ParamInt_examples1.html

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Given a singularly-perturbed linear differential system d/dx F = A(x,epsilon) F, ParamInt provides funcationalities for constructing locally a fundamental matrix of formal solutions and studying the properties of the system. In what follows, we give examples on three of these functionalities.

ParamInt is a growing package and more functionalitities are to be added.

It is based on the follwoing two papers:

- Barkatou, Moulay, Suzy S. Maddah, and Hassan Abbas. "On the reduction of singularly-perturbed linear differential systems." Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation. ACM, 2014.

- Barkatou, Moulay, Suzy S. Maddah. "On the Reduction of Singularly-Perturbed Linear Differential Systems." arXiv preprint arXiv:1401.5438 (2014).

xi-Rank Reduction (Generalization of Moser-Barkatou algorithm)

Splitting Lemma

Resolution of Turning Point

Generalisation of Levelt's Rank Reduction Algorithm

Comparison between xi-Rank Reduction and Generalization of Levelt's algorithm (sigma=0)

For all the examples, we run xi-Rank Reduction, Levelt's generalization with Moser's criterion adjoined, and Levelt's generalization without adjoining Moser's criterion, respectively. For the output, we first display the time of computaiton, the transformation matric computed, and the resulting equivalent system, respecting the order of the algorithms.

Matrix(%id = 18446744078376819758)

Matrix(%id = 18446744078367358478)

Matrix(%id = 18446744078364093486)

Matrix(%id = 18446744078375558430)

Matrix(%id = 18446744078367357158)

Matrix(%id = 18446744078364090846)

(5.1)

Matrix(%id = 18446744078375416526)

Matrix(%id = 18446744078364076382)

Matrix(%id = 18446744078374372646)

Matrix(%id = 18446744078364075302)

(5.2)

Matrix(%id = 18446744078379582518)

Matrix(%id = 18446744078386413190)

Matrix(%id = 18446744078380434966)

Matrix(%id = 18446744078386411990)

Matrix(%id = 18446744078380432550)

(5.3)

Matrix(%id = 18446744078369109766)

Matrix(%id = 18446744078366361110)

Matrix(%id = 18446744078376361126)

Matrix(%id = 18446744078364091806)

Matrix(%id = 18446744078366360870)

Matrix(%id = 18446744078376360406)

Matrix(%id = 18446744078364090486) (5.4)

> A := Matrix(%id = 18446744078364093006); -1; M1, T1, t1 := MoserParam(A, x, epsilon, 0, 10, 10); -1; M2, T2, t2 := LeveltParamM(A, x, epsilon, 10, 10); -1; M3, T3, t3 := LeveltParam(A, x, epsilon, 10,...

Matrix(%id = 18446744078378465990)

Matrix(%id = 18446744078362754094)

Matrix(%id = 18446744078359954966)

Matrix(%id = 18446744078362753614)

(5.5)

> A := Matrix(%id = 18446744078359955926); -1; M1, T1, t1 := MoserParam(A, x, epsilon, 0, 10, 10); -1; M2, T2, t2 := LeveltParamM(A, x, epsilon, 10, 10); -1; M3, T3, t3 := LeveltParam(A, x, epsilon, 10,...

Matrix(%id = 18446744078377898102)

Matrix(%id = 18446744078362100430)

Matrix(%id = 18446744078365182422)

Matrix(%id = 18446744078379048702)

Matrix(%id = 18446744078362099950)

Matrix(%id = 18446744078365181822) (5.6)

> A := Matrix(%id = 18446744078365183382); -1; M1, T1, t1 := MoserParam(A, x, epsilon, 0, 10, 10); -1; M2, T2, t2 := LeveltParamM(A, x, epsilon, 10, 10); -1; M3, T3, t3 := LeveltParam(A, x, epsilon, 10,...

Matrix(%id = 18446744078385846862)

Matrix(%id = 18446744078364074342)

Matrix(%id = 18446744078379046662)

Matrix(%id = 18446744078364094206)

Matrix(%id = 18446744078379045702) (5.7)

> A := Matrix(%id = 18446744078379047742); -1; M1, T1, t1 := MoserParam(A, x, epsilon, 0, 10, 10); -1; M2, T2, t2 := LeveltParamM(A, x, epsilon, 10, 10); -1; M3, T3, t3 := LeveltParam(A, x, epsilon, 10,...

Matrix(%id = 18446744078374974638)

Matrix(%id = 18446744078367757246)

Matrix(%id = 18446744078380432070)

Matrix(%id = 18446744078367756766)

(5.8)

Matrix(%id = 18446744078380434726)

Matrix(%id = 18446744078365102798)

Matrix(%id = 18446744078370985014)

Matrix(%id = 18446744078379582638)

Matrix(%id = 18446744078364404902)

(5.9)

Matrix(%id = 18446744078379585398)

Matrix(%id = 18446744078361173518)

Matrix(%id = 18446744078369894022)

Matrix(%id = 18446744078374272646)

Matrix(%id = 18446744078361172918)

Matrix(%id = 18446744078369893542)

(5.10)

Matrix(%id = 18446744078376798198)

Matrix(%id = 18446744078376859758)

Matrix(%id = 18446744078369118078)

Matrix(%id = 18446744078376861934)

Matrix(%id = 18446744078369116638)

(5.11)

Matrix(%id = 18446744078376864694)

Matrix(%id = 18446744078374374206)

Matrix(%id = 18446744078365104958)

Matrix(%id = 18446744078368628974)

Matrix(%id = 18446744078374373606)

Matrix(%id = 18446744078365103638)

(5.12)

> A := Matrix(%id = 18446744078368630294); -1; M1, T1, t1 := MoserParam(A, x, epsilon, 0, 10, 10); -1; M2, T2, t2 := LeveltParamM(A, x, epsilon, 10, 10); -1; M3, T3, t3 := LeveltParam(A, x, epsilon, 10,...

Matrix(%id = 18446744078368631014)

Matrix(%id = 18446744078376858078)

Matrix(%id = 18446744078369118918)

Matrix(%id = 18446744078368630534)

Matrix(%id = 18446744078376865654)

(5.13)

Matrix(%id = 18446744078369109286)

Matrix(%id = 18446744078365181222)

Matrix(%id = 18446744078386398262)

Matrix(%id = 18446744078380433030)

Matrix(%id = 18446744078365180982)

Matrix(%id = 18446744078386397422)

Matrix(%id = 18446744078380429550) (5.14)

Matrix(%id = 18446744078380435686)

Matrix(%id = 18446744078367756286)

Matrix(%id = 18446744078371033446)

Matrix(%id = 18446744078374976558)

(5.15)

> A := Matrix(%id = 18446744078387601646); -1; M1, T1, t1 := MoserParam(A, x, epsilon, 0, 10, 10); -1; M2, T2, t2 := LeveltParamM(A, x, epsilon, 10, 10); -1; M3, T3, t3 := LeveltParam(A, x, epsilon, 10,...

Matrix(%id = 18446744078376861814)

Matrix(%id = 18446744078362101030)

Matrix(%id = 18446744078386400422)

Matrix(%id = 18446744078362099830)

(5.16)