Comparing high-order multivariate AD methods

Neidinger, Richard D.; Altman, B.

doi:10.1080/10556788.2018.1472256

Cited by 2 publications

(4 citation statements)

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“…The effect of node placement can be studied through the condition number of the corresponding matrix, specifically G in the proof of Theorem 2 and V in section 6. In an application using a corner of nodes in [13], condition numbers help explain the loss of some significance in computing multivariate interpolating polynomial coefficients of degree 9 and total loss of accuracy in some coefficients of degree 25. Numerical experiments on arbitrary nodes in [18] used an algorithm equivalent to section 6 on hundreds of random nodes in two variables in [0, 1] 2 , and found mean error around 10 - 7 and largest error around 10 - 3 in (reproducing) node values of a smooth function with maximum 1.…”

Section: Discussionmentioning

confidence: 99%

“…The desired identity blocks in W are shrunken on each deletion, so they correspond to the degree of the monomials involved. Thus, our previous five rows of (13) 12 (x + 1)(x -1)(x + 2). This extended algorithm will work on any number of nodes to produce a polynomial subspace of minimal degree that always contains the unique interpolating polynomial for values on the nodes.…”

Section: Polynomials For Arbitrary Nodesmentioning

confidence: 90%

“…, where s m = \lambda m + \lambda m+1 + \cdot \cdot \cdot + \lambda n -1, as is argued in [13] for a different application. Our bivariate example \lambda = (2, 1) has…”

Section: General Divided-difference Algorithmsmentioning

confidence: 97%

“…This is precisely what was done in the application to automatic differentiation [12] that originally motivated this study. There, interpolation was used to find a multivariate Taylor polynomial (or multivariate derivative values), though stability problems were presented in [13].…”

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confidence: 99%

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Multivariate Polynomial Interpolation in Newton Forms

Neidinger¹

2019

SIAM Rev.

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Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain regions. While this idea, known as the tensor product case, has been around for over a century, this exposition uses modern multi-index notation to provide generality, concise algorithms, and rigor, appropriate as a topic in introductory numerical analysis. Node configurations where the tensor product case applies are called lower sets, and they include n-dimensional rectangles (boxes) and triangles (corners), the latter having unique interpolation over polynomials of fixed degree. Arbitrary distinct nodes do not ensure unique interpolating polynomials but, when possible, a different basis generalization, which we call Newton--Sauer, results in a nice triangular linear system for the coefficients. For the right number of distinct nodes, an algorithm based on Gaussian elimination produces either this Newton--Sauer basis or a polynomial that is zero on all nodes, showing that unique interpolation is impossible. The algorithm may also be used on any distinct nodes to produce a polynomial subspace of minimal degree where unique interpolation is possible. A variation of this algorithm using matrix block operations produces another basis generalization that we call Newton--Olver, which exactly agrees with the classic one for a corner of nodes and computes an interpolating polynomial using formulas similar to divided differences.

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Section: Discussionmentioning

confidence: 99%

Section: Polynomials For Arbitrary Nodesmentioning

confidence: 90%

“…, where s m = \lambda m + \lambda m+1 + \cdot \cdot \cdot + \lambda n -1, as is argued in [13] for a different application. Our bivariate example \lambda = (2, 1) has…”

Section: General Divided-difference Algorithmsmentioning

confidence: 97%

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confidence: 99%

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