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  • Engineering General  (9,884)
  • Wiley-Blackwell  (9,884)
  • Periodicals Archive Online (PAO)
  • Nature Publishing Group
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  • Wiley-Blackwell  (9,884)
  • Periodicals Archive Online (PAO)
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  • 1
    Electronic Resource
    Electronic Resource
    A new plate shell element of 16 nodes and 40 degrees of freedom by relative displacement method (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 15-20 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An isoparametric plate shell element of 16 nodes and 40 degrees of freedom is developed in this paper directly from 3-D elastic theories. In order to overcome the difficulties caused by straightforward use of a 3-D formulation, the relative displacement method is introduced. For curved shell elements, the use of the orthogonal curvilinear co-ordinate system greatly reduces the computer effort. The new element can conveniently solve shell problems with variable thicknesses and also easily link with other finite elements using displacement d.o.f. such as 8-node to 21-node 3-D elements. Numerical examples demonstrate the efficiency of the element.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090105
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  • 2
    Electronic Resource
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    Performance of a Hermitian element for a beam with rotational constraints (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 27-34 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The behaviour of a Hermitian two-node element based on the Bernoulli beam equation is examined. The assumed constraints generate rotation-dependent distributed moments. It is shown that for these moments a potential exists, and that a rigid translation is the only rigid body mode of the element. The analysis of the Bernoulli equation demonstrates that very large values of α= e/EI enforce the condition w,x = 0, resulting in displacements equal to zero. The element is examined for two types of constraints. The first type of constraint, diminishing rotations of a beam (α 〈 0), yields regular solutions which, however, seem to have a non-differentiality near the end of the beam. A special procedure is developed to evaluate analytical solutions for long beams or stiff constraints, for which computer accuracy is exceeded. For the second type of constraint, enlarging rotations of a beam (α 〉 0), a highly oscillatory nature of the solution is proven.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090107
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  • 3
    Electronic Resource
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    Quickly convergent integration methods for plane stress plasticity (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 293-305 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Explicit finite-element programs, such as DYNA3D, are frequently used to solve large structural problems, such as automobile impacts, that involve tens of thousands of shell elements. Since the explicit finite-element programs do not use a stiffness matrix, the major cost of the calculation is the evaluation of the force vector. A significant part of that cost is the evaluation of the constitutive models. The standard algorithm for integrating J2 plasticity at the stress points is the radial return method. An iterative variation of radial return is commonly used to impose the zero normal stress condition within the shell elements. In this paper we explore several different non-iterative methods that impose the zero normal stress condition in an approximate manner. We show that the cost of the constitutive evaluation can be reduced by 25 per cent without introducing significant errors.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090403
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  • 4
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    On bicubic transformation for the numerical evaluation of Cauchy principal value integrals (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 307-311 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Recently a bicubic transformation was introduced to numerically compute the Cauchy principal value (CPV) integrals. Numerical results show that this new method converges faster than the conventional Gauss-Legendre quadrature rule when the integrand contains different types of singularity. Assume η is the singular point of a CPV integral. The point η divides the interval [-1, 1] into two parts: [-1, η] and [η, 1]. The bicubic transformation maps the intervals [-1, η] and [η, 1] to the interval [-1, 1] with the following constraints: it maps the point η - ∊ to μn, and η + ∊ to -μn, where μn is the largest Gaussian point of an n-point Gauss-Legendre quadrature rule, and ∊ is a user-supplied constant. The n-point Gauss-Legendre quadruture rule is then applied. In contrast to ordinary expectation, further numerical experiment shows that smaller ∊ does not always produce better results. In this paper we are concerned with the selection of ∊ to yield rapid convergence of numerical integration when the bicubic transformation method is applied.
    Additional Material: 3 Tab.
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    URL: http://dx.doi.org/10.1002/cnm.1640090404
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  • 5
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    Incompressible viscoplastic flow analysis using a quadrilateral 2-D element based on mixed interpolation of tensorial components (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 157-164 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the Eulerian description of metal forming operations usually rigid/incompressible viscoplastic material models are used. In this paper a quadrilateral 2-D finite-element formulation for incompressible viscoplastic flow analysis is developed. The element (QMITC) is based on mixed interpolation of tensorial components, and the incompressibility constraint is imposed via the penalty method. It is shown that the resulting element does not lock either in plane strain rate flows or in axisymmetric flows, that the oscillations in the pressure predictions (checkerboard) tend to zero when the mesh is refined and that the solutions for some typical benchmark problems are very accurate.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090209
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  • 6
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    Conference diary (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 177-179 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090211
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  • 7
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    On perturbation behaviour in non-linear dynamics (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 165-175 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The static bifurcation criterion, i.e. singularity of the tangential stiffness matrix, is discussed in the sense of non-linear dynamics since it is sometimes used in the engineering literature. In this paper the connection between static stability, i.e. that the tangential stiffness matrix is positive-definite, and insensitivity, i.e. local damping of small perturbations, is proved. Since, in general, there is no connection between insensitivity and asymptotic stability, the concept of sensitivity cannot replace the classical stability theory of motion.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090210
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  • 8
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    Masthead (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993) 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090301
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  • 9
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    On a numerical solution of a class of partial differential equations of mixed type, not adding artificial terms (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 181-187 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A relaxation-like method for the numerical solution of a class of partial differential equations of mixed type is presented. The equations of this class have not yet been solved by functional outlines. The numerical method does not add artificial terms to the original equation.
    Additional Material: 3 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090302
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  • 10
    Electronic Resource
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    Wave distribution in fibre members subjected to kinematic forcing (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 189-196 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper is concerned with the solution of non-linear wave distribution in structural fibre members subjected to dynamic kinematic forcing. The development of reliable and efficient techniques for numerical handling of wave distribution based on the application of combined finite-element and transfer matrix approaches is emphasized. Illustrative numerical solutions are presented.
    Additional Material: 10 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090303
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  • 11
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    Solution of the advection-diffusion equation using the spline function and finite elements (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 197-206 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper deals with the solution of the one-dimensional advection-diffusion equation. The method of solution is based on the process-splitting technique. The transport equation is split up into the advection part and the diffusion part at every time step. The advection transport equation is solved by a method of characteristics using a spline function for interpolation. This solution approach proves to be very effective because it reduces numerical dispersion and wiggles. The diffusion equation is solved by the standard Galerkin finite-element method. A comparison of calculations with the results of Holly and Preissmann and with analytical solutions demonstrates the great accuracy of the proposed method.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090304
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  • 12
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    Generalized conforming quadrilateral plate element by using semi-loof constraints (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 417-426 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A 12-DOF quadrilateral element for plate bending is formulated based on the concept of generalized compatibility and semi-Loof constraints. The element DOFs are defined with the conventional displacements at corner nodes and the semi-Loof constraints are used in the formulation. This element is a generalized conforming one which passes Irons' patch test and can be used to solve problems with complicated boundaries. When overall considerations of simplicity, accuracy and reliability are taken into account, this new element is one of the best 12-DOF quadrilateral plate-bending elements currently available.
