This problem deals with the instability of polynomial root finding. First, note how MATLAB handles polynomial operations by doing some simple computations. A polynomial, in MATLAB is an dimensional...

This problem deals with the instability of polynomial root finding. First, note how MATLAB handles polynomial operations by doing some simple computations. A polynomial,
in MATLAB is an
dimensional vector, where
is the coefficient of
is the coefficient of

ⁿ⁻²
is the coefficient of

⁰.

Show how to represent the polynomial

⁴

³
in MATLAB.

To multiply two polynomials, use the MATLAB function conv. For instance, to compute the coefficients of (
²
proceed as follows:

Using conv find the coefficients of
⁴
and assign them to the vector

Using the MATLAB function roots, compute the roots of

⁴
Explain the results.

Let

_i
be the coefficient of

_i
in polynomial
and
be a simple root of
Suppose roundoff error perturbs

_i
by an amount

_i
so the root now becomes
Does a small

_i
result in a small
We can answer that question if we have a formula for the condition number,