program3 - k = α 2 β cos ± kπ n 1 ²(verify this...

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Program 3 Numerical Linear Algebra 1 Fall 2010 Due date: via email by 11:59PM on Monday, 10/15/10 1. Implement the Jacobi method for solving the symmetric eigenvalue problem. 2. Use the classical Jacobi sweep and other strategies that sweep through all off-diagonal elements in a specified order. (See the textbook for a discussion of alternatives.) 3. Explore using Jacobi algorithms on matrices where you have a good approximation of the eigenvectors, e.g., consider a series of eigenvalue problems where the matrix A varies slowly. 4. Explore using Jacobi algorithms on matrices that are near diagonal or have significant structure in their nonzero element pattern. Be sure to discuss in you solution how you generated the test problems used. There are forms of tridiagonal matrices that have known eigenvalues in addition to examples you can find in various text books. For example, if T R n and the diagonals are constant, i.e., a typical row has only β,α,β as the three nonzero elements then
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Unformatted text preview: λ k = α + 2 β cos ± kπ n + 1 ² (verify this identity if you use it). You can also use Gershgorin’s three theorems do generate a matrix with various distributions on eigenvalues. One such theorem is in the text on p. 320. Numerical Mathematics by Quarteroni, Sacco and Saleri has a statement of all three theorems. You can also generate a symmetric matrix A = Q Λ Q T by choosing the eigenvalues and then constructing an orthogonal matrix Q , e.g., random orthogonal. The solution should include the subroutine or Mfile as well as a test driver and the appropriate documentation. The test driver and documentation should include a description of the testing you did and the necessary code to repeat it. Most important in the solution is, of course, you discussion of how you evaluated your code and what it indicated about using Jacobi methods to solve the symmetric eigenvalue problem. 1...
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