The Matrix Padé Approximation in Systems of Differential ... - WSEAS

WSEAS TRANSACTIONS on MATHEMATICSC. Pestano-Gabino, C. Gonz_Lez-Concepcion, M.C. Gil-Farina∞k(1), F(t)= ∑ Fk(t − t0) , F k ∈ Cnxmn , as follows (withoutk=0considering the equivalent system of first-order).SubstitutingF (h (t)=∞∑k=0k(k + h)(k + h − 1)...(k + 1)F (t − t ) , h=1, 2...m,k+h 0in F (m (t)=A 1 (t)F (m-1 (t)+A 2 (t)F (m-2 (t)+ +A m-1 (t)F'(t)+A m (t)F(t). Later on, we will explain moreexplicitly the recurrence relation for the specificexamples we are going to study.2 Matrix Padé Approximants andRational FunctionsOnce the formal power series for a fundamental matrix(or for a particular solution) is obtained, it is ofpractical interest to get the theoretic functionassociated to the series. It is evident that this aim isunattainable in general and it may only be obtain incertain problems.In this section we will consider Matrix PadéApproximation (MPA) results to determine it there is arational solution and, if so, obtain minimum degrees -in certain sense- of the polynomials involved.We denote as F any formal power series, with matrixcoefficients as follows:∞kmxnF(t) = fk t fk∈ tk=0∑ C ∈C (4)Suppose that there exist matrix polynomials:Q (z) =hhi∑bizi=0hiN h(z) = ∑n iz n ii=0Pg(z) =Dg(z) =g∑i=0g∑i=0ia izidizq ∈Cip ∈Ci∈Cdi∈Cmxnmxnnxnmxmpara i = 0,1,..,hpara i = 0,1,..,hpara i = 0,1,..,gpara i = 0,1,..,gwhere p 0 =I nxn , d 0 =I mxm , F(t)-Q h (t)P -1 g (t)=O(t h+g+1 ) andF(t)-D -1 g (t)N h (t)=O(t h+g+1 ), Q h P -1 g is therefore said to bea right Matrix Padé Approximant (right MPA) which isdenoted R [h/g] F ; similarly, D -1 g N h is said to be a leftMatrix Padé Approximant (left MPA), which is denotedL [h/g] F . We shall use { L [h/g] F } ({ R [h/g] F }) to denote theset of all possible approximants L [h/g] F ( R [h/g] F ).As a consequence of the definition we can say that:* R [h/g] F exists, i.e., { R [h/g] F }≠∅, if and only if, thefollowing system has a solution:f h-g+k p g +f h-g+k+1 p g-1 +...+f h+k-1 p 1 =-f h+k , k=1,2...g RS(h,g)* L [h/g] F exists, i.e., { L [h/g] F }≠∅, if and only if thefollowing system can be solved:d g f h-g+k +d g-1 f h-g+k+1 +...+d 1 f h+k-1 =-f h+k k=1,2...g LS(h,g)In both cases it is assumed that f i =0, i0, letjM1( i, j) = ( f i− j+ h+ k− 1) h , k=1,LLM4 (i, j) = (f ) ,j,s+ r−isr i− j+ h+ k− 1 h,k=1M5 (i, j) = (f ) ,Rj+ 1,s+ r−isr i− j+ h+ k− 1 h,k=1M4 (i, j) = (f )s+ r−i,jsr i− j+ h+ k− 1 h,k=1M5 (i, j) = (f )R s+ r− i,j+1sr i− j+ h+ k− 1 h,k=1By convention, if j=0 the rank of these matrices is zerofor any i∈N.Definition 4: For any nonnegative integers i, j, letT1(i,j)=rank(M1(i,j)). We will display these quantities inan infinite two-dimensional table, Table 1, where i and jserve to enumerate the columns and the rowsrespectively.Definition 5: We define the staired block R1 to be thefollowing subset of N 2 :R1={(i,j)∈ N 2 /rank(M1(g,h))=rank(M1(g+k,h+k)) forany k∈N, g≥i and h≥j}..ISSN: 1109-2769 345 Issue 6, Volume 7, June 2008