Talk:Polyharmonic spline

Untitled
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)

Thats why there is a polynomial, namely to avoid the PSD matrices. See "Spline Models for Observational Data" by Wahba, Page 31. — Preceding unsigned comment added by Hannes36743 (talk • contribs) 17:13, 3 December 2013 (UTC)

What's T?
In the definition section, which is otherwise quite good, what is "T"? As in, the term superscripted on so many of the matrices....? — Preceding unsigned comment added by 50.26.246.186 (talk) 17:02, 19 February 2016 (UTC)

Matrix transpose, this is now explicitly stated in definition section Jrheller1 (talk) 19:02, 20 February 2016 (UTC)

O.K. Maybe it threw me off because it seems strange to declare B as a transpose, then to also transpose it in the constraint equation. In any case, definitely a good addition to the text! — Preceding unsigned comment added by 50.26.246.186 (talk) 22:55, 22 February 2016 (UTC)


 * That's a common notation in Mathematics. It saves a lot of space if the matrix is long and thin. Column vectors are also usually declared as the transpose of a row vector for the same reason. 93.132.186.56 (talk) 10:42, 9 March 2023 (UTC)

Clarification on additional constraints
In the section 'additional constraints' two systems of linear equations are derived: $$ A(A\mathbf{w} + B\mathbf{v} - \mathbf{f} +\lambda C \mathbf{w}) = 0 $$ and $$ B^{\textrm{T}}(A\mathbf{w} + B\mathbf{v} - \mathbf{f}) = 0.$$ Then it is stated that $$A$$ is invertible. Is this actually true? Why?