## Faster Numerical Algorithms via Exception Handling (1994)

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Venue: | IEEE Trans. Comput |

Citations: | 46 - 7 self |

### BibTeX

@ARTICLE{Demmel94fasternumerical,

author = {James W. Demmel and Xiaoye Li},

title = {Faster Numerical Algorithms via Exception Handling},

journal = {IEEE Trans. Comput},

year = {1994},

volume = {43},

pages = {983--992}

}

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### Abstract

An attractive paradigm for building fast numerical algorithms is the following: (1) try a fast but occasionally unstable algorithm, (2) test the accuracy of the computed answer, and (3) recompute the answer slowly and accurately in the unlikely event it is necessary. This is especially attractive on parallel machines where the fastest algorithms may be less stable than the best serial algorithms. Since unstable algorithms can overflow or cause other exceptions, exception handling is needed to implement this paradigm safely. To implement it efficiently, exception handling cannot be too slow. We This is an expanded version of a paper published in Proceedings of the 11th Symposium on Computer Arithmetic, Windsor, Ontario, June 29-July 2 1993, pp. 234-241, copyright c fl1993, IEEE. y Computer Science Division and Mathematics Department, University of California, Berkeley CA 94720. Email: demmel@cs.berkeley.edu. The author was supported by NSF grant ASC-9005933, DARPA contract DAAL03-9...

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Citation Context ... solve triangular systems of linear equations which are possibly very ill-conditioned. Triangular system solving is one of the matrix operations found in the Basic Linear Algebra Subroutines, or BLAS =-=[9, 10, 18]-=-. The BLAS, which include related operations like dot product, matrix-vector multiplication, and matrix-matrix multiplication, occur frequently in scientific computing. This has led to their standardi... |

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Citation Context ... solve triangular systems of linear equations which are possibly very ill-conditioned. Triangular system solving is one of the matrix operations found in the Basic Linear Algebra Subroutines, or BLAS =-=[9, 10, 18]-=-. The BLAS, which include related operations like dot product, matrix-vector multiplication, and matrix-matrix multiplication, occur frequently in scientific computing. This has led to their standardi... |

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Citation Context ... solve triangular systems of linear equations which are possibly very ill-conditioned. Triangular system solving is one of the matrix operations found in the Basic Linear Algebra Subroutines, or BLAS =-=[9, 10, 18]-=-. The BLAS, which include related operations like dot product, matrix-vector multiplication, and matrix-matrix multiplication, occur frequently in scientific computing. This has led to their standardi... |

115 |
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Citation Context ...st flags to detect when exceptions occur, and recover when exceptions occur. We report performance results on a "fast" DECstation 5000 and a "slow" DECstation 5000 (both have a MIP=-=S R3000 chip as CPU [17]), a Sun 4-=-/260 (which has a SPARC chip as CPU [15]), a DEC Alpha [11], a CRAY-C90 and a SPARCstation 10 with a Viking microprocessor. The "slow" DEC 5000 correctly implements IEEE arithmetic, but does... |

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Citation Context ...HSEIN (which now calls SLATRS) to compute the eigenvectors of a real upper Hessenberg matrix. In complex division, gradual underflow instead of flush to zero can guarantee a more accurate result, see =-=[5]-=-. This requires fast arithmetic with denormalized numbers. Floating point parallel prefix is a useful operation for various linear algebra problems. Its robust implementation with the protection again... |

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Citation Context ...cision. The division involved in the recurrence for t may cause division by zero or overflow. Again, to prevent the occurrence of the exceptions, a more careful scheme was first developed by W. Kahan =-=[16]-=- and later used in LAPACK SSTEBZ routine [2]. There, the algorithm first computes a threshold pivmin, which is the smallest number that can divide b 2 i without overflow. Inside the inner loop the div... |

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Citation Context ...ection algorithms on the CM-5. ing messages. The floating-point arithmetic on the CM-5 conforms to IEEE standard, and infinity arithmetic is as fast as conventional arithmetic. Inderjit Dhillon et al.=-=[8]-=- have designed a parallel bisection algorithm on the CM-5, where the whole spectrum is divided into 64 subintervals, and each processing node is responsible for finding the eigenvalues within one subi... |

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Citation Context ... else support user-level trap handling. In this paper we will assume the first response to exceptions is available; this corresponds to the default behavior of IEEE standard floating point arithmetic =-=[3, 4]-=-. Our numerical methods will be drawn from the LAPACK library of numerical linear algebra routines for high performance computers [2]. In particular, we will consider condition estimation (error bound... |

8 |
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Citation Context ... the reciprocal of k 1 (A), which we call RCOND, to avoid overflow in k 1 (A). The inputs to these routines include the factors L and U from the factorization A = LU and kAk 1 . Higham's modification =-=[14]-=- of Hager's method [13] is used to estimate jjA \Gamma1 jj 1 . The algorithm is derived from a convex optimization approach, and is based on the observation that the maximal value of the function f(x)... |

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Citation Context ...wo, and disregards the possibility of over/underflow. The second scales carefully to avoid over/underflow, and is the one currently used in LAPACK for condition estimation and eigenvector computation =-=[1]-=-. We will solve Lx = b, where L is a lower triangular n-by-n matrix. We use the notation L(i : j; k : l) to indicate the submatrix of L lying in rows i through j and columns k through l of L. Similarl... |

2 |
1992a), `Specifications for robust parallel prefix operations
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(Show Context)
Citation Context ...ared only by writing a new value into the Control/Status register. The nonsticky exception bits might be used in other applications requiring finer grained exception handling, such as parallel prefix =-=[6]-=-. In the algorithms developed in this paper we need only manipulate the trap enable bits (set them to zero to disable software traps) and the sticky bits. Procedure exceptionreset() clears the sticky ... |