## Improving interval enclosures (2009)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Neumaier09improvinginterval,

author = {Arnold Neumaier},

title = {Improving interval enclosures},

year = {2009}

}

### OpenURL

### Abstract

This paper serves as background information for the Vienna proposal for interval standardization, explaining what is needed in practice to make competent use of the interval arithmetic provided by an implementation of the standard to be. Discussed are methods to improve the quality of interval enclosures of the range of a function over a box, considerations of possible hardware support facilitating the implementation of such methods, and the results of a simple interval challenge that I had posed to the reliable computing mailing list on November 26, 2008. Also given is an example of a bound constrained global optimization problem in 4 variables that has a 2-dimensional continuum of global minimizers. This makes standard branch and bound codes extremely slow, and therefore may serve as a useful degenerate test problem.

### Citations

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Citation Context ... theory and practice of rigorously working on a computer with certain (exactly known) and uncertain (i.e., possibly not exactly known) real numbers, represented as intervals; see, e.g., the textbooks =-=[25, 26, 29, 28]-=-. It involves the appropriate use of standard floating-point calculations (in round to nearest mode), directed floating-point calculations (in rounding modes up or down), and interval arithmetic, comb... |

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Citation Context ... theory and practice of rigorously working on a computer with certain (exactly known) and uncertain (i.e., possibly not exactly known) real numbers, represented as intervals; see, e.g., the textbooks =-=[25, 26, 29, 28]-=-. It involves the appropriate use of standard floating-point calculations (in round to nearest mode), directed floating-point calculations (in rounding modes up or down), and interval arithmetic, comb... |

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Citation Context ... does not need expensive trigonometric functions, its parameters have a geometric meaning independent of the coordinate system used, and it has significantly better interpolation properties (Shoemake =-=[58]-=-, Ramamoorthi & Barr [48]). Note that the projective identification mentioned above has to be taken into account when constructing smooth motions joining two close rotations Q[r] with nearly opposite ... |

240 | Semidefinite programming relaxations for semialgebraic problems
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Citation Context .... also Huyer & Neumaier [12]. Similarly, implied inequalities such as the Cauchy-Schwarz inequality can be found automatically and rigorously by 12SOS (sum of squares) techniques; see, e.g., Parillo =-=[41]-=- and Schichl & Neumaier [54] for general theory, and Domes [5] for a rigorous implementation of some SOS techniques. Though these methods are slow, they need to be applied only once to an expression, ... |

95 | Introduction to Interval Analysis
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Citation Context ... theory and practice of rigorously working on a computer with certain (exactly known) and uncertain (i.e., possibly not exactly known) real numbers, represented as intervals; see, e.g., the textbooks =-=[25, 26, 29, 28]-=-. It involves the appropriate use of standard floating-point calculations (in round to nearest mode), directed floating-point calculations (in rounding modes up or down), and interval arithmetic, comb... |

71 | Complete search in continuous global optimization and constraint satisfaction. Acta Numerica
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Citation Context ...tant current applications of interval analysis include but are not restricted to the following areas (only sample references are given): • global optimization (used in lots of different applications) =-=[31, 40]-=-, • solving equations with multiple solutions (e.g., in chemical engineering [9], computational geometry [7, 35], robotics [24]), • quantified constraint satisfaction [49] (e.g, finding safe work spac... |

37 |
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Citation Context ...imation is O(ε 2 ) rather than the typical O(ε). One way of computing centered forms is by using slopes. If f(x) is an arithmetic expression in the components of x, a slope for f (Krawczyk & Neumaier =-=[21]-=-) is a row vector expression f[z,x] such that f(x) = f(z) + f[z,x](x − z) for all x,z. (9) Inserting for x the box x and for z an element from x (often the midpoint or a suitable corner) results in an... |

