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The authors have been using a largely algebraic form of ``computational discovery in various undergraduate classes at their respective institutions for some decades now to teach pure mathematics, applied mathematics, and computational mathematics. This paper describes what we mean by ``computational discovery, what good it does for the students, and some specific techniques that we used.
We derive explicit formulas for calculating $e^A$, $cosh{A}$, $sinh{A}, cos{A}$ and $sin{A}$ for a given $2times2$ matrix $A$. We also derive explicit formulas for $e^A$ for a given $3times3$ matrix $A$. These formulas are expressed exclusively in te
We solve the difference equation with linear coefficients by the Momentenansatz to obtain explicit formulas for orthogonal polynomials.
We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are obtained for
We analyze the distribution of unitarized L-polynomials Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g <= 3 defined over Q. In the generic case, we find experimental agreement with a predicted correspondence (based on the Katz-Sar
Newtons method for polynomial root finding is one of mathematics most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge locally. Based