There are two computational aspects in computing the zeros of orthogonal polynomials(Gauss quadrature nodes)
1: Eigenvalue approach
2: Root finding Iterative methods like as Newton's method.
Which one of the above schemes do not suffer from round-off errors?
Which one of these two technique is suitable specially for large values of
approximation degree?
Best.
I would suggest interval arithmetic approach. It does not control, but let you know how big is your error.
Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to Interval Analysis. Philadelphia: SIAM.
I think that both methods have self-correction, which means that the round-off errors do not accumulate. These are iterative methods, and on the next iteration, the round-off errors of the current iteration will be corrected.
Some (obsolete) methods for eigenvalues include finding the polynomial roots. The modern advanced methods for eigenvalues are based on the matrix decomposition. Furthermore, if the polynomial roots are needed, the square matrix is built from the polynomial coefficients, such that is eigenvalues are polynomial roots (and Newton method is avoided).
As I mentioned, the round-off errors are not crucial because both methods are self-correcting, but the eigenvalue method has some other advantages. In particular, unlike the Newton method, the initial guess is not needed.
Every orthogonal polynomial can be expressed in terms of a tri-diagonal matrix. Form Sturm's sequence of the tridiagonal matrix and find its eigen values.
You can always get the C++ code off my web site to calculate any order GQ using whatever level of accuracy you want. I've already done 4096 and 8192 point using 128-byte, 256-byte, 512-byte, and 1024-byte floating point.
Gauss quadrature is a numerical method for approximating definite integrals using a weighted sum of function evaluations at specific points, known as the Gauss quadrature nodes. Rounding errors can affect the accuracy of the Gauss quadrature approximation. Here are some strategies to control rounding errors and improve the accuracy of Gauss quadrature nodes:
No numerical method can completely eliminate rounding errors, but the above strategies can help mitigate their impact and improve the accuracy of Gauss quadrature approximations. Selecting an appropriate quadrature rule and precision level depends on the specific problem and the desired trade-off between accuracy and computational cost.
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