Consider the problem of minimizing f and g using the Newton method (NM) with y = 1. (a) f(x) : = a + bTx + x™Cx, where a E R, b E R", and C > 0. (i) What is the minimizer x* of f? Is it unique? (ii) Show that NM converges to x* in just one iteration from any initialization x° e R". (b) f(x) = (x/a) – log(x), for a, x > 0, where log denotes the natural logarithm. (i) What is the minimizer x* of g? Is it unique? 1 and x° = 1/2? 1 and x° (ii) What are the values of the first 3 iterates generated by NM for a = (iii) What are the values of the first 3 iterates generated by NM for a = = 10? (iv) Find an interval I C R such that for a > 0 and x° e I, NM monotonically converges to x*.
Consider the problem of minimizing f and g using the Newton method (NM) with y = 1. (a) f(x) : = a + bTx + x™Cx, where a E R, b E R", and C > 0. (i) What is the minimizer x* of f? Is it unique? (ii) Show that NM converges to x* in just one iteration from any initialization x° e R". (b) f(x) = (x/a) – log(x), for a, x > 0, where log denotes the natural logarithm. (i) What is the minimizer x* of g? Is it unique? 1 and x° = 1/2? 1 and x° (ii) What are the values of the first 3 iterates generated by NM for a = (iii) What are the values of the first 3 iterates generated by NM for a = = 10? (iv) Find an interval I C R such that for a > 0 and x° e I, NM monotonically converges to x*.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the problem of minimizing f and g using the Newton method (NM) with y = 1.
(a) f(x) :
= a + bTx + x™Cx, where a E R, b E R", and C > 0.
(i) What is the minimizer x* of f? Is it unique?
(ii) Show that NM converges to x* in just one iteration from any initialization x° e R".
(b) f(x) = (x/a) – log(x), for a, x > 0, where log denotes the natural logarithm.
(i) What is the minimizer x* of g? Is it unique?
1 and x° = 1/2?
1 and x°
(ii) What are the values of the first 3 iterates generated by NM for a =
(iii) What are the values of the first 3 iterates generated by NM for a =
= 10?
(iv) Find an interval I C R such that for a > 0 and x° e I, NM monotonically converges to x*.
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