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Simultaneous Determination of all Zeros of a Linear Combination of Legendre Polynomial.
Nazir Ahmad Mir, Aman Ullah Khan and Zahida aKram
Centre for Advanced Studies in Pure & Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.
Abstract: An algorithm based on modified improved Ehrlich method is developed which using real arithmetic could find all the zeros of Legendre-, Shifted Legendre-, and Doubly Shifted Legendre polynomials, simultaneously, using a three term recurrence relation. The only information required is degree of the polynomial. The algorithm also finds simultaneously all the zeros of a linear combination of these polynomials, requiring information about the number of terms and the coefficients in the linear combination. The same algorithm works for finding simultaneously all the zeros of realy polynomials requiring information about its degree and coeffients.
Keywords: Zeros, Polynomials, Generalised Basis, Simultaneous Methods.