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The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate

1973-04-01
Gene H. Golub , Víctor Pereyra
For given data $(t_i ,y_i ),i = 1, \cdots ,m$, we consider the least squares fit of nonlinear models of the form \[ \eta ({\bf a},{\boldsymbol \alpha} ;t) = \sum … For given data $(t_i ,y_i ),i = 1, \cdots ,m$, we consider the least squares fit of nonlinear models of the form \[ \eta ({\bf a},{\boldsymbol \alpha} ;t) = \sum _{j = 1}^n {a_j \varphi _j ({\boldsymbol \alpha} ;t),\qquad {\bf a} \in \mathcal{R}^n ,\qquad {\boldsymbol \alpha} \in \mathcal{R}^k .} \] For this purpose we study the minimization of the nonlinear functional \[ r({\bf a},{\boldsymbol \alpha} ) = \sum\limits_{i = 1}^m {\left( {y_i - \eta \left( {{\bf a},{\boldsymbol \alpha} ,t_i } \right)} \right)^2 } . \] It is shown that by defining the matrix $\{ {\bf \Phi} ({\boldsymbol \alpha} )\} _{i,j} = \varphi _j ({\boldsymbol \alpha} ;t_i )$, and the modified functional $r_2 ({\boldsymbol \alpha} ) = \| {\bf y} - {\bf \Phi} ({\boldsymbol \alpha} ){\bf \Phi} ^ + ({\boldsymbol \alpha} ){\bf y} \|_2^2 $, it is possible to optimize first with respect to the parameters ${\boldsymbol \alpha} $, and then to obtain, a posteriors, the optimal parameters $\bf {\hat a}$. The matrix ${\bf \Phi} ^ + ({\boldsymbol{\alpha}} )$ is the Moore–Penrose generalized inverse of ${\bf \Phi} ({\boldsymbol{\alpha}} )$. We develop formulas for the Frechet derivative of orthogonal projectors associated with ${\bf \Phi} ({\boldsymbol{\alpha}} )$ and also for ${\bf \Phi} ^ + ({\boldsymbol{\alpha}} )$, under the hypothesis that ${\bf \Phi} ({\boldsymbol{\alpha}} )$ is of constant (though not necessarily full) rank. Detailed algorithms are presented which make extensive use of well-known reliable linear least squares techniques, and numerical results and comparisons are given. These results are generalizations of those of H. D. Scolnik [20] and Guttman, Pereyra and Scolnik [9].

Solution of Vandermonde systems of equations

1970-01-01
Åke Björck , Víctor Pereyra
We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. … We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. This is achieved by observing that a part of the earlier algorithm is equivalent to Newton’s interpolation method. This allows also to produce a progressive algorithm which is significantly more efficient than previous available methods. Algol-60 programs and numerical results are included. Confluent Vandermonde systems are also briefly discussed.
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Solution of Vandermonde Systems of Equations

1970-10-01
Åke Björck , Víctor Pereyra

An Adaptive Finite Difference Solver for Nonlinear Two-Point Boundary Problems with Mild Boundary Layers

1977-03-01
M. Lentini , Víctor Pereyra
A variable order finite difference solver for first order nonlinear systems subject to two-point boundary conditions is described. The method uses deferred corrections and adaptive meshes are automatically produced in … A variable order finite difference solver for first order nonlinear systems subject to two-point boundary conditions is described. The method uses deferred corrections and adaptive meshes are automatically produced in order to detect and resolve mild boundary layers and other sharp gradient situations. A set of numerical examples solved with an implementation of the algorithm is presented, together with comparisons with several other codes.
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Mesh selection for discrete solution of boundary problems in ordinary differential equations

1974-06-01
Víctor Pereyra , E.G. Sewell

Pasva3: An adaptive finite difference fortran program for first order nonlinear, ordinary boundary problems

1979-01-01
Víctor Pereyra

On improving an approximate solution of a functional equation by deferred corrections

1966-06-01
Víctor Pereyra

Iterated deferred corrections for nonlinear operator equations

1967-11-01
Víctor Pereyra

A variable order finite difference method for nonlinear multipoint boundary value problems

1974-01-01
M. Lentini , Víctor Pereyra
An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied … An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied earlier by H. B. Keller. Variable order is provided through deferred corrections, while a built-in natural asymptotic estimator is used to automatically refine the mesh in order to achieve a required tolerance. Extensive numerical experimentation and a FORTRAN program are included.
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Accelerating the Convergence of Discretization Algorithms

1967-12-01
Víctor Pereyra
Acceleration techniques f o r discretization algorithms u s e d in t h e approximate solution of nonlinear operator equations are considered.Practical problems arising in the solution of large … Acceleration techniques f o r discretization algorithms u s e d in t h e approximate solution of nonlinear operator equations are considered.Practical problems arising in the solution of large systems of nonlinear algebraic equations are discussed.These techniques are applied t o t h e approximate solution of mildly nonlinear elliptic equations by finite differences, and s e v e r a l numerical examples are given.
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Iterative Methods for Solving Nonlinear Least Squares Problems

1967-03-01
Víctor Pereyra
In many technical applications it is desired to f i t a nonlinear model t o a s e t of observations.t o determine a b e s t s … In many technical applications it is desired to f i t a nonlinear model t o a s e t of observations.t o determine a b e s t s e t of parameters in the l e a s t squares s e n s e . Several iterative techniques have been devised i n orderIn t h i s paper w e d i s c u s s conditions for convergence, and give error e s t i m a t e s for a c l a s s of methods, which includes as particular c a s e s some well known techniques.modified Newton's iterations for a suitable functional equation, and then a general theorem, first indicated by Bartle, is proved and applied to this particular case.The hypotheses are s e t in such a w a y that their checking by a n automatic computer i s made possible.It is shown t h a t those methods c a n be considered a s Some numerical examples are given.The main aim is to show that t h e automatic error estimation procedure works, rather than attempting t o optimize the computational s c h e m e .
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Efficient computer manipulation of tensor products with applications to multidimensional approximation

1973-01-01
Víctor Pereyra , G. Scherer
The objective of this paper is twofold: (a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n dot … The objective of this paper is twofold: (a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n dot p squared"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">n \cdot {p^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> words of information for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="circled-times upper A Subscript i Baseline comma upper A Subscript i Baseline element-of script upper L left-parenthesis upper E Superscript p Baseline comma upper E Superscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\otimes _{i = 1}^n{A_i},{A_i} \in \mathcal {L}({E^p},{E^p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) instead of the tensor-product operators themselves (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p squared right-parenthesis Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{({p^2})^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> words of information). (b) To produce efficient algorithms for solving systems of linear equations with coefficient matrices being tensor products of nonsingular matrices, with special application to the approximation of multidimensional linear functionals.
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Symbolic generation of finite difference formulas

1978-01-01
Herbert B. Keller , Víctor Pereyra
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 exact rational weights are generated … 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 exact rational 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.
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A Method for Separable Nonlinear Least Squares Problems with Separable Nonlinear Equality Constraints

1978-02-01
Linda Kaufman , Víctor Pereyra
Recently several algorithms have been proposed for solving separable nonlinear least squares problems which use the explicit coupling between the linear and nonlinear variables to define a new nonlinear least … Recently several algorithms have been proposed for solving separable nonlinear least squares problems which use the explicit coupling between the linear and nonlinear variables to define a new nonlinear least squares problem in the nonlinear variables only whose solution is the solution to the original problem. In this paper we extend these techniques to the separable nonlinear least squares problem subject to separable nonlinear equality constraints.
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High order fast Laplace solvers for the Dirichlet problem on general regions

1976-05-01
Víctor Pereyra , Wlodek Proskurowski , Olof B. Widlund
Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in R/sup n/. A second order accurate scheme is combined with … Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in R/sup n/. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least O(h/sup 5/./sup 5/) in L/sub 2/. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.
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High order fast Laplace solvers for the Dirichlet problem on general regions

1977-01-01
Víctor Pereyra , Włodzimierz Proskurowski , Olof B. Widlund
Highly accurate finite difference schemes are developed for Laplace’s equation with the Dirichlet boundary condition on general bounded regions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow … Highly accurate finite difference schemes are developed for Laplace’s equation with the Dirichlet boundary condition on general bounded regions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size <italic>h</italic>, one of the methods has an accuracy of at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis h Superscript 5.5 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>5.5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O({h^{5.5}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.
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Least Squares Estimation for a Class of Non-Linear Models

1973-05-01
Irwin Guttman , Víctor Pereyra , Hugo D. Scolnik
A new method for determining least squares estimators for certain classes of non- linear models is discussed. The method is an extension of a variable projection method of Scolnik (1970), … A new method for determining least squares estimators for certain classes of non- linear models is discussed. The method is an extension of a variable projection method of Scolnik (1970), and involves the minimization of a modified functional. The feature of minimizing this modified functional is that for a certain class of non-linear models, called the constant-coefficients case, only one half the parameters are involved initially. To find the estimators of the remaining parameters is straight forward and relatively easy. This new two step-procedure is shown to be equivalent to the over-all least squares procedure. We also discuss the case of a class of models called the variable coefficients class. For this case, we formulate a new algorithm for determining the estimators which makes use of approximate confidence regions for the parameters.

