is an example.

Here we do not actually conduct a sampling process when recruiting a new observation; instead, we only draw observations from a data storage device. The sampling cost here is reduced; however, the computation cost is still noticeable, since we need to iteratively conduct the estimation process for each procedure time until its stopping criterion is fulfilled. That is, we need to compute the determinant of the matrix, repeatedly. To this end, we consider the following two computation strategies.

### Iterative formulas {#iterative-formulas .unnumbered}

It is clear that in such a sequential estimation procedure, the calculations of the determinant and inverse of the design matrix demand considerable computation power. However, with help of simple algebra, both can be calculated based on their predecessors. The following properties allow us to update the determinant of the new design matrix, when we sequentially add a new observation, with simple linear algebra. These two formulas are common in textbooks, such as [@rao1973]; for convenience, we describe these below.

Let $\mathbf A$ be a nonsingular $p\times p$ matrix and $\mathbf U \in R^p$ be a column vector. Then $$\begin{aligned}
\label{inverse}
(\mathbf A + \mathbf U \mathbf U^T)^{-1} = \mathbf A ^{-1} - \frac{ 