[NLRS] The best polynomial root finder:

[Dr. Gerald N. Johnson]

The best polynomial root finder that I know of uses an Eigen-Value 
program on a matrix with real coefficients. One loads the polynomial 
into a zeroed matrix, seems like coefficients along the top row and 1s 
on the diagonal or those positions swapped. Then solves for the Eigen 
values of the matrix and double precision Fortran could reliably solve 
30th order polynomials without iteration even if the polynomials were 
ill conditioned when evaluated.

Back about 1972, I built a matrix handling language that I called MATLAN 
around Eigen Value code the EE department already had. MATLAN 
manipulated matrices with real coefficients like we'd do numbers, with a 
programmable or scriptable sequence. It didn't have looping or 
conditional branching, so it was a limited language. It did take in data 
for a polynomial and did that solution as one of its commands.

Another language I created those days I called PWRMAT and it worked with 
matrices with complex coefficients up to 10 x 10 and included 
transformations characteristic of three phase power line transitions and 
couplings of parallel lines. It also didn't loop or have condition 
branching. It could invert matrices and in that way solve systems of 
equations when solvable. It could compute the mutual impedances of 
parallel three phase high voltage power lines in about 3 cards of input 
where a program that specialized in that took pages of code. It ran in 
about 128K on an IBM 360, though I made a version that would run on an 
IBM 1100 mini computer with only 16K of RAM. When an Iowa consulting 
engineering company compiled it on their mini, it only took 8 hours, but 
it ran a little faster than that, spending much time swapping 
subroutines into working memory. I actually ran that version on the 360 
to prove it worked. Source code was about a box of cards. I have kept 
some of those source codes and have a couple card readers that I've not 
made work. I've needed to analyze few coupled power lines since graduate 
school so I've not moved it to a PC, though there were enhanced versions 
at ISU at one time.

73, Jerry, K0CQ