Ray tracing methods for inverse problems

2000-11-21
Víctor Pereyra
We discuss the origin, use and implementation of ray tracing methods for nonlinear inverse modelling problems associated with wave propagation phenomena. These methods have a long tradition in acoustic and … We discuss the origin, use and implementation of ray tracing methods for nonlinear inverse modelling problems associated with wave propagation phenomena. These methods have a long tradition in acoustic and elastodynamic wave propagation problems for various important applications, and they can surely be helpful in other realms, including electromagnetic wave propagation and diffusion dominated phenomena.

Highly accurate numerical solution of casilinear elliptic boundary-value problems in 𝑛 dimensions

1970-01-01
Víctor Pereyra
The method of Iterated Deferred Corrections, whose theory was developed by the author, is applied to the problems of the title. The necessary asymptotic expansions are obtained and the way … The method of Iterated Deferred Corrections, whose theory was developed by the author, is applied to the problems of the title. The necessary asymptotic expansions are obtained and the way in which the corrections are produced by means of numerical differentiation is described in detail. Numerical results and comparisons with the variationalsplines methods are given.
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Excited-state effective masses in lattice QCD

2009-10-13
George Fleming , Saul D. Cohen , Huey-Wen Lin +1 more
We apply black-box methods, i.e. where the performance of the method does not depend upon initial guesses, to extract excited-state energies from Euclidean-time hadron correlation functions. In particular, we extend … We apply black-box methods, i.e. where the performance of the method does not depend upon initial guesses, to extract excited-state energies from Euclidean-time hadron correlation functions. In particular, we extend the widely used effective-mass method to incorporate multiple correlation functions and produce effective-mass estimates for multiple excited states. In general, these excited-state effective masses will be determined by finding the roots of some polynomial. We demonstrate the method using sample lattice data to determine excited-state energies of the nucleon and compare the results to other energy-level finding techniques.
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BOUNDARY PROBLEM SOLVERS FOR FIRST ORDER SYSTEMS BASED ON DEFERRED CORRECTIONS

1975-01-01
M. Lentini , Víctor Pereyra

Stability of general systems of linear equations

1969-06-01
Víctor Pereyra

Large scale least squares scattered data fitting

2002-12-27
Víctor Pereyra , G. Scherer

Numerical differentiation and the solution of multidimensional Vandermonde systems

1970-01-01
Giuliano Galimberti , Víctor Pereyra
We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices. These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit … We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices. These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit algorithm is produced to solve systems of linear equations with MV matrices of coefficients. This is an extension of work of Stenger for the two-dimensional case. Numerical results for three-dimensional numerical differentiation are given.
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Computational methods for inverse problems in geophysics: inversion of travel time observations

1980-02-01
Víctor Pereyra , Herbert B. Keller , W.H.K. Lee

Numerical methods for inverse problems in three-dimensional geophysical modeling

1988-03-01
Víctor Pereyra

Asynchronous distributed solution of large scale nonlinear inversion problems

1999-05-01
Víctor Pereyra

Variable order variable step finite difference methods for nonlinear boundary value problems

1974-01-01
Víctor Pereyra

Adaptation of a Two-Point Boundary Value Problem Solver to a Vector-Multiprocessor Environment

1990-05-01
Stephen J. Wright , Víctor Pereyra
Systems of linear equations arising from finite-difference discretization of two-point boundary value problems have coefficient matrices that are sparse, with most or all of the nonzeros clustered in blocks near … Systems of linear equations arising from finite-difference discretization of two-point boundary value problems have coefficient matrices that are sparse, with most or all of the nonzeros clustered in blocks near the main diagonal. Some efficiently vectorizable algorithms for factorizing these types of matrices and solving the corresponding linear systems are described. The relative effectiveness of the different algorithms varies according to the distribution of initial, final, and coupled end conditions. The techniques described can be extended to handle linear systems arising from other methods for two-point boundary value problems, such as multiple shooting and collocation. An application to seismic ray tracing is discussed.

High Order Fast Laplace Solvers for the Dirichlet Problem on General Regions

1977-01-01
Víctor Pereyra , Włodzimierz Proskurowski , Olof B. Widlund
Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in Rn.A second order accurate scheme is combined with a deferred … Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in Rn.A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy.The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss.A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least 0(h ' ) in ¿j.The linear systems of algebraic equations are solved by a capacitance matrix method.The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods. Introduction.It is the purpose of this paper to develop some highly accurate finite difference methods for the Dirichlet problem for a general bounded region Í2 in Rn.The most accurate of these has an L. error of order at most h5S, see Section 4. Our basic schemes use the standard (2n + l)-point formula for the interior mesh points and are therefore only second order accurate.The increased accuracy is achieved by two steps of a deferred correction or Richardson extrapolation procedure.We also discuss the computer implementation of these methods in some detail.The use of deferred correction and Richardson extrapolation methods is justified by finding asymptotic expansions of the error.Wasow [20] has shown that no useful expansions of this kind exist if the boundary condition is approximated to a low order of accuracy.An obvious remedy for this problem, already mentioned by Wasow, is to use higher order interpolation or extrapolation formulas at any irregular mesh point, i.e. a mesh point in the open set £2 which fails to have all its next neighbors in the closure of Í2.Volkov [19] proposed the use of high order one-dimensional Lagrange polynomials for this purpose.Because of the change of sign of the interpolation coefficients the matrix representing the difference scheme will then, in general, not be of positive type.The standard convergence proof based on a discrete maximum principle, (see Forsythe and Wasow [71) will therefore generally not apply.But by allowing the use of values of the mesh functions many mesh lengths away from the boundary, Volkov succeeded in designing schemes with diagonally dominant matrices.His
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Deferred Corrections Software and Its Application to Seismic Ray Tracing

1984-01-01
Víctor Pereyra

Highly Accurate Numerical Solution of Casilinear Elliptic Boundary-Value Problems in n Dimensions

1970-10-01
Víctor Pereyra

Least Squares Estimation for a Class of Non-Linear Models

1973-05-01
Irwin Guttman , Víctor Pereyra , Hugo D. Scolnik
A new method for determining least squares estimators for certain classes of non- linear models is discussed. The method is an extension of a variable projection method of Scolnik (1970), … A new method for determining least squares estimators for certain classes of non- linear models is discussed. The method is an extension of a variable projection method of Scolnik (1970), and involves the minimization of a modified functional. The feature of minimizing this modified functional is that for a certain class of non-linear models, called the constant-coefficients case, only one half the parameters are involved initially. To find the estimators of the remaining parameters is straight forward and relatively easy. This new two step-procedure is shown to be equivalent to the over-all least squares procedure. We also discuss the case of a class of models called the variable coefficients class. For this case, we formulate a new algorithm for determining the estimators which makes use of approximate confidence regions for the parameters.

Wave Equation Simulation Using a Compressed Modeler

2013-01-01
Víctor Pereyra
Repeated simulations of large scale wave propagation problems are prevalent in many fields. In oil exploration earth imaging problems, the use of full wave simulations is becoming routine and it … Repeated simulations of large scale wave propagation problems are prevalent in many fields. In oil exploration earth imaging problems, the use of full wave simulations is becoming routine and it is only hampered by the extreme computational resources required. In this contribution, we explore the feasibility of employing reduced-order modeling techniques in an attempt to significantly decrease the cost of these calculations. We consider the acoustic wave equation in two-dimensions for simplicity, but the extension to three-dimensions and to elastic or even anysotropic problems is clear. We use the proper orthogonal decomposition approach to model order reduction and describe two algorithms: the traditional one using the SVD of the matrix of snapshots and a more economical and flexible one using a progressive QR decomposition. We include also two a posteriori error estimation procedures and extensive testing and validation is presented that indicates the promise of the approach.
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On the Construction of Discrete Approximations to Linear Differential Expressions

1967-07-01
C. Ballester , Víctor Pereyra
An algorithm for generating d i s c r e t e approximations i n terms of ordinates for linear differential expressions is d e scribed.A s a n application … An algorithm for generating d i s c r e t e approximations i n terms of ordinates for linear differential expressions is d e scribed.A s a n application a complement to a table of numerical differentiation by Bickley is presented.
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Least squares collocation solution of elliptic problems in general regions

2006-07-27
Víctor Pereyra , G. Scherer

Nonlinear two-point boundary value problems with multiple solutions

1971-03-01
Seymour V. Parter , Víctor Pereyra
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Stabilizing linear least squares problems.

1968-01-01
Víctor Pereyra

The correction difference method for non linear boundary value problems of class M

1965-01-01
Víctor Pereyra

Stability of general systems of linear equations

1969-06-01
Víctor Pereyra

Acoustic Ray Tracing In Complex 3D Media

1970-01-01
Víctor Pereyra
The numerical solution of ray tracing problems in complex 3D media is considered in this paper that reviews some of the work and applications in which the author and his … The numerical solution of ray tracing problems in complex 3D media is considered in this paper that reviews some of the work and applications in which the author and his collaborators have been engaged in the past few years. The paper is not a survey but rather tries to point out what are the main issues common to many applications and how we have attacked the problem and implemented computational tools with general applicability to such disparate fields as earthquake seismology, oil prospecting, and nondestructive evaluation of complex mechanical parts.

