Polynomial Algebra by Values refers to doing polynomial algebra solely given the values of the polynomial---that is, working in the Lagrange basis. Several popular codes do this already, in certain circumstances (ode45 and ode15s in Matlab, for example, represent the piecewise polynomial interpolants they use in this way). The point of this research thread is to see how much we can do directly in the Lagrange basis, and why this is a good idea.

- The very first paper (unpublished, not even submitted): Generalized Companion Matrices for Polynomials not expressed in Monomial Bases . I wrote this paper in about 2000, and tried to get my masters student Gurjeet Litt interested in this; we used it in the Mapstone Lecture workshop at SUNY Geneseo in 2001. It was never finished, but a Maple implementation of the Lagrange basis matrices made it in to Maple 7. This implementation was changed to reflect the "improved" formulation I arrived at later, as discussed below.
- The first paper (on the ORCCA Tech Reports page): Polynomial Algebra By Values at the ORCCA Tech Report page.
- The next three, presented at EACA held June, 2004 in Santander, Spain,
and appearing in the Proceedings (edited by Laureano Gonzalez-Vega and Tomas
Recio).
- Generalized Companion Matrices in the Lagrange Basis , pages 317--322.
- The Bezout Matrix in the Lagrange Basis (by Azar Shakoori), pages 295--299.
- Dividing Polynomials when you only know their values (by Amir Amiraslani), pages 5--10.

- Bernstein bases are optimal, but, sometimes, Lagrange bases are better with Stephen Watt, published in the SYNASC 2004 proceedings (Timisoara), MITRON press, pages 141--153. Stephen's talk slides are Here .
- On a Generalized Companion Matrix Pencil for Matrix Polynomials Expressed in the Lagrange Basis , in Proceedings of the 1st SNC meeting, Xi'an, China, July 2005, D. Wang and L. Zhi, eds, pages 1--18. This paper discusses the geometry of the conditioning of the eigenvalues of the companion matrix pencil, and proves two theorems about placement of interpolation nodes and their relationship to the conditioning of the eigenvalues. An explicit formula for the resolvent is given.
- The Nearest Polynomial with a given zero, Revisited , with Nargol Rezvani. This paper will appear in the Sigsam Bulletin (Communications in Computer Algebra) September 2005, vol 134, no. 3, pp 71--76. Nargol's Masters' thesis extends this work to weighted norms (and gives a dual norm for a weighted $p$-norm) and also implements Karmarkar and Lakshman's algorithm in the degree-1 case, for any basis handled by MatrixPolynomialObjectImplementation, in Maple: this includes monomial, many orthogonal, the Bernstein (Bezier), Newton, and Lagrange bases.
- (On the ORCCA Tech Reports Page) The Nearest Polynomial with Lower Degree Robert M. Corless and Nargol Rezvani, submitted June 30, 2006 (actually this has been rejected and needs revision---the main thing is that we did not make connection with the CAGD literature, or explain ourselves very well. I think that the results are right and useful, but more needs to be clarified. Soon, I hope.) In the meantime, here is an Extended Abstract accepted to SNC , London, 2007.
- (Accepted April 17, 2007, to Mathematics in Computer Science) Pseudospectra of Matrix Polynomials that are Expressed in Alternative Bases , with Nargol Rezvani and Amir Amiraslani. (Current revised version) This grew out of the poster for MITACS/CAIMS June 2006. Here are some images: a still colour pseudospectrum of a six by six matrix polynomial of degree 7 (an essentially scalar matrix polynomial, so we know its spectrum easily) defined by P(lambda) = p( lambda*T^(-1) ) where T is tridiagonal, has 1/2 on the superdiagonal, 1/2 on the subdiagonal except for the T[2,1] entry which is 1, and zeros on the main diagonal. The eigenvalues of T are the roots of the Chebyshev polynomial of degree 6. The scalar polynomial p is just t^7 - 1. The still image (zipped) is symcolour.zip and the animation is an interesting mandala.
- (submitted) Rayleigh Quotient Iteration
for finding polynomial eigenvalues (with Amir Amiraslani and Dhavide Aruliah).
[Hint: Never use alphabetical order with
*either*of these two authors. Rare for Dhavide to get beat, though.] - (accepted to Theoretical Computer Science) LU factoring of generalized companion matrix pencils with the same authors.
- (accepted Nov 2007 to IMA J Numer Anal) Linearization of Matrix Polynomials Expressed in Polynomial Bases (original title: Interpolation of Matrix-valued functions using polynomial bases) with Amir Amiraslani and Peter Lancaster. This version was revised according to a reviewer's suggestions, and the title changed.
- (on the ORCCA Tech Reports page) Geometric Applications of the Bezout Matrix in the Bivariate Tensor-Product Lagrange basis Dhavide Aruliah, Robert M. Corless, Laureano Gonzalez-Vega, and Azar Shakoori, accepted to SNC 2007 .
- Companion Matrix Pencils for Hermite Interpolants , extended abstract accepted for SNC 2007 . This is joint work with Dhavide Aruliah, Robert M. Corless, Laureano Gonzalez-Vega, and Azar Shakoori.
- Files for my talk at the Oxford Summer School on Solving Polynomial Equations.