xi-Rank Reduction (Generalization of Moser-Barkatou algorithm)
MoserParam computes an equivalent system to that of the input matrix, which has the lowest possible rank of singularity in the parameter (epsilon or xi) and algebraic rank for the leading coefficient w.r.t. this parameter epsilon (or xi). The output is the equivalent system, the transformation, and computation time.
Function call: MoserParam(Input matrix with the poles in epsilon (or xi) and possibly in x, the variable, the parameter, sigma, truncation in the variable (in case of power series entries), truncation in parameter)
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![A := `+`(`/`(`*`(Matrix(4, 4, [0, 0, 0, 0, `*`(`^`(x, 2)), 0, 0, 0, `+`(`-`(x)), 0, 0, 0, 0, 2, 0, 0])), `*`(`^`(epsilon, 3))), `/`(`*`(Matrix(4, 4, [1, `+`(`-`(`*`(`^`(x, 3)))), `+`(1, x), 0, 0, x, 0...](images/ParamInt_examples_16.gif)
![A := `+`(`/`(`*`(Matrix(4, 4, [0, 0, 0, 0, `*`(`^`(x, 2)), 0, 0, 0, `+`(`-`(x)), 0, 0, 0, 0, 2, 0, 0])), `*`(`^`(epsilon, 3))), `/`(`*`(Matrix(4, 4, [1, `+`(`-`(`*`(`^`(x, 3)))), `+`(1, x), 0, 0, x, 0...](images/ParamInt_examples_17.gif) |
![Matrix(%id = 18446744078186322630)](images/ParamInt_examples_18.gif) |
![Matrix(%id = 18446744078186281190), Matrix(%id = 18446744078186281550), .155](images/ParamInt_examples_19.gif) |
(1.1) |
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![A := Matrix(%id = 18446744078457781654); 1; M, T, t := MoserParam(A, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_20.gif) |
![Matrix(%id = 18446744078174855166)](images/ParamInt_examples_21.gif) |
![Matrix(%id = 18446744078163913294), Matrix(%id = 18446744078163913774), .193](images/ParamInt_examples_22.gif) |
(1.2) |
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![A := Matrix(%id = 18446744078457774198); -1; M, T, t := MoserParam(A, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_23.gif)
![A := Matrix(%id = 18446744078457774198); -1; M, T, t := MoserParam(A, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_24.gif) |
![Matrix(%id = 18446744078163897270), Matrix(%id = 18446744078163891134), .224](images/ParamInt_examples_25.gif) |
(1.3) |
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![A := Matrix(%id = 18446744078457766246); 1; M, T, t := MoserParam(A, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_26.gif) |
![Matrix(%id = 18446744078164731534)](images/ParamInt_examples_27.gif) |
![Matrix(%id = 18446744078170607734), Matrix(%id = 18446744078170608454), .174](images/ParamInt_examples_28.gif) |
(1.4) |
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![AA := Matrix(%id = 18446744078386400662); 1; M, T, t := MoserParam(AA, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_29.gif) |
![Matrix(%id = 18446744078168270118)](images/ParamInt_examples_30.gif) |
![Matrix(%id = 18446744078168525270), Matrix(%id = 18446744078168525630), .117](images/ParamInt_examples_31.gif) |
(1.5) |
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![AA := Matrix(%id = 18446744078376819278); 1; M, T, t := MoserParam(AA, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_32.gif) |
![Matrix(%id = 18446744078186321550)](images/ParamInt_examples_33.gif) |
![Matrix(%id = 18446744078164732374), Matrix(%id = 18446744078164732614), 0.70e-1](images/ParamInt_examples_34.gif) |
(1.6) |
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![AA := Matrix(%id = 18446744078364075062); 1; M, T, t := MoserParam(AA, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_35.gif) |
![Matrix(%id = 18446744078168490582)](images/ParamInt_examples_36.gif) |
![Matrix(%id = 18446744078171335966), Matrix(%id = 18446744078171336206), 0.74e-1](images/ParamInt_examples_37.gif) |
(1.7) |
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![AA := Matrix(%id = 18446744078375556990); 1; M, T, t := MoserParam(AA, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_38.gif) |
![Matrix(%id = 18446744078171361022)](images/ParamInt_examples_39.gif) |
![Matrix(%id = 18446744078171640166), Matrix(%id = 18446744078171640406), 0.66e-1](images/ParamInt_examples_40.gif) |
(1.8) |
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![AA := Matrix(%id = 18446744078374377822); 1; M, T, t := MoserParam(AA, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_41.gif) |
![Matrix(%id = 18446744078171174422)](images/ParamInt_examples_42.gif) |
![Matrix(%id = 18446744078170632430), Matrix(%id = 18446744078170632790), 0.98e-1](images/ParamInt_examples_43.gif) |
(1.9) |
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![A := Matrix(%id = 18446744078379582878); 1; M, T, t := MoserParam(A, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_44.gif) |
![Matrix(%id = 18446744078170550630)](images/ParamInt_examples_45.gif) |
![Matrix(%id = 18446744078177347694), Matrix(%id = 18446744078177348414), .117](images/ParamInt_examples_46.gif) |
(1.10) |
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![A := Matrix(%id = 18446744078369108806); 1; M, T, t := MoserParam(A, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_47.gif) |
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![A := Matrix(%id = 18446744078376359326); 1; M, T, t := MoserParam(A, x, epsilon, 0, 10, 10); 1](images/ParamInt_examples_52.gif) |
![Matrix(%id = 18446744078164730574)](images/ParamInt_examples_53.gif) |
![Matrix(%id = 18446744078168524670), Matrix(%id = 18446744078168525030), 0.52e-1](images/ParamInt_examples_54.gif) |
(1.12) |
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![A := Matrix(%id = 18446744078362751454); 1; M, T, t := MoserParam(A, x, xi, -2, 10, 10); 1](images/ParamInt_examples_55.gif) |
![Matrix(%id = 18446744078160281958)](images/ParamInt_examples_56.gif) |
![Matrix(%id = 18446744078175143686), Matrix(%id = 18446744078175145486), 0.57e-1](images/ParamInt_examples_57.gif) |
(1.13) |