Finite difference solution of two-point boundary value problems and symbolic manipulation

1978-01-01
Víctor Pereyra

Model Order Reduction for Wave Propagation

2007-05-01
Víctor Pereyra

Wave Equation Simulation Using a Compressed Modeler

2013-01-01
Víctor Pereyra
Repeated simulations of large scale wave propagation problems are prevalent in many fields. In oil exploration earth imaging problems, the use of full wave simulations is becoming routine and it … Repeated simulations of large scale wave propagation problems are prevalent in many fields. In oil exploration earth imaging problems, the use of full wave simulations is becoming routine and it is only hampered by the extreme computational resources required. In this contribution, we explore the feasibility of employing reduced-order modeling techniques in an attempt to significantly decrease the cost of these calculations. We consider the acoustic wave equation in two-dimensions for simplicity, but the extension to three-dimensions and to elastic or even anysotropic problems is clear. We use the proper orthogonal decomposition approach to model order reduction and describe two algorithms: the traditional one using the SVD of the matrix of snapshots and a more economical and flexible one using a progressive QR decomposition. We include also two a posteriori error estimation procedures and extensive testing and validation is presented that indicates the promise of the approach.
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Excited-state effective masses in lattice QCD

2009-10-13
George Fleming , Saul D. Cohen , Huey-Wen Lin +1 more
We apply black-box methods, i.e. where the performance of the method does not depend upon initial guesses, to extract excited-state energies from Euclidean-time hadron correlation functions. In particular, we extend … We apply black-box methods, i.e. where the performance of the method does not depend upon initial guesses, to extract excited-state energies from Euclidean-time hadron correlation functions. In particular, we extend the widely used effective-mass method to incorporate multiple correlation functions and produce effective-mass estimates for multiple excited states. In general, these excited-state effective masses will be determined by finding the roots of some polynomial. We demonstrate the method using sample lattice data to determine excited-state energies of the nucleon and compare the results to other energy-level finding techniques.
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Model Order Reduction for Wave Propagation

2007-05-01
Víctor Pereyra

Least squares collocation solution of elliptic problems in general regions

2006-07-27
Víctor Pereyra , G. Scherer

Large scale least squares scattered data fitting

2002-12-27
Víctor Pereyra , G. Scherer

Ray tracing methods for inverse problems

2000-11-21
Víctor Pereyra
We discuss the origin, use and implementation of ray tracing methods for nonlinear inverse modelling problems associated with wave propagation phenomena. These methods have a long tradition in acoustic and … We discuss the origin, use and implementation of ray tracing methods for nonlinear inverse modelling problems associated with wave propagation phenomena. These methods have a long tradition in acoustic and elastodynamic wave propagation problems for various important applications, and they can surely be helpful in other realms, including electromagnetic wave propagation and diffusion dominated phenomena.

Asynchronous distributed solution of large scale nonlinear inversion problems

1999-05-01
Víctor Pereyra

Adaptation of a Two-Point Boundary Value Problem Solver to a Vector-Multiprocessor Environment

1990-05-01
Stephen J. Wright , Víctor Pereyra
Systems of linear equations arising from finite-difference discretization of two-point boundary value problems have coefficient matrices that are sparse, with most or all of the nonzeros clustered in blocks near … Systems of linear equations arising from finite-difference discretization of two-point boundary value problems have coefficient matrices that are sparse, with most or all of the nonzeros clustered in blocks near the main diagonal. Some efficiently vectorizable algorithms for factorizing these types of matrices and solving the corresponding linear systems are described. The relative effectiveness of the different algorithms varies according to the distribution of initial, final, and coupled end conditions. The techniques described can be extended to handle linear systems arising from other methods for two-point boundary value problems, such as multiple shooting and collocation. An application to seismic ray tracing is discussed.

Numerical methods for inverse problems in three-dimensional geophysical modeling

1988-03-01
Víctor Pereyra

Deferred Corrections Software and Its Application to Seismic Ray Tracing

1984-01-01
Víctor Pereyra

Computational methods for inverse problems in geophysics: inversion of travel time observations

1980-02-01
Víctor Pereyra , Herbert B. Keller , W.H.K. Lee

Pasva3: An adaptive finite difference fortran program for first order nonlinear, ordinary boundary problems

1979-01-01
Víctor Pereyra

A Method for Separable Nonlinear Least Squares Problems with Separable Nonlinear Equality Constraints

1978-02-01
Linda Kaufman , Víctor Pereyra
Recently several algorithms have been proposed for solving separable nonlinear least squares problems which use the explicit coupling between the linear and nonlinear variables to define a new nonlinear least … Recently several algorithms have been proposed for solving separable nonlinear least squares problems which use the explicit coupling between the linear and nonlinear variables to define a new nonlinear least squares problem in the nonlinear variables only whose solution is the solution to the original problem. In this paper we extend these techniques to the separable nonlinear least squares problem subject to separable nonlinear equality constraints.
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Symbolic generation of finite difference formulas

1978-01-01
Herbert B. Keller , Víctor Pereyra
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 exact rational weights are generated … 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 exact rational 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.
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Finite difference solution of two-point boundary value problems and symbolic manipulation

1978-01-01
Víctor Pereyra

An Adaptive Finite Difference Solver for Nonlinear Two-Point Boundary Problems with Mild Boundary Layers

1977-03-01
M. Lentini , Víctor Pereyra
A variable order finite difference solver for first order nonlinear systems subject to two-point boundary conditions is described. The method uses deferred corrections and adaptive meshes are automatically produced in … A variable order finite difference solver for first order nonlinear systems subject to two-point boundary conditions is described. The method uses deferred corrections and adaptive meshes are automatically produced in order to detect and resolve mild boundary layers and other sharp gradient situations. A set of numerical examples solved with an implementation of the algorithm is presented, together with comparisons with several other codes.
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High Order Fast Laplace Solvers for the Dirichlet Problem on General Regions

1977-01-01
Víctor Pereyra , Włodzimierz Proskurowski , Olof B. Widlund
Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in Rn.A second order accurate scheme is combined with a deferred … Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in Rn.A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy.The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss.A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least 0(h ' ) in ¿j.The linear systems of algebraic equations are solved by a capacitance matrix method.The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods. Introduction.It is the purpose of this paper to develop some highly accurate finite difference methods for the Dirichlet problem for a general bounded region Í2 in Rn.The most accurate of these has an L. error of order at most h5S, see Section 4. Our basic schemes use the standard (2n + l)-point formula for the interior mesh points and are therefore only second order accurate.The increased accuracy is achieved by two steps of a deferred correction or Richardson extrapolation procedure.We also discuss the computer implementation of these methods in some detail.The use of deferred correction and Richardson extrapolation methods is justified by finding asymptotic expansions of the error.Wasow [20] has shown that no useful expansions of this kind exist if the boundary condition is approximated to a low order of accuracy.An obvious remedy for this problem, already mentioned by Wasow, is to use higher order interpolation or extrapolation formulas at any irregular mesh point, i.e. a mesh point in the open set £2 which fails to have all its next neighbors in the closure of Í2.Volkov [19] proposed the use of high order one-dimensional Lagrange polynomials for this purpose.Because of the change of sign of the interpolation coefficients the matrix representing the difference scheme will then, in general, not be of positive type.The standard convergence proof based on a discrete maximum principle, (see Forsythe and Wasow [71) will therefore generally not apply.But by allowing the use of values of the mesh functions many mesh lengths away from the boundary, Volkov succeeded in designing schemes with diagonally dominant matrices.His
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High order fast Laplace solvers for the Dirichlet problem on general regions

1977-01-01
Víctor Pereyra , Włodzimierz Proskurowski , Olof B. Widlund
Highly accurate finite difference schemes are developed for Laplace’s equation with the Dirichlet boundary condition on general bounded regions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow … Highly accurate finite difference schemes are developed for Laplace’s equation with the Dirichlet boundary condition on general bounded regions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size <italic>h</italic>, one of the methods has an accuracy of at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis h Superscript 5.5 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>5.5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O({h^{5.5}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.
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High order fast Laplace solvers for the Dirichlet problem on general regions

1976-05-01
Víctor Pereyra , Wlodek Proskurowski , Olof B. Widlund
Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in R/sup n/. A second order accurate scheme is combined with … Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in R/sup n/. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least O(h/sup 5/./sup 5/) in L/sub 2/. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.
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BOUNDARY PROBLEM SOLVERS FOR FIRST ORDER SYSTEMS BASED ON DEFERRED CORRECTIONS

1975-01-01
M. Lentini , Víctor Pereyra

Mesh selection for discrete solution of boundary problems in ordinary differential equations

1974-06-01
Víctor Pereyra , E.G. Sewell

Variable order variable step finite difference methods for nonlinear boundary value problems

1974-01-01
Víctor Pereyra

A variable order finite difference method for nonlinear multipoint boundary value problems

1974-01-01
M. Lentini , Víctor Pereyra
An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied … An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied earlier by H. B. Keller. Variable order is provided through deferred corrections, while a built-in natural asymptotic estimator is used to automatically refine the mesh in order to achieve a required tolerance. Extensive numerical experimentation and a FORTRAN program are included.
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Least Squares Estimation for a Class of Non-Linear Models

1973-05-01
Irwin Guttman , Víctor Pereyra , Hugo D. Scolnik
A new method for determining least squares estimators for certain classes of non- linear models is discussed. The method is an extension of a variable projection method of Scolnik (1970), … A new method for determining least squares estimators for certain classes of non- linear models is discussed. The method is an extension of a variable projection method of Scolnik (1970), and involves the minimization of a modified functional. The feature of minimizing this modified functional is that for a certain class of non-linear models, called the constant-coefficients case, only one half the parameters are involved initially. To find the estimators of the remaining parameters is straight forward and relatively easy. This new two step-procedure is shown to be equivalent to the over-all least squares procedure. We also discuss the case of a class of models called the variable coefficients class. For this case, we formulate a new algorithm for determining the estimators which makes use of approximate confidence regions for the parameters.

