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Polynomials are algebraic expressions of terms with variables raised to non-negative integer powers. A central aspect of analyzing polynomial functions is determining their real zeros—values of the variable for which the polynomial evaluates to zero. These values represent the x-intercepts of the polynomial’s graph.The Rational Zeros Theorem lists possible rational solutions for a polynomial equation with integer coefficients. If f(x)=anxn+....+a0​, then every rational zero is...
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Integral-based parallel algorithm for the fast generation of the Zernike polynomials.

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    Riemann sums accurately approximate Zernike radial functions with minimal error. A parallel GPU algorithm significantly accelerates Zernike radial polynomial generation for optics applications.

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    Area of Science:

    • Optics and Computational Mathematics

    Background:

    • Zernike radial functions are crucial in optical metrology and aberration analysis.
    • Efficient computation of these functions is essential for advanced optical system design.

    Purpose of the Study:

    • To evaluate the accuracy of Riemann sums for approximating Zernike radial functions.
    • To develop and assess a parallel algorithm for rapid Zernike radial polynomial generation.

    Main Methods:

    • Approximation of the integral representation of Zernike radial functions using Riemann sums.
    • Development of a parallel algorithm utilizing graphics processing units (GPUs) for computation.

    Main Results:

    • Riemann sums demonstrate high accuracy, with errors below 3.3×10-14 for radial orders up to 50.
    • The parallel GPU algorithm achieves up to a 200-fold speedup compared to traditional CPU methods.
    • Demonstrated rapid generation of Zernike radial polynomials.

    Conclusions:

    • Riemann sums provide a highly accurate and efficient method for Zernike radial function approximation.
    • The proposed parallel algorithm offers significant computational advantages for generating Zernike radial polynomials.
    • The accelerated generation is beneficial for applications in aberration analysis and pattern recognition.