This root-counting theorem was produced by the French mathematician Jacques Sturm in 1829.
For any number , let denote the number of sign changes in the sequence .
For real numbers and that are both not roots of ,
In particular, we can count the total number of distinct real roots by looking at the limits as and . The total number of distinct real roots will depend only on the leading terms of the Sturm sequence polynomials.
Note that deg deg , and so the longest possible Sturm sequence has deg terms.
Also, note that this sequence is very closely related to the sequence of remainders generated by the Euclidean Algorithm; in fact, the term is the exact same except with a sign changed when or . Thus, the Half-GCD Algorithm may be used to compute this sequence. Be aware that some computer algebra systems may normalize remainders from the Euclidean Algorithm which messes up the sign.
For a proof, see Wolpert, N., “ http://web.archive.org/web/20050412175929/http://www.mpi-sb.mpg.de/ nicola/Vorlesung/sturm.psProof of Sturm’s Theorem”
|Date of creation||2013-03-22 14:28:49|
|Last modified on||2013-03-22 14:28:49|
|Last modified by||rspuzio (6075)|