Least Squares Estimation for a Class of Non-Linear Models

1973-05-01
Irwin Guttman , Víctor Pereyra , Hugo D. Scolnik
A new method for determining least squares estimators for certain classes of non- linear models is discussed. The method is an extension of a variable projection method of Scolnik (1970), … A new method for determining least squares estimators for certain classes of non- linear models is discussed. The method is an extension of a variable projection method of Scolnik (1970), and involves the minimization of a modified functional. The feature of minimizing this modified functional is that for a certain class of non-linear models, called the constant-coefficients case, only one half the parameters are involved initially. To find the estimators of the remaining parameters is straight forward and relatively easy. This new two step-procedure is shown to be equivalent to the over-all least squares procedure. We also discuss the case of a class of models called the variable coefficients class. For this case, we formulate a new algorithm for determining the estimators which makes use of approximate confidence regions for the parameters.

The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate

1973-04-01
Gene H. Golub , Víctor Pereyra
For given data $(t_i ,y_i ),i = 1, \cdots ,m$, we consider the least squares fit of nonlinear models of the form \[ \eta ({\bf a},{\boldsymbol \alpha} ;t) = \sum … For given data $(t_i ,y_i ),i = 1, \cdots ,m$, we consider the least squares fit of nonlinear models of the form \[ \eta ({\bf a},{\boldsymbol \alpha} ;t) = \sum _{j = 1}^n {a_j \varphi _j ({\boldsymbol \alpha} ;t),\qquad {\bf a} \in \mathcal{R}^n ,\qquad {\boldsymbol \alpha} \in \mathcal{R}^k .} \] For this purpose we study the minimization of the nonlinear functional \[ r({\bf a},{\boldsymbol \alpha} ) = \sum\limits_{i = 1}^m {\left( {y_i - \eta \left( {{\bf a},{\boldsymbol \alpha} ,t_i } \right)} \right)^2 } . \] It is shown that by defining the matrix $\{ {\bf \Phi} ({\boldsymbol \alpha} )\} _{i,j} = \varphi _j ({\boldsymbol \alpha} ;t_i )$, and the modified functional $r_2 ({\boldsymbol \alpha} ) = \| {\bf y} - {\bf \Phi} ({\boldsymbol \alpha} ){\bf \Phi} ^ + ({\boldsymbol \alpha} ){\bf y} \|_2^2 $, it is possible to optimize first with respect to the parameters ${\boldsymbol \alpha} $, and then to obtain, a posteriors, the optimal parameters $\bf {\hat a}$. The matrix ${\bf \Phi} ^ + ({\boldsymbol{\alpha}} )$ is the Moore–Penrose generalized inverse of ${\bf \Phi} ({\boldsymbol{\alpha}} )$. We develop formulas for the Frechet derivative of orthogonal projectors associated with ${\bf \Phi} ({\boldsymbol{\alpha}} )$ and also for ${\bf \Phi} ^ + ({\boldsymbol{\alpha}} )$, under the hypothesis that ${\bf \Phi} ({\boldsymbol{\alpha}} )$ is of constant (though not necessarily full) rank. Detailed algorithms are presented which make extensive use of well-known reliable linear least squares techniques, and numerical results and comparisons are given. These results are generalizations of those of H. D. Scolnik [20] and Guttman, Pereyra and Scolnik [9].

Efficient computer manipulation of tensor products with applications to multidimensional approximation

1973-01-01
Víctor Pereyra , G. Scherer
The objective of this paper is twofold: (a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n dot … The objective of this paper is twofold: (a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n dot p squared"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">n \cdot {p^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> words of information for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="circled-times upper A Subscript i Baseline comma upper A Subscript i Baseline element-of script upper L left-parenthesis upper E Superscript p Baseline comma upper E Superscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\otimes _{i = 1}^n{A_i},{A_i} \in \mathcal {L}({E^p},{E^p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) instead of the tensor-product operators themselves (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p squared right-parenthesis Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{({p^2})^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> words of information). (b) To produce efficient algorithms for solving systems of linear equations with coefficient matrices being tensor products of nonsingular matrices, with special application to the approximation of multidimensional linear functionals.
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Nonlinear two-point boundary value problems with multiple solutions

1971-03-01
Seymour V. Parter , Víctor Pereyra
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Solution of Vandermonde Systems of Equations

1970-10-01
Åke Björck , Víctor Pereyra

Highly Accurate Numerical Solution of Casilinear Elliptic Boundary-Value Problems in n Dimensions

1970-10-01
Víctor Pereyra

Solution of Vandermonde systems of equations

1970-01-01
Åke Björck , Víctor Pereyra
We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. … We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. This is achieved by observing that a part of the earlier algorithm is equivalent to Newton’s interpolation method. This allows also to produce a progressive algorithm which is significantly more efficient than previous available methods. Algol-60 programs and numerical results are included. Confluent Vandermonde systems are also briefly discussed.
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Numerical differentiation and the solution of multidimensional Vandermonde systems

1970-01-01
Giuliano Galimberti , Víctor Pereyra
We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices. These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit … We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices. These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit algorithm is produced to solve systems of linear equations with MV matrices of coefficients. This is an extension of work of Stenger for the two-dimensional case. Numerical results for three-dimensional numerical differentiation are given.
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Highly accurate numerical solution of casilinear elliptic boundary-value problems in 𝑛 dimensions

1970-01-01
Víctor Pereyra
The method of Iterated Deferred Corrections, whose theory was developed by the author, is applied to the problems of the title. The necessary asymptotic expansions are obtained and the way … The method of Iterated Deferred Corrections, whose theory was developed by the author, is applied to the problems of the title. The necessary asymptotic expansions are obtained and the way in which the corrections are produced by means of numerical differentiation is described in detail. Numerical results and comparisons with the variationalsplines methods are given.
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Acoustic Ray Tracing In Complex 3D Media

1970-01-01
Víctor Pereyra
The numerical solution of ray tracing problems in complex 3D media is considered in this paper that reviews some of the work and applications in which the author and his … The numerical solution of ray tracing problems in complex 3D media is considered in this paper that reviews some of the work and applications in which the author and his collaborators have been engaged in the past few years. The paper is not a survey but rather tries to point out what are the main issues common to many applications and how we have attacked the problem and implemented computational tools with general applicability to such disparate fields as earthquake seismology, oil prospecting, and nondestructive evaluation of complex mechanical parts.

Stability of general systems of linear equations

1969-06-01
Víctor Pereyra

Stability of general systems of linear equations

1969-06-01
Víctor Pereyra

Stabilizing linear least squares problems.

1968-01-01
Víctor Pereyra

Accelerating the Convergence of Discretization Algorithms

1967-12-01
Víctor Pereyra
Acceleration techniques f o r discretization algorithms u s e d in t h e approximate solution of nonlinear operator equations are considered.Practical problems arising in the solution of large … Acceleration techniques f o r discretization algorithms u s e d in t h e approximate solution of nonlinear operator equations are considered.Practical problems arising in the solution of large systems of nonlinear algebraic equations are discussed.These techniques are applied t o t h e approximate solution of mildly nonlinear elliptic equations by finite differences, and s e v e r a l numerical examples are given.
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Iterated deferred corrections for nonlinear operator equations

1967-11-01
Víctor Pereyra

On the Construction of Discrete Approximations to Linear Differential Expressions

1967-07-01
C. Ballester , Víctor Pereyra
An algorithm for generating d i s c r e t e approximations i n terms of ordinates for linear differential expressions is d e scribed.A s a n application … An algorithm for generating d i s c r e t e approximations i n terms of ordinates for linear differential expressions is d e scribed.A s a n application a complement to a table of numerical differentiation by Bickley is presented.
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Iterative Methods for Solving Nonlinear Least Squares Problems

1967-03-01
Víctor Pereyra
In many technical applications it is desired to f i t a nonlinear model t o a s e t of observations.t o determine a b e s t s … In many technical applications it is desired to f i t a nonlinear model t o a s e t of observations.t o determine a b e s t s e t of parameters in the l e a s t squares s e n s e . Several iterative techniques have been devised i n orderIn t h i s paper w e d i s c u s s conditions for convergence, and give error e s t i m a t e s for a c l a s s of methods, which includes as particular c a s e s some well known techniques.modified Newton's iterations for a suitable functional equation, and then a general theorem, first indicated by Bartle, is proved and applied to this particular case.The hypotheses are s e t in such a w a y that their checking by a n automatic computer i s made possible.It is shown t h a t those methods c a n be considered a s Some numerical examples are given.The main aim is to show that t h e automatic error estimation procedure works, rather than attempting t o optimize the computational s c h e m e .
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On improving an approximate solution of a functional equation by deferred corrections

