4.11 Scaling Strategies

Some algorithms attempt to reduce the roundoff error generated during LU decomposition by preprocessing A. The most common roundoff control strategy is row scaling where each equation is normalized (so that its largest coefficient has a value of one) before a pivot is chosen. Pivoting strategies which employ scaling techniques usually are implemented in two stages:

  1. Scan the equations to determine a set of scale factors.

  2. Choose the pivot elements taking the scale factors into account.

However, a word of caution is in order. Scaling techniques do not solve the fundamental roundoff problem of adding a large quantity to a small one. In fact, problems may arise where scaling techniques exacerbate rather than ameliorate roundoff error.