ParamAlg_examples1.html

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Given a one-parameter perturbation A(x) of a constant matrix, this package offers functions to recover leading terms and orders (or their bounds) of its eigenvalues from a few low order coefficients of a matrix similar to A(x). For more info, please refer to Chapter 2 of my thesis or to the following articles:

- Jeannerod, C-P., and Eckhard Pflügel. "A reduction algorithm for matrices depending on a parameter." Proceedings of the 1999 international symposium on Symbolic and algebraic computation. ACM, 1999.

- Jeannerod, Claude-Pierre. "An algorithm for the eigenvalue perturbation problem: reduction of a κ-matrix to a Lidskii matrix." Proceedings of the 2000 international symposium on Symbolic and algebraic computation. ACM, 2000.

AlgMoser(A,x,truncation) computes a transformation T(x) such that B = T^{-1} A T is in CRV form (has both a maximal valuation v and a minimal number of nonzero volumns of valuation v).

AlgTurnPt1(A,x,truncation) where A(x) is a non-nilpotent matrix, makes use of AlgMoser to compute a transformation T1(x), an integer s (computed using the Newton polygon associated to the characteristic polynomial), and a transformation T2(x), such that applying T1(x), the ramification x =t^s, and T2(x), yields a matrix B(x) whose leading coefficient is not nilpotent (i.e. the coefficient of lowest order).

AlgTurnPt2(A,x,truncation) computes s and a transformation T(x) for the same purpose as above. In other words, it takes T1(x) to be the identity matrix and T2(x) to be T(x).

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Matrix(%id = 18446744078177826206)

1, Matrix(%id = 18446744078171994582), Matrix(%id = 18446744078171986510), 0.37e-1 (5)

> B := Matrix(%id = 18446744078171980254); 1; Eigenvalues(B); 1; AlgTurnPt1(B, x, 10); 1

Matrix(%id = 18446744078171980254)

1, Matrix(%id = 18446744078171976278), Matrix(%id = 18446744078171967606), 0.12e-1 (6)

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> A := Matrix(%id = 18446744078165025846); 1; Eigenvalues(A); 1; alphaa, T2, BB, tim := AlgTurnPt1(A, x, 10); 1

Matrix(%id = 18446744078165025846)

Vector[column](%id = 18446744078167462366)

2, Matrix(%id = 18446744078167034446), Matrix(%id = 18446744078167026374), 0.42e-1 (8)

Matrix(%id = 18446744078167462486)

Matrix(%id = 18446744078159139774), Matrix(%id = 18446744078159139534) (9)

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