Formal series solutions of singular systems of linear differential equations and singular matrix pencils

A system of n linear differential equations xy'{x) =x~8B{x)y(x) is considered, in which s is a positive integer and B(x) is a formal power series. In a preceding paper [6] the author has developed a method for evaluating a formal fundamental matrix solution to the system. This method leads to the algebraic treatment of certain singular matrix pencils. In the present paper the author provides some transformation algorithms for singular matrix pencils which are used in all parts of the method for constructing the formal fundamental matrix solution. Introduction. L e t an n by n system of f o rmal l inear di f ferential equations xyf [x)—x~9B{x)y{x) be g iven, i n which s is a positive integer and B{x) a formal power series in x. The problem is how to calculate a formal fundamental m a t r i x solution of the form Y{x) —H{x)xJeQ[x). Here the exponent Q(x) is a diagonal m a t r i x containing polynomials i n negative f ract ional powers of x, J is a constant Jordan m a t r i x commuting w i t h Q{x), and H(x) is a formal power series in a positive f ract ional power of x. In a preceding paper [6] the author has developed the theoretical foundations for a new method by which a formal fundamental solution is computed by columns In the first step of this method the exponential part is determined. A f t e r this , the problem can be reduced to the computation of single f ormal logarithmic solutions as treated i n the author's paper [5]. B o t h parts of this method are based on the algebraic t reatment of certain matrices [X) containing the leading coefficients of B(x) and a linear complex parameter. The fundamental properties of matrices which are polynomials i n X can be read f rom their Smith 's canonical f o rm as i t is done i n the paper [5]. In fact, Smith 's canonical f o rm can a lgor i thmica l ly be evaluated — see [1], for instance—. F r o m a pract ical point of view, however, the evaluation of Smith 's canonical f o rm is not recommendable because the canonical f o rm as w e l l as the corresponding transformation matrices contain polynomials of h igher degree: this w i l l lead to severe storage requirements. A s the parameter X occurs only l inearly , the A^(X) are so-called s ingular m a t r i x pencils which can be transformed into Kronecker 's canonical f o rm by means of an a lgor i thm developed by van Dooren [4]. W e w i l l be contented w i t h some weaker properties than Kronecker 's canonical form. In Section 1 we report on some present results concerning the exponential part i n the f o rmal fundamental solution: i t turns out that we need the defect numbers and generalized characteristic polynomials of the A^(X). I n Section 2 we provide the a lgor i thm announced by which the defect numbers and characterist ic polynomials can be determined. A s compared w i t h van Dooren [4], we introduce two modifications. F i r s t we use rat ional transformation matrices instead of uni tary ones, because we intend to realize the a lgor i thm by a computer algebra system. Secondly, whenever we have constructed a transformation of an Ai^t(X) we want to use this result for a transformat ion of the subsequent A%li{X): this is desirable w i t h regard to the large size of the matrices. On the other hand, the la t ter modifications give rise to weaker structures of the transformed matrices than obtained by van Dooren's method. We w i l l ca l l these structures "property (C)" and "property (R)" suggesting the words " co lumns" and " rows" , respectively. Section 3 is devoted to the computation of the formal logarithmic part i n the f o rmal fundamental solution. W e refer to a constructional method presented i n the author's paper [5]. This method is based on an appropriate t reatment of certain systems of l inear homogeneous algebraic equations the coefficient matrices of which are derived f rom the A^(X). Here we describe the method i n a formal ly simplif ied way by considering polynomial identities instead of us ing augmented coefficient matrices. The main subject of this section, however, is the pract ical treatment of the l inear algebraic systems. I t turns out that the transformations from Section 2 are very useful for the evaluation both of the formal monodromy J and of the power series coefficients of H{x). 1. A surview on former results. The system considered has the f orm (1.1) xy'[x) = &B;^y(x), where sG N+, the Bv are constant n by ?i-matrices. The formal fundamental m a t r i x to be determined has the f o rm Y{x)=H{x)xJeQ{x\ where w i t h a certain integer p G N+ sp Q{x) = diag(^(x), • Qk(x)= E oik,u(px~ulPf V = I (1.2) J a constant n by w-Jordan m a t r i x commuting w i t h Qf H{x) a formal power series i n xllP. The matrices Ai^(X) [re {Ot1, •••,s}, m£N) are defined in the fo l lowing way—where I is the n by n ident i ty |matrix, X is a complex parameter—: For r G {0,1, • • •, s}Jwe define (1.3) dir> := min{dim3?(AL r ) W) : ^ G C } ( m 6 i V ) , ^ : = 0, where the symbol Jl denotes the null-space, and fur ther (1.4) yJmW :=greatest common divisor of a l l subdeterminants of the order {m + Vjn-d^ i n A^(X) (meN). Xi-IW : = 1. For each r the sequence of the d{*] ( m G / V U { l ) ) has the fo l lowing behaviour—see [2] and [5]: (1.5) L E M M A . There exists a unique AT R E N U { — 1} such that Vm 0 , the polynomials x{TM{Z) for m=Nr and i V r + 1 serve for the evaluation of the coefficients aktr i n the qk f rom (1.2). T h e b a s i c result is comprehended in the fo l lowing theorem proved by Schafke and Vo lkmer [3]: (1.6) T H E O R E M . Let re {If •••,s} be given. Then for all m^Nr is a polynomial independent of m. Further deg% ( r ) = #{fc€{l, •••,W} : dqk^r}. The zeros of %(r) are — —<*kr for those qk with dqk^r. r Here dqk denotes the max imal rat ional index a w i t h N0 The zeros of %(0) modulo Z are the diagonal elements of J in the columns corresponding to Qrjfe=O in Q. The definition of % ( 0 ) is extended to systems (1.1) w i t h a formal power series i n xllP [p£N+) as i t has been done for the % ( r ), r > 0 . L e t g be a polynomial i n a negative f ract ional power of x. Then Theorem (1.8) , applied to the system can serve for a test whether q occurs as a qk in Q or not, and is used in the a lgor i thm for the m i n i m a l choices of p. 2. Singular matrix pencils. L e t A and C be some rectangular n by m matrices w i t h complex coefficients, n and m being arb i t rary positive integers at the moment. Then the fami ly (A-XC)xec is cal led a s ingular m a t r i x pencil. F o r each X fixed, A-XC corresponds to a l inear map from C w into Cn and the transposed m a t r i x (A-XC)1 to a l inear map from Cn into Cm. W e define is a polynomial independent of m. Further, deg%<0) = #{A;e{l,2, •••,n}: qk=0}.