Hyman’s method revisited

Tridiagonal test matrices for eigenvalue computations: Two-parameter extensions of the Clement matrix

“…In the case where n is odd, similar results hold for positive parameter values for a and b, as shown in figure 6 for example. For negative parameter values we have a different situation, as the eigenvalues (8) can become imaginary if the two factors have opposite sign. When a < −n and b < −n, the eigenvalues (8) are real again and the behavior mimics that of the positive values of a and b.…”

“…Recall that both for n odd and n even, for the special case H n (a) the square roots in the expressions for the eigenvalues cancel, yielding real eigenvalues for every real value of the parameter a. In the general case, the eigenvalues ( 6) are real when a + b > −2 and those in (8) are real when a > −1 and b > −1 or a < −n and b < −n. A first remark is that when we compute the spectrum of C n using eig() in MATLAB, eigenvalues with imaginary parts are found when n exceeds 116, but not for lower values of n. Therefore, for the extensions, we have chosen n = 100 and n = 101 (for the even and the odd case respectively) for most of our tests, as this gives reasonably large matrices but is below the bound of 116.…”

“…It would be interesting future work if these new eigenvalue test matrices were to be used to test also numerical algorithms for computing eigenvalues designed specifically for matrices having multiple eigenvalues [8], being tridiagonal [15], or symmetric and tridiagonal [4].…”

Solution of the real and complex eigenvalue problems in the ABS class

“…Algorithms were developed already in the early computer age See for example [18], [24], [25] and [11]. Or later [23], [10] and so on.…”

“…The testing problems are the 12 small dimensions test matrices from [10] which are hereinafter referred to as T problems. Furthermore, we selected 4 problems whose matrices and eigenvalues are complex from G. Karney's book of test problems [16].…”