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Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
Treatment Of NearBreakdown In The Cgs Algorithm
 Numer. Algorithms
, 1994
"... Lanczos method for solving the system of linear equations Ax = b consists in constructing a sequence of vectors (x k ) such that r k = b \Gamma Ax k = P k (A)r 0 where r 0 = b \Gamma Ax 0 . P k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) ..."
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Cited by 9 (6 self)
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Lanczos method for solving the system of linear equations Ax = b consists in constructing a sequence of vectors (x k ) such that r k = b \Gamma Ax k = P k (A)r 0 where r 0 = b \Gamma Ax 0 . P k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in taking r k = P k (A)r 0 . In the recurrence relation for P k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can affect seriously the algorithm, a situation known as nearbreakdown. In this paper it is shown how to avoid nearbreakdown in the CGS algorithm in order to obtain a more stable method.
On Numerical Accuracy of GaussChebyshev Integration Rules Using the Stochastic Arithmetic
, 2008
"... Abstract—In this paper, the evaluation of I = f(x) dx is proposed by using the opened and −1 1−x2 closed Gauss Chebyshev integration rules in the stochastic arithmetic. For this purpose, a theorem is proved to show the accuracy of the GaussChebyshev rules. Then, the CESTAC 1 method and the stochas ..."
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Abstract—In this paper, the evaluation of I = f(x) dx is proposed by using the opened and −1 1−x2 closed Gauss Chebyshev integration rules in the stochastic arithmetic. For this purpose, a theorem is proved to show the accuracy of the GaussChebyshev rules. Then, the CESTAC 1 method and the stochastic arithmetic are used to validate the results and implement the numerical example.
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"... Stepsize control of the finite difference method for solving ordinary differential equations ..."
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Stepsize control of the finite difference method for solving ordinary differential equations