Instructions Discuss The Proof The Following As Clear And As Concise As Possible Use Complete Sentences Cite Any Theo 1 (91.7 KiB) Viewed 27 times
Instructions Discuss The Proof The Following As Clear And As Concise As Possible Use Complete Sentences Cite Any Theo 2 (69.86 KiB) Viewed 27 times
INSTRUCTIONS: Discuss the proof the following as clear and as concise as possible. Use complete sentences. Cite any theorems, lemmas, corollaries, or axioms used appropriately. Problem 1. Consider the quadratic function f(x) = 3x:"Az – bºx, where A is an n x n symmetric positive definite matrix. When the steepest dsecent method applies to this function, the iteration scheme becomes gk gk Xk+1 = k - gk with a 9. Aga where gk := f(xx). The aim of this problem is to derive the convergence speed of {ck} to the unique minimizer I. of f. (i) Derive the following identities: Ar. = b 9x = A** - 2.) f(xx) – f(xx) = 3(2x – xx)*A(mk – Ox), f(xx) - f(I) 296 () f(xx+1)-f(x) = (1 (9 Age)(9A-9) (f(xx) - f(1.)) = 97 A-9k –f(v.)
2 Cu FD')*($(w) – 5(v.) )* 2k (ii) Derive that f(3x+1) - f(x) < In + and And f(xk) - f(x) Antti (f(x0)-f(x)), using the following result known as the Kantorovich inequality: let 11 and An be the smallest and largest eigenvalues of A respectively. Then for any nonzero x € Rr there holds (272) 4ληλα > (27 Ax)(x+ A-) (11 + 12)2 (iii) Let k:= And which is called the condition number of A. Show that || ** -0. SVR *6+1) * | 20 – 2.ll.
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