| 1. |
- Arévalo, Carmen, et al.
(författare)
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Improving the accuracy of BDF methods for index 3 dierential algebraic equations
- 1995
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Ingår i: BIT. - 0006-3835 .- 1572-9125. ; 35:3, s. 297-308
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Tidskriftsartikel (refereegranskat)abstract
- Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, and O(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct to O(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct the O(1) errors in the algebraic variables appearing after a change of order.
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| 2. |
- Barrera, Tony, et al.
(författare)
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An integer based square root algorithm
- 1993
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Ingår i: BIT Numerical Mathematics. - 0006-3835 .- 1572-9125. ; 33, s. 254-261
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Tidskriftsartikel (refereegranskat)
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| 3. |
- Burdakov, Oleg, 1953-
(författare)
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A greedy algorithm for the optimal basis problem
- 1997
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Ingår i: BIT Numerical Mathematics. - : Springer. - 0006-3835 .- 1572-9125. ; 37:3, s. 591-599
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Tidskriftsartikel (refereegranskat)abstract
- The following problem is considered. Given m+1 points {x i }0 m in R n which generate an m-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors of the form x i −x j . This problem is present in, e.g., stable variants of the secant and interpolation methods, where it is required to approximate the Jacobian matrix f′ of a nonlinear mappingf by using values off computed at m+1 points. In this case, it is also desirable to have a combination of finite differences with maximal linear independence. As a natural measure of linear independence, we consider the hadamard condition number which is minimized to find an optimal combination of m pairs {x i ,x j }. We show that the problem is not NP-hard, but can be reduced to the minimum spanning tree problem, which is solved by the greedy algorithm in O(m 2) time. The complexity of this reduction is equivalent to one m×n matrix-matrix multiplication, and according to the Coppersmith-Winograd estimate, is below O(n 2.376) for m=n. Applications of the algorithm to interpolation methods are discussed.
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| 4. |
- Crouzeix, Michel, et al.
(författare)
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The stability of rational approximations of analytic semigroups
- 1993
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Ingår i: BIT Numer. Math.. - 0006-3835 .- 1572-9125. ; 33:1, s. 74-84
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Tidskriftsartikel (refereegranskat)abstract
- This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL p , 1
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| 5. |
- Edlund, Ove
(författare)
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Linear M-estimation with bounded variables
- 1997
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Ingår i: BIT Numerical Mathematics. - 0006-3835 .- 1572-9125. ; 37:1, s. 13-23
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Tidskriftsartikel (refereegranskat)abstract
- A subproblem in the trust region algorithm for non-linear M-estimation by Ekblom and Madsen is to find the restricted step. It is found by calculating the M-estimator of the linearized model, subject to anL 2-norm bound on the variables. In this paper it is shown that this subproblem can be solved by applying Hebden-iterations to the minimizer of the Lagrangian function. The new method is compared with an Augmented Lagrange implementation.
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| 6. |
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| 7. |
- Hansbo, Anita
(författare)
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Error estimates for the numerical solution of a time-periodic linear parabolic problem
- 1991
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Ingår i: BIT Numerical Mathematics. - 0006-3835 .- 1572-9125. ; 31:4, s. 664-685
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider the numerical solution of a time-periodic linear parabolic problem. We derive optimal order error estimates in Lz(f2) for approximate solutions obtained by discretizing in space by a Galerkin finite-element method and in time by single-step finite difference methods, using known estimates for the associated initial value problem. We generalize this approach and obtain error estimates for more general discretization methods in the norm of a Banach space B c L2(Ω), e.g., B = H10(Ω) or L∞(Ω). Finally, we consider some computational aspects and give a numerical example.
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