# An Algorithmic Approach to Nonlinear Analysis and by E J Beltrami By E J Beltrami

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Y") /I A. "),1') s,,. C'ovipf. R e d . 5. 4. 5. 6. A Og(z) = 2 by 2 < u11a22 A A A. 7. 8. = A. A-l A-l A. 9. z(t) z A-l. a? = f =0 En. 10. 5. C1 v(t)-+ V 0 t+ + co. LEAST SQUARES APPROXIMATION f= be C1 on En (u)= p, 46 1. ITERATIVE METHODS O N NORMED L I N E A R SPACES p E Em. )u,) m = n - p = 0. 1) Jf {all} m Jf m

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.

3. 4. 39 GRADIENT Err i n b> Line X OIIX,t 0 12 2 FIG. 4. :bl of Ax = b. by A A - aI2 = A a11a22- A A aZ2 A = 1 %Q12 a,, 0. 2 =d a 1 1 + azz) = 0; - a12 aI2 = 0 40 1. )I. = 0, A A < u11a22 = A 1. C:I b A A ill conditioned. h - XI) = 2 by 2 h A 13 7~ el, = - 2h A = 0, 2, h,/A, Ax 0, A, - A, = A, = ~ h, 4 - 0 hl/X2 A = 1 1. + E2 ,4 pp. -{- h,x,Ae, . x h2 = 0 ( A + = A . 4. 4, x 6 b no f ( x ) = (x, Ax) = A,/X, on on -A-l Of A A -kl Of - no A A A (x,Ax): by Ax = A,/& by by b A x - by 0, by by A. 42 1.