1966-06-01
Víctor Pereyra

The correction difference method for non linear boundary value problems of class M

1965-01-01
Víctor Pereyra
Coauthor Papers Together
Olof B. Widlund 3
M. Lentini 3
G. Scherer 3
Włodzimierz Proskurowski 2
Herbert B. Keller 2
Hugo D. Scolnik 2
Irwin Guttman 2
Åke Björck 2
George Fleming 1
Giuliano Galimberti 1
Linda Kaufman 1
Wlodek Proskurowski 1
C. Ballester 1
Seymour V. Parter 1
Huey-Wen Lin 1
E.G. Sewell 1
Gene H. Golub 1
W.H.K. Lee 1
Stephen J. Wright 1
Saul D. Cohen 1

Iterated deferred corrections for nonlinear operator equations

1967-11-01
Víctor Pereyra

Iterated deferred corrections for nonlinear boundary value problems

1968-02-01
Victor Pereyna

Accurate Difference Methods for Nonlinear Two-Point Boundary Value Problems

1974-04-01
Herbert B. Keller
We show that each isolated solution, $y(t)$, of the general nonlinear two-point boundary value problem $( * ):y' = f(t,y),a < t < b,g(y(a),y(b)) = 0$ can be approximated by … We show that each isolated solution, $y(t)$, of the general nonlinear two-point boundary value problem $( * ):y' = f(t,y),a < t < b,g(y(a),y(b)) = 0$ can be approximated by the (box) difference scheme $( * * ):{{[u_j - u_{j - 1} ]} / {h_j }} = f(t_{{{j - 1} 2}} ,{{[u_j + u_{j - 1} ]} / 2}),\, 1 \leqq j \leqq J,\, g(u_0 ,u_J ) = 0$. For $h = \max _{1 \leqq j \leqq J} h_j $ sufficiently small, the difference equations (**) are shown to have a unique solution $\{ u_j \} _0^J $ in some sphere about $\{ y(t_j )\} _0^J $, and it can be computed by Newton's method which converges quadratically. If $y(t)$ is sufficiently smooth, then the error has an asymptotic expansion of the form $u_j - y(t_j ) = \sum _{v = 1}^m {h^{2v} e_v (t_j ) + O(h^{2m + 2} )} $, so that Richardson extrapolation is justified. The coefficient matrices of the linear systems to be solved in applying Newton's method are of order $n(J + 1)$ when $y(t) \in \mathbb{R}^n $. For separated endpoint boundary conditions: $g_1 (y(a)) = 0,\, g_2 (y(b)) = 0$ with $\dim g_1 = p,\dim g_2 = q$ and $p + q = n$, the coefficient matrices have the special block tridiagonal form $A \equiv [B_j ,A_j ,C_j ]$ in which the $n \times n$ matrices $B_j (C_j )$ have their last q (first p) rows null. Block elimination and band elimination without destroying the zero pattern are shown to be valid. The numerical scheme is very efficient, as a worked out example illustrates.
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Solution of Vandermonde systems of equations

1970-01-01
Åke Björck , Víctor Pereyra
We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. … We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. This is achieved by observing that a part of the earlier algorithm is equivalent to Newton’s interpolation method. This allows also to produce a progressive algorithm which is significantly more efficient than previous available methods. Algol-60 programs and numerical results are included. Confluent Vandermonde systems are also briefly discussed.
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An Adaptive Finite Difference Solver for Nonlinear Two-Point Boundary Problems with Mild Boundary Layers

1977-03-01
M. Lentini , Víctor Pereyra
A variable order finite difference solver for first order nonlinear systems subject to two-point boundary conditions is described. The method uses deferred corrections and adaptive meshes are automatically produced in … A variable order finite difference solver for first order nonlinear systems subject to two-point boundary conditions is described. The method uses deferred corrections and adaptive meshes are automatically produced in order to detect and resolve mild boundary layers and other sharp gradient situations. A set of numerical examples solved with an implementation of the algorithm is presented, together with comparisons with several other codes.
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Solution of Vandermonde Systems of Equations

1970-10-01
Åke Björck , Víctor Pereyra

Numerical Methods for Two-Point Boundary Value Problems

2019-08-29
Herbert B. Keller
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A variable order finite difference method for nonlinear multipoint boundary value problems

1974-01-01
M. Lentini , Víctor Pereyra
An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied … An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied earlier by H. B. Keller. Variable order is provided through deferred corrections, while a built-in natural asymptotic estimator is used to automatically refine the mesh in order to achieve a required tolerance. Extensive numerical experimentation and a FORTRAN program are included.
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Accelerating the Convergence of Discretization Algorithms

1967-12-01
Víctor Pereyra
Acceleration techniques f o r discretization algorithms u s e d in t h e approximate solution of nonlinear operator equations are considered.Practical problems arising in the solution of large … Acceleration techniques f o r discretization algorithms u s e d in t h e approximate solution of nonlinear operator equations are considered.Practical problems arising in the solution of large systems of nonlinear algebraic equations are discussed.These techniques are applied t o t h e approximate solution of mildly nonlinear elliptic equations by finite differences, and s e v e r a l numerical examples are given.
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NUMERICAL METHODS FOR TWO-POINT BOUNDARY-VALUE PROBLEMS

1969-11-01
L. Fox
By Herbert B. Keller: pp. viii, 184. (Blaisdell Publishing Co. Waltham, Massachusets. 1968). By Herbert B. Keller: pp. viii, 184. (Blaisdell Publishing Co. Waltham, Massachusets. 1968).

Asymptotic expansions for the error of discretization algorithms for non-linear functional equations

1965-02-01
Hans J. Stetter

Discrete Variable Methods in Ordinary Differential Equations.

1962-11-01
A. B. Farnell , Peter Henrici

<i>Analysis of Numerical Methods</i>

1967-07-01
Eugene Isaacson , Herbert B. Keller , George H. Weiss
Keywords: analyse ; methodes : numeriques ; equations : lineaires ; calcul : integral ; equations : differentielles Reference Record created on 2005-11-18, modified on 2016-08-08 Keywords: analyse ; methodes : numeriques ; equations : lineaires ; calcul : integral ; equations : differentielles Reference Record created on 2005-11-18, modified on 2016-08-08

Mesh selection for discrete solution of boundary problems in ordinary differential equations

1974-06-01
Víctor Pereyra , E.G. Sewell

Good approximation by splines with variable knots. II

1974-01-01
Carl de Boor

Discrete approximations to elliptic differential equations

1955-03-01
Wolfgang Wasow

Highly Accurate Numerical Solution of Casilinear Elliptic Boundary-Value Problems in n Dimensions

1970-10-01
Víctor Pereyra

On a Differential Equation of Boundary Layer Type

1968-04-01
Carl E. Pearson

Highly accurate numerical solution of casilinear elliptic boundary-value problems in 𝑛 dimensions

1970-01-01
Víctor Pereyra
The method of Iterated Deferred Corrections, whose theory was developed by the author, is applied to the problems of the title. The necessary asymptotic expansions are obtained and the way … The method of Iterated Deferred Corrections, whose theory was developed by the author, is applied to the problems of the title. The necessary asymptotic expansions are obtained and the way in which the corrections are produced by means of numerical differentiation is described in detail. Numerical results and comparisons with the variationalsplines methods are given.
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Numerical methods of high-order accuracy for nonlinear boundary value Problems

1967-04-01
Philippe G. Ciarlet , Martin H. Schultz , R. S. Varga

Accurate Difference Methods for Linear Ordinary Differential Systems Subject to Linear Constraints

