Algorithms for the Nonsymmetric Eigenvalue Problem

Abstract

In this chapter, we will present the main algorithms for solving the nonsymmetric eigenvalue problem using direct methods. Direct methods are usually applied to dense matrices, and iterative methods such as the Rayleigh–Ritz method and Lanczos’s algorithm are applied to sparse matrices. Iterative methods usually can compute not all of the eigenvalues and eigenvectors, but only some subset, and their convergence depends on the structure of the matrix.

The original version of this chapter was revised. An Erratum to this book can be found at https://doi.org/10.1007/978-3-319-57304-5_10

Questions

Do the following exercises, explaining the results that you have obtained.

10.1

(Programming)

Use the MATLAB$^\circledR$ program PowerM.m of Section 1.9 to test the power method and to compute the largest eigenvalue of the matrix A.²⁰ Try the following examples when the matrix A and the tolerance $\varTheta$ in Algorithm 10.1, p. 345, are defined as follows:

• 1. A $=$ randn(5) and tolerance $\varTheta = \{1e-5,1e-4,1e-3,1e-2,0.1 \}$.

• 2. $A =\text {diag}(\text {ones}(2n, 1)) + \text {diag}(\text {ones}(2n-1,1), 1) + \text {diag}(\text {ones}(2n-1,1),-1)$ for each number $n=3,4,5$ and tolerance $\varTheta {=} \{1e{-}12,1e-10, 1e-8, 1e-7$}.

• 3.
  $$\begin{aligned} A = \left( \begin{array}{cccc} 1 &{} 1e6 &{} 0 &{} 0 \\ 0 &{} 2 &{} 1 &{} 0 \\ 1 &{} 2 &{} 3 &{} 10 \\ 0 &{} 0 &{} -1 &{} 4 \end{array}\right) \end{aligned}$$

  (10.14)

  and tolerance $\varTheta = \{ 1e-12,1e-10, 1e-8\}$.

• 4.
  $$\begin{aligned} A = \left( \begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 2 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 2 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 3 &{} 1e2 &{} 1e4 \\ 0 &{} 0 &{} 0 &{} 0 &{} 3 &{} 1e2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3 \end{array}\right) \end{aligned}$$

  (10.15)

  and tolerance $\varTheta = \{1e-10,1e-8, 1e-6, 1e-4, 1e-3 \}$.

10.2

(Programming)

Use the MATLAB$^\circledR$ program InverseIteration.m of Section 1.10 to test the inverse iteration method for the computation of the eigenvalue of the matrix A that is closest to the shift $\sigma$. Try all examples of matrix A and tolerance $\varTheta$ as in Question 10.1. Choose also different shifts $\sigma$. For a stopping criterion using tolerance $\varTheta$, we refer to Algorithm 10.2, p. 348.

10.3

(Programming)

Use the MATLAB$^\circledR$ program MethodOrtIter.m of Section 1.11 to test the method of orthogonal iteration for the computation of the eigenvalues of the matrix A.²¹ Try all examples of matrix A and tolerance $\varTheta$ defined in Question 10.1. For a stopping criterion using tolerance $\varTheta$, we refer to Algorithm 10.3, p. 351.

10.4

(Programming)

Use the MATLAB$^\circledR$ program MethodQR_iter.m of Section 1.12 to test the method of QR iteration for the computation of the eigenvalues of the matrix A. Try all examples of matrix A and tolerance $\varTheta$ defined in Question 10.1. For a stopping criterion using tolerance $\varTheta$, we refer to Algorithm 10.4, p. 355.

10.5

(Programming)

Use the MATLAB$^\circledR$ program MethodQR_shift.m of Section 1.13 to test the method of QR iteration with the shift $\sigma =A(n, n)$ for the computation of the eigenvalues of the matrix A. Try all examples of matrix A and tolerance $\varTheta$ defined in Question 10.1. For a stopping criterion using tolerance $\varTheta$, we refer to Algorithm 10.5, p. 356.

10.6

(Programming)

Use the MATLAB$^\circledR$ program MethodQR_Wshift.m of Section 1.14 to test the method of QR iteration with Wilkinson’s shift for the computation of the eigenvalues of the matrix A. Try all examples of matrix A and tolerance $\varTheta$ defined in Question 10.1. For a stopping criterion using tolerance $\varTheta$, we refer to Algorithm 10.5, p. 356.

10.7

(Programming)

Use the MATLAB$^\circledR$ program HessenbergQR.m of Section 1.15 to test the reduction of the matrix A to an upper Hessenberg matrix. Try the following examples when the matrix A and the tolerance $\varTheta$ are defined thus:

• 1. A $=$ randn(5) and the tolerance is $\varTheta = \{1e-7,1e-5,1e-4,1e-3,1e-2,0.1 \}$.

• 2. $A =\text {diag}(\text {ones}(2n, 1)) + \text {diag}(\text {ones}(2n-1,1), 1) + \text {diag}(\text {ones}(2n-1,1),-1)$ for each number $n=3,4,5$ and tolerance $\varTheta = \{1e-12,1e-10, 1e-8, 1e-7 \}$.

• 3.
  $$\begin{aligned} A = \left( \begin{array}{cccc} 1 &{} 1e6 &{} 0 &{} 0 \\ 0 &{} 2 &{} 1e-3 &{} 0\\ 0 &{} 0 &{} 3 &{} 10 \\ 0 &{} 0 &{} -1 &{} 4 \end{array}\right) \end{aligned}$$

  (10.16)

  and tolerance $\varTheta = \{ 1e-12,1e-10, 1e-8\}$.

• 4. Each tolerance from the set $\varTheta = \{1e-10,1e-8, 1e-6, 1e-4, 1e-3 \}$ and the matrix $A=(1,0,0,0,0,0;0,2,1,0,0,0;0,0,2,0,0,0;0,0,0,1e4,1,1;0,0,0,0,1e2,1; 0,0,0,0,0,1e4) + \text {diag}(\text {ones}(5,1),-1)$.