confluent interpolation polynomials

Lest it seem that all these results are too good to be true, be assured of that; this form of Lagrange interpolation assumes that all points - that is, eigenvalues - are distinct. In the contrary case, confluent forms of the polynomials may be required; and even then, considerable discretion is required.

When there are multiple roots, they should be grouped together in factoring the characteristic polynomial. Suppose that $m_i$ is the multiplicity of $\lambda_i$ and that there are $k$ distinct roots:

$$\begin{eqnarray*} \chi(M) & = & (\lambda_1 I - M)^{m_1} (\lambda_2 I - M)^{m_2} (\lambda_3 I - M)^{m_3} \ldots (\lambda_k I - M)^{m_k}. \end{eqnarray*}$$

This time the Gee's should be defined by

$$\begin{eqnarray*} G_i & = & \prod_{j\ne i}\frac{(\lambda_j I - M)^{m_j}} {(\lambda_j - \lambda_i)^{m_i}}. \end{eqnarray*}$$

They should be accompanied by the additional polynomials

$$\begin{eqnarray*} N_i^p & = & \frac{1}{p!}(\lambda_i I- M)^p \prod_{j \ne i}\f... ...mbda_i)^{m_i}} \\ & = & \frac{1}{p!}(\lambda_i I- M)^p G_i(M). \end{eqnarray*}$$

The factorial and reduced power attest their confluent origin, because these interpolating polynomials are specified by their values and $p < n_i$ derivatives at each interpolation point $\lambda_i$. Note that $G_i = N_i^0$, and also that the superscript $p$ in $N_i^p$ is not quite an exponent, although it behaves like one.

Complications arise for the multiplication table for the $N_i^p$ although it is still true that

$$\begin{eqnarray*} N_i^p N_j^q & = & \left\{ \begin{array}{ccc} O & i \ne j & \\ O & i = j & p + q \ge m_i \end{array} \right. . \end{eqnarray*}$$

Consequently the $G$'s are still orthogonal idempotents, but they might not quite resolve the unit matrix. In principle, the remainder of the multiplication table could be worked out by calculating and evaluating the derivatives of the products to get the data required for the confluent interpolation.

Nevertheless their sum,

$$\begin{eqnarray*} G & = & \sum_{i=1}^k G_i \end{eqnarray*}$$

by failing to vanish at any of the interpolation points, has no zeroes in common with any of the $G_i$, nor with the characteristic polynomial $\chi(M)$. That means that the greatest common denominator of G(m) and $\chi(M)$ is the constant $1$, thanks to which the Euclidean algorithmEuclidean algorithm assures the existence of polynomials $\sigma(\lambda)$ and $\rho(\lambda)$ for which

$$\begin{eqnarray*} \chi(\lambda) \rho(\lambda) + G(\lambda) \sigma(\lambda) & = & 1. \end{eqnarray*}$$

Recognizing that $\chi(M) = O$ because of the Cayley-Hamilton theorem, the matrix version of the equation requires $G(M)$ to be invertible given that

$$\begin{eqnarray*} G(\lambda) \sigma(\lambda) & = & I. \end{eqnarray*}$$

In order to get the confluent version of Sylvester's formula, consider a Taylor's series for the function $f$, but written in the more complicated form

$$\begin{eqnarray*} f(\lambda_i+\lambda-\lambda_i) & = & \sum_{n=0}^\infty \frac{f^{(n)}(\lambda_i)}{n!}(\lambda-\lambda_i)^n; \end{eqnarray*}$$

it has the immediate matrix counterpart

$$\begin{eqnarray*} f(M) & = & \sum_{n=0}^\infty \frac{f^{(n)}(\lambda_i)}{n!}(M... ...)^n \\ & = & f_{m_i}(M) + (M - \lambda_i I)^{s_i} t_{m_i}(M). \end{eqnarray*}$$

where

$$\begin{eqnarray*} f_{m_i}(M) & = & \sum_{n=0}^{m_i} \frac{f^{(n)}(\lambda_i)}{... ...}^\infty \frac{f^{(n)}(\lambda_i)}{n!}(M - \lambda_i I)^{n-m_i} \end{eqnarray*}$$

are respectively the head, and the tail with a common factor removed, of the Taylor's series based on each eigenvalue. This splitting foresees applying the sum of the Gees, combined with its inverse in the form of the unit matrix, to $f(M)$.

$$\begin{eqnarray*} f(M) & = & G^{-1}\sum_{i=1}^k G_i(M) f(M) \\ & = & G^{-1}\s... ..._i} \left( \sum_{n=0}^{m_i-1} f^{(n)}(\lambda_i) N_i^n \right) \end{eqnarray*}$$

An inconvenient aspect of this formula is the presence of the factor $G^{-1}$, which compensates for the simple form of the basis polynomials $N_i^p$. Alternativley, a resolution of the identity could be used which would avoid having to find and use the overall factor $G$. Consider the identity

$$\begin{eqnarray*} \frac{\chi(\lambda)}{\chi(\lambda)} & = & 1 \end{eqnarray*}$$

in which the quotient $1/\chi(\lambda)$ is first resolved into partial fractions, and then multiplied by the numerator $\chi(\lambda)$ to get a sum of terms similar to the $N_i^p$. The difference between the formulations lies in relations between the basis polynomials and their dual basis. In one case, the use of powers complicates the dual, while in the other the dual is formed elegantly by just values and derivatives, but the basis no longer consists of simple power products.