R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....5 to give the Newton operator N(x, x) x J(x) 1 f(x) In practice this operator is not often used since an interval linear system has to be solved for each iteration. Also J(x) is likely to contain singular matrices unless the width of x is small. An alternative is to follow Krawczyk [11] and use what is now called the Krawczyk operator. The advantage of this operator is that no interval linear equations have to be solved at any stage, thus increasing speed and reliability. Substituting the terms of the linear system (14) into the Krawczyk iteration given in Section 5.3 leads to ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....5 to give the Newton operator N( x; x) x J(x) f( x) In practice this operator is not often used since an interval linear system has to be solved for each iteration. Also J(x) is likely to contain singular matrices unless the width of x is small. An alternative is to follow Krawczyk [11] and use what is now called the Krawczyk operator. The advantage of this operator is that no interval linear equations have to be solved at any stage, thus increasing speed and reliability. Substituting the terms of the linear system (14) into the Krawczyk iteration given in Section 5.3 leads to ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187-201, 1969.

....The mean value interval extension of a function is a monotonic interval extension. It is interesting to note that box consistency on the mean value interval extension of a system of constraints is closely related to the Hansen Sengupta operator [8] which is an improvement over Krawczyk s operator [13]. Hansen and Smith also argue that these operators are more effective when the interval Jacobian of the system is diagonally dominant [9] and they suggest conditioning the system S. For the purpose of this paper, we simply assume that we have a conditioning operator cond(S; I) and use the ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

.... zeros of polynomials [Boe83, Gru87, Fra87, Loh88a, Fis89, Fra89, Pet89, Geo90, Kra90a, Dim90, Kra91b, Kra91c, Kra91e, Pet91, Sch90a, Ham92b, Ata92b, Pet92a, Ata93, Ata93a, Yam94, Kan94] automatic differentiation [Ral80, Ral81, Cor88, Cor91a, Fis90, Gri89, Gri91, Shi92b, Fis93, Shi94] other [Kra69, Mir86a, Boc89, Boc90, Rum91a, Mar91b, Cuy92, Ott93, Jah93, Hei93] ffl Expert system for numerical verification: Koe91, Koe92, Koe92a, Koe93] verification in computer algebra systems [Ste93] ffl Parallel programming: PASCAL XSC on transputers [Boh92a, Boh92b, Ker94] basic arithmetic ....

Krawczyk, R.: Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4, pp. 187-201, 1969.

....and the pruning is relatively weak due to the decomposition process, making this approach unpractical on many applications. By contrast, interval research in numerical analysis has focused, among other things, on producing reliable and reasonably fast methods to solve the above applications (e.g. [4,6,7,9,11 14,20,26,33]) Many of these techniques use ideas behind Newton root nding method, exploit properties such as di erentiability, and de ne various pruning operators, many of which extend the seminal work of Krawczyk [14] These algorithms can often be viewed as an iteration of two steps, constraint ....

.... and reasonably fast methods to solve the above applications (e.g. 4,6,7,9,11 14,20,26,33] Many of these techniques use ideas behind Newton root nding method, exploit properties such as di erentiability, and de ne various pruning operators, many of which extend the seminal work of Krawczyk [14]. These algorithms can often be viewed as an iteration of two steps, constraint propagation and splitting, although they are rarely presented this way and it is not always clear what the constraint propagation step computes. The key contribution of Newton is the notion of box consistency, an ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187-201, 1969.

....complexity (i.e. it is NP hard) and due to the numerical issues involved to guarantee correctness (i.e. nding all solutions) and to ensure termination. Several interesting methods have been proposed in the past for this task, including two fundamentally di erent methods: interval methods (e.g. [4, 5, 7, 8, 11, 13, 14, 15, 19, 25, 29]) and continuation methods (e.g. 24, 35] Continuation methods have been shown to be e ective for problems for which the total degree is not too high, since the number of paths explored depends on the estimation of the number of solutions. Interval methods are generally robust but tend to be ....

....system, the partial derivatives are computed numerically using automatic di erentiation [27] 4. 3 Conditioning It is interesting to note that box consistency on the Taylor interval extension is closely related to the Hansen Segupta s operator [8] which is an improvement over Krawczyk s operator [15]. Hansen and Smith [9] also argued that these operators are more e ective for a system ff 1 = 0; f n = 0g wrt a box hI 1 ; I n i when the interval Jacobian M ij = d f i x j (I 1 ; I n ) 1 i; j n) is diagonally dominant, i.e. mig(M i;i ) n X j=1;j 6=i mag(M ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187-201, 1969.

