Note on osculatory rational interpolation

Salzer, Herbert E.

doi:10.1090/S0025-5718-1962-0149648-7

Note on osculatory rational interpolation
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by Herbert E. Salzer PDF

Math. Comp. 16 (1962), 486-491 Request permission

Abstract:

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

References

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