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The local evaluation of the derivative of a determinant. (English) Zbl 0593.65028
The nonlinear lambda matrix problem is studied. When computing eigenvalues (points for which the matrix is singular), it is suggested that a Newton method finding the zeros of the determinant is used. It is described how to find values of the derivative of the determinant by means of diagonalization of the constant term in a Taylor series expansion of the $\lambda$ matrix.
Reviewer: A.Ruhe
##### MSC:
 65F40 Determinants (numerical linear algebra) 65H17 Eigenvalue and bifurcation problems of nonlinear algebraic equations (numerical methods) 15A54 Matrices over function rings 65F15 Eigenvalues, eigenvectors (numerical linear algebra)
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##### References:
 [1] Gantmacher, F. R.: The theory of matrices. (1959) · Zbl 0085.01001 [2] Lancaster, P.: Lambda matrices and vibrating systems. (1966) · Zbl 0146.32003 [3] Bridges, T. J.; Morris, P. J.: AIAA paper 84-0437, aerospace sciences meeting. (Jan. 1984) [4] Bridges, T. J.: Math. res. Center report 2839. (1985) [5] Lancaster, P.: Int. schr, numer. Math. (Birkhauser). 38, 43 (1977) [6] Rube, A.: SIAM J. Numer. anal.. 10, 674 (1973) [7] Bridges, T. J.; Morris, P. J.: J. comput. Phys.. 55, 437 (1984) [8] T. J. Bridges and P. J. Morris, A statistical analysis of the effect of freestream turbulence on the Blasius boundary layer, in preparation. [9] Kublanovskaya, V. N.: SIAM J. Numer. anal.. 7, 532 (1970) [10] Elishakoff, I.: Probabilistic methods in the theory of structures. (1983) · Zbl 0572.73094 [11] Fox, L.; Parker, I.: Chebyshev polynomials in numerical analysis. (1968) · Zbl 0153.17502 [12] Taylor, B.: Philos. trans. Royal soc. London. 30, 610 (1717) [13] Conte, S. D.: Elemantary numerical analysis. (1965) · Zbl 0213.41501 [14] Traub, J. F.: Iterative methods for the solution of equations. (1964) · Zbl 0121.11204