Talk:Polyharmonic spline: Difference between revisions

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I might be mistaken, but I didn't think that Polyharmonic splines actually do guarantee that the linear system matrix is positive definite, just that it's nonsingular. For example, consider phi(r) = r with centers 0 and 1; the matrix is

$M\,=\,{\begin{bmatrix}0&&1&&1&&0\\1&&0&&1&&1\\1&&1&&0&&0\\0&&1&&0&&0\end{bmatrix}}$

which is not positive definite:

${\begin{bmatrix}1&&0&&-1&&0\end{bmatrix}}M{\begin{bmatrix}1\\0\\-1\\0\end{bmatrix}}=-2$

Can somebody with more experience than I verify this?

128.143.137.224 (talk) 18:10, 1 October 2009 (UTC)[reply]

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 I might be mistaken, but I didn't think that Polyharmonic splines actually do guarantee that the linear system matrix is positive definite, just that it's nonsingular. For example, consider phi(r) = r with centers 0 and 1; the matrix is