## 1 Introduction
In our case we take the mean to be 2 and variance to be 4, though the specific values the mean and variance parameter take do not matter. We first standardize the initial matrix \(\mathbf{X}^{(\mathbf{0})}\) at the level of each by row, _i.e._, first subtracting the row mean from each entry and then dividing each entry in a given row by its row standard deviation. The matrix is then standardized at the level of each column, _i.e._, by first subtracting the column mean from each entry and then by dividing each entry by the respective column standard deviation. Row mean and standard deviation polishing followed by column mean and standard deviation polishing is defined as one iteration in the process of attempting to row and column standardize the matrix. The resulting matrix is defined as \(\mathbf{X^{(1)}}\). Now the same process is repeated with \(\mathbf{X^{(1)}}\) and repeated until successive renormalization eventually yields a row and column standardized matrix. The successive normalizations are repeated until \"convergence\" which for our purposes is defined as the difference in the squared Frobenius norm between two consecutive iterations being less than \(10^{-8}\). The figures (see Figures 1 and 2) are plots of the log of the ratios of the squared Frobenius norms of the differences between consecutive iterates. In particular, they capture the type of convergence patterns that are observed in the \(5\times 5\) case from different starting values. In [1] it was proved that regardless of the starting value (provided the dimensions are at least 3) the process of successive normalization always converges, in the sense that it leads to a doubly standardized matrix. In addition, it was noted that the convergence is very rapid. We can see empirically from Figures 1-11 that eventually the log of the  
Figure 1: Examples of patterns of convergence. Figure 2: Examples of patterns of convergence. successive squared differences tends to decay in a straight line, _i.e._, the rate of convergence is perhaps exponential. This phenomenon of linear decay between successive iterations in the log scale is observed in all the diagrams. Hence, one is led naturally to ask whether this is always true in theory, and if so under what conditions. The rate of convergence of successive normalization and related questions are addressed in this paper. A natural question to ask is whether the convergence phenomenon observed will still occur if simultaneous normalization is undertaken as compared to successive normalization. In other words the row and column mean polishing and row and column standard deviation polishing are all done at once:  
\[X_{ij}^{t+1}=\frac{X_{ij}^{t}-\bar{X}_{i.}^{t}-\bar{X}_{.j}^{t}}{s_{i.}s_{.j}}\]  
In this case the simultaneous normalization algorithm does not converge. In fact it is shown to diverge. Figures 3-8 illustrate this phenomenon. Figure 4: (Left) Example of simultaneous normalization, (Right) Same example zoomed in. Figure 3: Two examples of simultaneous normalization: these illustrate that simultaneous normalization does not lead to convergence. Figure 5: (Left) Example of simultaneous normalization, (Right) Same example zoomed in. Figure 6: (Left) Example of simultaneous normalization, (Right) Same example zoomed in. Figure 7: Two further examples of simultaneous normalization: these illustrate that simultaneous normalization does not lead to convergence.