1969-03-01
Herbert B. Keller
Previous article Next article Accurate Difference Methods for Linear Ordinary Differential Systems Subject to Linear ConstraintsHerbert B. KellerHerbert B. Kellerhttps://doi.org/10.1137/0706002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Earl A. Coddington and … Previous article Next article Accurate Difference Methods for Linear Ordinary Differential Systems Subject to Linear ConstraintsHerbert B. KellerHerbert B. Kellerhttps://doi.org/10.1137/0706002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Earl A. Coddington and , Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955xii+429 MR0069338 0064.33002 Google Scholar[2] L. Fox, The numerical solution of two-point boundary problems in ordinary differential equations, Oxford University Press, New York, 1957xi+371 MR0102178 0077.11202 Google Scholar[3] William B. Gragg, On extrapolation algorithms for ordinary initial value problems, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 384–403 MR0202318 0135.37803 LinkGoogle Scholar[4] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons Inc., New York, 1962xi+407 MR0135729 0112.34901 Google Scholar[5] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944viii+558 MR0010757 0063.02971 Google Scholar[6] Eugene Isaacson and , Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons Inc., New York, 1966xv+541 MR0201039 0168.13101 Google Scholar[7] Herbert B. Keller, Numerical methods for two-point boundary-value problems, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968viii+184 MR0230476 0172.19503 Google Scholar[8] Milton Lees, J. H. Bramble, Discrete methods for nonlinear two-point boundary value problemsNumerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965), Academic Press, New York, 1966, 59–72 MR0202323 0148.39206 Google Scholar[9] V. L. Pereyra, Highly accurate discrete methods for nonlinear problems, MRC Rep. 749, Mathematics Research Center, U.S. Army, University of Wisconsin, Madison, 1967 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Well-posedness and analyticity of solutions to a water wave problem with viscosityJournal of Differential Equations, Vol. 265, No. 10 Cross Ref Heat Transfer in MHD Flow due to a Linearly Stretching Sheet with Induced Magnetic FieldAdvances in Mathematical Physics, Vol. 2018 Cross Ref On analyticity of linear waves scattered by a layered mediumJournal of Differential Equations, Vol. 263, No. 8 Cross Ref A high-order perturbation of surfaces method for scattering of linear waves by periodic multiply layered gratings in two and three dimensionsJournal of Computational Physics, Vol. 345 Cross Ref MHD flow due to a linearly stretching sheet with induced magnetic field21 May 2016 | Acta Mechanica, Vol. 227, No. 10 Cross Ref MHD Flow due to the Nonlinear Stretching of a Porous SheetAdvances in Mathematical Physics, Vol. 2016 Cross Ref My Limiting Behavior of MHD Flow with Hall Current, Due to a Porous Stretching SheetJournal of Applied Mathematics and Physics, Vol. 02, No. 05 Cross Ref Limiting behavior of MHD flow over a porous rotating disk with Hall currents17 December 2012 | ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 93, No. 9 Cross Ref MHD Flow with Hall Current and Ion-Slip Effects due to a Stretching Porous DiskJournal of Applied Mathematics, Vol. 2013 Cross Ref Comment on "Group solution of a time dependent chemical convective process" by M.M. 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Adaptive Mesh Selection Strategies for Solving Boundary Value Problems

1978-02-01
Robert D. Russell , J. Christiansen
Various adaptive mesh selection strategies for solving two-point boundary value problems are brought together and a limited comparison is made. The mesh strategies are applied using collocation methods, and a … Various adaptive mesh selection strategies for solving two-point boundary value problems are brought together and a limited comparison is made. The mesh strategies are applied using collocation methods, and a careful error analysis is made for some of them. A number of the observations and conclusions are valid for the other noninitial value type methods (and for more general differential equations). The effects of removing regions where the solution has been accurately computed and then separately solving on those regions for which the solution is still poor are investigated. After presenting several representative examples, some conclusions are drawn.

Variable order variable step finite difference methods for nonlinear boundary value problems

1974-01-01
Víctor Pereyra

BOUNDARY PROBLEM SOLVERS FOR FIRST ORDER SYSTEMS BASED ON DEFERRED CORRECTIONS

1975-01-01
M. Lentini , Víctor Pereyra

Numerical differentiation and the solution of multidimensional Vandermonde systems

1970-01-01
Giuliano Galimberti , Víctor Pereyra
We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices. These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit … We define multidimensional Vandermonde matrices (MV) to be certain submatrices of Kronecker products of standard Vandermonde matrices. These MV matrices appear naturally in multidimensional problems of polynomial interpolation. An explicit algorithm is produced to solve systems of linear equations with MV matrices of coefficients. This is an extension of work of Stenger for the two-dimensional case. Numerical results for three-dimensional numerical differentiation are given.
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Iterative Solution of Nonlinear Equations in Several Variables

1971-04-01
E. I. , J. M. Ortega , Werner C. Rheinboldt

Accelerated iteration schemes for transonic flow calculations using fast Poisson solvers

1975-03-01
A. Jameson
Results obtained by using a fast Poisson solver to accelerate the rate of convergence of the iterative scheme are described. A fast iterative method for transonic flow calculations is formulated … Results obtained by using a fast Poisson solver to accelerate the rate of convergence of the iterative scheme are described. A fast iterative method for transonic flow calculations is formulated first, and then application is made to the transonic potential flow equation in a mapped domain. It is concluded that the use of a fast elliptic solver in combination with relaxation is an effective way to accelerate the convergence of transonic flow calculations, particularly when a marching scheme can be used to treat the supersonic zone in the relaxation process. 3 figures (RWR)
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Numerical solution of nonlinear two-point boundary problems by finite difference methods

1964-06-01
James F. Holt
Solution of nonlinear two-point boundary-value problems is often an extremely difficult task. Quite apart from questions of reality and uniqueness, there is no established numerical technique for this problem. At … Solution of nonlinear two-point boundary-value problems is often an extremely difficult task. Quite apart from questions of reality and uniqueness, there is no established numerical technique for this problem. At present, shooting techniques are the easiest method of attacking these problems. When these fail, the more difficult method of finite differences can often be used to obtain a solution. This paper gives examples and discusses the finite difference method for nonlinear two-point boundary-value problems.
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High order fast Laplace solvers for the Dirichlet problem on general regions

1976-05-01
Víctor Pereyra , Wlodek Proskurowski , Olof B. Widlund
Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in R/sup n/. A second order accurate scheme is combined with … Highly accurate finite difference schemes are developed for Laplace's equation with the Dirichlet boundary condition on general bounded regions in R/sup n/. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size h, one of the methods has an accuracy of at least O(h/sup 5/./sup 5/) in L/sub 2/. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.
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A Generalized Conjugate Gradient Method for Nonsymmetric Systems of Linear Equations

1976-01-01
Paul Concus , Gene H. Golub
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High order fast Laplace solvers for the Dirichlet problem on general regions

1977-01-01
Víctor Pereyra , Włodzimierz Proskurowski , Olof B. Widlund
Highly accurate finite difference schemes are developed for Laplace’s equation with the Dirichlet boundary condition on general bounded regions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow … Highly accurate finite difference schemes are developed for Laplace’s equation with the Dirichlet boundary condition on general bounded regions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A second order accurate scheme is combined with a deferred correction or Richardson extrapolation method to increase the accuracy. The Dirichlet condition is approximated by a method suggested by Heinz-Otto Kreiss. A convergence proof of his, previously not published, is given which shows that, for the interval size <italic>h</italic>, one of the methods has an accuracy of at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis h Superscript 5.5 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>5.5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O({h^{5.5}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The linear systems of algebraic equations are solved by a capacitance matrix method. The results of our numerical experiments show that highly accurate solutions are obtained with only a slight additional use of computer time when compared to the results obtained by second order accurate methods.
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Numerical Solution of Two Point Boundary Value Problems

1986-01-01
Herbert B. Keller
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Efficient computer manipulation of tensor products with applications to multidimensional approximation

1973-01-01
Víctor Pereyra , G. Scherer
The objective of this paper is twofold: (a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n dot … The objective of this paper is twofold: (a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n dot p squared"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">n \cdot {p^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> words of information for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="circled-times upper A Subscript i Baseline comma upper A Subscript i Baseline element-of script upper L left-parenthesis upper E Superscript p Baseline comma upper E Superscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\otimes _{i = 1}^n{A_i},{A_i} \in \mathcal {L}({E^p},{E^p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) instead of the tensor-product operators themselves (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p squared right-parenthesis Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{({p^2})^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> words of information). (b) To produce efficient algorithms for solving systems of linear equations with coefficient matrices being tensor products of nonsingular matrices, with special application to the approximation of multidimensional linear functionals.
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On improving an approximate solution of a functional equation by deferred corrections

1966-06-01
Víctor Pereyra

Pasva3: An adaptive finite difference fortran program for first order nonlinear, ordinary boundary problems

1979-01-01
Víctor Pereyra

A conjugate gradient approach to nonlinear elliptic boundary value problems in irregular regions

