  ==Root sensitivity== Let p be a polynomial of degree n. We are looking for the condition number of the map  at the point u, where u is the vector of coefficients of the given polynomial p (in a certain polynomial basis q) and  is the vector of roots of p.  The condition number is known to be: : where D is the diagonal matrix with entries equal to  and V_q is the generalized Vandermonde matrix (each column is a basis vector - e.g.  if q is the power basis - evaluated at each root).  You can consult for example Gautschi, Questions of numerical condition related to polynomials, or calcultate it by evaluating the Jacobian of f (which equals the matricial product in the formula above).  Remark that D is a n times n matrix while the Vdmonde is a n times n+1.  If q is the power basis (say q=r),  becomes the traditionnal Vandermonde matrix.  If q is the Lagrange basis (say q=q), the generalized Vdmonde is the identity (and last column equals 0).  We have: : since D is diagonal (at least for the infinity norm).  Since we can choose arbitrary a point y distinct from each root of p as additionnal interpolation point, it is always possible to choose it such that: :  For example, if we consider the infinity norm, the point y=0 is always a such point unless it is already a root of p.  In this case, one can take a point y'' near zero.  All in all: :