1 V f ( x ) \ l c - l . y x I/ I I G , u En on = =I Taylor’s Theorem and Some Applications U on R1. C2 p. 961) on U , 4(x, 4 = df(x, 4 + ,f”(O, &, 4 =f(.

B As x,, xu, t {xn} no x, xn 2 linear by I<. illcG111 (1963). Proc. Internat. Astronaut. 6. 4 b b(4 = J f ( 4 4 , 4 44 + f ( u ( t ) , t). Solution of Linear Two-Point Boundary Value Problems A xl, x,, b on. on u(0) TI. n - K zi = A ( t ) u k d(0) 1 < k.

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