> restart; -1; with(LinearAlgebra); -1; with(algcurves); -1; with(PolynomialTools); -1; with(simplex); -1; libname; 1
 

/Library/Frameworks/Maple.framework/Versions/18/lib (1)
 

> libname :=
 

/Users/maddah/Documents/Work/Packages/packages/libraries/ParamAlg/ParamAlg.lib (2)
 

> march('list',
 

[[
[[
[[
[[
(3)
 

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). 

 

 

> A := Matrix(2, 2, [0, 1, `*`(`^`(x, 5)), 0]); 1; Eigenvalues(A); 1; AlgTurnPt1(A, x, 10); 1; AlgTurnPt2(A, x, 10); 1
 

 

 

 

Matrix(%id = 18446744078185943398)
Vector[column](%id = 18446744078185946518)
2, Matrix(%id = 18446744078185882078), Matrix(%id = 18446744078177882710), 0.84e-1
Matrix(%id = 18446744078177864870), 2, Matrix(%id = 18446744078177844150), Matrix(%id = 18446744078177836078), 0.48e-1 (4)
 

> A := Matrix(3, 3, [0, 1, 0, 0, 0, 1, 0, 0, x]); 1; Eigenvalues(A); 1; AlgTurnPt1(A, x, 10); 1
 

 

 

Matrix(%id = 18446744078177826206)
Vector[column](%id = 18446744078172016638)
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)
Vector[column](%id = 18446744078171982894)
1, Matrix(%id = 18446744078171976278), Matrix(%id = 18446744078171967606), 0.12e-1 (6)
 

> A := Matrix(2, 2, [0, 1, `*`(`^`(x, 5)), 0]); 1; Eigenvalues(A); 1; AlgTurnPt1(A, x, 10); 1
 

 

 

Matrix(%id = 18446744078171966150)
Vector[column](%id = 18446744078171959774)
2, Matrix(%id = 18446744078165033198), Matrix(%id = 18446744078165025126), 0.66e-1 (7)
 

> 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)
 

>
 

> A := Matrix(4, 4, [`+`(`-`(`/`(`*`(2), `*`(x)))), 0, `/`(1, `*`(`^`(x, 2))), 0, `*`(`^`(x, 2)), `/`(`*`(`+`(`-`(`*`(`^`(x, 2))), 1)), `*`(x)), `*`(`^`(x, 2)), `+`(`-`(`*`(`^`(x, 3)))), 0, `/`(1, `*`(`...
 

 

Matrix(%id = 18446744078167462486)
Matrix(%id = 18446744078159139774), Matrix(%id = 18446744078159139534) (9)
 

> A := Matrix(2, 2, [0, 1, `*`(`^`(x, 5)), 0]); 1; AlgMoser(A, x, 10); 1
 

 

Matrix(%id = 18446744078159153022)
Matrix(%id = 18446744078181085182), Matrix(%id = 18446744078159343358) (10)
 

> A := Matrix(2, 2, [0, 1, `*`(`^`(x, 2)), 0]); 1; s, T, B, t := AlgTurnPt1(A, x, 10); 1
 

 

Matrix(%id = 18446744078181318518)
1, Matrix(%id = 18446744078181330686), Matrix(%id = 18446744078185882318), 0.23e-1 (11)
 

>
 

>
 

>
 

>