.... propagation methods, CoSAc [22] which combines Maple, a Grobner bases module and a linear solver, the system presented in [20] based on Grobner bases, the simplex algorithm and interval propagation methods, TKIB [14] introducing an approximation notion called tightening and the Krawckzyk operator [16, 17]. In this paper, we propose to combine Grobner bases computations over subsets of the initial system, used as a preprocessing step, interval Newton methods to compute weak arc consistency and enumeration techniques used to separate the solutions. This combination was motivated by three main ....

....f be a real function continuously di#erentiable between x and y and consider that y is a zero of f . It can be deduced from the mean value theorem that y = x f(x) f # (a) The Newton method iterates this formula to approximate roots of f . This method has been extended to interval functions [24, 16, 12, 1, 11, 17, 25]. Let X be an interval containing x and y and suppose that F # is the natural interval extension of f # , F the natural interval extension of f and m(X) the approximation of the center of X . The Newton interval function is the function N(X) m(X) F (m(X) F # (X) From this definition one ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....box (left picture) and a set that is topologically equivalent to a 2 face (right picture) Both yield sharper enclosures than rF i [z] A similar operator ( u] v] w] allows Taylor based box cutting for inequalities F i (z) 0; see Lang, 1995. 2 Krawczyk type acceleration; see Krawczyk, 1969. If z 2 [z] C 2 R M M , and z is a solution in [z] then 0 = C F( z) F 0 ( i) z z) with a di erent intermediate point (i) for each component F i ) and therefore z z = I CF 0 ( i) z z) CF( z) 2 (I CF 0 [z] z] z) CF( ....

Krawczyk, R. (1969). Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187-201.

....and bound search techniques that have been developed for other NP complete problems. This paper describes a particular branch and cut search method based on interval techniques. Interval techniques manipulate upper and lower bounds on variables and have been explored by a variety of authors [12, 5, 4, 8]. The main innovation presented here consist of two local interval cuts that are effective in regions of large uncertainty, i.e. where the bounds associated with variables remain quite loose. Our two novel cuts are local in the sense that each cut examines only a single constraint. Our emphasis ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....k = k 1; X = Y E; Y = Z C X; ready = in0(Y,X) end if ready X = xs Y; else disp( no inclusion achieved for ) X = NaN; end Algorithm 4.4. Solution of dense linear systems 13 This is a well known algorithm for solving systems of linear equations based on the Krawczyk operator [17], 21] The algorithm computes an inclusion of the difference of the true solution and the approximate solution xs. It is identical to the one presented in [25] The 4th line Z = R (b intval(A) xs) is a typical statement to ensure interval calculation of the residual. Note that a statement like, ....

....possible by the Matlab profile function. The statements profile verifynlss; n=200; verifynlss( f ,10 ones(n,1) profile report; profile done produce the following output: Total time in c: matlab toolbox intlab intval verifynlss.m : 21.41 seconds 100 of the total time was spent on lines: [18 33 34 19 24 38 23 17 26 32] 16: xsold = xs; 0.10s, 0 17: x = initvar(xs) 10.47s, 49 18: y = feval(f,x) 1.26s, 6 19: xs = xs y.dx y.x; 20: end 22: interval iteration 0.15s, 1 23: R = inv(y.dx) 0.42s, 2 24: Z = R feval(f,intval(xs) 25: X = Z; 0.03s, 0 26: E = 0.1 rad(X) hull( 1,1) ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4, pages 187--201, 1969.

....(i.e. it is NP hard) and due to the numerical issues involved to guarantee correctness (i.e. finding all solutions) and to ensure termination. Several interesting methods have been proposed in the past for this task, including two fundamentally different methods: interval methods (e.g. [4, 5, 7, 8, 11, 13, 14, 15, 19, 25, 29]) and continuation methods (e.g. 24, 35] Continuation methods have been shown to be effective for problems for which the total degree is not too high, since the number of paths explored depends on the estimation of the number of solutions. Interval methods are generally robust but tend to be ....

....the partial derivatives are computed numerically using automatic differentiation [27] 4. 3 Conditioning It is interesting to note that box consistency on the Taylor interval extension is closely related to the Hansen Segupta s operator [8] which is an improvement over Krawczyk s operator [15]. Hansen and Smith [9] also argued that these operators are more effective for a system ff 1 = 0; f n = 0g wrt a box hI 1 ; I n i when the interval Jacobian M ij = d f i x j (I 1 ; I n ) 1 i; j n) is diagonally dominant, i.e. mig(M i;i ) n X j=1;j 6=i mag(M ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....The mean value interval extension of a function is a monotonic interval extension. It is interesting to note that box consistency on the mean value interval extension of a system of constraints is closely related to the Hansen Sengupta operator [8] which is an improvement over Krawczyk s operator [13]. Hansen and Smith also argue that these operators are more effective when the interval Jacobian of the system is diagonally dominant [9] and they suggest conditioning the system S. For the purpose of this paper, we simply assume that we have a conditioning operator cond(S; I) and use the ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....a NSF National Young Investigator Award, and European Esprit Basic Research project ACCLAIM no 7195. 2 Arc consistency itself cannot be computed exactly due to machine limitation. 1 differentiability, and define various pruning operators, many of them extending the seminal work of Krawczyk [10]. These algorithms can often be viewed as an iteration of two steps, constraint propagation and splitting, although they are rarely presented this way and it is not always clear what the constraint propagation step computes. The goal of our research is to reconcile as best as possible the ....

