diff --git a/books/bookvol10.4.pamphlet b/books/bookvol10.4.pamphlet index cf98a80..81ee93e 100644 --- a/books/bookvol10.4.pamphlet +++ b/books/bookvol10.4.pamphlet @@ -6989,6 +6989,37 @@ BezoutMatrix(R,UP,M,Row,Col): Exports == Implementation where Implementation ==> add +\end{chunk} +The Sylvester matrix is used to compute the {\bf resultant} of two +polynomials. The Sylvester matrix is formed from the coefficients of +the two polynomials. Given a polynomial with degree $m$ and another of +degree $n$ form an $(m+n)\times(m+n)$ matrix by filling the matrix +from the upper left corner with the coefficients of the first +polynomial then shifting down one row and one column to the right and +filling in the coefficients starting there until they hit the right +column. Starting at the next row, do the same process for the second +polynomial. The determinant of this matrix is the {\bf resultant} of +the two polynomials. + +For example, given $a_3x^3+a_2x^2+a_1x+a_0$ and $b_2x^2+b_1x+b_0$ +the Sylvester matrix is a $(3+2)\times(3+2)$ matrix: +$\left[\begin{array}{ccccc} +a_3 & a_2 & a_1 & a_0 & 0\\ +0 & a_3 & a_2 & a_1 & a_0\\ +b_2 & b_1 & b_0 & 0 & 0\\ +0 & b_2 & b_1 & b_0 & 0\\ +0 & 0 & b_2 & b_1 & b_0 +\end{array}\right]$ + +The resultant of these two polynomials (assuming a leading coefficient +of 1), is the product of the differences $p_i-q_i$ between the roots +of the polynomials. If there are roots in common then the product +will contain a 0 and the whole equation reduces to 0. This can be +used to determine if two polynomials have common roots. + +In symbolic form the resultant can show the multiplicity of roots. + +\begin{chunk}{package BEZOUT BezoutMatrix} sylvesterMatrix(p,q) == n1 := degree p; n2 := degree q; n := n1 + n2 sylmat : M := new(n,n,0) diff --git a/changelog b/changelog index 0c36fd7..8045dbd 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,5 @@ +20140927 tpd src/axiom-website/patches.html 20140927.02.tpd.patch +20140927 tpd books/bookvol10.4 document Sylvester matrix 20140927 tpd src/axiom-website/patches.html 20140927.01.tpd.patch 20140927 tpd books/bookvol0 add obituaries for Bronstein, O'Connor 20140927 tpd books/bookvolbib add bib refs for obits diff --git a/patch b/patch index 70d53a3..fc3c90a 100644 --- a/patch +++ b/patch @@ -1,4 +1,4 @@ -books/bookvol0 add obits for Bronstein, O'Connor +books/bookvol10.4 document Sylvester matrix -Manuel Bronstein and Christine O'Connor (nee Sundarasan while working -on Scratchpad) obituaries are added to the front of the Jenks book. +add inline documentation on the Sylvester matrix and its role in +computing the resultant. diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 8879766..a3dfefc 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4654,6 +4654,8 @@ books/bookvolbib add abstracts, rearrange, add new sections