Improved Algebra files

Comput roots and polynomial functions with high accuracy.

Description

Improved algorithms for finding the roots (direct method for first order quadratic, cubic and Quartic) are used for low order equations.
Fadeev algorithm for finding the characteristic polynomial is implemented to ensure high accuracy.
These files can be used as a replacement of original MATLAB files.
PDFversion of technical note is included to show the algorithm and examples.

• MATLAB release: MATLAB 6.1 (R12.1)
• Other requirements: Can be used in any version of MATLAB, although tested under 6.1

[Roger Crouch]

this is such a neat compact script! thank you!

[Marcelo Pisani]

There is an error for 3rd degree polynomials solving using ROOT1. I just tryied this:
---
a=[3 -2 3 -4];
r=roots(a)
r1=roots1(a)
polyval(a,r)
polyval(a,r1)
---

[Raymond Kan]

To fix the problem, change lines 50 and 51 to
u=sign(rr(1))*abs(rr(1))^(1/3);
v=sign(rr(2))*abs(rr(2))^(1/3);

[Ryan Bennink]

The algorithm switches to matlab's less accurate ROOTS function in the case of complex coefficients. If you take the switch out, the algorithms are not quite suitable for complex numbers because they don't always use the right sign of the root. The code below, which replaces lines 50 and 51, yields the correct sign in all cases:

if abs(rr(1))>abs(rr(2))
  u=rr(1)^(1/3);
else
  u=rr(2)^(1/3);
end
v=-p/u;

In theory, either rr(1) or rr(2) gives the same set of roots, but the larger one will contain relatively less roundoff error.

[Santhosh Kodipaka]

bug]: Try this:
% roots1.m is kind of buggy here
P = [ 0.9159 0.6020 0.2536 0.8735 0.5134];
roots1( P )
roots(P )

[Felix D]

With bugs....

See
u = [2.0000 0 0.0625 -0.0187]
roots(u)
roots1(u)