Kearfott, R. B., Interval Arithmetic Techniques in the Computational Solution of Nonlinear Systems of Equations: Introduction, Examples, and Comparisons, in "Computational Solution of Nonlinear Systems of Equations," (Lectures in Applied Mathematics, volume 26), American Mathematical Society, Providence, RI, 1990, pp. 337--358.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... to exclude a cell such as Sturm s method for univariate polynomials [9, 80, 92] or Descartes rule [86, 79] which can be adapted to any interval using Bernstein polynomials and de Casteljau bissection algorithm [72] or exclusion functions based on Taylor expansion [29] or interval arithmetic [54] or other test such as Turan test used for instance in [73] Analytic methods. These are methods which exploit the value of the functions (f i ) i=1; m (or their derivatives) in order to compute the zero level f 1 = f m = 0. A very common approach for solving (polynomial) ....

R. B. Kearfott, Interval arithmetic techniques in the computational solution of nonlinear systesm of equations: Introduction, examples and comparisons, Lectures in Applied Mathemetics, AMS Press, 1990, pp. 337-357.

.... box, a properly implemented interval Newton generalized bisection method can enclose with mathematical and computational certainty any and all solutions to a speci ed tolerance, or can determine with mathematical certainty that there are no solutions in the given box (Kearfott and Novoa, 1990; Kearfott, 1990). 3 Results and Discussion We discuss the results from one of our test problems below. This system contains 3 components (isobutene, methanol, and MTBE) and one reaction (isobutene methanol MTBE) This problem has also been considered by Okasinski and Doherty (1997) This system is of ....

Kearfott, R. B., Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics, 26, 337-357 (1990).

.... box, a properly implemented interval Newton generalized bisection method can enclose with mathematical and computational certainty any and all solutions to a speci ed tolerance, or can determine with mathematical certainty that there are no solutions in the given box (Kearfott and Novoa, 1990; Kearfott, 1990). The technique can also be readily applied in the context of global optimization (Hansen, 1992) 3 Discussion The e ectiveness of this methodology in solving high pressure phase behavior problems using equations of state with standard mixing rules has been demonstrated by Hua et al. 1998) ....

Kearfott, R. B., Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics, 26, 337-357 (1990).

.... of real solutions in the specified initial box, a properly implemented interval Newton generalized bisection method can find with mathematical certainty any and all such solutions to a specified tolerance, or can determine with mathematical certainty that there are no solutions in the given box (Kearfott, 1987,1990). The technique used here for computing N (k) is the preconditioned Gauss Seidel like technique developed by Hansen and Sengupta (1981) and Hansen and Greenburg (1983) Neumaier (1985,1990) has proven the existence and uniqueness test for this method of determining N (k) Since all variables ....

Kearfott, R. B. (1990). Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons.

.... the specified initial box, a properly implemented interval Newton generalized bisection method can find with mathematical certainty any and all solutions to a specified tolerance, or can determine with mathematical certainty that there are no solutions in the given box (Kearfott and Novoa, 1990; Kearfott, 1990). 3 Results and Discussion We discuss the results from one of our test problems below. It is a three component system consisting of ethanol, methyl ethyl ketone, and water. The activity coefficient model used is the Wilson equation. This system has also recently been studied by Fidkowski et al. ....

Kearfott, R. B., Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics, 26, 337-357 (1990).

.... the specified initial box, a properly implemented interval Newton generalized bisection method can find with mathematical certainty any and all solutions to a specified tolerance, or can determine with mathematical certainty that there are no solutions in the given box (Kearfott and Novoa, 1990; Kearfott, 1990). The system of equations (3) 4) and (6) that must be solved in this case study involves n 1 variables, the n component mole fractions x and the molar volume v. For the mole fraction variables, initial intervals of [0,1] are suitable. In practice the initial lower bound is set to an ....

Kearfott, R. B., Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics, 26, 337--357 (1990).

....monographs by Neumaier (1990) Hansen (1992) and Kearfott (1996) Of particular interest here are interval Newton generalized bisection (IN GB) methods. These techniques provide the power to find, with confidence, enclosures of all solutions of a system of nonlinear equations (Neumaier, 1990; Kearfott and Novoa, 1990), and to find with total reliability the global minimum of a nonlinear objective function (Hansen, 1992) provided only that upper and lower bounds are available for all variables. Efficient techniques for implementing IN GB are a relatively recent development, and thus such methods have not yet ....

....efficiency since intervals with these higher values of volume tend to be quickly eliminated based on the range test for the EOS. Our implementation of the IN GB method for the phase stability problem is based on appropriately modified double precision routines from the packages INTBIS (Kearfott and Novoa, 1990) and INTLIB (Kearfott et al. 1994) 3 Enhancements The efficiency of the method discussed above depends to a considerable extent on how tightly one can compute interval extensions F (X) of real expressions f(x) Denoting F R (X) ff(x) j x 2 Xg as the exact range of f(x) over the interval X, ....

Kearfott, R. B. Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics 1990, 26 , 337--357.

.... number of real solutions in the specified initial box, a properly implemented IN GB method can find with mathematical certainty any and all solutions to a specified tolerance, or can determine with mathematical certainty that there are no solutions in the given box (Kearfott and Novoa, 1990; Kearfott, 1990). The technique used here for computing N (k) from equation (13) is the preconditioned Gauss Seidel like technique developed by Hansen and Sengupta (1981) A detailed step by step description of the IN GB algorithm used here is given by Schnepper and Stadtherr (1996) It should be noted that, ....

Kearfott, R. B. 1990 Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics 26, 337--357.

.... number of real solutions in the specified initial box, a properly implemented IN GB method can find with mathematical certainty any and all solutions to a specified tolerance, or can determine with mathematical certainty that there are no solutions in the given box (Kearfott and Novoa, 1990; Kearfott, 1990). The technique used here for computing N (k) from (5) is the preconditioned Gauss Seidel like technique developed by Hansen and Sengupta (1981) A detailed step by step description of the IN GB algorithm used here is given by Schnepper and Stadtherr (1996) The set of equations (2) 4) that ....

Kearfott, R. B., Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics, 26, 337-357 (1990).

.... number of real solutions in the specified initial box, a properly implemented IN GB method can find with mathematical certainty any and all solutions to a specified tolerance, or can determine with mathematical certainty that there are no solutions in the given box (Kearfott and Novoa, 1990; Kearfott, 1990). The technique used here for computing N (k) from equation (7) is the preconditioned Gauss Seidel like technique developed by Hansen and Sengupta (1981) A detailed step by step description of the IN GB algorithm used here is given by Schnepper and Stadtherr (1996) The system of equations (3) ....

Kearfott, R. B., Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Lectures in Applied Mathematics, 26, 337--357 (1990).

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Kearfott, R. B., Interval Arithmetic Techniques in the Computational Solution of Nonlinear Systems of Equations: Introduction, Examples, and Comparisons, in "Computational Solution of Nonlinear Systems of Equations," (Lectures in Applied Mathematics, volume 26), American Mathematical Society, Providence, RI, 1990, pp. 337--358.

....(ibid. Interval algorithms for reliably finding all roots to nonlinear systems of equations have a somewhat similar structure to branch and bound optimization algorithms, but differ in some important respects. Nonlinear equation solvers also involve a subdivision process and a search; see [7] [9], 15] and others. However, without an objective function more boxes (containing all possible roots) must be considered. Also, nonlinear equation solvers do not need to consider subregions abutting the boundary of the original region specially, since only roots (i.e. critical points) and not ....

....quantities, while vectors and matrices will be denoted with upper case. 2. The Interval Gauss Seidel method Interval Newton methods are used in general to sharpen bounds on the solutions to systems of nonlinear equations, and in computational existence and uniqueness tests; see [14, ch. 5] [9], 15] or [20] among others. Here, we will use them to efficiently reduce the sizes of interior subregions containing critical points, and to reject subregions which do not contain critical points. Suppose we have a function F : R n R n , i.e. 2) F (X) f 1 (x 1 ; x 2 ; xn ) ....

Kearfott, R. B., Interval Arithmetic Techniques in the Computational Solution of Nonlinear Systems of Equations: Introduction, Examples, and Comparisons, in "Computational Solution of Nonlinear Systems of Equations," (Lectures in Applied Mathematics, volume 26), American Mathematical Society, Providence, RI, 1990, pp. 337--358.

No context found.

R. B. Kearfott, Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons, Lectures in Applied Mathematics, 26 (1990), pp. 337357.

No context found.

R. B. Kearfott, Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons, in "Lectures in Applied Mathematics," Vol. 26, pp. 337--357, Amer. Math. Soc., Providence, 1990.

No context found.

R. B. Kearfott, Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons, Lectures in Applied Mathematics, 26 (1990), pp. 337--357.

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