We show how an IEEE-754 conformant precision- p base-β arithmetic can be implemented based on some binary floating-point and/or integer arithmetic. This includes the four basic operations and square root subject to the five IEEE-754 rounding modes, namely the nearest roundings with roundTiesToEven and roundTiesToAway, the directed roundings downwards and upwards, as well as rounding towards zero. Exceptional values like ∞ of NaN are covered according to the IEEE-754 arithmetic standard. The results of the precision- p base-β operations are computed using some underlying precision- q binary arithmetic. We distinguish two cases. When using a precision- q binary integer arithmetic, the base-β precision p is limited for all operations by β 2 p ≤ 2 q , whereas using a precision- q binary floating-point arithmetic imposes stronger limits on the base-β precision, namely β 2 p ≤ 2 q for addition and multiplication, β 2 p ≤ 2 q-1 for division and β 2 p ≤ 2 q -3 for the square root. Those limitations cannot be improved. The algorithms are implemented in a Matlab/Octave flbeta-toolbox with the choice of using uint64 or binary64 as underlying arithmetic. The former allows larger precisions, the latter is advantageous for the square root, whereas computing times are similar. The flbeta-toolbox offers precision- p base-β scalar, vector and matrix operations including sparse matrices as well as corresponding interval operations. The base β can be chosen in the range β ∊ [2,64]. The flbeta-toolbox will be part of Version 13 of INTLAB [ 18 ], the Matlab/Octave toolbox for reliable computing.

We present a pair arithmetic for the four basic operations and square root. It can be regarded as a simplified, more-efficient double-double arithmetic. The central assumption on the underlying arithmetic is the first standard model for error analysis for operations on a discrete set of real numbers. Neither do we require a floating-point grid nor a rounding to nearest property. Based on that, we define a relative rounding error unit u and prove rigorous error bounds for the computed result of an arbitrary arithmetic expression depending on u, the size of the expression, and possibly a condition measure. In the second part of this note, we extend the error analysis by examining requirements to ensure faithfully rounded outputs and apply our results to IEEE 754 standard conform floating-point systems. For a class of mathematical expressions, using an IEEE 754 standard conform arithmetic with base β , the result is proved to be faithfully rounded for up to 1 / √ β u - 2 operations. Our findings cover a number of previously published algorithms to compute faithfully rounded results, among them Horner’s scheme, products, sums, dot products, or Euclidean norm. Beyond that, several other problems can be analyzed, such as polynomial interpolation, orientation problems, Householder transformations, or the smallest singular value of Hilbert matrices of large size.

This note summarizes recent progress in error bounds for compound operations performed in some computer arithmetic. Given a general set of real numbers together with some operations satisfying the first standard model, we identify three types A, B, and C of weak sufficient assumptions implying new results and sharper error estimates. Those include linearized error estimates in the number of operations, faithfully rounded and reproducible results. All types of assumptions are satisfied for an IEEE-754 p-digit base-β floating-point arithmetic.

Suppose an m -digit floating-point arithmetic in base β ≥ 2 following the IEEE754 arithmetic standard is available. We show how a k -digit arithmetic with k < m can be inherited solely using m -digit operations. This includes the rounding into k digits, the four basic operations and the square root, all for even or odd base β. In particular, we characterize the relation between k and m so that no double rounding occurs when computing in m digits and rounding the result into k digits. We discuss rounding to nearest as well as directed rounding, and our approach covers exceptional values including signed zero. For binary arithmetic, a Matlab toolbox based on binary64 including k -bit scalar, vector and matrix operations as well as k -bit interval arithmetic is part of Version 8 of INTLAB, the Matlab toolbox for reliable computing.

The classical mathematical proof is performed by pencil and paper. However, there are many ways in which computers may be used in a mathematical proof. But "proofs by computers" or even the use of computers in the course of a proof are not so readily accepted (the December 2008 issue of the Notices of the American Mathematical Society is devoted to formal proofs by computers).

The High-Accuracy Arithmetic Subroutine Library (ACRITH) is a program product for engineering / scientific application — It consists of a subroutine library for solving problems in numerical computation. All results obtained have algorithmically verified accuracy.

The floating-point arithmetic on computers is designed to approximate the corresponding operations over the real numbers as close as possible. In this paper it is shown by means of counterexamples that this need not to be true for existing machines. For achieving good numerical results a floating-point arithmetic approximating the real operations as close as possible is probably best. For achieving verifications on computers, at least a precisely defined computer arithmetic is indispensable. In this paper we first introduce the Kulisch/Miranker theory, which represents a sound basis for computer arithmetic. Each operation is precisely defined and, moreover, is of maximum accuracy. That means, the, computed result is the floating-point number of the working precision closest to the infinite precise result. The theory also covers directed roundings allowing computations with intervals. These properties hold true for the floating-point numbers of single and double precision as well as for the vectors, matrices and complex extensions over those. In the second part of the paper we demonstrate the theoretical basis for what we call ‘Higher Order Computer Arithmetic’. This is an inclusion theory allowing the development of algorithms to compute bounds for the solution of various problems in numerical analysis. These bounds are automatically verified to be correct and they are of high accuracy. Very often they are of maximum accuracy, that means the left and right bounds of all components of the solution are adjacent in the floating-point screen. Moreover existence and uniqueness of a solution within the computed bounds is automatically verified by the algorithm. If this verification is not possible, a respective message is given. We develop the theory and give algorithms for the solution of systems of linear and nonlinear equations. As demonstrated by examples even for extremely ill-conditioned problems existence and uniqueness of the solution is verified within bounds of least significant bit accuracy.