Editing Vandermonde matrix (section)

== Confluent Vandermonde matrices ==
{{see also|Hermite interpolation}}
As described before, a Vandermonde matrix describes the linear algebra [[polynomial interpolation|interpolation problem]] of finding the coefficients of a polynomial <math>p(x)</math> of degree <math>n - 1</math> based on the values <math> p(x_1),\, ...,\, p(x_n)</math>, where <math>x_1,\, ...,\, x_n</math> are ''distinct'' points. If <math>x_i</math> are not distinct, then this problem does not have a unique solution (and the corresponding Vandermonde matrix is singular). However, if we specify the values of the derivatives at the repeated points, then the problem can have a unique solution. For example, the problem

:<math>\begin{cases}
   p(0) = y_1 \\
  p'(0) = y_2 \\
   p(1) = y_3 
\end{cases}</math>

where <math>p(x) = ax^2+bx+c</math>, has a unique solution for all <math>y_1,y_2,y_3</math> with <math>y_1\neq y_3</math>. In general, suppose that <math>x_1, x_2, ..., x_n</math> are (not necessarily distinct) numbers, and suppose for simplicity that equal values are adjacent:

:<math>
  x_1 = \cdots = x_{m_1},\ 
  x_{m_1+1} = \cdots = x_{m_2},\ 
  \ldots,\ 
  x_{m_{k-1}+1} = \cdots = x_{m_k}
</math>

where  <math>m_1 < m_2 < \cdots < m_k=n,</math> and <math>x_{m_1}, \ldots ,x_{m_k}</math> are distinct. Then the corresponding interpolation problem is

:<math>\begin{cases}
  p(x_{m_1}) = y_1, & p'(x_{m_1}) = y_2, & \ldots, & p^{(m_1-1)}(x_{m_1}) = y_{m_1}, \\
  p(x_{m_2}) = y_{m_1+1}, & p'(x_{m_2})=y_{m_1+2}, & \ldots, & p^{(m_2-m_1-1)}(x_{m_2}) = y_{m_2}, \\
  \qquad \vdots & & & \qquad\vdots \\
  p(x_{m_k}) = y_{m_{k-1}+1}, & p'(x_{m_k}) = y_{m_{k-1}+2}, & \ldots, & p^{(m_k-m_{k-1}-1)}(x_{m_k}) = y_{m_k}.
\end{cases}</math>

The corresponding matrix for this problem is called a '''confluent Vandermonde matrix''', given as follows.<ref>{{Cite journal | volume = 57| issue = 1| pages = 15–21| last = Kalman| first = D.| title = The Generalized Vandermonde Matrix| journal = Mathematics Magazine| date = 1984| doi = 10.1080/0025570X.1984.11977069}}</ref> If <math>1 \leq i,j \leq n</math>, then <math>m_\ell < i \leq m_{\ell + 1} </math> for a unique <math>0 \leq \ell \leq k-1</math> (denoting <math>m_0 = 0</math>). We let

:<math>V_{i,j} = \begin{cases}
                                                            0 & \text{if } j < i - m_\ell, \\[6pt]
  \dfrac{(j-1)!}{(j - (i - m_\ell))!} x_i^{j-(i-m_\ell)} & \text{if } j \geq i - m_\ell.
\end{cases}</math>

This generalization of the Vandermonde matrix makes it [[Invertible matrix|non-singular]], so that there exists a unique solution to the system of equations, and it possesses most of the other properties of the Vandermonde matrix. Its rows are derivatives (of some order) of the original Vandermonde rows. 

Another way to derive the above formula is by taking a limit of the Vandermonde matrix as the <math>x_i</math>'s approach each other. For example, to get the case of <math>x_1 = x_2</math>, take subtract the first row from second in the original Vandermonde matrix, and let <math>x_2\to x_1</math>: this yields the corresponding row in the confluent Vandermonde matrix. This derives the generalized interpolation problem with given values and derivatives as a limit of the original case with distinct points: giving <math>p(x_i), p'(x_i)</math> is similar to giving <math>p(x_i), p(x_i + \varepsilon)</math> for small <math>\varepsilon</math>. Geometers have studied the problem of tracking confluent points along their tangent lines, known as [[Configuration space (mathematics)|compacitification of configuration space]].