    Additional Material: 6 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090507
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  • 13
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    The Gauss-Laguerre quadrature rule for finite-part integrals (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 439-450 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The classical Gauss-Laguerre quadrature rule for the semi-infinite integration interval [0,∞] is modified and applied to the case of the weight function exp(-x)/x corresponding to finite-part (or, equivalently, hypersingular) integrals. The new set of orthogonal polynomials is constructed and it is seen to consist of linear combinations of the classical Laguerre polynomials with appropriately determined coefficients. The zeros of these modified Laguerre polynomials are seen to be distinct, but one of these lies outside the integration interval. Formulae for the corresponding weights are also given and numerical values and results are presented. The present results generalize the corresponding results for the finite interval [0,1] to semi-infinite intervals and they are applicable to a variety of applied mechanics and related problems, where finite-part integrals appear in a natural way.
    Additional Material: 2 Tab.
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    URL: http://dx.doi.org/10.1002/cnm.1640090509
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  • 14
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    The virtual contact loading method for elastic contact problems (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 455-461 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The virtual contact loading method (VCLM), a highly efficient numerical method to solve elastic contact problems, is proposed in this paper. Having applied virtual loads on the possible contact region (PCR) of a structure system, we carried out the usual finite-element calculations only once and made a local iteration analysis in PCR then the contact region (shape and size), contact state (stick, slide or mixed), contact normal and shear stress distributions, and the stress and displacement fields in contact bodies were all obtained simultaneously. This method is very simple and easy to use.The elastic contact problems for cylinder-cylinder are solved in this paper and the results agree well with those in previous work.
    Additional Material: 5 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090602
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  • 15
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    Method to determine optimum number of knots for cubic splines (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 483-488 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this paper a new method has been presented to determine the optimum number of knots for cubic splines. The knot-finding process is based on the numerical integration of the input curve. The number and the location of the knots is determined automatically. The method has been applied to a test case and the performance has been compared with two other existing methods. It is shown that fewer knots are retained for the portions of the curve having small curvature whereas a larger number of knots is retained for highly curved portions. The computer time required by our method depends only upon the number of points in the input curve and does not depend upon the shape of the curve or the desired accuracy.
    Additional Material: 3 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090605
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  • 16
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    Error study on numerical approximation of radiation boundary condition for one-dimensional wave equation (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 475-482 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The radiation boundary condition is frequently used at the artificial outflow boundary created in the numerical computation of fluid flow problems to allow outgoing waves to leave the domain undisturbed. However, the numerical approximation of the radiation boundary condition will produce errors which will affect the interior solution. In this work the errors due to two different numerical approximations of the radiation boundary condition are studied, with the one-dimensional wave equation being used as the model equation. A theoretical analysis shows that: (a) if the numerical approximation of the boundary condition generates a celerity error, a reflected wave with amplitude proportional to the size of the celerity error is produced; (b) if the numerical approximation of the boundary condition generates a numerical diffusion and hence an amplitude damping error, a reflected wave with amplitude proportional to the magnitude of the numerical diffusion is produced. A modification of the Sommerfeld radiation boundary condition to account for the linear bottom friction effect is investigated. Numerical experiments show that the use of the backward characteristics scheme with quadratic interpolation is able to reduce significantly the reflection of the wave at the outflow boundary and that the scheme is unconditionally stable.
    Additional Material: 4 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090604
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  • 17
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    Standard eigenvalue analysis by boundary-element method (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 489-495 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A method for analysing eigenvalues of the Helmholtz equation using a standard existing subroutine for eigenvalue determination is developed in this paper. It is based on the boundary integral equation formulation known as the multiple reciprocity method (MRM). The above-mentioned formulation, having polynomial matrices in terms of the eigenvalue as the coefficient matrices, is transformed into the standard-type eigenvalue problem. The resulting formulation makes it possible to determine the required eigenvalues only by boundary discretisation without any initial rough estimation.
    Additional Material: 1 Tab.
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    URL: http://dx.doi.org/10.1002/cnm.1640090606
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  • 18
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    A posteriori error bounds for numerical solutions of regularized compressible flow problems (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 511-523 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We give a posteriori error bounds for numerical solutions of regularized compressible flow problems. To verify the efficiency of our method, numerical experiments are performed.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090609
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  • 19
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    Flow development in entrance region of ducts (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 503-509 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The integral transform method is used to compute numerically the development of the velocity profile in the hydrodynamic entrance region for laminar flow inside a parallel plate channel. The results have user-prescribed accuracy and can be used to test other numerical methods.
    Additional Material: 1 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090608
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  • 20
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    Appropriate boundary conditions for FE-modelled elastic half-plane (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 497-501 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The completely adhesive contact between an FE-modelled orthotropic half-plane and a vertically loaded rigid flat punch is studied. Interface normal and shear-stress distributions between the punch and half-plane are FE-calculated and compared for some sets of half-plane boundary conditions frequently used in the literature for contact problems with friction. These results are compared with analytical solutions, and appropriate half-plane boundary conditions are recommended for frictional contact problems.
    Additional Material: 3 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090607
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  • 21
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    Conference diary (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 543-545 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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    URL: http://dx.doi.org/10.1002/cnm.1640090612
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  • 22
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    Masthead (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993) 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090701
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  • 23
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    Corrective technique for numerical prediction of liquid flow rate in annular flow study (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 533-542 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The computer code FIDAS-3DT, developed by Power Reactor & Nuclear Fuel Development Corporation of Japan, has been used to solve by the finite-difference method an air-water horizontal annular flow problem based on two-phase, three-fluid and two-component velocity modelling. In an attempt to improve the water flow rate results, as compared with those of a reported experiment, several corrections applied to the code are investigated, and the recommended technique is presented.
