A Newton Method for Shape-Preserving Spline Interpolation
Summary (1 min read)
If the integral over [a, b] of the piecewise linear function (
- Furthermore, in this case quadratic convergence of the Newton method would follow directly from [13] .
- The following example of dimension 2 shows that such an argument does not work.
- Hence the function above is not piecewise smooth.
- Then the positive semidefiniteness of the elements of ∂F − (λ) follows from the fact that any matrix in the generalized Jacobian of the gradient of a convex function must be symmetric and positive semidefinite.
- By combining the above lemmas and applying Theorem 1.1, the authors obtain the main result of this paper which settles the question posed in [11] .
4. Numerical results.
- In Figures 1-5 , the dashed line is for the resulting shape-preserving cubic spline (using the data obtained with the starting point λ 0 = sign(d)); the solid line is for the natural spline (using the MATLAB SPLINE function), and "o" stands for the original given data.
- In Table 1 for results of the numerical experiments the authors use the following notation: 1 , they observe that Algorithm 3.2 converges rapidly to the solution from both starting points for all problems except Example 4.4 (y 9 = 4), to which the algorithm within 30 iterations failed to produce an approximate solution meeting the required accuracy.
- These observations indicate that how far away from zero each divided difference is may make a big difference in the numerical performance of the algorithm.
- The problem is, however, nonsmooth, and here the authors are entering a new territory.
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Citations
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Additional excerpts
...Doob, J. (1953). Stochastic Processes....
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Cites background from "A Newton Method for Shape-Preservin..."
..., [10, 12, 16, 23, 41], and many critical questions remain open in analysis and computation when general dynamics and control constraints are taken into account....
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...Hence the computation of the smoothing spline boils down to determining parameters of a polynomial on each interval, which can be further reduced to a quadratic or semidefinite program that attains efficient algorithms [10, 12]....
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Additional excerpts
...[13] D ow nl oa de d 11 /1 6/ 14 to 1 29 ....
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References
88 citations
"A Newton Method for Shape-Preservin..." refers methods in this paper
...For more discussion of the inexact Newton method, we refer to [3, 7, 14]....
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83 citations
"A Newton Method for Shape-Preservin..." refers background in this paper
...[15] is as follows: minimize ‖f ′′‖2 (3)...
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...[15], the attention of a number of researchers has been attracted to spline interpolation problems with constraints....
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81 citations
"A Newton Method for Shape-Preservin..." refers background or methods or result in this paper
...We report results with the starting point e, the vector of all ones in R , which is commonly selected as a starting point in algorithms for convex best interpolations; see [11, 6]....
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...We approach the problem of Irvine, Marin, and Smith [11] in a new way, by using recent advances in optimization....
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...1, we obtain the main result of this paper which settles the question posed in [11]....
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...As for related results, the conjecture of Irvine, Marin, and Smith [11] was proved in [1] under an additional condition which turned out to be equivalent to smoothness of the function F in (5)....
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...It was also observed in [11] that the Newton-type iteration (7) reduces to M(λ)λ = d; that is, no evaluations of the function F are needed during iterations....
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54 citations
"A Newton Method for Shape-Preservin..." refers methods in this paper
...For more discussion of the inexact Newton method, we refer to [3, 7, 14]....
[...]
42 citations