Calculations of the roots of classical orthogonal polynomials: an application to gaussian quadrature

Shaidolda, Gulnaz
This thesis focuses on classical orthogonal polynomials namely Jacobi, Laguerre and Hermite polynomials and a method to calculate the roots of these polynomials is constructed. The roots are expressed as the eigenvalues of a tridiagonal matrix whose coefficients depend on the recurrence formula for the classical orthogonal polynomials. These approximations of roots are used as method of computation of Gaussian quadratures. Then the discussion of the numerical results are then introduced to deduce the efficiency of the method.


Randomness properties of some vector sequences generated by multivariate polynomial iterations
Gürkan Balıkçıoğlu, Pınar; Diker Yücel, Melek; Department of Cryptography (2016)
We examine the randomness properties of the sequences generated by the multivariate polynomial iterations method proposed by Ostafe and Shparlinski, by using the six different choices of polynomials given by the same authors. Our analysis is based on two approaches: distributions of the periods and linear complexities of the produced vector sequences. We define the efficiency parameters, PE for “period efficiency” and LCE for “linear complexity efficiency”, so that the actual values of the period and linear com...
A New Combinatorial Identity for Catalan Numbers
Aker, Kursat; Gursoy, Aysin Erkan (2017-10-01)
In this article, we prove a conjecture about the equality of two generating functions described in "From Parking Functions to Gelfand Pairs (Aker, Can 2012)" attached to two sets whose cardinalities are given by Catalan numbers: We establish a combinatorial bijection between the two sets on which the two generating functions were based on.
Classification of skew-symmetric forms corresponding to cluster algebras with principal coefficients
Mazı, Sedanur; Seven, Ahmet İrfan; Department of Mathematics (2016)
In this thesis, we study algebraic and combinatorial properties of the skew-symmetric forms that correspond to cluster algebras with principal coefficients. We obtain a classification of these forms under congruence and compute the Arf invariants for finite types. 
HFE based multi-variate quadratic cryptosystems and Dembowski Ostrom polynomials
Alam, Bilal; Akyıldız, Ersan; Kırlar, Barış Bülent; Department of Cryptography (2013)
Harayama and Friesen proposed linearised binomial attack for multivariate quadratic cryptosystems and introduced weak Dembowski Ostrom(DO) polynomials in this framework over the finite fi eld F2. They conjecture about the existence of infi nite class of weak DO polynomials and presented the open problem of enumerating their classes. We extend linearised binomial attack to multivariate quadratic cryptosystems over Fp for any prime p and redefi ne the weak DO polynomials for general case. We identify an in fi...
On the computation of generalized division polynomials
Küçüksakallı, Ömer (2015-01-01)
We give an algorithm to compute the generalized division polynomials for elliptic curves with complex multiplication. These polynomials can be used to generate the ray class fields of imaginary quadratic fields over the Hilbert class field with no restriction on the conductor.
Citation Formats
G. Shaidolda, “Calculations of the roots of classical orthogonal polynomials: an application to gaussian quadrature,” M.S. - Master of Science, Middle East Technical University, 2019.