Quoting Fred lunnon <fred.lunnon@gmail.com>:
I have never seen this done in a textbook. But Bob Churchhouse in his Numerical Analysis course at Cardiff always used to discuss the behaviour of the coefficients as functions of the roots. I thought it a most useful exercise, and did not hesitate to appropriate it for my own course when the time came. WFL
Yes, the other way around exposes the instability better. The derivative formula is nice, but the picture is more visual ([sic]). This also leads to a nice graphical solution of cubic equations if the coefficients are normalized so that the sum of the roots is one (or any non-zero). Contours of the linear term intersect contours of the constant term (both as functions of the roots) whereby the roots can then be read off. Tangencies again reveal instabilities and non-intersections approximations to complex roots. (Yes - complex, real part from close approach and imaginary from how close. Not precise, but informative). -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos