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Let Sp ( 3, &Dgr; ) be the set of cubic C 1-splines on the grid &Dgr; : x 0 ≡ 0 < x 1 < x 2 < . . . < x n - 1 x n≡ 1 of the interval [ 0, 1 ]. Let the points ( x 0 , y 0 ) , ( x 1 , y1 ) , . . . , ( x n , y n ) be in convex position, i. e., &tgr; 1 ≤ &tgr; 2 ≤ . . . ≤ &tgr; n, where &tgr; i = ( y i - y i - 1 ) / ( x i - x i - 1 ). The problem of finding a function s ( x ) ∈ Sp ( 3,&Dgr; ) such that s ( x i ) = y i , i = 0 , 1 , . . . , n, and s is convex on [ 0,1 ], may be not solvable. If a solution does exist, the function s is not in general uniquely determined. In this case, the selection of a cubic spline which has minimal curvature leads to a quadratic programming problem of special structure. The authors derive a corresponding dual problem without constraints which can be solved effectively. Interesting algorithms for the numerical treatment of the problem are clearly described and discussed. The possibility of extension to some splines of higher degree is considered.
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Reviewer:
L. Gatteschi |
Review #: CR110255 |
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Minkowski matrices.
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