AppSing_examples_presentation1.html

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Desingp(Matrix,variable,p) tests and performs desingularization at an irreducible polynomial p

Desing(Matrix,variable) identifies, tests, and performs desingularization at all irreducible polynomials p

The output is the change of basis T, its inverse invT, the matrix B of the equivalent desingularized system, and the execution time t, respectively.

DesingpSteps(Matrix,variable,p) and DesingSteps(Matrix,variable) give the same output and print key intermediate steps as well

For more information, please refer to our paper entitled "Removing Apparent Singularities of Systems of Linear Differential Equations with Rational Function Coefficients" and/or the description of the package.

Example 1 (Example 1 in the paper)

Input Matrix

Matrix(%id = 18446744078273212534) (1.1.1)

Desingularization at p= z^2 +2

Matrix(%id = 18446744078273205662), Matrix(%id = 18446744078273206382), Matrix(%id = 18446744078273207102), .138 (1.2.1)

Details of Desingularization at p= z^2 +2 ( intermediate steps)

This is our matrix and its residue matrix at

Matrix(%id = 18446744078273198790), Matrix(%id = 18446744078273199510), Matrix(%id = 18446744078273192054), 0.63e-1 (1.3.1)

Desingularization at p= z

Matrix(%id = 18446744078254402366), Matrix(%id = 18446744078254403086), Matrix(%id = 18446744078254395630), .262 (1.4.1)

Details of Desingularization at p= z ( intermediate steps)

This is our matrix and its residue matrix at

Matrix(%id = 18446744078273179886), Matrix(%id = 18446744078273180606), Matrix(%id = 18446744078273182286), .269 (1.5.1)

Desingularization

Matrix(%id = 18446744078270509774), Matrix(%id = 18446744078270510494), Matrix(%id = 18446744078270511214), .282 (1.6.1)

Details of Desingularization

Matrix(%id = 18446744078270664942), Matrix(%id = 18446744078270665662), Matrix(%id = 18446744078270666382), .341 (1.7.1)

Example 2: Shaoshi Chen, M. Jaroschek, M. Kauers, M. F.Singer, Desingularization Explains Order-Degree Curves for Ore Operators, Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, pp. 157-164, ACM, U.S.A. (2013))

Input Matrix

Matrix(%id = 18446744078271055262) (2.1.1)

Desingularization at p= 2*x^3 -x^2 -20*x +23

Matrix(%id = 18446744078254390918), Matrix(%id = 18446744078254375270), Matrix(%id = 18446744078254375990), .283
(2.2.1)

Matrix(%id = 18446744078254373094) (2.2.2)

Details of Desingularization at p= 2*x^3 -x^2 -20*x +23 (intermediate steps)

This is our matrix and its residue matrix at

Matrix(%id = 18446744078273196390), Matrix(%id = 18446744078273197110), Matrix(%id = 18446744078273197830), .225
(2.3.1)

Example 3 (the introductory example in the paper)

Input Matrix

Matrix(%id = 18446744078273199030) (3.1.1)

Desingularization

Matrix(%id = 18446744078252940334), Matrix(%id = 18446744078252941054), Matrix(%id = 18446744078252966366), .275 (3.2.1)

Example 4: Shaoshi Chen, M. Kauers, M. F. Singer, Desingularization of Ore Operators, Available at: arXiv:1408.5512v1, (2014))

Input Matrix

Matrix(%id = 18446744078252968766) (4.1.1)

Desingularization at p = x^2 - 3*x +3

Matrix(%id = 18446744078255271438), Matrix(%id = 18446744078096939718), Matrix(%id = 18446744078096940678), .202 (4.2.1)

Desingularization

Matrix(%id = 18446744078254378750), Matrix(%id = 18446744078254371294), Matrix(%id = 18446744078254372014), .567 (4.3.1)

Example 5: A. Bostan, S. Boukraa, S. Hassani, M. van Hoeij, J.-M. Maillard, J.-A. Weil, and N. Zenine, The Ising model: From Elliptic Curves to ModularForms and Calabi-Yau equations. J. Phys. A: Math. Theor., 44(4):045204, 44, (2011))

Example 5.2 : DE from http://www.unilim.fr/pages_perso/jacques-arthur.weil/L3tilde.mpl

Input Matrix