## Toward accurate polynomial evaluation in rounded arithmetic (2008)

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Citations: | 4 - 1 self |

### BibTeX

@MISC{Demmel08towardaccurate,

author = {James Demmel and Ioana Dumitriu and Olga Holtz},

title = {Toward accurate polynomial evaluation in rounded arithmetic},

year = {2008}

}

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### Abstract

### Citations

2531 |
The design and analysis of computer algorithms
- Aho, Hopcroft, et al.
- 1975
(Show Context)
Citation Context ...th nonempty Zariski open sets, 16which are all of full measure. Finally, it is worth noting that the Zariski sets we will work with are algorithm-dependent. Finally, we represent any algorithm as in =-=[11, 1]-=- by a directed acyclic graph (DAG) with input nodes, branching nodes, and output nodes. For simplicity in dealing with negation (given that negation is an exact operation), we define a special type of... |

916 |
Accuracy and Stability of Numerical Algorithms
- Higham
- 1996
(Show Context)
Citation Context ... number. The constant ɛ is called the machine precision, by analogy to floating point computation, since this model is is the traditional one used to analyze the accuracy of floating point algorithms =-=[34, 63]-=-. To illustrate the obstacles to accurate evaluation that this model poses, consider evaluating p(x) = x1 + x2 + x3 in the most straightforward way: add (and round) x1 and x2, and then add (and round)... |

876 | Enumerative combinatorics - Stanley - 1999 |

865 |
Symmetric functions and Hall polynomials
- Macdonald
- 1979
(Show Context)
Citation Context ...zed Vandermonde matrix. A generalized Vandermonde matrix is known to have determinant of the form sλ(x) ∏ i<j (xi − xj) where sλ(x) is a polynomial of degree |λ| = ∑ i λi, and called a Schur function =-=[43]-=-. In infinitely many variables (not our situation) the Schur function is irreducible [31], but in finitely many variables, the Schur function is sometimes irreducible and sometimes not [57, Exer. 7.30... |

587 |
A Decision Method for Elementary Algebra and Geometry
- Tarski
- 1951
(Show Context)
Citation Context ...ed relative error η) or exhibits a proof that none exists. To be more precise, we must say what our set of possible algorithms includes. The above decision question is apparently not Tarski-decidable =-=[7, 10]-=- despite its appearance, because we see no way to express “there exists an algorithm” in that format. A more formal description of the algorithms that we consider is as follows. 1. We insist that the ... |

568 | Applied Numerical Linear Algebra
- Demmel
- 1997
(Show Context)
Citation Context ...ut x exactly means that at best (i.e., in the absence of any further error) we could only hope to compute the exact value of p(ˆx) for some ˆx ≈ x, an algorithmic property known as backward stability =-=[18, 34]-=-. Since we insist that zero outputs be computed exactly in order to have bounded relative error, this means there is no way to guarantee that p(ˆx) = 0 when p(x) = 0, for nonconstant p. This is true e... |

565 |
Methods and Applications of Interval Analysis
- Moore
- 1979
(Show Context)
Citation Context ... to one with a small support would only distinguish between positive definite polynomials, the easy case discussed in section 3, and polynomials that are not positive definite. In interval arithmetic =-=[46, 47, 2]-=- one represents each number by a floating point interval guaranteed to contain it. To do this one rounds interval endpoints “outward” to ensure that, for example, the sum c = a + b of two intervals yi... |

547 |
Interval Methods for Systems of Equations
- Neumaier
- 1990
(Show Context)
Citation Context ... to one with a small support would only distinguish between positive definite polynomials, the easy case discussed in section 3, and polynomials that are not positive definite. In interval arithmetic =-=[46, 47, 2]-=- one represents each number by a floating point interval guaranteed to contain it. To do this one rounds interval endpoints “outward” to ensure that, for example, the sum c = a + b of two intervals yi... |

487 |
Introduction to Interval Computations
- Alefeld, Herzberger
- 1983
(Show Context)
Citation Context ...re a common perturbation theory, that the condition number grows proportionally to the reciprocal of the distance to the smallest problem with an infinite condition number [3]. 6. Interval arithmetic =-=[1]-=- represents numbers by intervals, and does arithmetic with them by rounding the endpoints “outward” so as to provably include the true answer. It is natural to ask whether accurate evaluability of p(x... |

387 |
On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines
- Blum, Shub, et al.
- 1989
(Show Context)
Citation Context ...th nonempty Zariski open sets, 16which are all of full measure. Finally, it is worth noting that the Zariski sets we will work with are algorithm-dependent. Finally, we represent any algorithm as in =-=[11, 1]-=- by a directed acyclic graph (DAG) with input nodes, branching nodes, and output nodes. For simplicity in dealing with negation (given that negation is an exact operation), we define a special type of... |

378 |
Lectures on polytopes
- Ziegler
- 1995
(Show Context)
Citation Context ...ional rays of −N(P )∩R k + (which are normal to facets of the Newton polytope), we intersect −N(P ) ∩ Rk + with, say, the hyperplane ˜x1 + · · · + ˜xk = 1. Perform the Voronoi tesselation (see, e.g., =-=[64]-=-) of the simplex ˜x1 +· · ·+ ˜xk = 1, ˜xj ≥ 0, j = 1, . . . , k relative to the intersection points of −N(P )∩R k + with the hyperplane ˜x1 +· · ·+˜xk = 1. Connecting each Voronoi cell of the tesselat... |

362 |
Complexity and Real Computation
- Blum, Cucker, et al.
- 1997
(Show Context)
Citation Context ...ow are not exhaustive, but illustrative): • Are numbers (and any errors) represented discretely (e.g., as bit strings such as floating point numbers) [23, 34, 63], or as a (real or complex) continuum =-=[10, 13]-=-? • Is arithmetic exact [10, 8] or rounded [14, 15, 34, 63]? If it is rounded, is the error bounded in a relative sense [34], absolute sense [10], or something else [42, 17, 16] [34, Sec. 2.9]? • In w... |

267 | Basic Algebraic Geometry - Shafarevich - 1974 |

193 | Enumerative Combinatorics, Volume 2 - Stanley - 1997 |

175 |
Complexity theory of real functions
- Ko
- 1991
(Show Context)
Citation Context ...ithmetic, and explains the advantages of our model. 62.1 Exact or Rounded Inputs We must decide whether we assume that the arguments are given exactly [9, 14, 51, 58] or are known only approximately =-=[15, 30, 39, 49]-=-. Not knowing the input x exactly means that at best (i.e., in the absence of any further error) we could only hope to compute the exact value of p(ˆx) for some ˆx ≈ x, an algorithmic property known a... |

174 |
On the computational complexity and geometry of the first-order theory of the reals, parts I–III
- Renegar
- 1992
(Show Context)
Citation Context ...ed relative error η) or exhibits a proof that none exists. To be more precise, we must say what our set of possible algorithms includes. The above decision question is apparently not Tarski-decidable =-=[7, 10]-=- despite its appearance, because we see no way to express “there exists an algorithm” in that format. A more formal description of the algorithms that we consider is as follows. 1. We insist that the ... |

145 |
Rounding errors in algebraic processes
- Wilkinson
- 1994
(Show Context)
Citation Context ... number. The constant ɛ is called the machine precision, by analogy to floating point computation, since this model is is the traditional one used to analyze the accuracy of floating point algorithms =-=[34, 63]-=-. To illustrate the obstacles to accurate evaluation that this model poses, consider evaluating p(x) = x1 + x2 + x3 in the most straightforward way: add (and round) x1 and x2, and then add (and round)... |

143 | Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates
- Shewchuk
- 1997
(Show Context)
Citation Context ...takes any expression and identifies whether it can be evaluated accurately, and provides the algorithm if it exists. The impact would be both to formalize the process of accurate algorithm generation =-=[9]-=- and to systematize recent results [5] identifying apparently disparate classes of structured matrices for which efficient and accurate linear algebra algorithms exist. We give some examples to illust... |

121 |
Total positivity
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- 1968
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Citation Context ... numbers, then the situation changes: The nonnegativity of the coefficients of the Schur functions shows that they are positive in D, and indeed the generalized Vandermonde matrix is totally positive =-=[38]-=-. Combined with the homogeneity of the Schur function, Theorem 3.2 implies that the Schur function, and so determinants (and minors) of totally positive generalized Vandermonde matrices can be evaluat... |

98 | Some concrete aspects of Hilbert’s 17th problem
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- 2000
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Citation Context ...ition on the real variety VR(Mjk), the set of real x where Mjk(x) = 0, see Theorem 4.4. When k/j = 3, i.e., on the boundary between the above two cases, Mjk(x) is a multiple of the Motzkin polynomial =-=[8]-=-. The real variety VR(Mjk) = {x : |x1| = |x2| = |x3|} of this polynomial satisfies the necessary condition of Theorem 4.4, and the simplest accurate algorithm to evaluate it that we know of has 8 case... |

95 |
On condition numbers and the distance to the nearest ill-posed problem
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Citation Context ...rithmetic so far seem to share a common perturbation theory, that the condition number grows proportionally to the reciprocal of the distance to the smallest problem with an infinite condition number =-=[3]-=-. 6. Interval arithmetic [1] represents numbers by intervals, and does arithmetic with them by rounding the endpoints “outward” so as to provably include the true answer. It is natural to ask whether ... |

69 | A Fortran-90 based multiprecision system
- Bailey
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Citation Context ... by representing a high precision number y as a sum y = ∑k i=1 yi of numbers satisfying |yi| ≫ |yi+1|, the idea being that each yi represents (nearly) disjoint parts of the binary expansion of y (see =-=[5, 6, 22, 50]-=- and the references therein). This technique can be modeled by the correct choice of black-box operations as we now illustrate. Suppose we include the enumerable set of black-box operations ∑n i=1 pi,... |

69 | Algorithms for arbitrary precision floating point arithmetic
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Citation Context ... by representing a high precision number y as a sum y = ∑k i=1 yi of numbers satisfying |yi| ≫ |yi+1|, the idea being that each yi represents (nearly) disjoint parts of the binary expansion of y (see =-=[5, 6, 22, 50]-=- and the references therein). This technique can be modeled by the correct choice of black-box operations as we now illustrate. Suppose we include the enumerable set of black-box operations ∑n i=1 pi,... |

67 |
Computability in Analysis and Physics
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Citation Context ...ithmetic, and explains the advantages of our model. 62.1 Exact or Rounded Inputs We must decide whether we assume that the arguments are given exactly [9, 14, 51, 58] or are known only approximately =-=[15, 30, 39, 49]-=-. Not knowing the input x exactly means that at best (i.e., in the absence of any further error) we could only hope to compute the exact value of p(ˆx) for some ˆx ≈ x, an algorithmic property known a... |

65 | Combinatorial commutative algebra - Miller, Sturmfels - 2005 |

57 | Computing the singular value decomposition with high relative accuracy
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Citation Context ...is expressed as a product of simple nonnegative bidiagonal matrices arising from its Neville factorization), acyclic matrices, suitably discretized elliptic partial differential operations, and so on =-=[20, 24, 23, 21, 19, 25, 27, 26, 40, 41]-=-. It has been recently shown that all the matrices on the above list (except Toeplitz and nontotally-positive generalized Vandermonde matrices) admit accurate algorithms in rounded arithmetic for many... |

55 | Safe and effective determinant evaluation - Clarkson - 1992 |

51 | A floating point technique for extending the available precision - Dekker - 1971 |

50 |
Numerical inverting matrices of high order
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Citation Context ...ut some positive attributes of our model: 1. The model rnd(op(a, b)) = op(a, b)(1 + δ) has been the most widely used model for floating point error analysis [34] since the early papers of von Neumann =-=[62]-=-, Turing [60] and Wilkinson [63]. 2. The extension to include black-boxes includes widely used floating point techniques for extending the precision. 3. Though the model is for real (or complex) arith... |

41 |
Smoothed analysis of algorithms
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Citation Context ... [12, 30, 39, 49, 52] (and if rounded, again how is the error bounded)? • Do we want a “worst case” error analysis [34, 63], or by modeling rounding errors as random variables, a statistical analysis =-=[61, 37, 56]-=- [34, Sec. 2.8]? Does a condition number appear explicitly in the complexity of the problem [15]? First we consider floating point arithmetic itself, i.e., where real numbers are represented by a pair... |

37 | Underflow and the reliability of numerical software - Demmel - 1984 |

36 | Correction d’une somme en arithmétique à virgule flottante - Pichat - 1972 |

35 |
Rounding-off errors in matrix processes
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Citation Context ...ive attributes of our model: 1. The model rnd(op(a, b)) = op(a, b)(1 + δ) has been the most widely used model for floating point error analysis [34] since the early papers of von Neumann [62], Turing =-=[60]-=- and Wilkinson [63]. 2. The extension to include black-boxes includes widely used floating point techniques for extending the precision. 3. Though the model is for real (or complex) arithmetic, it can... |

33 |
Complexity estimates depending on condition and round-off error
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Citation Context ...ithmetic, and explains the advantages of our model. 62.1 Exact or Rounded Inputs We must decide whether we assume that the arguments are given exactly [9, 14, 51, 58] or are known only approximately =-=[15, 30, 39, 49]-=-. Not knowing the input x exactly means that at best (i.e., in the absence of any further error) we could only hope to compute the exact value of p(ˆx) for some ˆx ≈ x, an algorithmic property known a... |

29 | Multiprecision translation and execution of FORTRAN programs
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Citation Context ... by representing a high precision number y as a sum y = ∑k i=1 yi of numbers satisfying |yi| ≫ |yi+1|, the idea being that each yi represents (nearly) disjoint parts of the binary expansion of y (see =-=[5, 6, 22, 50]-=- and the references therein). This technique can be modeled by the correct choice of black-box operations as we now illustrate. Suppose we include the enumerable set of black-box operations ∑n i=1 pi,... |

24 |
Accurate and efficient floating point summation
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24 | Stability analysis of algorithms for solving confluent Vandermonde-like systems
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Citation Context ...rix [48] [4, 3] [23] [25] [25] [29] Cauchy n2 n2 n2 ≤ n3 n3 n3 Displace- [19] [19] [29] ment Vandermonde n2 No No No n3 n3 Rank One [23] [23] [23] [19, 27] [29] Polynomial n2 No No No ∗ ∗ Vandermonde =-=[33]-=- Section 6.1 Section 6.1 Section 6.1 [27] [29] Sym Thus, if the determinants pn(x) = detM n×n (x) of a class of n-by-n structured matrices M do not satisfy the necessary conditions described in Theore... |

23 | Accurate eigenvalues and SVDs of totally nonnegative matrices
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Citation Context ...is expressed as a product of simple nonnegative bidiagonal matrices arising from its Neville factorization), acyclic matrices, suitably discretized elliptic partial differential operations, and so on =-=[20, 24, 23, 21, 19, 25, 27, 26, 40, 41]-=-. It has been recently shown that all the matrices on the above list (except Toeplitz and nontotally-positive generalized Vandermonde matrices) admit accurate algorithms in rounded arithmetic for many... |

20 | The Accurate and Efficient Solution of a Totally Positive Generalized Vandermonde Linear System
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Citation Context ...is expressed as a product of simple nonnegative bidiagonal matrices arising from its Neville factorization), acyclic matrices, suitably discretized elliptic partial differential operations, and so on =-=[20, 24, 23, 21, 19, 25, 27, 26, 40, 41]-=-. It has been recently shown that all the matrices on the above list (except Toeplitz and nontotally-positive generalized Vandermonde matrices) admit accurate algorithms in rounded arithmetic for many... |

18 | Several complex variables with connections to algebraic geometry and Lie groups, Graduate - Taylor - 2002 |

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Citation Context ...d Matrices Any Type of matrix det A A−1 minor LDU SVD EVD Acyclic n n2 n ≤ n2 n3 N/A (bidiagonal and other) [21] [21] [21] [21] [21] Total Sign Compound n n3 n n4 n4 n4 (TSC) [21] [21] [21] [21] [21] =-=[29]-=- Diagonally Scaled Totally n3 n5 n3 n3 n3 n3 Unimodular (DSTU) [21] [21] [21] [21] [29] Weakly diagonally n3 n3 No n3 n3 n3 dominant M-matrix [48] [4, 3] [23] [25] [25] [29] Cauchy n2 n2 n2 ≤ n3 n3 n3... |

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Citation Context ...trative): • Are numbers (and any errors) represented discretely (e.g., as bit strings such as floating point numbers) [23, 34, 63], or as a (real or complex) continuum [10, 13]? • Is arithmetic exact =-=[10, 8]-=- or rounded [14, 15, 34, 63]? If it is rounded, is the error bounded in a relative sense [34], absolute sense [10], or something else [42, 17, 16] [34, Sec. 2.9]? • In which of these metrics is the fi... |

10 |
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Citation Context ...e of n addition, subtraction, multiplication or division (by nonzero) operations can increase the largest exponent e by at most O(n) bits, and so can be done in time polynomial in the input size. See =-=[23]-=- for further discussion. This is in contrast to repeated squaring in the BSS model [7] which can lead to exponential time simulations. 10Models of arithmetic may be categorized according to several c... |

10 |
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Citation Context ...crease the largest exponent e by at most O(n) bits, and so can be done in time polynomial in the input size. See [23] for further discussion. This is in contrast to repeated squaring in the BSS model =-=[7]-=- which can lead to exponential time simulations. 10Models of arithmetic may be categorized according to several criteria (the references below are not exhaustive, but illustrative): • Are numbers (an... |

9 |
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Citation Context ...real or complex) continuum [10, 13]? • Is arithmetic exact [10, 8] or rounded [14, 15, 34, 63]? If it is rounded, is the error bounded in a relative sense [34], absolute sense [10], or something else =-=[42, 17, 16]-=- [34, Sec. 2.9]? • In which of these metrics is the final error assessed? • Is the input data exact [10] or considered “rounded” from its true value [12, 30, 39, 49, 52] (and if rounded, again how is ... |

8 | Accurate SVDs of structured matrices
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8 | Quasi double precision in floating-point arithmetic - Møller - 1965 |

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