36 |
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Citation Context |

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Citation Context ...uations with multiple solutions (e.g., in chemical engineering [9], computational geometry [7, 35], robotics [24]), • quantified constraint satisfaction [49] (e.g, finding safe work spaces for robots =-=[23, 37]-=-), • design under uncertainty [34, 38], • computer-assisted proofs [33], • verified linear and semidefinite programming [14, 16]. 2Many applications (e.g., all global problems) need good bounds for v... |

30 | Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis
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Citation Context ...ven): • global optimization (used in lots of different applications) [31, 40], • solving equations with multiple solutions (e.g., in chemical engineering [9], computational geometry [7, 35], robotics =-=[24]-=-), • quantified constraint satisfaction [49] (e.g, finding safe work spaces for robots [23, 37]), • design under uncertainty [34, 38], • computer-assisted proofs [33], • verified linear and semidefini... |

29 | Efficient solving of quantified inequality constraints over the real numbers
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Citation Context ... different applications) [31, 40], • solving equations with multiple solutions (e.g., in chemical engineering [9], computational geometry [7, 35], robotics [24]), • quantified constraint satisfaction =-=[49]-=- (e.g, finding safe work spaces for robots [23, 37]), • design under uncertainty [34, 38], • computer-assisted proofs [33], • verified linear and semidefinite programming [14, 16]. 2Many applications... |

27 | Rigorous Analysis of Nonlinear Motion in Particle Accelerators - Makino - 1998 |

27 | Fast construction of accurate quaternion splines
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Citation Context ...trigonometric functions, its parameters have a geometric meaning independent of the coordinate system used, and it has significantly better interpolation properties (Shoemake [58], Ramamoorthi & Barr =-=[48]-=-). Note that the projective identification mentioned above has to be taken into account when constructing smooth motions joining two close rotations Q[r] with nearly opposite r of length close to 1. T... |

27 |
Systems of linear interval equations
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Citation Context ...monotonicity, many other kinds of rigorous computations require such a mix, for example, • the optimal solution of linear systems with an interval M-matrix by Barth & Nuding [2], • algorithms by Rohn =-=[52]-=- (see also [29, Chapter 7]) for the hull of interval linear systems, drastically reducing the number of endpoint combinations to be considered, • the Hansen-Bliek method [30] for solving linear system... |

25 | T.: A comparison of complete global optimization solvers
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(Show Context)
Citation Context ...tant current applications of interval analysis include but are not restricted to the following areas (only sample references are given): • global optimization (used in lots of different applications) =-=[31, 40]-=-, • solving equations with multiple solutions (e.g., in chemical engineering [9], computational geometry [7, 35], robotics [24]), • quantified constraint satisfaction [49] (e.g, finding safe work spac... |

24 | Safe bounds in linear and mixed-integer programming
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- 2003
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Citation Context ...educing the number of endpoint combinations to be considered, • the Hansen-Bliek method [30] for solving linear systems with interval coefficients, • optimal a posteriori bounds in linear programming =-=[14, 16, 39]-=-, • Makino’s range reduction technique, mentioned towards the end of Section 3. Thus, implementations of interval arithmetic should be such that mixed interval calculations and real calculations invol... |

20 | Linear systems with large uncertainties with applications to truss structures
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Citation Context ... in chemical engineering [9], computational geometry [7, 35], robotics [24]), • quantified constraint satisfaction [49] (e.g, finding safe work spaces for robots [23, 37]), • design under uncertainty =-=[34, 38]-=-, • computer-assisted proofs [33], • verified linear and semidefinite programming [14, 16]. 2Many applications (e.g., all global problems) need good bounds for very wide intervals, some (e.g., roundi... |

17 |
Interval linear systems with symmetric matrices, skewsymmetric matrices and dependencies in the right hand side
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- 1991
(Show Context)
Citation Context ...arameters is linear, the right-hand side is multilinear or fractional multilinear (depending on how the transformation to fixed-point form is done), and the above theorems apply. (As shown in Jansson =-=[13]-=-, the affine-linear dependence also allows a representation in which each interval parameter appears only once.) For a nonlinear parameter dependence, the bicentered form from Theorem 3.1 may be used.... |

14 | Automatic computation of a linear interval enclosure
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(Show Context)
Citation Context ...cursive way of computing centered forms, often with less overestimation. Related work on so-called G-intervals, together with applications to the solution of nonlinear systems, has been done by Kolev =-=[18, 19, 20]-=-. The most general and precise result about the quadratic approximation property is in Neumaier [32, Theorem 8.1], and applies to traditional centered forms (and can be easily adapted to the bicentere... |

13 | A simple derivation of the Hansen-Bliek-Rohn-NingKearfott enclosure for linear interval equations, Reliable Computing 5
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Citation Context ...[2], • algorithms by Rohn [52] (see also [29, Chapter 7]) for the hull of interval linear systems, drastically reducing the number of endpoint combinations to be considered, • the Hansen-Bliek method =-=[30]-=- for solving linear systems with interval coefficients, • optimal a posteriori bounds in linear programming [14, 16, 39], • Makino’s range reduction technique, mentioned towards the end of Section 3. ... |

12 |
Use of interval slopes for the irrational part of factorable functions
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Citation Context ...idpoint or a suitable corner) results in an enclosure with the quadratic approximation property. For the computation of slopes, see [21], Neumaier [29, Sections 2.2 and 2.3], Shen & Wolfe [57], Kolev =-=[17]-=-, and Schnurr [55, 56]. For wider intervals, the fact that the slope itself is evaluated naively may result in unnecessarily wide enclosures. To improve upon this, 10Stolfi & de Figueiredo [59] discu... |

11 | forms – use and limits, Reliable Computing 9 - Neumaier, Taylor - 2003 |

10 |
Optimal centered forms
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Citation Context ...expressions, checking monotonicity usually requires the prior enclosure of partial derivatives and checking their sign. If the signs are not constant, one may use the optimal centered form of Baumann =-=[3]-=- that uses the interval derivatives in the best possible way to get an enclosure. However, there is little computational advantage compared to bicentered forms, and the latter’s wider applicability (a... |

8 | GloptLab – a configurable framework for the rigorous global solution of quadratic constraint satisfaction problems
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(Show Context)
Citation Context ...uch as the Cauchy-Schwarz inequality can be found automatically and rigorously by 12SOS (sum of squares) techniques; see, e.g., Parillo [41] and Schichl & Neumaier [54] for general theory, and Domes =-=[5]-=- for a rigorous implementation of some SOS techniques. Though these methods are slow, they need to be applied only once to an expression, and provide – when successful – important information that may... |

7 | Optimale Lösung von Intervallgleichungsystemen - Barth, Nuding - 1974 |

7 |
On the solution of parametrised linear systems
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- 2001
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Citation Context ...can be found out by looking at the signs of the components of F(x). With y fixed at a bound, one can use half a bicentered form in x for each bound to get good enclosures for the range. Popova et al. =-=[42, 43, 44, 45]-=- (see also Tonon [60]) use Kaucher interval arithmetic for bounding ranges of monotone rational functions. In [43, 44] range enclosure is combined with a general-purpose technique for guaranteed enclo... |

7 | Interval arithmetic with containment sets
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- 2006
(Show Context)
Citation Context ...osed at (x,y,z) by monotonicity arguments using w = y2 + z2 and the enclosure [ f = x/ √ x2 + w , x/ √ x2 ] + w . (5) Similarly, the function f(x,y) := x 2 /(x 2 + y 2 ) considered by Pryce & Corliss =-=[47]-=- can be optimally handled by computing u := x 2 , v := y 2 , and (3) f = [u/(u + v),u/(u + v)]. (6) A more complicated instance is the following. To enclose the function f(a,q,e) := (a(1 − q) + e √ kq... |

7 |
On interval enclosures using slope arithmetic
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(Show Context)
Citation Context ...(often the midpoint or a suitable corner) results in an enclosure with the quadratic approximation property. For the computation of slopes, see [21], Neumaier [29, Sections 2.2 and 2.3], Shen & Wolfe =-=[57]-=-, Kolev [17], and Schnurr [55, 56]. For wider intervals, the fact that the slope itself is evaluated naively may result in unnecessarily wide enclosures. To improve upon this, 10Stolfi & de Figueired... |

6 |
Solving linear systems whose input data are rational functions of interval parameters, in: T. Boyanov et al
- Popova
- 2007
(Show Context)
Citation Context ...can be found out by looking at the signs of the components of F(x). With y fixed at a bound, one can use half a bicentered form in x for each bound to get good enclosures for the range. Popova et al. =-=[42, 43, 44, 45]-=- (see also Tonon [60]) use Kaucher interval arithmetic for bounding ranges of monotone rational functions. In [43, 44] range enclosure is combined with a general-purpose technique for guaranteed enclo... |

5 |
Hentenryck, “Standardized notation in interval analysis”, http://www.mat.univie.ac.at/ neum/software/int/notation.ps.gz
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(Show Context)
Citation Context ... global minimizers. This makes standard branch and bound codes extremely slow, and therefore may serve as a useful degenerate test problem. Our notation follows the recommendations in Kearfott et al. =-=[15]-=- for standardized notation in interval analysis. In particular, boldface letters such as x denote intervals or interval vectors; x and x denote their lower and upper bound, respectively. Acknowledgmen... |

4 | Efficient method of solution of large scale engineering problems with interval parameters
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- 2004
(Show Context)
Citation Context ...e bicentered form from [29, p.59]. Suitable values for x l and x u can be found by the endpoints minimizing and maximizing a linear Taylor expansion at the midpoint of x, a choice suggested by Pownuk =-=[46]-=- to get a good inner approximation (10) of the range that frequently is optimal. Alternatively, a more sophisticated local search may be employed to find good x l and x u . With exact arithmetic and P... |

4 | Transposition theorems and qualification-free optimality conditions
- Schichl, Neumaier
(Show Context)
Citation Context .... Similarly, implied inequalities such as the Cauchy-Schwarz inequality can be found automatically and rigorously by 12SOS (sum of squares) techniques; see, e.g., Parillo [41] and Schichl & Neumaier =-=[54]-=- for general theory, and Domes [5] for a rigorous implementation of some SOS techniques. Though these methods are slow, they need to be applied only once to an expression, and provide – when successfu... |

3 |
Groebner Bases: Theory and Applications
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Citation Context ...enge below. In general, implied equations such as the Gaussian product formula (12) can be found automatically and rigorously by symbolic algebra, using Gröbner basis techniques (Buchberger & Winkler =-=[4]-=-); cf. also Huyer & Neumaier [12]. Similarly, implied inequalities such as the Cauchy-Schwarz inequality can be found automatically and rigorously by 12SOS (sum of squares) techniques; see, e.g., Par... |

3 | Certi¯ed error bounds for uncertain elliptic equations
- Neumaier
(Show Context)
Citation Context ... in chemical engineering [9], computational geometry [7, 35], robotics [24]), • quantified constraint satisfaction [49] (e.g, finding safe work spaces for robots [23, 37]), • design under uncertainty =-=[34, 38]-=-, • computer-assisted proofs [33], • verified linear and semidefinite programming [14, 16]. 2Many applications (e.g., all global problems) need good bounds for very wide intervals, some (e.g., roundi... |

3 | Computer graphics, linear interpolation, and nonstandard intervals,” http://www.mat.univie.ac.at/~neum/ms/nonstandard.pdf
- Neumaier
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(Show Context)
Citation Context ... references are given): • global optimization (used in lots of different applications) [31, 40], • solving equations with multiple solutions (e.g., in chemical engineering [9], computational geometry =-=[7, 35]-=-, robotics [24]), • quantified constraint satisfaction [49] (e.g, finding safe work spaces for robots [23, 37]), • design under uncertainty [34, 38], • computer-assisted proofs [33], • verified linear... |

3 |
2003b), Constraint satisfaction and global optimization in robotics. Manuscript: www.mat.univie.ac.at/∼neum/papers.html#rob A. Neumaier and H. Schichl (2003), Sharpening the Karush–John optimality conditions. Manuscript: www.mat.univie.ac.at/∼neum/papers.
- Neumaier
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(Show Context)
Citation Context ...uations with multiple solutions (e.g., in chemical engineering [9], computational geometry [7, 35], robotics [24]), • quantified constraint satisfaction [49] (e.g, finding safe work spaces for robots =-=[23, 37]-=-), • design under uncertainty [34, 38], • computer-assisted proofs [33], • verified linear and semidefinite programming [14, 16]. 2Many applications (e.g., all global problems) need good bounds for v... |

3 | Computing Slope Enclosures by Exploiting a Unique Point of
- Schnurr
- 2007
(Show Context)
Citation Context ... outside the convex hull of the endpoints. Also, it is quite basic in that it enables one to get good enclosures for polynomials and splines given in control polygon form; see the discussion in Hayes =-=[10]-=- and Neumaier [35], where algorithms for optimal interpolation operations are described (by Hayes in terms of modal interval arithmetic, by me in terms of monotonicity and directed rounding only). 4 O... |

2 | Interval Methods for Nonlinear Equation Solving Applications. Handbook of Granular Computing
- CR, Lin, et al.
(Show Context)
Citation Context ...following areas (only sample references are given): • global optimization (used in lots of different applications) [31, 40], • solving equations with multiple solutions (e.g., in chemical engineering =-=[9]-=-, computational geometry [7, 35], robotics [24]), • quantified constraint satisfaction [49] (e.g, finding safe work spaces for robots [23, 37]), • design under uncertainty [34, 38], • computer-assiste... |

2 |
System and Method to Compute Narrow Bounds on a Modal Interval Polynomial Function,” Pub. No. WO/2007/041523. Page 57 of 61
- Hayes
(Show Context)
Citation Context ... outside the convex hull of the endpoints. Also, it is quite basic in that it enables one to get good enclosures for polynomials and splines given in control polygon form; see the discussion in Hayes =-=[10]-=- and Neumaier [35], where algorithms for optimal interpolation operations are described (by Hayes in terms of modal interval arithmetic, by me in terms of monotonicity and directed rounding only). 4 O... |

2 |
A new method for global solution of nonlinear equations
- Kolev
- 1998
(Show Context)
Citation Context ...cursive way of computing centered forms, often with less overestimation. Related work on so-called G-intervals, together with applications to the solution of nonlinear systems, has been done by Kolev =-=[18, 19, 20]-=-. The most general and precise result about the quadratic approximation property is in Neumaier [32, Theorem 8.1], and applies to traditional centered forms (and can be easily adapted to the bicentere... |

2 | Automatic Slope Computation and its Application - Ratz - 1998 |

1 |
Roots of Quadratic Interval Polynomials
- Alolyan, Real
(Show Context)
Citation Context ...reasing) in some variable t attains its minimum over an interval at the lower (upper) bound and its maximum at the upper (lower) bound of the interval. ⊓⊔ For the example e(x,y,z) = x ∗ y − z and x = =-=[0, 1]-=-, we get the enclosure [−1, 1] of the range [−0.25, 0], showing that, in general, the theorem need not give the exact range. To apply the theorem to a given expression f(x), one splits the occurrences... |

1 |
Quadratic constraint propagation. Manuscript (2008). http://www.mat.univie.ac.at/~dferi/publ/Propag.pdf [10] 23 J.E. Flores Diaz, Improvement in the ray tracing of implicit surfaces based on interval arithmetic
- Domes, Neumaier
- 2008
(Show Context)
Citation Context ... of work in a branch and bound method for minimizing f(x). The optimal handling of general univariate quadratic expressions with interval coefficients is discussed in Alolyan [1] and Domes & Neumaier =-=[6]-=-. For multivariate polynomials, Moore [27] presents a number of rearrangement techniques. The frequently useful representation of a univariate or multivariate polynomial in Bezier form and its interva... |

