Note on osculatory rational interpolation

Abstract

In n-point osculatory interpolation of order r i − 1 {r_i}\, - \,1 at points x i {x_i} , i = 1 , 2 , ⋯ , n , i\, = \,1,\,2,\, \cdots \,,\,n, , by a rational expression N ( x ) / D ( x ) {N(x)}/{D(x)} , where N ( x ) N(x) and D ( x ) D(x) are polynomials ∑ a j x j \sum {{a_j}{x^j}} and ∑ b j x j \sum {{b_j}{x^j}} , We use the lemma that the system (1) { N ( x i ) / D ( x i ) } ( m ) = f ( m ) ( x i ) , m = 0 , 1 , ⋯ , r i − 1 {{\{N({{x}_{i}})/D({{x}_{i}})\}}^{(m)}}\,=\,{{f}^{(m)}}({{x}_{i}}),\,m=0,1,\cdots ,{{r}_{i}}-1 , is equivalent to (2) N ( m ) ( x i ) = { f ( x i ) D ( x i ) } ( m ) , m = 0 , 1 , ⋯ , r i − 1 , D ( x i ) ≠ 0 {N^{(m)}}({{x_i}})\, = \,{\{{f({{x_i}})D({{x_i}})}\}^{(m)}},\,m\, = \,0,\,1,\, \cdots \,,\,{r_i}\, - \,1,\,D({{x_i}})\, \ne \,0 . This equivalence does not require N ( x ) N(x) or D ( x ) D(x) to be a polynomial or even a linear combination of given functions. The lemma implies that (1), superficially non-linear in a j {a_j} and b j {b_j} , being the same as (2), is actually linear. For the n-point interpolation problem, the linear system, of order ∑ i = 1 n r i \sum \limits _{i = 1}^n {{r_i}} , which might be large, is replaceable by separate linear Systems of orders r i {r_i} (or even r i + r i + 1 + ⋯ + r i + j {r_i}\, + \,{r_{i + 1}}\, + \, \cdots \, + \,{r_{i + j}} when conveniently small) by applying the lemma to the continued fraction (3) N ( x ) / D ( x ) = a 1 , 0 + x − x 1 | | a 1 , 1 + x − x 1 | | a 1 , 2 + ⋯ + x − x 1 | | a 1 , r 1 − 1 + x − x 1 | | a 2 , 0 + x − x 1 | | a 2 , 1 + ⋯ + x − x 1 | | a 2 , r 2 − 1 + x − x 1 | | a 3 , 0 + ⋯ + x − x n − 1 | | a n , 0 + x − x n | | a n , 1 + ⋯ + x − x n | | a n , r n − 1 {N(x)}/{D(x)\,=\,{{a}_{1,0}}\,+\,\tfrac {x- {{x}_{1}}|}{| {{a}_{1,1}} }\,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{1,2}} }\,+\,\cdots \,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{1,{{r}_{1}}-1}} }\,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{2,0}} }\,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{2,1}} }\,+\,\cdots \,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{2,{{r}_{2}}-1}} }\,+\,\tfrac {x- {{x}_{1}} |}{| {{a}_{3,0}} }\,+\,\cdots \,+\,\tfrac {x- {{x}_{n-1}} |}{| {{a}_{n,0}} }\,+\,\tfrac {x-\ {{x}_{n}} |}{| {{a}_{n,1}} }\,+\,\cdots \,+\tfrac {x- {{x}_{n}} |}{| {{a}_{n,{{r}_{n}}-1}} }}\; . In (3), which has the property (proven in two ways) that the determination of a i , m {a_{i,m}} is independent of all a’s that follow, we find a i , m {a_{i,m}} stepwise, but several at a time (instead of singly which is more tedious), retrieving them readily from the solutions of those lower-order linear systems.