1. For the 5 x 5 tridiagonal matrix 2 0 0 0 -1 2 -1 0 0 T=10-1 2-1 0 0 0 -1 2-1 0 0 02 use Sturm ...
8. Let An be the following n x n tridiagonal matrix ab 0 0 0 Cab 00 0 0 Oca 0 0 0. 0 0 0 C a Show that AnalAn- 1l-bc|A,-21 for n 2 3. If a = 1+bc, show that |An 1+bc+ (be)2 ++(bc)" If a 2 cos with 0 <0<T and b c 1 then show that sin (n+1)0 |An = sin 0 nn change
8. Let An be the following n x n tridiagonal matrix ab...
II. 1. Find the eigenvalues and the eigenfunctions for the following Sturm-Liouville problem X"+AX=0, x(0)=0, X'(TT) = 0
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without partial pivoting. (10 marks)
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without...
Consider a 2 x 2 matrix A that has eigenvalues 11 -2 and A2 = 5. Find the eigenvalues of A², A- and A - 21. Is the matrix A + 21 invertible? Explain. Suppose that A is a 10 x 10 matrix and that Avi V1 Av2 = 202, x = 2v1 - 02 Find real numbers a, 8 such that A’x = av. + 802
The vector-matrix form of the system model is: (18 0 18000 72001 x f(t) or Mx + Kx = f(t) 08.3. -7200 8000 | X, 3. (1) X= M = (18 0 08 K 18000 -7200 7200 8000 and f(t) = (1) 12(0)] [x₂(t) The system's eigenvalues, natural frequencies, and eigenvectors are: 1 2 = 400, 0, = 20 s', and v, 1.5 1.) = 1600, 0), = 40 s', and v, = -1.5 1 1 The inverse of modal...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
5. Consider the problem a2y"y _2.J 0 x1 = 0, y(0) 0, y(1= 0. (a) Put the problem in Sturm-Liouville form and explain the nature of any singular points. (b) State the appropriate modified boundary conditions (c) Find all eigenvalues and eigenfunctions for the modified problem
5. Consider the problem a2y"y _2.J 0 x1 = 0, y(0) 0, y(1= 0. (a) Put the problem in Sturm-Liouville form and explain the nature of any singular points. (b) State the appropriate modified...
Show that the matrix is not diagonalizable. 1-42 13 0 02 STEP 1: Use the fact that the matrix is triangular to write down the eigenvalues. (Enter your answers from smallest to largest.) (11.22) = STEP 2: Find the eigenvectors Xi and X2 corresponding to 1, and 12, respectively. X1 = X2 - STEP 3: Since the matrix does not have ---Select-- linearly independent eigenvectors, you can conclude that the matrix is not diagonalizable.
2) For the Sturm-Liouville eigenvalue problem + λφ-0, dt2 do 0, dc (a) 0 verify the following properties: a) The nth eigenfunction has (n-1) zeros on the open interval 0<x<a b) There are an infinite number of eigenvalues with a smallest, but no largest. c) What does the Rayleigh Quotient say about negative and zero eigenfunctions.
(1) Use the Bisection method to find solutions accurate to within 10-2 for x3 – 7x2 + 14x – 6 = 0 on the interval [3.2, 4]. Using 4-digit rounding arithmatic. (2) Consider the function f(x) = cos X – X. (a). Approximate a root of f(x) using Fixed- point method accurate to within 10-2 . (b). Approximate a root of f(x) using Newton's method accurate to within 10-2. Find the second Taylor polynomial P2(x) for the function f(x) =...