....not apply Newton interval method to the interval extension of a real function but rather to an interval function coming from the projection of an interval function. It is also interesting to look at all constraints as a whole and to compare box consistency with Krawczyk s method and its successors [10, 7, 6]. On the one hand, Krawczyk s methods achieve more pruning since they manipulate all constraints as a whole, allowing a more precise evaluation of the interval functions. On the other hand, boxconsistency provides additional pruning by obtaining tight bounds on the intervals. Both methods seem ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1985.

.... intervals in numerical computations is hardly new, since it originated from Moore s thesis in 1966 [ Moore, 1966 ] and is a very active research area (e.g. Hammer et al. 1993; Hansen, 1992; Hansen and Greenberg, 1983; Hansen and Sengupta, 1981; Hong and Stahl, 1994; Kearfott, 1990; 1991; 1997; Krawczyk, 1969; Moore, 1966; Neumaier, 1990; Rump, 1988 ] What distinguishes the constraint solving algorithm of Numerica is the combination of techniques from numerical analysis and artificial intelligence to obtain effective pruning techniques (for many problems) At a very abstract level, Numerica can be ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

.... (i.e. it is NP hard) and due to the numerical issues involved to ensure termination and guarantee correctness (i.e. finding all solutions) Several interesting methods have been proposed in the past for the former task, including two fundamentally different methods: interval methods (e.g. [4, 5, 7, 8, 11, 14, 15, 18, 23, 30, 35]) and continuation methods (e.g. 29, 43] Continuation methods have been shown to be effective for problems for which the total degree is not too high, since the number of paths explored depends on the estimation of the number of solutions. Interval methods are generally robust and have been ....

....system, the partial derivatives are computed numerically using automatic differentiation [32] 6. 3 Conditioning It is interesting to note that box consistency on the Taylor interval extension is closely related to the Hansen Segupta s operator [8] which is an improvement over Krawczyk s operator [18]. Hansen and Smith [9] also argued that these operators are more effective for a system ff 1 = 0; f n = 0g wrt a box hI 1 ; I n i when the interval Jacobian M ij = d f i x j (I 1 ; I n ) 1 i; j n) is diagonally dominant, i.e. mig(M i;i ) n X j=1;j 6=i mag(M ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....operations use Algorithms 3.1 to 3.4, and the algorithms for complex operations including vector and matrix operations can be written in a similar way. 4. Computational speed. The core of many numerical algorithms is the solution of systems of linear equations. The popular Krawczyk type iteration [14, 24] for the veri ed solution of dense linear systems requires computation of an approximate inverse R as preconditioner, and the veri ed computation of the residual I RA. For sparse systems of linear equations [25] an approximation of the smallest singular value (for example, by inverse iteration) ....

....and whether it perpetuates for composed operations. All computations in this section are performed using IEEE 754 double precision corresponding to approximately 16 decimal digits. The rst test problem is the solution of systems of linear equations. We use the popular Krawczyk type iteration [14] with improvements such as inclusion of the error with respect to an approximate solution [24] For di erent values of n we generate (in Matlab notation) A = 2 rand(n) 1) 1 e) b = A (2 rand(n; 1) 1) 22) that means the midpoint of A ij is a random number uniformly distributed in [ ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4, pages 187-201, 1969.

....a constraint, or a system of constraints is a monotonic interval extension. It is interesting to note that box consistency on the mean value interval extension of a system of constraints is closely related to the Hansen Sengupta s operator [7] which is an improvement over Krawczyk s operator [11]. Hansen and Smith also argue that these operators are more effective when the interval Jacobian of the system is diagonally dominant [8] and they suggest to condition the system S. For the purpose of this paper, it is sufficient to abstract the notion of conditioning by the following definition. ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....regions X are taken to be boxes, i.e. interval vectors or, geometrically, rectangular parallelepipeds. Along these lines, implementations generally involve some form of interval Newton method for steps 2, 3 and 5. The most common such interval Newton methods appear to be the Krawczyk method (from [20] and explained in [24] and [26] and the interval Gauss Seidel method (such as in [9] and [17] although preconditioned interval Gaussian elimination could also be used (see [29] Theory and practice indicate that the interval Gauss Seidel method is somewhat better than Krawczyk s method for ....

