By Samuel Daniel Conte

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Z = z(t) are continuously differentiable functions of t, then g(t) = f(x(t), y(t), . . , z(t)) is also continuously differentiable, and From this theorem, one obtains an expression for f(x, y, . . , z) in terms of the value and the partial derivatives at (a, b, . . , c) by introducing the function and then evaluating its Taylor series expansion around t = 0 at t = 1. 37) for all (x,y) in D, where for some depending on (x,y), and the subscripts on f denote partial differentiation. 38) Finally, in the discussion of eigenvalues of matrices and elsewhere, we need the following theorem.

For this purpose and others, the Newton form of the interpolating polynomial is much better suited. Indeed, write the interpolating polynomial p,(x) in its Newton form, using the interpolation points x0, . . 9) in the form for some polynomial r(x) of no further interest. The point is that this last term (x - x0) · · · (x - xk)r(x) vanishes at the points x0, . . , xk, hence qk(x) itself must already interpolate f(x) at x0, . . , xk [since pn(x) does]. , q k (x) must be the unique polynomial of degree < k which interpolates f(x) at x0, .

Zk, then p(x) = q(x) identically. 1, be written in the form with r(x) some polynomial. Suppose that Then some coefficients c0, . . , cm with for which is nonsense. Hence, r(x) = 0 identically, and so p(x) = q(x). ” These considerations concerning zeros of polynomials can be refined through the notion of multiplicity of a zero. This will be of importance to us later on, in the discussion of osculatory interpolation. We say that the point z is a zero of (exact) multiplicity j, or of order j, of the function f(x) provided Example For instance, the polynomial has a zero of multiplicity j at z.