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

WSEAS TRANSACTIONS on MATHEMATICSC. Pestano-Gabino, C. Gonz_Lez-Concepcion, M.C. Gil-FarinaAlthough this set seems rather abstract, the meaning of"staired" becomes clear observing Tables 1 in examplesbelow.To begin with, we can propose a method, using Table 1,to determine whether or not a matrix series stems from arational function. This can be set out as follows.2.1.1.1 RationalityF is rational if and only if in the bottom right part ofTable 1 we can mark at least one NW-SE diagonal ofinfinite size where all its boxes have the same value. It isimportant to state that, there are several such diagonals,with their union coinciding with R1. Note that, withinR1, boxes of different diagonals can have differentvalues. Furthermore, it is assured that if (a,b)∈R1 thenthe formal power series of F and [a/b] F is the same.However, (a,b) are not necessarily left or right m.d.In practice, the Table 1 that we can construct is "finite".However, in some applications where a rule for theformation of the coefficients of the series F is known,certain relations between the matrices that define theelements of Table 1 can help validate the properties toan infinite size. Otherwise, we can only say the availablecoefficients of F coincide with the coefficients of arational function of certain degrees.2.1.1.2 MinimalityThe boxes (i,j) such that T1(i,j)∈R1, T1(i-1,j)∉R1 andT1(i,j-1)∉R1 are called corners of R1. In particularsituations we can say that a corner of R1 correspondswith a pair of left or right m.d., for instance:Property 1: If (i,j)∈R1, (i-1,j)∉R1 and T1(i,j)=jn then(i-u,j-v) is not a pair of right m.d. for any u, v such that1≤u≤i and 0≤v≤j.Property 2: If (i,j)∈R1, (i-u,j)∉R1, (i,j-v)∉R1, for anyu, v such that 1≤u≤i and 1≤v≤j, and they are not pairs ofleft (right) m.d., then (i,j) is a pair of left (right) m.d.2.1.2 Table 2: MinimalityIf F is rational, there are (and we can find) certaindegrees r and s associated to two pairs of matrixpolynomials which represent the function in rationalform in two ways, that is to say, with the invertedpolynomial of degree r, multiplied to the right or left.However, it may be that r and s are neither left nor rightm.d. for representing F since there are cases in which,for instance, the function is identical to an approximantL [i/j] F but the approximant R [i/j] F does not exist, orviceversa; in this case, rank(M1(i,j))≠rank(M1(i+1,j+1))and thus Table 1 would not provide informationconcerning the left or right approximant. For this reason,we present Table 2 -one for the left and one for the rightapproximant-, whose structure reflects the possiblecombinations of m.d. (left and right, respectively) forrepresenting F.Firstly we present the following definitions.Definition 6: Given (s,r)∈R1, for integer i, j, such that0≤i≤s and 0≤j≤r, let L T2 sr (i,j)=0, if rank( L M4 sr (i,j))=rank( L M5 sr (i,j)), or otherwise L T2 sr (i,j)=1. We willdisplay these quantities in a finite two-dimensional table,Table 2 for left approximant, which has s+1 columnsand r+1 rows.Definition 7: We denote as a staired block L R2 sr , with(s,r)∈R1, the following subset of N 2 :L R2 sr ={(i,j)∈ N 2 /0≤i≤s, 0≤j≤r andrank( L M4 sr (i,j))=rank( L M5 sr (i,j))}.To define Table 2 for right approximant we mustconsider R M4 sr (i,j) and R M5 sr (i,j) instead of L M4 sr (i,j) andL M5 sr (i,j), respectively. We only expound the theory forthe left case taking into account that for the right case itis similar.Property 3: F is a rational function identical to L [q/p] Fwhere q and p are left m.d., q≤s and p≤r, if and only if,L T2 sr (q,p)=0, L T2 sr (q-1,p)=1 and L T2 sr (q,p-1)=1, that is,(q,p) is a corner of L R2 sr .The system corresponding to the denominator of theelement of { L [q/p] F } that coincides with F is:d p f q-p+i +d p-1 f q-p+i+1 +...+d 1 f q+i-1 =-f q+i i=1,2...s+r-qwhose associated matrix is L M4 sr (q,p).Then we will consider the results of MPA to studypossible rational solutions of differential equationssystems and of partial differential equations.3 Rational Solutions for Systems ofOrdinary Differential EquationsIn order to obtain a clear understanding, it is importantto read the following remarks relating to possiblerational solutions of the system (1).- Even if A 1 (t), A 2 (t)...A m (t) are not rational there mayexist rational solutions. For instance, the systemtt⎛ e − e + 2 ⎞⎜⎟Y'(t)= ⎜ 2t + 4 2t + 4 ⎟ Y(t)sint − sin t + 2⎜⎟⎝ 2t + 4 2t + 4 ⎠⎛has a rational solution, Y(t)= 2t + 4 ⎞⎜ ⎟⎝ 2t + 4.⎠- The fact that there exists rational solution does notimply the fundamental matrix is rational. Note thatgenerally the linear combination of no rationalfunctions could be a rational function, for instance, ifx(t)= cos t−and y(t)= 2cos t + 8 the linear3t − 23t − 216combination 4x(t)+2y(t) is the rational function3t − 2.- The system (1) may have a fundamental matrix withISSN: 1109-2769 346 Issue 6, Volume 7, June 2008