If p is the MATLAB representation of a polynomial, the function polyder computes the derivative of p.
Using polyder, compute the derivative of the function in part
of Problem 10.32.
Compute the condition number for the roots
2
and
3
of the polynomial in part
of Problem 10.32 at the coefficient of x5. Do your results confirm your explanation in part
of
Problem 10.32
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,
ai
for the computation. By applying Theorem 2.1 in Ref. [30],
Use this result in Problems 10.33 and 10.34.