Scientific Computing

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
CS205 Homework #5 Problem 1 [Heath 5.5, p.248] 1. Show that the iterative method x k +1 = x k - 1 f ( x k ) - x k f ( x k - 1 ) f ( x k ) - f ( x k - 1 ) is mathematically equivalent to the secant method for solving a scalar nonlinear equa- tion f ( x ) = 0. 2. When implemented in finite-precision floating-point arithmetic, what advantages or disadvantages does the formula given in part (1) have compared with the formula for the secant method (given in the notes and in Heath, section 5.5.4)? Problem 2 [Heath 5.6, p.249] Suppose we wish to develop an iterative method to compute the square root of a given positive number y , i.e., to solve the nonlinear equation f ( x ) = x 2 - y = 0 given the value of y . Each of the functions g 1 and g 2 listed next gives a fixed-point problem that is equivalent to the equation f ( x ) = 0. For each of these functions, determine whether the corresponding fixed-point iteration scheme x k +1 = g i ( x k ) is locally convergent to y if y = 3. Explain your reasoning in each case. 1. g 1 ( x ) = y + x - x 2 . 2. g 2 ( x ) = 1 + x - x 2 /y . 3. What is the fixed-point iteration function given by Newton’s method for this particular
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: problem? Problem 3 [Heath 5.11, p.249] Suppose you are using the secant method to find a root x * of a nonlinear equation f ( x ) = 0. Show that if at any iteration it happens to be the case that either x k = x * or x k-1 = x * (but not both), then it will also be true that x k +1 = x * . 1 Problem 4 [Heath 6.8, p.302] Consider the function f : R 2 → R defined by f ( x ) = 1 2 ( x 2 1-x 2 ) 2 + 1 2 (1-x 1 ) 2 1. At what point does f attain a minimum? 2. Perform one iteration of Newton’s method for minimizing f using as starting point x = ± 2 2 ² T 3. In what sense is this a good step? 4. In what sense is this a bad step? [Note: The Newton method for optimization of a function of multiple variables f : R n → R gives the update step s k = x k +1-x k as the solution to the system H ( x k ) s k =-∇ f ( x k ), where H ( x k ) is the Hessian matrix of f evaluated at x k . See also Heath, section 6.5.3.] 2...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern