Finding $A^TA = B^TB + I$ where $B$ is known and $I$ is identity. Does it exist any explicit formula for $A$, if $B$ is known and I is identity? But without using decomposition of $B^TB$, since numeric error is too large for using $B^TB = U\Lambda U^T$ or without using $B^TB + I = U\Lambda U^T$ and take $A = \Lambda^{\frac{1}{2}} U^T$. 

We can note that $B^TB + I = B^TB + I^TI$. Thus, we can take $A = (B \quad I)$. It can be easily verified that $A^TA = B^TB + I^TI$. This is the cheapest solution, since we don't need to do any additional computation.
