lternative approach, the ID algorithm, for implementing fault tolerance in matrix multiplication is based upon the generation of syndromes for intermediate results. The algorithm detects faults with single iteration fault latency as opposed to the full problem run-time fault detection latency for the checksum algorithm. The penalty for this implementation is an increase in matrix size by a factor of logn. While the error coverage of the algorithm is not complete, it is better than the checksum algorithm. Another classical problem in the computing environment is the sorting of lists and database elements. Due to the nature of the sorting problem, there is no opportunity to identify errors in the intermediate phases of the sorting process; the executable assertion is only valid after completion of the entire process. As a solution to this condition, the bitonic sort algorithm was introduced as a parallel sorting algorithm. The general idea of a bitonic sort is to build up longer bitonic sequences which will eventually lead to a sorted sequence while providing the ability to apply executable assertions to the intermediate results. Researchers have examined ABFT techniques for fast Fourier transforms (FFT) using the modified recomputing with shifted operands technique and an extra stage in the standard FFT butterfly network. Others propose an encoding based upon the superposition and shift properties of the FFT. The more general class of signal processing applications have been addressed by utilizing orthogonal transformations such as the properties of applications in which the sum-of- squares of the input and that of the output are proportional. The FFT is one of the problems in which the sum-of-squares of the output equals the sum-of-squares of the input times the length of the input vector. Another signal processing algorithm in which sum-of-squares encoding applies is QR factorization, which is the factorization of a matrix into two matrices, an orthogonal and an upper triangular matrix. A number of problems, e.g