# A Newton Method for Shape-Preserving Spline Interpolation

TL;DR: This paper proves local quadratic convergence of their proposed Newton-type method for shape-preserving interpolation by viewing it as a semismooth Newton method and presents a modification of the method which has global quadRatic convergence.

Abstract: In 1986, Irvine, Marin, and Smith proposed a Newton-type method for shape-preserving interpolation and, based on numerical experience, conjectured its quadratic convergence. In this paper, we prove local quadratic convergence of their method by viewing it as a semismooth Newton method. We also present a modification of the method which has global quadratic convergence. Numerical examples illustrate the results.

## 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

59 citations

### 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

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### "A Newton Method for Shape-Preservin..." refers background in this paper

...As a side result, from Lemma 2.3 and the Clarke inverse function theorem [2, Theorem 7.1.1], we obtain that the solution of the problem (3) is a Lipschitz continuous function of the interpolation values yi....

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...For a locally Lipschitz continuous function G : Rn → Rn, the generalized Jacobian ∂G(x) of G at x in the sense of Clarke [2] is the convex hull of all limits obtained along sequences on which G is differentiable: ∂G(x) = co { lim xj→x ∇G(xj) | G is differentiable at xj ∈ Rn } ....

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...For a locally Lipschitz continuous function G : R → R, the generalized Jacobian ∂G(x) of G at x in the sense of Clarke [2] is the convex hull of all limits obtained along sequences on which G is differentiable:...

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1,361 citations

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### "A Newton Method for Shape-Preservin..." refers methods in this paper

...Furthermore, in this case quadratic convergence of the Newton method would follow directly from [13]....

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116 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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