The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
Reducing truncation errors by programming
Communications of the ACM
A description and comparison of subroutines for computing Euclidean inner products on the IBM 360
A description and comparison of subroutines for computing Euclidean inner products on the IBM 360
Rounding Errors in Algebraic Processes
Rounding Errors in Algebraic Processes
Best “ordering” for floating-point addition
ACM Transactions on Mathematical Software (TOMS)
Algorithm 550: Solid Polyhedron Measures [Z]
ACM Transactions on Mathematical Software (TOMS)
An annotated bibliography on microprogramming: late 1969 -- early 1972
MICRO 5 Conference record of the 5th annual workshop on Microprogramming
A comparison of methods for accurate summation
ACM SIGSAM Bulletin
Loss of Significance in Floating Point Subtraction and Addition
IEEE Transactions on Computers
Floating-Point Computation of Functions with Maximum Accuracy
IEEE Transactions on Computers
Additive preconditioning and aggregation in matrix computations
Computers & Mathematics with Applications
A new error-free floating-point summation algorithm
Computers & Mathematics with Applications
Accurate Matrix Factorization: Inverse LU and Inverse QR Factorizations
SIAM Journal on Matrix Analysis and Applications
Comments on fast and exact accumulation of products
PARA'10 Proceedings of the 10th international conference on Applied Parallel and Scientific Computing - Volume 2
Verified Bounds for Least Squares Problems and Underdetermined Linear Systems
SIAM Journal on Matrix Analysis and Applications
Accurate solution of dense linear systems, part I: Algorithms in rounding to nearest
Journal of Computational and Applied Mathematics
| Hi-index | 48.23 |
cumulation of floating-point sums is considered on a computer which performs t-digit base &bgr; floating-point addition with exponents in the range —m to M. An algorithm is given for accurately summing n t-digit floating-point numbers. Each of these n numbers is split into q parts, forming q·n t-digit floating-point numbers. Each of these is then added to the appropriate one of &eegr; auxiliary t-digit accumulators. Finally, the accumulators are added together to yield the computed sum. In all, q·n + &eegr; - 1 t-digit floating-point additions are performed. Let &ngr; = ⌈(M + m + 1)/(&eegr; + 1)⌉. If n ≤ (1/q)&bgr;⌈((q-1)/q)t⌈-&ngr;+1 (*), then the relative error in the computed sum is at most ⌈(t + 1)/&ngr;⌉&bgr;1-t. Further, with an additional q + &eegr; - 1 t-digit additions, the computed sum can be corrected to full t-digit accuracy.For example, for the IBM/360 (&bgr; = 16, t = 14, M = 63, m = 64), typical values for q and &eegr; are q = 2 and &eegr; = 32. In this case, (*) becomes n ≤ 1/2 × 164 = 32,768, and we have ⌈(t + 1)/&ngr;⌉&bgr;1-t = 4 × 16-13.