Methods and applications of error-free computation
Methods and applications of error-free computation
A new polynomial-time algorithm for linear programming
Combinatorica
A variation on Karmarkar's algorithm for solving linear programming problems
Mathematical Programming: Series A and B
Pi and the AGM: a study in the analytic number theory and computational complexity
Pi and the AGM: a study in the analytic number theory and computational complexity
Scientific American
A polynomial-time algorithm, based on Newton's method, for linear programming
Mathematical Programming: Series A and B
Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi
American Mathematical Monthly
Vectors versus matrices: p-inversion, cryptographic application, and vector implementation
Neural, Parallel & Scientific Computations
Solving linear programming problems exactly
Applied Mathematics and Computation
Fast Multiple-Precision Evaluation of Elementary Functions
Journal of the ACM (JACM)
O(n3) noniterative heuristic algorithm for linear programs with error-free implementation
Applied Mathematics and Computation
Arthur Norberg, the Charles Babbage Institute, and the History of Computing
IEEE Annals of the History of Computing
Interpolation for nonlinear BVP in circular membrane with known upper and lower solutions
Computers & Mathematics with Applications
Golden ratio versus pi as random sequence sources for Monte Carlo integration
Mathematical and Computer Modelling: An International Journal
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We present here the best k-digit rational bounds for a given irrational number, where the numerator has k digits. Of the two bounds, either the upper bound or the lower bound, will be the best k-digit rational approximation for the given irrational number. The rational bounds derived from the corresponding k-digit decimal bounds are not often the best rational bounds for an irrational number. Such bounds not only allow a possible introduction of irrational numbers such as @p, e, and log"e2 but also to compute error-bounds in an error-free computational problem. We have also focused on the importance of twenty-first century supercomputers with steadily increasing computing power-both sequential and parallel-in computing the best bounds as well as in determining error-bounds for a problem in error-free computational environment. We have also focused on the tremendous activities during/after pre-historic era on obtaining rational approximation/bounds of famous irrational numbers to justify the relevance and possible importance of this study in the current ultra-high speed computing age.