    Additional Material: 5 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090611
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    Mesh enrichment against mesh regeneration using quadrilateral elements (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 547-554 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An adaptive refinement procedure using a new automatic quadrilateral element mesh generator is presented. The performance of the adaptive procedure in achieving a given accuracy with different mesh refinement strategies is compared. It is found that when using bilinear or biquadratic quadrilateral elements refinement strategies of both mesh enrichment and mesh regeneration can lead to a prescribed accuracy being achieved by the Zienkiewicz-Zhu adaptive procedure1 with the optimal rate of convergence. However, fewer mesh refinement steps are needed if the strategy of mesh regeneration is employed with the newly developed automatic quadrilateral element mesh generator2.
    Additional Material: 5 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090702
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    Geometrically based, automatic generator for irregular triangular networks (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 555-566 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The network generation method described herein produces two-dimensional irregular triangular grids in which element area is proportional to some scalar density function defined over the domain. This density function can reflect arbitrarily chosen properties of the geometry or flow. After the initial generation, various adjustment utilities and an interactive (graphic-based) editor ensures that the resultant triangular elements are well shaped. Some of these utilities can be used with adaptive refinement schemes.
    Additional Material: 10 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090703
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    Editorial (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 105-105 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010202
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    Equivalence of conditions for convergence of iterative methods for singular equations (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 151-154 
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    ISSN: 1070-5325
    Keywords: Linear systems ; Singular systems ; Iterative methods ; Convergent methods ; Index of the iteration operator ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Equivalence is shown between different conditions for convergence of iterative methods for consistent singular systems of linear equations on Banach spaces. These systems appear in many applications, such as Markov chains and Markov processes. The conditions considered relate the range and null spaces of different operators.
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    URL: http://dx.doi.org/10.1002/nla.1680010206
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  • 28
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    Conference Diary (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 211-212 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/nla.1680010209
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    New convergence results and preconditioning strategies for the conjugate gradient method (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 179-210 
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    ISSN: 1070-5325
    Keywords: Linear system solvers ; Conjugate gradients ; Iteration number estimates ; Explicit preconditionings ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Instead of the standard estimate in terms of the spectral condition number we develop a new CG iteration number estimate depending on the quantity B = 1/ntr M/(det M)1/n, where M is an n × n preconditioned matrix. A new family of iterative methods for solving symmetric positive definite systems based on B-reducing strategies is described. Numerical results are presented for the new algorithms and compared with several well-known preconditioned CG methods.
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    Diagonally compensated reduction and related preconditioning methods (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 155-177 
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    ISSN: 1070-5325
    Keywords: Preconditioning ; Diagonal compensation ; Eigenvalue bounds ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.
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    Masthead (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994) 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    On the convergence of difference schemes for a heat conduction equation (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 237-245 
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    ISSN: 1070-5325
    Keywords: Korovkin theorem ; Heat conduction equation ; Lax theorem ; Difference scheme ; Convergence ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Sufficient conditions are obtained for the convergence of difference schemes for the numerical solution of the Cauchy problem for a heat conduction equation in two space variables. The sufficient conditions are derived in a form similar to those for the convergence of a sequence of linear positive operators in the Korovkin theorem. As an application it is shown that difference schemes that are widely used in practice can easily be checked for convergence by these conditions.
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    Algebraic multilevel iteration method for Stieltjes matrices (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 213-236 
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    ISSN: 1070-5325
    Keywords: Optimal order preconditioners ; Algebraic multilevel ; Chebyshev polynomial approximation ; Diagonal compensation ; Approximate inverses ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The numerical solution of elliptic selfadjoint second-order boundary value problems leads to a class of linear systems of equations with symmetric, positive definite, large and sparse matrices which can be solved iteratively using a preconditioned version of some algorithm. Such differential equations originate from various applications such as heat conducting and electromagnetics. Systems of equations of similar type can also arise in the finite element analysis of structures.We discuss a recursive method constructing preconditioners to a symmetric, positive definite matrix. An algebraic multilevel technique based on partitioning of the matrix in two by two matrix block form, approximating some of these by other matrices with more simple sparsity structure and using the corresponding Schur complement as a matrix on the lower level, is considered.The quality of the preconditioners is improved by special matrix polynomials which recursively connect the preconditioners on every two adjoining levels. Upper and lower bounds for the degree of the polynomials are derived as conditions for a computational complexity of optimal order for each level and for an optimal rate of convergence, respectively.The method is an extended and more accurate algebraic formulation of a method for nine-point and mixed five- and nine-point difference matrices, presented in some previous papers.
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    Masthead (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994) 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Editorial (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 1-1 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Low-rank revealing QR factorizations (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 33-44 
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    ISSN: 1070-5325
    Keywords: Rank revealing QR factorization ; Column pivoting ; Numerical rank ; Subset selection ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Rank revealing factorizations are used extensively in signal processing in connection with, for example, linear prediction and signal subspace algorithms. We present an algorithm for computing rank revealing QR factorizations of low-rank matrices. The algorithm produces tight upper and lower bounds for all the largest singular values, thus making it particularly useful for treating rank deficient problems by means of subset selection, truncated QR, etc. The algorithm is similar in spirit to an algorithm suggested earlier by Chan for matrices with a small nullity, and it can also be considered as an extension of ordinary column pivoting.
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    A modified direct preconditioner for indefinite symmetric Toeplitz systems (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 415-429 
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    ISSN: 1070-5325
    Keywords: indefinite ; inverse ; preconditioning ; symmetric ; Toeplitz ; Trench algorithm ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A modification is presented of the classical O(n2) algorithm of Trench for the direct solution of Toeplitz systems of equations. The Trench algorithm is stable for symmetric matrices that are positive definite, but can be unstable for the indefinite case (and for general matrices). Our modification permits extension of the algorithm to compute approximate inverses of indefinite symmetric Toeplitz matrices, for which the unmodified algorithm breaks down when principal submatrices are singular. As a preconditioner, the approximate inverse has an advantage that only matrix - vector multiplications are required for the solution of a linear system, without forward and backward solves. The approximate inverse so obtained can be sufficiently accurate, moreover, that, when it is used as a preconditioner for the applications investigated, subsequent iteration may not even be necessary. Numerical results are given for several test matrices. The perturbation to the original matrix that defines the modification is related to a perturbation in a quantity generated in the Trench algorithm; the associated stability of the Trench algorithm is discussed.
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    Conference diary (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 483-485 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Completely parallelizable preconditioning methods (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 447-465 
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    ISSN: 1070-5325
    Keywords: preconditioned iterative methods ; SSOR factorization ; parallelization ; connection machine ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A class of parallelizable preconditioned iterative methods for the solution of certain finite difference or finite element linear systems of equations is presented. The methods are based on calculation of approximate inverses of the SSOR factorization. The speed of the methods may be increased by making the approximation of the inverse more accurate. The construction of the preconditioning as well as the solution of the preconditioning systems (realized by matrix-vector multiplication) can be made in parallel over the total amount of meshpoints. The methods are suitable for implementation on massively parallel computers such as connection machines. Problems with constant as well as strongly varying orthotropy are examined and the methods are compared to other parallel techniques with respect to rate of convergence, computational complexity and consumed CM200 computing time. We report a small but significant decrease in computing time compared to the (until now) most frequently used completely parallel preconditioning, the Jacobi method.
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    A saxpy formulation for plane rotations (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 533-541 
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    Keywords: plane rotation ; shear product ; Saxpy operation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: An algorithm for pre- or post-multiplication of a matrix by a plane rotation, using only three vector saxpy operations instead of the four vector operations usually considered necessary, is described. No auxiliary storage for overwriting is required. The method is shown to be numerically stable.
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    Multilevel preconditioners for discretizations of the biharmonic equation by rectangular finite elements (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 487-505 
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    ISSN: 1070-5325
    Keywords: biharmonic equation ; rectangular finite elements ; preconditioning ; multilevel methods ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Recently, some new multilevel preconditioners for solving elliptic finite element discretizations by iterative methods have been proposed. They are based on appropriate splittings of the finite element spaces under consideration, and may be analyzed within the framework of additive Schwarz schemes. In this paper we discuss some multilevel methods for discretizations of the fourth-order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner-Fox-Schmit rectangle, the generalization of the Bramble-Pasciak-Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived.
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    Preconditioning Mixed Finite Element Saddle-point Elliptic Problems (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 1-20 
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    ISSN: 1070-5325
    Keywords: preconditioning saddle-point problems ; eigenvalue estimation ; mixed finite element method ; minimum residual method ; second-order elliptic problems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider saddle-point problems that typically arise from the mixed finite element discretization of second-order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form(δ is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditioner for the above bilinear form can be applied. We also show some numerical experiments that illustrate the convergence properties of the proposed technique.
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    Masthead (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996) 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    Multifrontal QR Factorization in a Multiprocessor Environment (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 275-300 
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    ISSN: 1070-5325
    Keywords: sparse QR factorization ; multifrontal method ; parallelism ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We describe the design and implementation of a parallel QR decomposition algorithm for a large sparse matrix A. The algorithm is based on the multifrontal approach and makes use of Householder transformations. The tasks are distributed among processors according to an assembly tree which is built from the symbolic factorization of the matrix ATA.We first address uniprocessor issues and then discuss the multiprocessor implementation of the method. We consider the parallelization of both the factorization phase and the solve phase. We use relaxation of the sparsity structure of both the original matrix and the frontal matrices to improve the performance. We show that, in this case, the use of Level 3 BLAS can lead to very significant gains in performance. We use the eight processor Alliant˜FX/80 at CERFACS to illustrate our discussion.
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    Two-stage Multisplitting Methods with Overlapping Blocks (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 113-124 
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    ISSN: 1070-5325
    Keywords: iterative methods ; linear systems ; multisplittings ; overlap ; parallel algorithms ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Parallel two-stage multisplitting methods with overlap for the solution of linear systems of algebraic equations are studied. It is shown that, under certain hypotheses, the method with overlap is asymptotically faster than that without overlap. Experiments illustrating this phenomenon are presented.
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    On Fast Computations with the Inverse to Harmonic Potential Operators via Domain Decomposition (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 91-111 
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    ISSN: 1070-5325
    Keywords: boundary integral operators ; domain decomposition ; interface operators ; fast elliptic problem solvers ; parallel algorithms ; preconditioning ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper a method for fast computations with the inverse to weakly singular, hypersingular and double layer potential boundary integral operators associated with the Laplacian on Lipschitz domains is proposed and analyzed. It is based on the representation formulae suggested for above-mentioned boundary operations in terms of the Poincare-Steklov interface mappings generated by the special decompositions of the interior and exterior domains. Computations with the discrete counterparts of these formulae can be efficiently performed by iterative substructuring algorithms provided some asymptotically optimal techniques for treatment of interface operators on subdomain boundaries. For both two- and three-dimensional cases the computation cost and memory needs are of the order O(N logp N) and O(N log2 N), respectively, with 1 ≤ p ≤ 3, where N is the number of degrees of freedom on the boundary under consideration (some kinds of polygons and polyhedra). The proposed algorithms are well suited for serial and parallel computations.
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    Short Communication: Two Properties of Diagonally Dominant Matrices (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 173-177 
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    Keywords: diagonally dominant matrix ; determinant ; norm ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A property of strictly diagonally dominant matrices and a generalization of a Varga's bound for ∥ A-1∥∞ to the case ∥ A-1B∥∞ are given and the two-sided bounds for the determinants of strictly diagonally dominant matrices are derived
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    Piecewise Polynomial Solutions Without a priori Break Points (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 513-524 
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    Keywords: regularization ; I-norm ; discontinuous solutions ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In certain inverse problems it is useful to be able to compute solutions which are, in some sense, as simple as possible. For example,k one may wish to compute solutions which are piecewise constant and with as few discontinuities as possible. Such solutions are suited to describe models, e.g., geological layers, where the coarse structure is more important than the fine structure. A natural generalization of piecewise constant functions is piecewise polynomial solutions. In this paper we present a new algorithm which is capable of computing solutions that are piecewise polynomials, without having to specify a priori the positions of the break points between the polynomial pieces.
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    On IOM(q): The Incomplete Orthogonalization Method for Large Unsymmetric Linear Systems (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 491-512 
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    ISSN: 1070-5325
    Keywords: unsymmetric ; Krylov subspace ; IOM(q) ; FOM ; truncated ; basis vector ; orthonormality ; convergence ; restarted ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The incomplete orthogonalization method (IOM(q)), a truncated version of the full orthogonalization method (FOM) proposed by Saad, has been used for solving large unsymmetric linear systems. However, no convergence analysis has been given. In this paper, IOM(q) is analysed in detail from a theoretical point of view. A number of important results are derived showing how the departure of the matrix A from symmetric affects the basis vectors generated by IOM(q), and some relationships between the residuals for IOM(q) and FOM are established. The results show that IOM(q) behaves much like FOM once the basis vectors generated by it are well conditioned. However, it is proved that IOM(q) may generate an ill-conditioned basis for a general unsymmetric matrix such that IOM(q) may fail to converge or at least cannot behave like FOM. Owing to the mathematical equivalence between IOM(q) and the truncated ORTHORES(q) developed by Young and Jea, insights are given into the convergence of the latter. A possible strategy is proposed for choosing the parameter q involved in IOM(q). Numerical experiments are reported to show convergence behaviour of IOM(q) and of its restarted version.
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    A simpler GMRES (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 571-581 
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    ISSN: 1070-5325
    Keywords: GMRES ; GMRES(m) ; Large-scale nonsymmetric linear systems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The generalized minimal residual (GMRES) method is widely used for solving very large, nonsymmetric linear systems, particularly those that arise through discretization of continuous mathematical models in science and engineering. By shifting the Arnoldi process to begin with Ar0 instead of r0, we obtain simpler Gram-Schmidt and Householder implementations of the GMRES method that do not require upper Hessenberg factorization. The Gram-Schmidt implementation also maintains the residual vector at each iteration, which allows cheaper restarts of GMRES(m) and may otherwise be useful.
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    A fast minimal residual algorithm for shifted unitary matrices (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 555-570 
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    ISSN: 1070-5325
    Keywords: Linear system of equations ; Iterative method ; Fast algorithm ; Arnoldi process ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρU + ζI, where ρ and ζ are constants, U is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.
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    Conference Diary (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 583-587 
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    Keywords: Engineering ; Engineering General
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    Matrix shapes invariant under the symmetric QR algorithm (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 87-93 
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    ISSN: 1070-5325
    Keywords: QR algorithm ; zero pattern ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The QR algorithm is a basic algorithm for computing the eigenvalues of dense matrices. For efficiency reasons it is prerequisite that the algorithm is applied only after the original matrix has been reduced to a matrix of a particular shape, most notably Hessenberg and tridiagonal, which is preserved during the iterative process. In certain circumstances a reduction to another matrix shape may be advantageous. In this paper, we identify which zero patterns of symmetric matrices are preserved under the QR algorithm.
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    Efficient reduction algorithms for bordered band matrices (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 95-113 
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    ISSN: 1070-5325
    Keywords: arrowhead matrix ; band matrix ; inverse eigenvalue problem ; givens rotations ; singular value decomposition ; updating ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Various plane rotation patterns are presented, which provide stable algorithms for reducing a b-band matrix bordered by p rows and/or columns to (b + p)-band form. These schemes generalize previously presented O(N2) reduction algorithms for matrices of order N, b = 1, and p = 1 to the reduction of more general b-band, p-bordered matrices where b ≥ 1 and p ≥ 1. Moreover, by splitting the matrix into two similarly structured submatrices and chasing nonzeros to the corners in two directions, the newly proposed patterns reduce the number of required rotations and hence the computational cost by one half compared to the other existing one-way chasing algorithms. Symmetric, as well as more general matrices, are considered. An example of the first type is the symmetric arrowhead matrix that arises in solving inverse eigenvalue problems. Examples of the second type are found in updating the singular value decomposition (SVD) and the partial SVD.
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    Block Two-stage Methods for Singular Systems and Markov Chains (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 413-426 
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    ISSN: 1070-5325
    Keywords: iterative methods ; linear systems ; singular matrices ; block methods ; multisplitting ; two-stage ; non-stationary ; Markov chains ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The use of block two-stage methods for the iterative solution of consistent singular linear systems is studied. In these methods, suitable for parallel computations, different blocks, i.e., smaller linear systems, can be solved concurrently by different processors. Each of these smaller systems are solved by an (inner) iterative method. Hypotheses are provided for the convergence of non-stationary methods, i.e., when the number of inner iterations may vary from block to block and from one outer iteration to another. It is shown that the iteration matrix corresponding to one step of the block method is convergent, i.e., that its powers converge to a limit matrix. A theorem on the convergence of the infinite product of matrices with the same eigenspace corresponding to the eigenvalue 1 is proved, and later used as a tool in the convergence analysis of the block method. The methods studied can be used to solve any consistent singular system, including discretizations of certain differential equations. They can also be used to find stationary probability distribution of Markov chains. This last application is considered in detail.
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    Editorial (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 349-350 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: No Abstract
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    Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 427-453 
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    ISSN: 1070-5325
    Keywords: mixed method ; nonconforming method ; multilevel preconditioner ; condition number ; second order elliptic problem ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A new approach for constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners for this system are then constructed based on a triangulation of the domain into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory.
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    Conference diary (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 455-457 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680030502
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    A Comparison of Three Basic Conjugate Direction Methods (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 473-489 
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    ISSN: 1070-5325
    Keywords: conjugate gradients ; biconjugate gradients ; conjugate directions ; non-symmetric linear systems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper three basic conjugate direction methods for solving non-symmetric linear systems are described and compared. All three have entirely different basic structures and, consequently, completely different characteristics. The motivation for this work is the comparative assessment of the methods in order to decide which of the three is the most suitable for further development.
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    Masthead (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996) 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680030601
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    Conference diary (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 543-545 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680030602
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    On Parlett's matrix norm inequality for the Cholesky decomposition (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 243-250 
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    ISSN: 1070-5325
    Keywords: Cholesky ; norm inequality ; perturbation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We show that a certain matrix norm ratio studied by Parlett has a supermum that is O(\documentclass{article}\pagestyle{empty}\begin{document}$\mathop \[\sqrt n \] $\end{document}) when the chosen norm is the Frobenius norm, while it is O(log n) for the 2-norm. This ratio arises in Parlett's analysis of the Cholesky decomposition of an n by n matrix.
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    Optimal backward perturbation bounds for the linear least squares problem (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 271-286 
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    ISSN: 1070-5325
    Keywords: linear least squares ; backward perturbations ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let A be an m × n matrix, b be an m-vector, and x̃ be a purported solution to the problem of minimizing ‖b  -  Ax‖2. We consider the following open problem: find the smallest perturbation E of A such that the vector x̃ exactly minimizes ‖b  -  (A+E)x‖2. This problem is completely solved when E is measured in the Frobenius norm. When using the spectral norm of E, upper and lower bounds are given, and the optimum is found under certain conditions.
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    On the solution of block Hessenberg systems (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 287-296 
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    ISSN: 1070-5325
    Keywords: block Hessenberg matrix ; block Toeplitz matrix ; linear system ; queue ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper describes a divide-and-conquer strategy for solving block Hessenberg systems. For dense matrices the method is as efficient as Gaussian elimination; however, because it works almost entirely with the original blocks, it is much more efficient for sparse matrices or matrices whose blocks can be generated on the fly. For Toeplitz matrices, the algorithm can be combined with the fast Fourier transform.
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    Announcement (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. ii 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/nla.1680020602
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    Implicit Cholesky algorithms for singular values and vectors of triangular matrices (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 507-531 
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    ISSN: 1070-5325
    Keywords: Cholesky decomposition ; singular values ; eigenvalues ; null spaces ; noise spaces ; URV and ULV factorizations ; QR algorithm ; LR algorithm ; Jacobi methods ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The implicit Cholesky algorithm has been developed by several authors during the last 10 years but under different names. We identify the algorithm with a special version of Rutishauser's LR algorithm. Intermediate quantities in the transformation furnish several attractive approximations to the smallest singular value.The paper extols the advantages of using shifts with the algorithm. The nonorthogonal transformations improve accuracy.
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    Conference diary (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 543-545 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1002/nla.1680020606
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    Toeplitz Preconditioners with Block Structure for First-order PDEs (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 21-44 
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    ISSN: 1070-5325
    Keywords: first-order PDEs ; iterative methods ; preconditioners ; Toeplitz ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Preconditioners to nonsymmetric, nondiagonally dominant systems of equations are constructed and examined numerically. The preconditioners are based on a Toeplitz approach with a certain symmetry that we define. The inversion of the preconditioners is defined through a Fast Modified Sine Transform. As a model problem we study the systems of equations arising from a implicit time-discretization with a large time-step of a scalar hyperbolic PDE.
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    Generalization of Strang's Preconditioner with Applications to Toeplitz Least Squares Problems (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 45-64 
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    ISSN: 1070-5325
    Keywords: Toeplitz least squares problems ; circulant preconditioned conjugate gradient method ; deconvolution ; image restoration ; atmospheric imaging ; medical imaging ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The th column of our circulant preconditioner Sn is equal to the th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. We show that if the matrix An has decaying coefficients away from the main diagonal, then is a good preconditioner for An. Comparisons of our preconditioner with other circulant-based preconditioners are carried out for some 1-D Toeplitz least squares problems: min ∥ b - Ax∥2. Preliminary numerical results show that our preconditioner performs quite well, in comparison to other circulant preconditioners. Promising test results are also reported for a 2-D deconvolution problem arising in ground-based atmospheric imaging.
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    Editorial (1993)
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    Communications in Numerical Methods in Engineering 9 (1993), S. i 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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    URL: http://dx.doi.org/10.1002/cnm.1640090102
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    Suitability of three-dimensional finite elements for modelling material incompressibility using exact integration (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 313-329 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper examines the suitability of three-dimensional finite elements to model accurately problems involving material incompressibility, using the displacement finite element method and exact numerical integration. The previously used method for classification of element suitability is presented and extended to the three-dimensional case. However, an alternative approach for examining suitability, quantified in terms of free degrees of freedom (equal to the degrees of freedom minus the incompressibility constraints), is introduced. This is used to examine Lagrangian cubic, serendipity cubic and tetrahedral three-dimensional elements configured in a regular cubic arrangement. The findings of this paper are substantiated by a number of three-dimensional numerical experiments and comparison with a separate two-dimensional study.All serendipity cubic elements are found to be unsuitable. The linear strain tetrahedron is on the borderline of suitability in a 6-tetrahedra-per-cube arrangement, and is thought to be only suitable if the mesh boundary nodes are not over-constrained. The same element in a 5-tetrahedra-per-cube arrangement, and higher order tetrahedra (quadratic strain etc), are suitable. The Lagrangian cube elements of higher order than the 27-node cube are suitable, but are probably not as efficient computationally as the tetrahedral elements of the same order.
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    Incorporation of strain-hardening in the implicit elasto-viscoplasticity algorithm (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 331-336 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: An algorithm is presented for the inclusion of general strain-hardening/softening behaviour into the direct method of finite-element analysis of elasto-viscoplasticity using implicit time discretization.
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    Evaluation of field singularities by an iterative boundary-element method (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 337-343 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A numerical method is presented for the evaluation of field singularities of the type (1/r1-λ) using iterative solutions of standard boundary-element analyses in a region surrounding the origin of singularity. Results for λ are presented for both two and three-dimensional problems, and representative asymptotic fields for these examples are discussed.
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    Variational validation of a membrane finite element with drilling rotations (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 345-351 
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    Keywords: Engineering ; Engineering General
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    Notes: It is shown that a triangular membrane finite element with vertex ‘drilling rotations’ which was developed in earlier publications is, in fact, a correct implementation of a variational principle. The existence of a variational principle is a crucial factor which places the earlier work on a secure theoretical foundation. It implies superior computational characteristics of the finite-element equations and provides valuable guidance on practical issues such as the consistent treatment of boundary conditions.
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    Conference diary (1993)
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    Communications in Numerical Methods in Engineering 9 (1993), S. 361-363 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    URL: http://dx.doi.org/10.1002/cnm.1640090410
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    Masthead (1993)
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    Communications in Numerical Methods in Engineering 9 (1993) 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    An algorithm for the numerical inversion of a tridiagonal matrix (1993)
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    Communications in Numerical Methods in Engineering 9 (1993), S. 353-359 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: This paper presents an algorithm for obtaining the inverse of a tridiagonal matrix numerically. The algorithm does not require diagonal dominance in the matrix and is also computationally efficient.
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    An element error indicator and a related stress-improvement procedure (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 207-217 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A posteriori error indicators for strain energy error in finite-element solutions have recently been shown to be effective. A popular error indicator, usually denoted by η, makes use of ‘error energy’, i.e. strain energy associated with the difference between the conventional element stresses σ and a smoothed stress field constructed from σ. It is intended that η be constructed by summing element contributions and that it be applied at the global level. The present study suggests that an error indicator applicable at the element level can be obtained by combining an element value of η with information about stress gradient and strain energy density. It is also shown how the stress gradient information can be used to obtain more accurate stresses at mesh boundaries.
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    Balancing domain decomposition (1993)
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    Communications in Numerical Methods in Engineering 9 (1993), S. 233-241 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: The Neumann-Neumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finite-element discretizations of difficult problems with strongly discontinuous coefficients (De Roeck and Le Tallec, 1991). However, this algorithm suffers from the need to solve in each iteration an inconsistent singular problem for every subdomain, and its convergence deteriorates with increasing number of subdomains due to the lack of a coarse problem to propagate the error globally. We show that the equilibrium conditions for the singular problems on subdomains lead to a simple and natural construction of a coarse problem. The construction is purely algebraic and applies also to systems such as those that arise in elasticity. A convergence bound independent of the number of subdomains is proved and results of computational tests are reported.
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    How to deal with hypersingular integrals in the symmetric BEM (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 219-232 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The symmetric boundary-element method for mixed boundary-value problems is currently gaining increasing interest as it permits the derivation of symmetric matrices. It is even possible to obtain positive-definite finite-element stiffness matrices for BEM subregions, and therefore the method provides a very straightforward coupling to the finite-element method (FEM). The BEM subregions can be handled like FEM substructures in a domain decomposition context. The symmetric BEM uses the full set of integral representations for the 2n types of boundary data for self-adjoint problems of order 2n and solves them in a weighted residual (Galerkin) sense. The main difficulty of the method is caused by the need to compute hypersingular integrals arising from the integral representations of the Neumann boundary data. It is the object of the paper to show that, for 2-D problems on polygonal domains, these hypersingular integrands pose no problem at all, even for higher-order approximations. The present approach makes use of analytical integrations as far as possible. The integrals of the hypersingular expressions are derived completely within the framework of simple engineering mathematics. The problem is reduced to the computation of weakly singular integrals even for C0 continuous approximations.
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    Self-adaptive finite-element simulation of bipolar, strongly reverse-biased pn-junctions (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 243-250 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Notes: The potential distribution of bipolar, strongly reverse-biased pn-junctions can be described by a double obstacle problem for the Laplacian. The problem is solved by a self-adaptive finite-element method involving automatic termination criteria for the iterative solver, local error estimation and local mesh refinement. Special attention is paid to the efficient resolution of the geometries typically arising in semiconductor device simulation. The algorithm is applied to a reverse-biased pn-junction with multistep field plate and stop electrode to illustrate its efficiency and reliability.
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    Superconvergent patch recovery techniques - some further tests (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 251-258 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A modification has been introduced to the new superconvergent patch derivative recovery technique recently developed (Zienkiewicz and Zhu, 1991). By use of a locally normalized co-ordinate system, the application of this technique can now be applied to higher-order finite-element shape functions. Numerical studies are carried out for up to 6th-order elements in one dimension and for up to quartic elements in two dimensions, and superconvergent results are obtained for all examples tested.
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    Conference diary (1993)
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    Communications in Numerical Methods in Engineering 9 (1993), S. 269-271 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Masthead (1993)
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    Communications in Numerical Methods in Engineering 9 (1993) 
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    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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    Bandwidth reduction of rectangular matrices (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 259-267 
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    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Two graph-theoretical algorithms are developed for bandwidth reduction of rectangular matrices. A K-total graph is defined for a selected cycle basis, generalized cycle basis and cut set basis of a graph. This transforms a simultaneous ordering of the elements of a basis and members of a graph into that of a nodal numbering of its K-total graph.The first algorithm uses the K-total graph and results in banded equilibrium and self-stress matrices. The second employs a nodal ordering algorithm followed by a member ordering process. This algorithm can be used for optimizing the bandwidth of member-cocycle incidence matrices, resulting in banded equilibrium matrices.
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    A simple local error estimator and an adaptive time-stepping procedure for direct integration method in dynamic analysis (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 273-292 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A simple a posteriori local error estimator for time discretization in structural dynamic analysis is presented. It is derived from the difference of the solutions between an ordinary integration method (the Newmark scheme) and another higher-order one which assumes that the derivatives of accelerations vary linearly within each time step. It may be obtained directly without resolving new equations, so the additional computational cost is small and the implementation is convenient. Furthermore, it is shown that this error estimator may also be obtained by Taylor expansion or by a post-processing technique. Accordingly, an adaptive time-stepping procedure, which automatically adjusts the time-step size so that the local error at each time step is within a prescribed accuracy, is described. Numerical examples, including two single-DOF problems, a two-DOF problem and a multi-DOF model, are presented. The results show that the presented local error estimator is simple, reliable and accurate.
    Additional Material: 9 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090402
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    Numerical solution of regularized long-wave equation (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 579-586 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A combination of the splitting method and the cubic spline technique has been used to solve a non-linear regularized long wave (RLW) equation. The accuracy and the stability of the proposed method are discussed. Then, two numerical examples are solved to illustrate the robustness of the algorithm. The results are in good agreement with the available results.
    Additional Material: 6 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090705
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    Iterative solution of the stream function-vorticity equations using a multigrid solver with finite elements (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 587-594 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The solution of the stream function-vorticity equations is developed using a multigrid method to improve efficiency in a finite-element p-method. The equations are discretized using 2-D Lagrange finite elements (linear, quadratic and cubic) and solved in iteratively decoupled form using successive approximation and continuation methods. The comparative performance of several multigrid and direct matrix solvers is investigated with respect to convergence characteristics and time performance on the Cray Y-MP.
    Additional Material: 5 Ill.
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    URL: http://dx.doi.org/10.1002/cnm.1640090706
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  • 89
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    Properties of generalized conjugate gradient methods (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 45-63 
    Details
    ISSN: 1070-5325
    Keywords: Conjugate gradients ; Convergence ; Linear systems ; Acceleration of conjugate gradients ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The solution of linear systems has considerable importance for the computation of problems resulting from engineering, physics, chemistry, computer science, mathematics, medicine and economics. The calculation of costly and time-consuming problems, e.g. crash tests, simulation of the human lung and skin, calculation of electrical and magnetical fields, thermal analysis and fluid dynamics, to name only a few, has become possible with the recent developments of advanced computer architectures and iterative solvers. Generalized conjugate gradient (CG) methods are the most important iterative solvers because they converge very quickly under certain conditions. Therefore they are widely used and in a rapid further development.The purpose of this paper is to present new results for the convergence of generalized CG methods. A convergence result for non-symmetric and non-positive definite matrices is given that includes the classical theory for symmetric, positive definite matrices as a special case.The norm of the residuals resulting from CG methods may oscillate heavily. Different remedies for smoothing this sequence have been proposed, for example by van der Vorst. Schönauer introduced in the 1980s a smoothing algorithm to get a norm nonincreasing function of the iteration index. For this algorithm a complete theoretical analysis is given. A surprising result is obtained showing that the smoothing algorithm is in a sense optimal. Convergence estimates are derived therefrom. A geometric interpretation of the smoothing algorithm is given showing the propagation of the errors.It should be stressed that a smooth convergence of the residuals is not equivalent to a smooth convergence of the errors which is the proper aim. A class of error minimizing methods can be easily derived from the theory.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010106
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  • 90
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    Displacement decomposition - incomplete factorization preconditioning techniques for linear elasticity problems (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 107-128 
    Details
    ISSN: 1070-5325
    Keywords: Incomplete factorization ; Preconditioning ; Linear elasticity problems ; Displacement decomposition ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Two preconditioning techniques for the numerical solution of linear elasticity problems are described and studied. Both techniques are based on spectral equivalence approach. The first technique consists in an incomplete factorization of the separate displacement component part of the stiffness matrix. The second technique uses an incomplete factorization of the isotropic approximation to the stiffness matrix. Results concerning existence, stability and efficiency of these preconditioning techniques are presented. The efficiency and robustness of the described techniques are illustrated by numerical experiments.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/nla.1680010203
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    The concept of special inner products for deriving new conjugate gradient-like solvers for non-symmetric sparse linear systems (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 129-139 
    Details
    ISSN: 1070-5325
    Keywords: Conjugate gradient method ; Nonsymmetric linear system ; Krylov subspace method ; Conjugate residual method ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Yserentant's and Bramble/Pasciak/Xu's hierarchical preconditioners only require the hierarchical node connection of a finite element grid. So they could be applied to nonsymmetric FE-systems too, having a symmetric preconditioner. We investigate the question whether this fact could be useful in the CG-like iterative methods. The main key is the consideration of an appropriate choice of the inner product defined in the N-vector space. It is shown, how the inner product influences the formulas of the methods, the rate of convergence and some other properties.
    Additional Material: 7 Tab.
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    URL: http://dx.doi.org/10.1002/nla.1680010204
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    Numerical calculation of invariant manifolds for maps (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 141-150 
    Details
    ISSN: 1070-5325
    Keywords: Invariant manifolds ; Error estimates ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Many processes in the sciences and in engineering are modelled by dynamical systems and - in discretized version - by nonlinear maps. To understand the often complicated dynamical behaviour it is a well established tool to use the concept of invariant manifolds of the system. In this way it is often possible to reduce the dimension of the system considerably. In this paper we propose a new method to calculate numerically invariant manifolds near fixed points of maps. We prove convergence of our procedure and provide an error estimation. Finally, the application of the method is illustrated by examples.
    Additional Material: 1 Ill.
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    URL: http://dx.doi.org/10.1002/nla.1680010205
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    Editorial (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 335-335 
    Details
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010402
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    GMRESR: a family of nested GMRES methods (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 369-386 
    Details
    ISSN: 1070-5325
    Keywords: GMRES ; Unsymmetric linear systems ; Iterative solver ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Recently Eirola and Nevanlinna have proposed an iterative solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest variants of GMRES, in which a preconditioner is constructed per iteration step by a suitable approximation process, e.g., by GMRES itself. Our numerical experiments indicate that this may lead to considerable savings in CPU-time and memory requirements in typical CFD applications.
    Additional Material: 10 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010404
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    Local refinement techniques for elliptic problems on cell-centered grids; II. Optimal order two-grid iterative methods (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 337-368 
    Details
    ISSN: 1070-5325
    Keywords: Local refinement ; Preconditioning ; Two-grid method ; Cell-centered grids ; Finite differences ; Elliptic problem ; Optimal rate of convergence ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Two preconditioning techniques for solving difference equations arising in finite difference approximation of elliptic problems on cell-centered grids are studied. It is proven that the BEPS and the FAC preconditioners are spectrally equivalent to the corresponding finite difference schemes, including a nonsymmetric one, which is of higher-order accuracy. Numerical experiments that demonstrate the fast convergence of the preconditioned iterative methods (CG and GCG-LS in the nonsymmetric case) are presented.
    Additional Material: 3 Ill.
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    URL: http://dx.doi.org/10.1002/nla.1680010403
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    Conference diary (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 421-425 
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    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010407
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    On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 403-420 
    Details
    ISSN: 1070-5325
    Keywords: Quasi-minimal residual iteration ; Non-Hermitian matrix ; Singular linear system ; Markov chain modeling ; Krylov-subspace method ; Convergence ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Recently, Freund and Nachtigal proposed the quasi-minimal residual algorithm (QMR) for solving general nonsingular non-Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix - vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose-free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov-subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.
    Additional Material: 2 Ill.
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    URL: http://dx.doi.org/10.1002/nla.1680010406
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    Masthead (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994) 
    Details
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010501
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  • 99
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    ILUT: A dual threshold incomplete LU factorization (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 387-402 
    Details
    ISSN: 1070-5325
    Keywords: Preconditioning ; Incomplete LU ; Threshold strategies ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we describe an Incomplete LU factorization technique based on a strategy which combines two heuristics. This ILUT factorization extends the usual ILU(O) factorization without using the concept of level of fill-in. There are two traditional ways of developing incomplete factorization preconditioners. The first uses a symbolic factorization approach in which a level of fill is attributed to each fill-in element using only the graph of the matrix. Then each fill-in that is introduced is dropped whenever its level of fill exceeds a certain threshold. The second class of methods consists of techniques derived from modifications of a given direct solver by including a dropoff rule, based on the numerical size of the fill-ins introduced, traditionally referred to as threshold preconditioners. The first type of approach may not be reliable for indefinite problems, since it does not consider numerical values. The second is often far more expensive than the standard ILU(O). The strategy we propose is a compromise between these two extremes.
    Additional Material: 5 Ill.
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    URL: http://dx.doi.org/10.1002/nla.1680010405
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    On the convergence behavior of the restarted GMRES algorithm for solving nonsymmetric linear systems (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 427-447 
    Details
    ISSN: 1070-5325
    Keywords: Nonsymmetric linear systems ; Conjugate gradient type iterative methods ; GMRES ; Restarting ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The solution of nonsymmetric systems of linear equations continues to be a difficult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s〉0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choice of this parameter can lead to reduced solution time, while an improper choice may hinder or preclude convergence. An adaptive procedure is also presented for determining automatically when to restart. The results of numerical experiments are presented.
    Additional Material: 4 Tab.
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    URL: http://dx.doi.org/10.1002/nla.1680010502
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