Hansen, E.: Interval Arithmetic in Matrix Computation, Part II. SIAM J. Numer. Anal. 4, pp. 1-9, 1967.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... general subject of interval arithmetic for more references see the topic literature lists below; introduction and proceedings of meetings [Moo66, Nic67, Han69, Ale74, Ale83, Nic75, Nic80, Nic85, May89, Zju89, Neu90, Aci91, Her94a, Ull90d, Ull94] matrix arithmetic and linear systems [Han65, Han67, Apo68a, Han69a, Kri75, Han81, Rat82, Roh82, Roh84, Roh85, Sch87a, Roh88] nonlinear equations [Nic67a, Sch85] extended interval arithmetic [Kau73, Dim92, Jah92] comparison of packages [Cor93] journal [Nes92] complexity of interval methods and NP hardness [Gag85, Roh94, Sha, Sho, Kre96] ....

Hansen, E.: Interval Arithmetic in Matrix Computation, Part II. SIAM J. Numer. Anal. 4, pp. 1-9, 1967.

.... on the general subject of interval arithmetic for more references see the topic literature lists below; introduction and proceedings of meetings [Moo66, Nic67, Han69, Ale74, Ale83, Nic75, Nic80, Nic85, May89, Zju89, Neu90, Aci91, Her94a, Ull90d, Ull94] matrix arithmetic and linear systems [Han65, Han67, Apo68a, Han69a, Kri75, Han81, Rat82, Roh82, Roh84, Roh85, Sch87a, Roh88] nonlinear equations [Nic67a, Sch85] extended interval arithmetic [Kau73, Dim92, Jah92] comparison of packages [Cor93] journal [Nes92] complexity of interval methods and NP hardness [Gag85, Roh94, Sha, Sho, ....

Hansen, E.: Interval Arithmetic in Matrix Computation, Part I. SIAM J. Numer. Anal. 2, pp. 308-320, 1965.

....a transformation. Preconditioning is usually done by multiplying a suitable point matrix P to the original system. Instead of solving A I Omega X I = b I , we deal with the following system: P Omega A I Omega X I = P Omega b I : 7) We call P the preconditioner . Hansen [22] suggests an inverse mid point matrix as preconditioner, which is shown to be optimal [12] in the sense that the preconditioned system gives the tightest bounds of the solutions of the original system. Let A denote the mid point matrix of A I . We define a ij = l ij u ij ) 2 where A I = ....

E.R. Hansen. Interval arithmetic in matrix computations. SIAM Journal on Numerical Analysis, 2:308--320, 1965.

.... Gamma0:79975469: In the following, we show that better results can be obtained by adopting operators from generalized interval arithmetic in Gaussian elimination. 5.1. 2 Generalized Interval Gaussian Elimination The interval Gaussian elimination procedure is well studied [23, 4, 22, 25]. A well known algorithm, known as preconditioned interval Gaussian elimination, is proposed by Hansen [23] Preconditioned Gaussian elimination transforms the coefficient matrix A I to a near identity matrix I I before applying naive interval Gaussian elimination. Since all subdiagonal ....

....from generalized interval arithmetic in Gaussian elimination. 5.1. 2 Generalized Interval Gaussian Elimination The interval Gaussian elimination procedure is well studied [23, 4, 22, 25] A well known algorithm, known as preconditioned interval Gaussian elimination, is proposed by Hansen [23]. Preconditioned Gaussian elimination transforms the coefficient matrix A I to a near identity matrix I I before applying naive interval Gaussian elimination. Since all subdiagonal elements are nearly zeroes, the variable dependency effect are Chapter 5 The Linear Solver highly reduced. We ....

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E.R. Hansen. Interval arithmetic in matrix computations. SIAM Journal on Numerical Analysis, 2:308--320, 1965.

....solver in a CLP(Interval) language. The interval Gauss Seidel method is an iterative method for solving interval linear systems. Convergence of the method to the approximations of the hull of the solution set 1 is guaranteed, however, for only a certain class of linear systems. Preconditioning [9] is a process to transform a linear system into another equivalent system satisfying the convergence criteria. To incorporate the preconditioned interval Gauss Seidel method into the constraint solver of a CLP language, we have to tackle two problems. First, the method, as originally designed, ....

....a transformation. Preconditioning is usually done by multiplying a suitable point matrix P to the original system. Instead of solving A I Omega X I = b I , we deal with the following system: P Omega A I Omega X I = P Omega b I : 1) We call P the preconditioner . Hansen [9] suggests an inverse mid point matrix as preconditioner, which is shown to be optimal [4] in the sense that the preconditioned system gives the tightest bounds of the solutions of the original system. Let A denote the mid point matrix of A I . We define a ij = l ij u ij ) 2 where A I = ....

E.R. Hansen. Interval arithmetic in matrix computations. SIAM Journal on Numerical Analysis, 2:308--320, 1965.

....guarantees termination of the algorithm. 5 Alternative Approaches Preconditioning. A linear inequality can be viewed as a particular linear equation with interval coefficients. For linear interval equation systems, preconditioning methods can improve efficiency significantly, see for example [6], 9] 10] However, the problem is that the interval equation system which arises from the inequality system has unbounded interval coefficients on the right side. This could be handled by enclosing the polytope in a larger polytope which has pairs of parallel faces. Alternative box selection ....

E. Hansen and R. R. Smith. Interval Arithmetic in Matrix Computations, Part 2. SIAM J. Numerical Analysis, 4(1):1--9, 1967.

....division in the interval arithmetic, which causes numerical error propagations. We notice here that the left hand side of (8) may be rewritten as f a k = a k 0 ; a k 1 ; a k n ) j a k = X k ) 01 r k ; X k 2 [X k ] r k 2 [r k ] g: We shall remember Hansen s method (Hansen, 1965) which is a technique for computing an interval matrix U such that f (X k ) 01 j X k 2 [X k ] g U . The merit of this method is to be able to estimate the bound of an interval matrix H defined by U = f (X k ) 01 j X k 2 [X k ] g H: Hansen s method enables us to compute U [r k ....

Hansen, E. 1965. Interval arithmetic in matrix computations. part I. SIAM J. Numerical Analysis., 2:308--320.

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Hansen, E.; Smith, R. Interval Arithmetic in Matrix Computations, Part II. J. SIAM Numer. Anal. 1967, 4,1. 24

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Hansen, E. Interval Arithmetic in Matrix Computations, Part I. J. SIAM Numer. Anal. 1965, 2, 308.

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