1 |
Rigorous affine lower bounds for multivariate polynomials and their use in global optimization
- Garloff, Smith
- 2008
(Show Context)
Citation Context ...a number of rearrangement techniques. The frequently useful representation of a univariate or multivariate polynomial in Bezier form and its interval evaluation is discussed, e.g., in Garloff & Smith =-=[8]-=- and Neumaier [35]. Using centered forms. Centered forms are traditionally used for the enclosure of functions over fairly narrow intervals since they enjoy the quadratic approximation property that g... |

1 |
System and Method to Compute Narrow Bounds on a
- Hayes
(Show Context)
Citation Context ... from a monotonicity analysis can also be obtained by using modal coercion theorems together with Kaucher interval arithmetic. For example, the modal procedure described in the recent patent by Hayes =-=[11]-=- gives an alternative optimal enclosure for (4). In practice, the use of the coercion theorems for range enclosures is restricted to cases where the expression to be enclosed is totally monotonic, so ... |

1 |
Integral approximation of rays and verification of feasibility, Reliable Computing 10
- Huyer, Neumaier
- 2004
(Show Context)
Citation Context ...equations such as the Gaussian product formula (12) can be found automatically and rigorously by symbolic algebra, using Gröbner basis techniques (Buchberger & Winkler [4]); cf. also Huyer & Neumaier =-=[12]-=-. Similarly, implied inequalities such as the Cauchy-Schwarz inequality can be found automatically and rigorously by 12SOS (sum of squares) techniques; see, e.g., Parillo [41] and Schichl & Neumaier ... |

1 |
VSDP: Verified SemiDefinite Programming, Werb site
- Jansson
- 2008
(Show Context)
Citation Context ... constraint satisfaction [49] (e.g, finding safe work spaces for robots [23, 37]), • design under uncertainty [34, 38], • computer-assisted proofs [33], • verified linear and semidefinite programming =-=[14, 16]-=-. 2Many applications (e.g., all global problems) need good bounds for very wide intervals, some (e.g., rounding error control) need high accuracy for narrow intervals. Interval arithmetic is the coll... |

1 |
Lurupa – Rigorous Error Bounds
- Keil
- 2008
(Show Context)
Citation Context ... constraint satisfaction [49] (e.g, finding safe work spaces for robots [23, 37]), • design under uncertainty [34, 38], • computer-assisted proofs [33], • verified linear and semidefinite programming =-=[14, 16]-=-. 2Many applications (e.g., all global problems) need good bounds for very wide intervals, some (e.g., rounding error control) need high accuracy for narrow intervals. Interval arithmetic is the coll... |

1 |
On reducing overestimation of ranges of multinomials without splitting boxes, Manuscript
- Moore
- 2006
(Show Context)
Citation Context ... minimizing f(x). The optimal handling of general univariate quadratic expressions with interval coefficients is discussed in Alolyan [1] and Domes & Neumaier [6]. For multivariate polynomials, Moore =-=[27]-=- presents a number of rearrangement techniques. The frequently useful representation of a univariate or multivariate polynomial in Bezier form and its interval evaluation is discussed, e.g., in Garlof... |

1 |
Vienna proposal for interval standardization, Manuscript
- Neumaier
- 2008
(Show Context)
Citation Context ...ve three floating-point pipelines, each one dedicated to one particular rounding mode (up, down, nearest), with the directed modes following Part 7 of the Vienna proposal for interval standardization =-=[36]-=-. Then it will pay to adapt the algorithms (or the compilers) to optimally use the directed rounding pipelines. 15Alternatively, one could implement three different pipelines for floating-point opera... |