We adapt Victor Y. Pan's root-based algorithm for finding approximate GCD to the case where the polynomials are expressed in Bernstein bases. We use the numerically stable companion pencil of Guðbjörn Jónsson to compute the roots, and the Hopcroft-Karp bipartite matching method to find the degree of the approximate GCD. We offer some refinements to improve the process.

We look at two classical examples in the theory of numerical analysis, namely the Runge example for interpolation and Wilkinson's example (actually two examples) for rootfinding. We use the modern theory of backward error analysis and conditioning, as instigated and popularized by Wilkinson, but refined by Farouki and Rajan. By this means, we arrive at a satisfactory explanation of the puzzling phenomena encountered by students when they try to fit polynomials to numerical data, or when they try to use numerical rootfinding to find polynomial zeros. Computer algebra, with its controlled, arbitrary precision, plays an important didactic role.

We give an apparently new proof of Stirling's original asymptotic formula for the behavior of ln z! for large z. Stirling's original formula is not the formula widely known as "Stirling's formula", which was actually due to De Moivre. We also show by experiment that this old formula is quite effective for numerical evaluation of ln z! over C, when coupled with the sequence acceleration method known as Levin's u-transform. As an homage to Stirling, who apparently used inverse symbolic computation to identify the constant term in his formula, we do the same in our proof.

No abstract

One useful standard method to compute eigenvalues of matrix polynomials ( ) ∈ C × (︀ ⌋︀ of degree at most ℓ in (denoted of grade ℓ, for short) is to first transform ( ) to an equivalent linear matrix polynomial ( ) = − , called a companion pencil, where and are usually of larger dimension than ( ) but ( ) is now only of grade 1 in . The eigenvalues and eigenvectors of ( ) can be computed numerically by, for instance, the QZ algorithm. The eigenvectors of ( ), including those for infinite eigenvalues, can also be recovered from eigenvectors of ( ) if ( ) is what is called a "strong linearization" of ( ). In this paper we show how to use algorithms for computing the Hermite Normal Form of a companion matrix for a scalar polynomial to direct the discovery of unimodular matrix polynomial cofactors ( ) and ( ) which, via the equation ( ) ( ) ( ) = diag( ( ), , . . . , ), explicitly show the equivalence of ( ) and ( ). By this method we give new explicit constructions for several linearizations using different polynomial bases. We contrast these new unimodular pairs with those constructed by strict equivalence, some of which are also new to this paper. We discuss the limitations of this experimental, computational discovery method of finding unimodular cofactors.