5.3. Consider the quadratic functionf (x) = x'Qx – c'x,where Q is a positive-definite matrix. Let p be a direction of descent for f at thepoint x. Prove that the solution of the exact line search problemminimize f(x+ap)a>0isp'V f (x)p'Qpa = -

Given- Consider the quadratic function fx = 12xTQx - cTx, where Q is a positive-definite matrix. Let…

Answered: 5.3. Consider the quadratic function f…

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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The equations xy² + 6xzcos(u) + ye" = 12 and x³yz + xe" - 8u²v² = 24 are solved for u and v as functions of x, y and z near the point P where (x,y,z)=(1,1,1) and (u, v)=(,0). Find ()zy at P. Türkçe: xy² + 6xzcos(u) + ye" = 12 ve x³yz + xe" - 8u²v² = 24 denklemleri P noktası (x,y,z)=(1,1,1) ve (u, v) = (,0) civarında x,y ve z'nin fonksiyonu olmak üzere u ve v için çözümlü olsun. P'de ()zy 'yi hesaplayınız.) O-0,33 O -0,17 12,00 -0,50 64,00 -56,00 -30,00 O 0,83
1 Find the derivative of y = Vx4 –3x2+5 금 (24 -3a22 + 5) (4z3 - 6z) (a4 – 3a² + 5) This function is not differentiable 글 (24 - 3㎡2 + 5)호 (4az3 - 6x)
Although we used gradient descent to find the minimizer of ƒ in part (a), we can also use calculus to solve the problem directly. Show that the minimizer of f is â = (A" A + 71)-'A"b, where I is the 4 × 4 identity matrix. What is the exact numerical value of âî for the problem in (a) and how does it compare to your result from gradient descent? What to submnit: your working out.
7 6. Find the gradient of f(x, y) = ,x' - zy". grad f +
Two waves on a string are given by the following functions: yı (x, 1) = 4 cos(201 – 30x) (cm) y2(x, 1) = -4 cos(201 + 30x) (cm) where x is in centimeters. The waves are said to interfere constructively when their superposition lys| = ly1 + y2|maximum, and they interfere destructively when lys| is a minimum. (a) What are the directions of propagation of waves y1 (x.f) and y2(x, t1)? (b) At t = (= (T/50) s, at what x location y do the two waves interfere constructively, and what is the corresponding value of ys|? (c) At t = (7/50) s, at what locationx do the two waves interfere destructively, and what is the corresponding value of lys?
4) Determine whether the given set of functions is linearly independent on the interval (-x,). a) (z) = cos 2z, (2) = 1, 1,()= cos z b) (z)=z, 1,(2)=z-1, (z)=z+3 c) ()=e**, 1,(=)=e
Compute the gradients of each of the following functions: o = xyz & = ln [(x² - y²)-¹/2] r = || = √√√x² + y² + x² 1/r Note: Remember that your tutorial has Hints in it, at the end of each chapter. If you're stuck, they are often quite useful! • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, shift and - for subscripts, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx= x², Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {A₂, Ay, A₂}. Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. Vo= V = Vr = V= (2A, 2A, 2A₂).
3 2 2) Z³ - 3x ²y + 6 xyz = 0; use the method of partial differentiation 1 to find ax да да ду I
25.* For a linear function g show that (1) if for some r x', it holds that g(x) = 9(x'), then its kernel contains vectors other than the null vector (2) if for all r, r', g(x) = 9(x') implies x = r', then its kernel contains only the null vector
Q.1) Express your answer usi (s^(-6)t^(-1)*s^(4)t^(5))/(s^(9)t^(9))

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5.3. Consider the quadratic function f (x) = x"Qx -c'x, where Q is a positive-definite matrix. Let p be a direction of descent for f at the point x. Prove that the solution of the exact line search problem minimize f (x +ap) a>0 is p'Vf(x) p'Qp a = -