....paradigms in algorithms with automatic result verification: In the first, existence is assumed, and bounds on the solution are refined, while existence is verified a posteriori (after computation of an approximate solution) in the second. Verification of approximate solutions began with Krawczyk [20] and Moore [24] and continued with the introduction of fixed point theory by Rump [31] In global search algorithms for nonlinear systems, besides in [10] it has been used in the univariate global optimization algorithm proposed by Caprani and Madsen in [3] and in the multivariate global ....

R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlershranken, Computing, 4 (1969), pp. 187--201.

....(i.e. it is NP hard) and due to the numerical issues involved to guarantee correctness (i.e. finding all solutions) and to ensure termination. Several interesting methods have been proposed in the past for this task, including two fundamentally different methods: interval methods (e.g. [13, 5, 4, 8]) and continuation methods (e.g. 18, 26] Continuation methods have been shown to be effective for problems for which the total degree is not too high, since the number of paths explored depends on the estimation of the number of solutions. Interval methods are generally robust but tend to be ....

....D[ f 1 ; i] D[ f n ; i] n Theta f n Gamma1 Theta D[ f; i] D[ f) i] D[ f; i] 4. 3 Conditioning It is interesting to note that box consistency on the Taylor interval extension is closely related to Hansen Segupta s operator [5] which is an improvement over Krawczyk s operator [8]. Hansen and Smith [6] also argued that these operators are more effective for a system ff 1 = 0; f n = 0g wrt a box hI 1 ; I n i when the interval Jacobian M ij = DER(f i ; j) I 1 ; I n ) 1 i; j n) is diagonally dominant, i.e. mig(M i;i ) n X j=1;j 6=i mag(M ....

[Article contains additional citation context not shown here]

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....to the numerical issues involved to guarantee correctness (i.e. finding all solutions or the global optimum) and to ensure termination. Newton [27] is a new constraint logic programming language designed to support this class of applications. It combines interval analysis from numerical analysis [2, 3, 4, 5, 7, 9, 10, 11, 12, 17, 22, 24]) with consistency techniques from artificial intelligence [14, 16] to produce one of most efficient constraint solvers in this area [26] As is traditional with constraint programming languages, Newton allows a short development time of its applications. However, Newton programs, although short ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

....(i.e. it is NP hard) and due to the numerical issues involved to guarantee correctness (i.e. finding all solutions) and to ensure termination. Several interesting methods have been proposed in the past for this task, including two fundamentally different methods: interval methods (e.g. [4, 5, 7, 8, 11, 13, 14, 15, 20, 26, 30]) and continuation methods (e.g. 25, 35] Continuation methods have been shown to be effective for problems for which the total degree is not too high, since the number of paths explored depends on the estimation of the number of solutions. Interval methods are generally robust but tend to be ....

....version of our system, the partial derivatives are computed using automatic differentiation. 4. 3 Conditioning It is interesting to note that box consistency on the Taylor interval extension is closely related to Hansen Segupta s operator [8] which is an improvement over Krawczyk s operator [15]. Hansen and Smith [9] also argued that these operators are more effective for a system ff 1 = 0; f n = 0g wrt a box hI 1 ; I n i when the interval Jacobian M ij = d f i x j (I 1 ; I n ) 1 i; j n) 6 The distributed version can easily be turned into a canonical ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

.... propagation methods, CoSAc [22] which combines Maple, a Grobner bases module and a linear solver, the system presented in [20] based on Grobner bases, the simplex algorithm and interval propagation methods, TKIB [14] introducing an approximation notion called tightening and the Krawckzyk operator [16, 17]. In this paper, we propose to combine Grobner bases computations over subsets of the initial system, used as a preprocessing step, interval Newton methods to compute weak arc consistency and enumeration techniques used to separate the solutions. This combination was motivated by three main ....

....a real function continuously differentiable between x and y and consider that y is a zero of f . It can be deduced from the mean value theorem that y = x Gamma f(x) f 0 (a) The Newton method iterates this formula to approximate roots of f . This method has been extended to interval functions [24, 16, 12, 1, 11, 17, 25]. Let X be an interval containing x and y and suppose that F 0 is the natural interval extension of f 0 , F the natural interval extension of f and m(X) the approximation of the center of X . The Newton interval function is the function N(X) m(X) Gamma F (m(X) F 0 (X) From this ....

R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1969.

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R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187--201, 1985.

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R. Krawczyk. Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187-- 201, 1969.

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Krawczyk R. Newton-algorithmen zur bestimmung von Nullstellen mit Fehlerschranken. Computing, 4:187{ 201, 1969.

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