1974-01-01
Richard H. Bartels , James W. Daniel

A Perturbation Technique for Nonlinear Two-Point Boundary Value Problems

1969-09-01
S. M. Roberts , J. S. Shipman , William J. Ellis
Previous article Next article A Perturbation Technique for Nonlinear Two-Point Boundary Value ProblemsS. M. Roberts, J. S. Shipman, and W. J. EllisS. M. Roberts, J. S. Shipman, and W. J. … Previous article Next article A Perturbation Technique for Nonlinear Two-Point Boundary Value ProblemsS. M. Roberts, J. S. Shipman, and W. J. EllisS. M. Roberts, J. S. Shipman, and W. J. Ellishttps://doi.org/10.1137/0706032PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Richard Bellman, Perturbation techniques in mathematics, physics, and engineering, Holt, Rinehart and Winston, Inc., New York, 1964viii+118 MR0161003 Google Scholar[2] Richard E. Bellman and , Robert E. Kalaba, Quasilinearization and nonlinear boundary-value problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3, American Elsevier Publishing Co., Inc., New York, 1965ix+206 MR0178571 0139.10702 Google Scholar[3] F. A. Ficken, The continuation method for functional equations, Comm. Pure Appl. Math., 4 (1951), 435–456 MR0045308 0043.32202 CrossrefISIGoogle Scholar[4] T. R. Goodman and , G. N. Lance, The numerical integration of two-point boundary value problems, Math. Tables Aids Comput., 10 (1956), 82–86 MR0081553 0071.34006 CrossrefGoogle Scholar[5] L. V. Kantorovich and , G. P. Akilov, Functional analysis in normed spaces, Translated from the Russian by D. E. Brown. Edited by A. P. Robertson. International Series of Monographs in Pure and Applied Mathematics, Vol. 46, The Macmillan Co., New York, 1964xiii+771 MR0213845 0127.06104 Google Scholar[6] S. M. Roberts and , J. S. Shipman, The Kantorovich theorem and two-point boundary value problems, IBM J. Res. Develop., 10 (1966), 402–406 MR0202325 0196.49704 CrossrefISIGoogle Scholar[7] S. M. Roberts and , J. S. Shipman, Continuation in shooting methods for two-point boundary value problems, J. Math. Anal. Appl., 18 (1967), 45–58 10.1016/0022-247X(67)90181-3 MR0207213 0155.47303 CrossrefISIGoogle Scholar[8] S. M. Roberts, , J. S. Shipman and , C. V. Roth, Continuation in quasilinearization, J. Optimization Theory Appl., 2 (1968), 164–178 10.1007/BF00926998 0176.15001 CrossrefGoogle Scholar[9] S. M. Roberts and , J. S. Shipman, Justification for the continuation method in two-point boundary value problems, J. Math. Anal. Appl., 21 (1968), 23–30 10.1016/0022-247X(68)90236-9 MR0220451 0179.22102 CrossrefISIGoogle Scholar[10] James F. Holt, Numerical solution of nonlinear two-point boundary problems by finite difference methods, Comm. ACM, 7 (1964), 366–373 10.1145/512274.512291 MR0168123 0123.11805 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Survey of time-optimal attitude maneuversJournal of Guidance, Control, and Dynamics, Vol. 17, No. 2 Cross Ref An Asymptotic Perturbation Method for Nonlinear Optimal Control Problems Cross Ref Role of continuation in engineering analysisChemical Engineering Science, Vol. 42, No. 6 Cross Ref Numerical solution of nonlinear equations by one-parameter imbedding methods26 June 2007 | Numerical Functional Analysis and Optimization, Vol. 3, No. 2 Cross Ref Solution of nonlinear boundary value problems—XIChemical Engineering Science, Vol. 34, No. 5 Cross Ref One-parameter imbedding techniques for the solution of nonlinear boundary-value problemsApplied Mathematics and Computation, Vol. 4, No. 4 Cross Ref A classification and survey of numerical methods for boundary value problems in ordinary differential equationsInternational Journal for Numerical Methods in Engineering, Vol. 11, No. 5 Cross Ref NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS: SURVEY AND SOME RECENT RESULTS ON DIFFERENCE METHODS Cross Ref BOUNDARY PROBLEM SOLVERS FOR FIRST ORDER SYSTEMS BASED ON DEFERRED CORRECTIONS Cross Ref Variable order variable step finite difference methods for nonlinear boundary value problems28 August 2006 Cross Ref A variable order finite difference method for nonlinear multipoint boundary value problems1 January 1974 | Mathematics of Computation, Vol. 28, No. 128 Cross Ref Extension of a perturbation technique for nonlinear two-point boundary-value problems3 August 2013 | Journal of Optimization Theory and Applications, Vol. 12, No. 5 Cross Ref The epsilon variation method in two-point boundary-value problems3 August 2013 | Journal of Optimization Theory and Applications, Vol. 12, No. 2 Cross Ref Numerical Solutions by the Continuation MethodE. Wasserstrom18 July 2006 | SIAM Review, Vol. 15, No. 1AbstractPDF (2435 KB)Solution of Troesch's two-point boundary value problem by a combination of techniquesJournal of Computational Physics, Vol. 10, No. 2 Cross Ref A survey of optimal structural design under dynamic constraintsInternational Journal for Numerical Methods in Engineering, Vol. 4, No. 4 Cross Ref Solving Boundary-Value Problems by ImbeddingJournal of the ACM, Vol. 18, No. 4 Cross Ref Volume 6, Issue 3| 1969SIAM Journal on Numerical Analysis History Submitted:27 February 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706032Article page range:pp. 347-358ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

On shooting methods for boundary value problems

1969-08-01
M. R. Osborne

On mildly nonlinear partial difference equations of elliptic type

1953-11-01
Lipman Bers
The use of t he fini te d ifferences met hod is in solving t he boundar y value problem of t he first kind for t he nonlinear elli … The use of t he fini te d ifferences met hod is in solving t he boundar y value problem of t he first kind for t he nonlinear elli ptic equation A</ > = F (X,y,</>, </>., cf>u) is justified by first showing t hat t h e problem of the corresp onding difference equat ion has a uniqu e solution, and t h en t hat t he solution of t he differe nce e quat ion tends t o that of t he different ial e quation when t he net uni t te nds t o zero.Also a numerical m cthod of t he Liebmann type for t he computa tion of t he solution of t he diffe re nce equat ion is deyeloped , and t hese result s are extended to more general n onlinear ellipt ic equat ions.

The correction difference method for non linear boundary value problems of class M

