  We reconsider the theory of Lagrange interpolation polynomials with multiple
interpolation points and apply it to linear algebra. For instance, $A$ be a
linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can
see that the evaluation of a meromorphic function $F$ at $A$ is equal to
$Q(A)$, where $Q$ is the degree $<n$ interpolation polynomial of $F$ with the
the set of interpolation points equal to the set of roots of the polynomial
$P$. In particular, for $A$ an $n \times n$ matrix, there is a common belief
that for computing $F(A)$ one has to reduce $A$ to its Jordan form. Let $P$ be
the characteristic polynomial of $A$. Then by the Cayley-Hamilton theorem,
$P(A)=0$. And thus the matrix $F(A)$ can be found without reducing $A$ to its
Jordan form. Computation of the Jordan form for $A$ involves many extra
computations. In the paper we show that it is not needed. One application is to
compute the matrix exponential for a matrix with repeated eigenvalues, thereby
solving arbitrary order linear differential equations with constant
coefficients.
