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3. (25 pts) Let fe C2[a, b], for a < b, and let {p,}0 be Newton's...
3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0 for all , and f(x) has a root at p satisfying f'(p) < 0. Let p, be Newton's method's sequence of approximations for initial guess po < p. Prove pi > po and pı < p Remember, Newton's method is Pn+1 = pn - f(pn)/f'(P/) and 1 f"(En P+1 P2 f(pP-p)2. between pn and p for some
3. (25 pts) Suppose f(x) is twice...
This problem is about Newton's method for solving 22 – A = 0, where A > O is a constant. The iteration scheme (formula) is 17 = pn-1 + Pn = A - ), Pn-1) n > 0. (a) Let po = A. Find the expressions for P1, P2, P3. (b) Suppose po > 0 and {Pn} is the sequence generated by the iteration. Prove Pn > 0 for all n, and for n >1, P.- VA = 207, (Pn-1...
3. Let (p) be a sequence of orthogonal functions on [a, b] having the property that the zero function is the only con- tinuous real-valued function f satisfying fo, dA ofor all nE N. Prove that the system (p.) is complete. (Hint: First use the hypothesis to prove that if fE P((a, b) satisfies fo, dA -0 for all n E N. then f = 0 a.e. Next use complteness of to prove that Parseval's equality holds for every fE...
(2) Let f : Rn → R be a C2 function. Suppose a sequence (zk) converges to x*, where the Hessian Hf(z.) is positive definite. Let ▽ := ▽f(xk)メ0, Hfk := H f(zk), dkー-Bİigfe, and :=-[Hfel-ı▽fk for each k, where each matrix Bk is ll(Be-Hfe)del = 0 if and only if ei adtive lim lidt dall =0. (11 points)
(2) Let f : Rn → R be a C2 function. Suppose a sequence (zk) converges to x*, where the Hessian...
gol The fixed-point iteration Pn+1 = g(P) converges to a fixed point p = 0 of g(x) = x for all 0 < po < 1. The order of convergence of the sequence {n} is a > 0 if there exists > O such that lim Pn+1-pl =X. -00 P -plº Use the definition (6) to find the order of convergence of the sequence in (5).
Let H be a separable Hilbert space with basis en]nen and define P as the orthogonal projection onto span(e,... ,en) (a) A sequence of operators T, E B(H) is said to converge strongly to T if |Th-Tnhl converges to 0 for all h EH (note that strong convergence is actually weaker than operator norm convergence-think of this as the difference between pointwise and uniform convergence). Show that, for any T E B(H), the sequence P,T Pn converges strongly to T....
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
Exercise 17: Let (an) be a sequence. a) Assume an> 0 for all n E N and lim nan =1+0. Show that an diverges. n=1 b) Assume an> 0 for all N EN and lim n'an=1+0. Show that an converges. nal
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose the sequence {f) converges uniformly on every compact subset of (a, b). Prove thatf is differen- tiable on (a, b) and that f'(x) = lim f(x) for all x E (a, b).
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose...