Solution of Vandermonde systems of equations

Björck, Åke; Pereyra, Víctor

doi:10.1090/s0025-5718-1970-0290541-1

Cited by 157 publications

(98 citation statements)

References 9 publications

(17 reference statements)

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“…However, accurate solutions to Vandermonde systems are still possible. Björck and Pereyra present some examples that can be solved with good accuracy by their algorithm [17]. Higham gives an error analysis of the Björck-Pereyra algorithm and shows why accurate solutions are possible [18].…”

Section: Vandermonde Matrices Consider the Vandermonde Matrixmentioning

confidence: 99%

“…Consider the 2D discrete Poisson matrices (17). Similarly, we look at the condition of matrix A which is the last Schur complement in nested dissection (Table VI).…”

Section: Example 10mentioning

confidence: 99%

“…We further consider the 2D problem (17). One step of the modified V-cycle (19) with zero initial vectors is used in the SCE method when each linear system is solved.…”

mentioning

confidence: 99%

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Applications of statistical condition estimation to the solution of linear systems

Laub¹,

Xia²

2008

Numerical Linear Algebra App

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SUMMARYThis paper discusses some applications of statistical condition estimation (SCE) to the problem of solving linear systems. Specifically, triangular and bidiagonal matrices are studied in some detail as typical of structured matrices. Such a structure, when properly respected, leads to condition estimates that are much less conservative compared with traditional non-statistical methods of condition estimation. Some examples of linear systems and Sylvester equations are presented. Vandermonde and Cauchy matrices are also studied as representative of linear systems with large condition numbers that can nonetheless be solved accurately. SCE reflects this. Moreover, SCE when applied to solving very large linear systems by iterative solvers, including conjugate gradient and multigrid methods, performs equally well and various examples are given to illustrate the performance. SCE for solving large linear systems with direct methods, such as methods for semi-separable structures, are also investigated. In all cases, the advantages of using SCE are manifold: ease of use, efficiency, and reliability.

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Section: Vandermonde Matrices Consider the Vandermonde Matrixmentioning

confidence: 99%

“…Consider the 2D discrete Poisson matrices (17). Similarly, we look at the condition of matrix A which is the last Schur complement in nested dissection (Table VI).…”

Section: Example 10mentioning

confidence: 99%

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Applications of statistical condition estimation to the solution of linear systems

Laub¹,

Xia²

2008

Numerical Linear Algebra App

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“…This type of system is of the well-known Vandermonde variety for the solution of which very efficient and robust algorithms exist [10]. It is worthwhile to note that the solution of this system of equations, consisting of a set of ðK þ L þ 1Þ numbers a jþm actually represents a whole row (the jth row) of the matrix representation of # P x M : This is easy to see from the following matrix element:…”

Section: Discretization Techniques For the Efficient Solutionmentioning

confidence: 99%

Discretization techniques for the efficient solution of the eigenvalue problem in heterostructures

Kougianos

Mohanty

2008

Int J Numerical Modelling

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SUMMARYA systematic development of efficient discretization schemes for the numerical evaluation of the eigenvalues of the single-band effective mass equation that describes the motion of electrons in an ideal periodic crystal is presented. The approach presented makes use of the translational symmetry of the crystal lattice and utilizes the quantum mechanical properties of the momentum operator as the generator of spatial translation. Boundary conditions satisfied at the heterointerfaces are explicitly incorporated in the discretization procedure and the effects of this approach in overall accuracy are evaluated by studying a prototype quantum well heterostructure.

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“…These approximations are particularly useful in applications of deferred corrections, and this was part of the motivation for the present work [5]. Fast weight generators have also.been used for this purpose [1], [2] but the current tables are more efficient. In addition, our tables are motivated by new schemes for solving higher order O.D.E.…”

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confidence: 99%

Symbolic Generation of Finite Difference Formulas

Keller¹,

Pereyra²

1978

Mathematics of Computation

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Abstract. Tables of coefficients for high order accurate, compact approximations to the first ten derivatives on and at the midpoints of uniform nets are presented.The exactrational weights are generated and tested by means of symbolic manipulation implemented through MACSYMA. These weights are required in the application of deferred corrections to new methods for solving higher order two point boundary value problems.1. Introduction. Compact difference schemes are, by definition, those which use the least number of net points to obtain consistent approximations (i.e. at least first order accurate). Extending this definition, higher order compact schemes are those which use the least number of net points to obtain higher order accurate approximations. In this paper we derive and present tables of the coefficients for higher order compact approximations, from accuracy h2 to h10, to the first 10 derivatives of smooth functions on uniform nets. These approximations are particularly useful in

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Solution of Vandermonde systems of equations

Cited by 157 publications

References 9 publications

Applications of statistical condition estimation to the solution of linear systems

Applications of statistical condition estimation to the solution of linear systems

Discretization techniques for the efficient solution of the eigenvalue problem in heterostructures

Symbolic Generation of Finite Difference Formulas

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