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Let G m×n be an m × n real random matrix whose elements are independent and identically distributed standard normal random variables, and let κ 2 (G m×n) be the 2-norm condition number of G m×n. We prove that, for any m ≥ 2, n ≥ 2, and x ≥ |n − m| + 1, κ 2 (G m×n) satisfies 1 √ 2π (c/x) |n−m|+1 < P (κ 2 (G m×n) n/(|n−m|+1) > x) < 1 √ 2π (C/x) |n−m|+1 ,(More)
Soft errors are one-time events that corrupt the state of a computing system but not its overall functionality. Large supercomputers are especially susceptible to soft errors because of their large number of components. Soft errors can generally be detected offline through the comparison of the final computation results of two duplicated computations, but(More)
The probability that a failure will occur before the end of the computation increases as the number of processors used in a high performance computing application increases. For long running applications using a large number of processors, it is essential that fault tolerance be used to prevent a total loss of all finished computations after a failure.(More)
As the number of processors in today's high performance computers continues to grow, the mean-time-to-failure of these computers are becoming significantly shorter than the execution time of many current high performance computing applications. Although today's architectures are usually robust enough to survive node failures without suffering complete(More)
Modeling and analysis of large scale scientific systems often use linear least squares regression, frequently employing Cholesky factorization to solve the resulting set of linear equations. With large matrices, this often will be performed in high performance clusters containing many processors. Assuming a constant failure rate per processor, the(More)
— Error correction codes defined over real-number and complex-number fields have been studied and recognized as useful in many applications. However, most real-number and complex-number codes in literature are quite suspect in their numerical stability. In this paper, we introduce a class of numerically stable real-number and complex-number codes that are(More)
Fail-stop failures in distributed environments are often tolerated by checkpointing or message logging. In this paper, we show that fail-stop process failures in ScaLAPACK matrix-matrix multiplication kennel can be tolerated without checkpointing or message logging. It has been proved in previous algorithm-based fault tolerance that, for matrix-matrix(More)
A simple checkpoint-free fault-tolerant scheme for parallel iterative methods is given. Assuming that when one processor fails, all its data is lost and the system is recovered with a new processor, this scheme computes a new approximate solution from the data of the non-failed system. The iterative method is then restarted from this new vector. The main(More)
  • Zizhong Chen
  • 2009
It has been demonstrated recently that single fail-stop process failure in ScaLAPACK matrix multiplication can be tolerated without checkpointing. Multiple simultaneous processor failures can be tolerated without checkpointing by encoding matrices using a real-number erasure correcting code. However, the floating-point representation of a real number in(More)
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