# SVD decomposition and the update problem of matrix differential equations

For a matrix $$Y(t) \in \mathbb{R}^{m \times n}$$, its rank-r approximation could be represented in a factorized SVD-like form. $$Y(t) = U(t) S(t) V^T(t),$$ where $$U^{T}U = I_m$$, $$V^{T}V = I_n$$ and $$S \in \mathbb{R}^{r \times r}$$.

I want to integrate the matrix $$Y$$ and have obtained the following iterative format: $$U^{k+1} S^{k+1} V^{k+1} = U^{k} S^{k} V^{k} + (1 - (U^{k} S^{k} V^{k}).^2). \qquad \tag{1}$$

where $$1-{}$$ and $$.^2$$ denote elementwise operations. Then, can $$U^{k+1}$$, $$S^{k+1}$$, $$V^{k+1}$$ be represented by some combination of $$U^{K}$$, $$S^{K}$$, and $$V^{k}$$?

In fact, equation (1) could easily be rewritten." $$Y^{k+1} = U^{k} S^{k} V^{k} + (1 - (U^{k} S^{k} V^{k}).^2). \tag{2}$$ then $$[U^{k+1}, S^{k+1}, V^{k+1}] = svd(Y^{k+1}).$$ However, I don't want to conduct SVD decomposition at every iteration step. Is it possible to directly obtain $$U^{k+1},S^{k+1},V^{k+1}$$?

• Have you considered the dynamic low-rank approximation? See doi.org/10.1137/050639703, eq 2.8 Commented Mar 13 at 4:29
• @StevenRoberts This is what I did. The low-rank approximation involves an iterative step, and I don't know if it is possible to update U, S, and V for equation (1) without an additional SVD decomposition. Commented Mar 13 at 4:42