Math 6800: QR-factorization & Eigenvalues of Symmetric Tridiagonal Matrices - Prof. Donald | Assignments Linear Algebra

Math 6800

D. Schwendeman Problem Set 7

Due:

Monday, 11/22/10

1. Text exercise 27.5, p.210.

2. Text exercise 28.1, p.218.

3. Text exercise 28.2, p.218.

4. Construct an algorithm that computes the QR-factorization of a real, symmetric, tridi-

agonal matrix Tusing Householder triangularization. Your algorithm will be similar to 10.1

in the text but should take advantage of the sparsity of T. In fact, your algorithm should

be based on orthogonal matrices of the form

Qk=













where I1and I2are (k−1)×(k−1) and (n−k−1)×(n−k−1) identity matrices, respectively,

and F=I−2vvT/kvk2is a 2 ×2 Householder reflector, instead of the more general form

given in equation (10.2) of the text. Having computed the QR-factorization of T, we are

interested in the product RQ (to find eigenvalues of Tusing the QR-algorithm). To do

this, we do not form Qexplicitly, but rather apply the v’s that define the 2 ×2F’s to R.

This product can be obtained following algorithm 10.2 or 10.3 with suitable modifications

given that the v’s are only 2-vectors and Ris a banded matrix for the case here. Write a

subprogram, called myQRtridiag say, that takes as input a symmetric tridiagonal matrix

Tand returns a tridiagonal matrix U=RQ. In writing your code, do not worry about

storing your matrices compactly (i.e. they can be saved as a full square matrix with lots of

zeros), but do take advantage of symmetry and sparsity when possible in order to reduce

the operation count. You should check your code using Matlab’s qr function.

5. Construct an algorithm that computes the eigenvalues of a real, symmetric, tridiagonal

matrix Tusing the QR-algorithm with shifts (see algorithm 28.2). Your algorithm should

use the Wilkinson shift discussed on page 222 and deflation. Write a subprogram, called

myEigTridiag say, that takes as input Tand returns a vector containing the eigenvalues

of T. The step involving the QR-factorization and the reverse product RQ can be done

using your subroutine myQRtridiag. Again, you need not worry about compact storage.

Test your code with random n×nmatrices Tof size n= 10 say. You might print out the

iterates Tkto see how the iteration behaves. (Here you really only need to print out the

diagonal and subdiagonal, say, of Tk.) You will need to come up with a test to decide when

to deflate during the iteration and when to stop. How quickly does Tkbecome diagonal for

your test matrices?

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Math 6800

D. Schwendeman Problem Set 7

Due: Monday, 11/22/

Text exercise 27.5, p.210.
Text exercise 28.1, p.218.
Text exercise 28.2, p.218.
Construct an algorithm that computes the QR-factorization of a real, symmetric, tridi- agonal matrix T using Householder triangularization. Your algorithm will be similar to 10. in the text but should take advantage of the sparsity of T. In fact, your algorithm should be based on orthogonal matrices of the form

Qk =

  

I 1

Fk I 2

  

where I 1 and I 2 are (k−1)×(k−1) and (n−k−1)×(n−k−1) identity matrices, respectively, and F = I − 2 vvT^ /‖v‖^2 is a 2 × 2 Householder reflector, instead of the more general form given in equation (10.2) of the text. Having computed the QR-factorization of T , we are interested in the product RQ (to find eigenvalues of T using the QR-algorithm). To do this, we do not form Q explicitly, but rather apply the v’s that define the 2 × 2 F ’s to R. This product can be obtained following algorithm 10.2 or 10.3 with suitable modifications given that the v’s are only 2-vectors and R is a banded matrix for the case here. Write a subprogram, called myQRtridiag say, that takes as input a symmetric tridiagonal matrix T and returns a tridiagonal matrix U = RQ. In writing your code, do not worry about storing your matrices compactly (i.e. they can be saved as a full square matrix with lots of zeros), but do take advantage of symmetry and sparsity when possible in order to reduce the operation count. You should check your code using Matlab’s qr function.

Construct an algorithm that computes the eigenvalues of a real, symmetric, tridiagonal matrix T using the QR-algorithm with shifts (see algorithm 28.2). Your algorithm should use the Wilkinson shift discussed on page 222 and deflation. Write a subprogram, called myEigTridiag say, that takes as input T and returns a vector containing the eigenvalues of T. The step involving the QR-factorization and the reverse product RQ can be done using your subroutine myQRtridiag. Again, you need not worry about compact storage. Test your code with random n × n matrices T of size n = 10 say. You might print out the iterates Tk to see how the iteration behaves. (Here you really only need to print out the diagonal and subdiagonal, say, of Tk.) You will need to come up with a test to decide when to deflate during the iteration and when to stop. How quickly does Tk become diagonal for your test matrices?

Math 6800: QR-factorization & Eigenvalues of Symmetric Tridiagonal Matrices - Prof. Donald, Assignments of Linear Algebra

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D. Schwendeman Problem Set 7

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