Error analysis in unnormalized floating point arithmetic

The need to construct architectures in VLSI has focused attention on unnormalized floating point arithmetic. Certain unnormalized arithmetics allow one to 'pipe on digits,' thus producing significant speed up in computation and making the input problems of special purpose devices such as systolic arrays easier to solve. We consider the error analysis implications of using unnormalized arithmetic in numerical algorithms. We also give specifications for its implementation. Our discussion centers on the example of Gaussian elimination. We show that the use of unnormalized arithmetic requires change in the analysis of this algorithm. We will show that only for certain classes of matrices that include diagonally dominant matrices (either row or column), Gaussian elimination is as stable in unnormalized arithmetic as in normalized arithmetic. However, if the diagonal elements of the upper triangular matrix are post normalized, then Gaussian elimination is as stable in unnormalized arithmetic as in normalized arithmetic for all matrices.