Example Usage of FFreduce and ModreduceHoward Cheng and George LabahnThis document shows how to use the FFreduce and Modreduce routines implemented in Maple. These routines implement algorithms in:Beckermann, B., Cheng, H. and Labahn, G. "Fraction-free Row Reduction of Matrices of Ore Polynomials." Journal of Symbolic Computation, 41(5), pages 513-543, 2006. Cheng, H. and Labahn, G. "Modular Computation for Matrices of Ore Polynomials." Computer Algebra 2006: Latest Advances in Symbolic Algorithms: Proceedings of the Waterloo Workshop in Computer Algebra 2006, pages 43-66, 2007.For both routines, the input consists of five parameters:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: a matrix of Ore polynomials over some domain D[Z;\317\203, \316\264] (D must be the integers for Modreduce)Z: the indeterminate in the domain D[Z;\317\203, \316\264]\317\203: the automorphism, implemented as a function (D, n) \342\206\222 D\316\264: the derivation, implemented as a function (D, n) \342\206\222 Dn: the variable in the domain D (e.g. Q[n, q^n])JSFHThe output consists of four components:M: the order basisR: the residual\317\211: the final order achieved\316\274: the final degree constraints achievedThe output satisfies the order condition:M\302\267F = R \302\267 Z\317\211with the trailing coefficient of R having the same rank as F.
Note that the implementation of FFreduce and Modreduce routines return the same result up to sign.
To use these routines, we first define the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy functions. For example, the following defines the Ore algebra for studying usual differential equations:Pkkmc2lnbWFHNiJmKjYkSSJ4R0YkSSJ0R0YkRiQ2JEkpb3BlcmF0b3JHRiRJJmFycm93R0YkRiQ5JEYkRiRGJA==print(); # input placeholderPkkmZGVsdGFHNiJmKjYkSSJ4R0YkSSJ0R0YkRiQ2JEkpb3BlcmF0b3JHRiRJJmFycm93R0YkRiQtSSVkaWZmRyUqcHJvdGVjdGVkRzYkOSQ5JUYkRiRGJA==print(); # input placeholder
We read in the routine definitions:JSFHQVEtZmZyZWR1Y2UubXBsNiI=QVEubW9kcmVkdWNlLm1wbDYiJSFHJSFHNow we define an input matrix F:QyQ+SSJGRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMvSSQlaWRHRiUiKSVRTiU9ISIiprint(); # input placeholderWe can call each routine by
FFreduce(F, Z, \317\203, \316\264, t) or Modreduce(F, Z, \317\203, \316\264, t)For example:QyQ+NiYmSSJNRzYiNiMiIiImSSJSR0YnRigmSSZvbWVnYUdGJ0YoJkkjbXVHRidGKC1JKUZGcmVkdWNlR0YnNidJIkZHRidJIlpHRidJJnNpZ21hR0YnSSZkZWx0YUdGJ0kidEdGJyEiIg==print(); # input placeholderQyQ+NiYmSSJNRzYiNiMiIiMmSSJSR0YnRigmSSZvbWVnYUdGJ0YoJkkjbXVHRidGKC1JKk1vZHJlZHVjZUdGJzYnSSJGR0YnSSJaR0YnSSZzaWdtYUdGJ0kmZGVsdGFHRidJInRHRichIiI=print(); # input placeholder
We can verify that the two results are the same up to sign:LUkkbWFwRyUqcHJvdGVjdGVkRzYkSSdleHBhbmRHRiQsJiZJIk1HNiI2IyIiIkYsJkYpNiMiIiNGLA==print(); # input placeholderLUkkbWFwRyUqcHJvdGVjdGVkRzYkSSdleHBhbmRHRiQsJiZJIlJHNiI2IyIiIkYsJkYpNiMiIiNGLA==print(); # input placeholderJSFHThe results are:JkkiTUc2IjYjIiIiprint(); # input placeholderJkkiUkc2IjYjIiIiprint(); # input placeholderJkkmb21lZ2FHNiI2IyIiIg==print(); # input placeholderJkkjbXVHNiI2IyIiIg==print(); # input placeholderJSFHTTdSMApJNVJUQUJMRV9TQVZFLzI1MDgzNDAwWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMmJSJtRzYkIiIiRikmRic2JCIiI0YpJkYnNiRGKUYsJkYnNiRGLEYsRiU=TTdSMApJNVJUQUJMRV9TQVZFLzE4NDM1OTg0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIiFGJkYmRiZGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzE4NDM2MjI0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIiFGJkYmRiZGJQ==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TTdSMApJNVJUQUJMRV9TQVZFLzQxNDE3MTQ0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIiImRiZGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzQxNDE3MjY0WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjIiImRiZGJQ==