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F(xo) global. xo on R1). 18. En on U x0 Vf(x0) 0. 13, Proof. 0 11 u 11 < 6. 11 u 11 < 6. (Vf(C), Vf(x0N Vf(xo) # 0, ( V f ( 0 ,Vf(x0N a: --f 0 a: - >0 3 0. II Vf(x0)I12 Cl), -a(Vf(S), V f ( X 0 ) ) < 0 01: = 0. En, 7 u f, = 0 = x + I x, u En NU x C1 (Y (Vf(v),u ) = 0. u /I u /IG, x u q 26 1. 1. 2. 3. 4. 5. 6. 7. f H, C1 on E2 f ( x ,y ) y 11. 4. 27 GRADIENT TECHNIQUES 11 YoZ:&f(x, d Y)) = 6(,F:&f(X, Y ) ) = 0. 4. GRADIENT T E CHN I Q U ES Newton’s Method on Of = 0. f :R1+ R1 = 0. 1) 28 g(x) 1.

6. m by n (V,W). 7. 6. on N ( A ) N(A)J-1. , n TI. ( x(t) = t on two-point boundary value problem). 6. 55 TWO-POINT BOUNDARY VALUE PROBLEMS x -+ A? ) = A? - f(x, B g 0 ) . B [B on TI g. of g C1 f u B V’i(x, x =) f i x], t t. TI, 11 u Ij + 0, 56 1. ). ) 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.

8) Vf. = Proof. ) u0 Vf. 11 Vf(zi,) = u0 01, = A,), = 1 a + ( a , ). 13), u0 - u0 = -01~ Vf(u0) = V f (u,) f ( u ) = f (uo) + Vf(u,) uo u = - u0 u,. u, I, yo = 1, 1. 36 n ---f E n by /I u ;1 = En, 2 2 Z/Cu pp. f ( u ) -= c g(2) + ( a , + A(u, U) -=f(dG-’z)= c Gu) ( Z / E ’ a , z ) -+ f ( 2 , 2). ~’~ g z,, zo ~ uo 2 N nO - uo - = ‘C,g(zo) Yf(uo) Z/cuo. 12 by Tg by Tf(u)]. \/cu, Z / c u ) -: “u “2 “ “z zi {Efr

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