I have a few questions regarding the (scaled) partial pivoting method.
In my lecture slides, it says that partial pivoting chooses the biggest pivot in magnitude to prevent round-off errors from accumulating. However, I fail to understand why choosing the biggest pivot is better, because what if this pivot is $10^{8}$ while the other entries are relatively small. The partial pivoting method would then choose $10^{8}$ as the pivot, but won't this pivot cause round-off errors too?
For the method scaled partial pivoting, we choose the relatively big pivot in magnitude for situations where the other elements of the pivot row are relatively large. But why? What about elements that are relatively small in the pivot row? Or is it really only about relatively large elements?
Thank you.