1965-01-01
Víctor Pereyra

On the Continuity of the Generalized Inverse

1969-01-01
G. W. Stewart
Previous article Next article On the Continuity of the Generalized InverseG. W. StewartG. W. Stewarthttps://doi.org/10.1137/0117004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Afriat, Orthogonal and oblique projectors and the … Previous article Next article On the Continuity of the Generalized InverseG. W. StewartG. W. Stewarthttps://doi.org/10.1137/0117004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Afriat, Orthogonal and oblique projectors and the characteristics of pairs of vector spaces, Proc. Cambridge Philos. Soc., 53 (1957), 800–816 MR0094880 (20:1389) CrossrefGoogle Scholar[2] A. Ben-Israel and , A. Charnes, Contributions to the theory of generalized inverses, J. Soc. Indust. Appl. Math., 11 (1963), 667–699 10.1137/0111051 MR0179192 (31:3441) 0116.32202 LinkISIGoogle Scholar[3] Adi Ben-Israel, On error bounds for generalized inverses, SIAM J. Numer. Anal., 3 (1966), 585–592 10.1137/0703050 MR0215504 (35:6344) 0147.13201 LinkGoogle Scholar[4] G. Golub and , W. Kahan, Calculating the singular values and pseudo-inverse of a matrix, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 205–224 MR0183105 (32:587) 0194.18201 LinkGoogle Scholar[5] G. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), 206–216 10.1007/BF01436075 MR0181094 (31:5323) 0142.11502 CrossrefGoogle Scholar[6] G. H. Golub and , J. H. Wilkinson, Note on the iterative refinement of least squares solution, Numer. Math., 9 (1966), 139–148 10.1007/BF02166032 MR0212984 (35:3849) 0156.16106 CrossrefISIGoogle Scholar[7] Alston S. Householder, Unitary triangularization of a nonsymmetric matrix, J. Assoc. Comput. Mach., 5 (1958), 339–342 MR0111128 (22:1992) 0121.33802 CrossrefISIGoogle Scholar[8] Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964xi+257 MR0175290 (30:5475) 0161.12101 Google Scholar[9] E. H. Moors, On the reciprocal of the general algebraic matrix, Bull. Amer. Math. Soc., 26 (1919-20), 394–395, Abstract Google Scholar[10] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406–413 MR0069793 (16,1082a) 0065.24603 CrossrefGoogle Scholar[11] R. Penrose, On best approximation solutions of linear matrix equations, Proc. Cambridge Philos. Soc., 52 (1956), 17–19 MR0074092 (17,536d) CrossrefGoogle Scholar[12] J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall Inc., Englewood Cliffs, N.J., 1963vi+161 MR0161456 (28:4661) 1041.65502 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Homogeneity tests for one-way models with dependent errors under correlated groupsTEST, Vol. 11 | 2 September 2022 Cross Ref Bayesian linear models for cardinal paired comparison dataComputational Statistics & Data Analysis, Vol. 172 | 1 Aug 2022 Cross Ref Blind inverse problems with isolated spikesInformation and Inference: A Journal of the IMA, Vol. 64 | 16 June 2022 Cross Ref Tension distribution algorithm based on graphics with high computational efficiency and robust optimization for two-redundant cable-driven parallel robotsMechanism and Machine Theory, Vol. 172 | 1 Jun 2022 Cross Ref Expressions and properties of weak core inverseApplied Mathematics and Computation, Vol. 415 | 1 Feb 2022 Cross Ref Acute perturbation for Moore-Penrose inverses of tensors via the T-ProductJournal of Applied Mathematics and Computing, Vol. 3 | 4 January 2022 Cross Ref Forward and inverse analysis for particle size distribution measurements of disperse samples: A reviewMeasurement, Vol. 187 | 1 Jan 2022 Cross Ref Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrixJournal of Mathematical Sciences, Vol. 259, No. 1 | 16 October 2021 Cross Ref Representations and properties for the MPCEP inverseJournal of Applied Mathematics and Computing, Vol. 67, No. 1-2 | 6 January 2021 Cross Ref Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrixUkrainian Mathematical Bulletin, Vol. 18, No. 3 | 9 September 2021 Cross Ref The local limit of uniform spanning treesProbability Theory and Related Fields, Vol. 12 | 22 June 2021 Cross Ref Representations for the weak group inverseApplied Mathematics and Computation, Vol. 397 | 1 May 2021 Cross Ref Continuity of the core-EP inverse and its applicationsLinear and Multilinear Algebra, Vol. 69, No. 3 | 24 April 2019 Cross Ref Matrix Completion with Deterministic Sampling: Theories and MethodsIEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 43, No. 2 | 1 Feb 2021 Cross Ref Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control ProblemsJournal of Optimization Theory and Applications, Vol. 188, No. 1 | 18 November 2020 Cross Ref Perturbation bounds for the W-weighted core-EP inverseJournal of Algebra and Its Applications, Vol. 19, No. 12 | 4 December 2019 Cross Ref A Simple Asymptotically F -Distributed Portmanteau Test for Diagnostic Checking of Time Series Models With Uncorrelated InnovationsJournal of Business & Economic Statistics, Vol. 15 | 25 November 2020 Cross Ref On an improved convergence analysis of a two-step Gauss-Newton type method under generalized Lipschitz conditionsCarpathian Journal of Mathematics, Vol. 36, No. 3 | 30 September 2020 Cross Ref On accurate error estimates for the quaternion least squares and weighted least squares problemsInternational Journal of Computer Mathematics, Vol. 97, No. 8 | 18 July 2019 Cross Ref Efficient GMM estimation with singular system of moment conditionsStatistical Theory and Related Fields, Vol. 4, No. 2 | 23 August 2019 Cross Ref On the perturbation of the Moore–Penrose inverse of a matrixApplied Mathematics and Computation, Vol. 374 | 1 Jun 2020 Cross Ref On the continuity and differentiability of the (dual) core inverse in C *-algebrasLinear and Multilinear Algebra, Vol. 68, No. 4 | 12 September 2018 Cross Ref Exploring implicit spaces for constrained sampling-based planningThe International Journal of Robotics Research, Vol. 38, No. 10-11 | 26 August 2019 Cross Ref Recovery map for fermionic Gaussian channelsJournal of Mathematical Physics, Vol. 60, No. 7 | 1 Jul 2019 Cross Ref BibliographyMatrix Differential Calculus with Applications in Statistics and Econometrics | 11 May 2019 Cross Ref Detection of block-exchangeable structure in large-scale correlation matricesJournal of Multivariate Analysis, Vol. 169 | 1 Jan 2019 Cross Ref Perturbation Analysis of the Moore-Penrose Inverse and the Weighted Moore-Penrose InverseGeneralized Inverses: Theory and Computations | 13 May 2018 Cross Ref Expanding the Applicability of Four Iterative Methods for Solving Least Squares ProblemsAnnals of West University of Timisoara - Mathematics and Computer Science, Vol. 55, No. 2 | 29 December 2017 Cross Ref Optimization of Mixed-ADC Multi-Antenna Systems for Cloud-RAN DeploymentsIEEE Transactions on Communications, Vol. 65, No. 9 | 1 Sep 2017 Cross Ref Chapter 8. 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StewartSIAM Review, Vol. 19, No. 4 | 17 February 2012AbstractPDF (2178 KB)On continuity of the Moore-Penrose and Drazin generalized inversesLinear Algebra and its Applications, Vol. 18, No. 1 | 1 Jan 1977 Cross Ref Differentiation of the Drazin InverseStephen L. CampbellSIAM Journal on Applied Mathematics, Vol. 30, No. 4 | 28 July 2006AbstractPDF (363 KB)Residue Arithmetic Algorithms for Exact Computation of g-Inverses of MatricesT. Mahadeva Rao, K. Subramanian, and E. V. KrishnamurthySIAM Journal on Numerical Analysis, Vol. 13, No. 2 | 14 July 2006AbstractPDF (1158 KB)Perturbations and Approximations for Generalized Inverses and Linear Operator EquationsGeneralized Inverses and Applications | 1 Jan 1976 Cross Ref Annotated Bibliography on Generalized Inverses and ApplicationsGeneralized Inverses and Applications | 1 Jan 1976 Cross Ref Continuity properties of the Drazin pseudoinverseLinear Algebra and its Applications, Vol. 10, No. 1 | 1 Feb 1975 Cross Ref On the solution of the linear least squares problems and pseudo-inversesComputing, Vol. 13, No. 3-4 | 1 Sep 1974 Cross Ref Approximations to Generalized Inverses of Linear OperatorsR. H. Moore and M. Z. NashedSIAM Journal on Applied Mathematics, Vol. 27, No. 1 | 12 July 2006AbstractPDF (1415 KB)On the Numerical Properties of an Iterative Method for Computing the Moore–Penrose Generalized InverseTorsten Söderström and G. W. StewartSIAM Journal on Numerical Analysis, Vol. 11, No. 1 | 14 July 2006AbstractPDF (1050 KB)Perturbation theory for pseudo-inversesBIT, Vol. 13, No. 2 | 1 Jun 1973 Cross Ref The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables SeparateG. H. Golub and V. PereyraSIAM Journal on Numerical Analysis, Vol. 10, No. 2 | 14 July 2006AbstractPDF (1690 KB)ReferencesRegression and the Moore-Penrose Pseudoinverse | 1 Jan 1972 Cross Ref Error analysis in a problem of approximation of a solution of perturbed signal space and its orthogonal complementProceedings of Thirtieth Southeastern Symposium on System Theory Cross Ref Extensions and applications of the Householder algorithm for solving linear least squares problemsMathematics of Computation, Vol. 23, No. 108 | 1 January 1969 Cross Ref Volume 17, Issue 1| 1969SIAM Journal on Applied Mathematics1-221 History Submitted:26 April 1968Published online:28 July 2006 InformationCopyright © 1969 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0117004Article page range:pp. 33-45ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares

1982-03-01
Christopher C. Paige , Michael A. Saunders
article Free AccessLSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares Authors: Christopher C. Paige School of Computer Science, McGill University, Montreal, P.Q., Canada H3C 3G1 School of … article Free AccessLSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares Authors: Christopher C. Paige School of Computer Science, McGill University, Montreal, P.Q., Canada H3C 3G1 School of Computer Science, McGill University, Montreal, P.Q., Canada H3C 3G1View Profile , Michael A. Saunders Department of Operations Research, Stanford University, Stanford, CA Department of Operations Research, Stanford University, Stanford, CAView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 8Issue 1pp 43–71https://doi.org/10.1145/355984.355989Published:01 March 1982Publication History 3,010citation6,538DownloadsMetricsTotal Citations3,010Total Downloads6,538Last 12 Months727Last 6 weeks183 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
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Alternate Row and Column Elimination for Solving Certain Linear Systems

1976-03-01
J. M. Varah
In this paper, we show how to solve linear systems with a certain block structure in a stable manner, without introducing any extraneous elements, by alternate row and column elimination. … In this paper, we show how to solve linear systems with a certain block structure in a stable manner, without introducing any extraneous elements, by alternate row and column elimination. The amount of work involved is compared to other known methods.

<i>Numerical Solution of Ordinary and Partial Differential Equations</i>

1963-10-01
L. Fox , J. Gillis

Note on the iterative refinement of least squares solution

1966-12-01
Gene H. Golub , J. H. Wilkinson

Numerical treatment of ordinary differential equations by extrapolation methods

1966-03-01
Roland Bulirsch , Josef Stoer

On finite-difference methods for the numerical solution of boundary-value problems

1961-07-04
W. G. Bickley , Sol M. Michaelson , M. R. Osborne
We consider the application of finite-difference methods to the numerical solution of boundary-value problems. In particular we are concerned to study the feasibility and con­vergence of the difference-correction method for … We consider the application of finite-difference methods to the numerical solution of boundary-value problems. In particular we are concerned to study the feasibility and con­vergence of the difference-correction method for the solution of partial differential equations of elliptic type. These topics form the subject matter for §§ 3 to 6. The material of the first two sections is intended to serve as a preliminary for the main discussion. The topics considered here are finite difference formulae for numerical differentiation, and finite difference methods for the solution of partial differential equations.

On Selection of Equidistributing Meshes for Two-Point Boundary-Value Problems

1979-06-01
Andrew B. White
A general theory is developed for calculating equidistributing meshes $\{ t_i \} $ for difference methods for boundary-value problems of the form \[ u' = f(u,t),\qquad b(u(0),u(1)) = 0. \] … A general theory is developed for calculating equidistributing meshes $\{ t_i \} $ for difference methods for boundary-value problems of the form \[ u' = f(u,t),\qquad b(u(0),u(1)) = 0. \] It is shown that the original problem and the equidistribution constraints on the mesh $\{ t_i \} $ can be replaced by a transformed boundary-value problem on a uniform mesh. Existence, uniqueness, and convergence of Newton's method for the discrete solution and the equidistributing mesh are proved. Equidistribution of arc length is given for boundary-layer problems. Five sample problems are solved with different methods of choosing the mesh $\{ t_i \} $.

Finite-Difference Methods for Partial Differential Equations

1962-02-01
Preston C. Hammer , George E. Forsythe , Wolfgang Wasow

The effect of quadrature errors in the numerical solution of boundary value problems by variational techniques

1969-10-01
R. J. Herbold , Martin H. Schultz , R. S. Varga