The large finite element global stiffness matrix is an algebraic, discreet, even-order, differential operator of zero row sums. Direct application of the, practically convenient, readily applied, Gershgorin’s eigenva...The large finite element global stiffness matrix is an algebraic, discreet, even-order, differential operator of zero row sums. Direct application of the, practically convenient, readily applied, Gershgorin’s eigenvalue bounding theorem to this matrix inherently fails to foresee its positive definiteness, predictably, and routinely failing to produce a nontrivial lower bound on the least eigenvalue of this, theoretically assured to be positive definite, matrix. Considered here are practical methods for producing an optimal similarity transformation for the finite-elements global stiffness matrix, following which non trivial, realistic, lower bounds on the least eigenvalue can be located, then further improved. The technique is restricted here to the common case of a global stiffness matrix having only non-positive off-diagonal entries. For such a matrix application of the Gershgorin bounding method may be carried out by a mere matrix vector multiplication.展开更多
In this note we consider some basic, yet unusual, issues pertaining to the accuracy and stability of numerical integration methods to follow the solution of first order and second order initial value problems (IVP). I...In this note we consider some basic, yet unusual, issues pertaining to the accuracy and stability of numerical integration methods to follow the solution of first order and second order initial value problems (IVP). Included are remarks on multiple solutions, multi-step methods, effect of initial value perturbations, as well as slowing and advancing the computed motion in second order problems.展开更多
In this note, we experimentally demonstrate, on a variety of analytic and nonanalytic functions, the novel observation that if the least squares polynomial approximation is repeated as weight in a second, now weighted...In this note, we experimentally demonstrate, on a variety of analytic and nonanalytic functions, the novel observation that if the least squares polynomial approximation is repeated as weight in a second, now weighted, least squares approximation, then this new, second, approximation is nearly perfect in the uniform sense, barely needing any further, say, Remez correction.展开更多
In this paper we examine single-step iterative methods for the solution of the nonlinear algebraic equation f (x) = x2 - N = 0 , for some integer N, generating rational approximations p/q that are optimal in the sense...In this paper we examine single-step iterative methods for the solution of the nonlinear algebraic equation f (x) = x2 - N = 0 , for some integer N, generating rational approximations p/q that are optimal in the sense of Pell’s equation p2 - Nq2 = k for some integer k, converging either alternatingly or oppositely.展开更多
This note is mainly concerned with the creation of oppositely converging and alternatingly converging iterative methods that have the added advantage of providing ever tighter bounds on the targeted root. By a slight ...This note is mainly concerned with the creation of oppositely converging and alternatingly converging iterative methods that have the added advantage of providing ever tighter bounds on the targeted root. By a slight parametric perturbation of Newton’s method we create an oscillating super-linear method approaching the targeted root alternatingly from above and from below. Further extension of Newton’s method creates an oppositely converging quadratic counterpart to it. This new method requires a second derivative, but for it, the average of the two opposite methods rises to become a cubic method. This note examines also the creation of high order iterative methods by a repeated specification of undetermined coefficients.展开更多
This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of init...This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.展开更多
Considered in this note are numerical tracking algorithms for the accurate following of implicit curves. We start with a fixed point on the curve, and then systematically place on it additional points, one after the o...Considered in this note are numerical tracking algorithms for the accurate following of implicit curves. We start with a fixed point on the curve, and then systematically place on it additional points, one after the other. In this note we first go over the basic procedure of moving forward tangentially from an already placed point then orthogonally returning to the curve. Next, we further consider higher order forward stepping procedures for greater accuracy. We note, however, that higher order methods, desirable for greater accuracy, may harbor latent instabilities. This note suggests ways of holding such instabilities in check, to have stable and highly accurate tracing methods. The note has several supporting numerical examples, including the rounding of a dynamical “snap-through” point.展开更多
文摘The large finite element global stiffness matrix is an algebraic, discreet, even-order, differential operator of zero row sums. Direct application of the, practically convenient, readily applied, Gershgorin’s eigenvalue bounding theorem to this matrix inherently fails to foresee its positive definiteness, predictably, and routinely failing to produce a nontrivial lower bound on the least eigenvalue of this, theoretically assured to be positive definite, matrix. Considered here are practical methods for producing an optimal similarity transformation for the finite-elements global stiffness matrix, following which non trivial, realistic, lower bounds on the least eigenvalue can be located, then further improved. The technique is restricted here to the common case of a global stiffness matrix having only non-positive off-diagonal entries. For such a matrix application of the Gershgorin bounding method may be carried out by a mere matrix vector multiplication.
文摘In this note we consider some basic, yet unusual, issues pertaining to the accuracy and stability of numerical integration methods to follow the solution of first order and second order initial value problems (IVP). Included are remarks on multiple solutions, multi-step methods, effect of initial value perturbations, as well as slowing and advancing the computed motion in second order problems.
文摘In this note, we experimentally demonstrate, on a variety of analytic and nonanalytic functions, the novel observation that if the least squares polynomial approximation is repeated as weight in a second, now weighted, least squares approximation, then this new, second, approximation is nearly perfect in the uniform sense, barely needing any further, say, Remez correction.
文摘In this paper we examine single-step iterative methods for the solution of the nonlinear algebraic equation f (x) = x2 - N = 0 , for some integer N, generating rational approximations p/q that are optimal in the sense of Pell’s equation p2 - Nq2 = k for some integer k, converging either alternatingly or oppositely.
文摘This note is mainly concerned with the creation of oppositely converging and alternatingly converging iterative methods that have the added advantage of providing ever tighter bounds on the targeted root. By a slight parametric perturbation of Newton’s method we create an oscillating super-linear method approaching the targeted root alternatingly from above and from below. Further extension of Newton’s method creates an oppositely converging quadratic counterpart to it. This new method requires a second derivative, but for it, the average of the two opposite methods rises to become a cubic method. This note examines also the creation of high order iterative methods by a repeated specification of undetermined coefficients.
文摘This paper considers practical, high-order methods for the iterative location of the roots of nonlinear equations, one at a time. Special attention is being paid to algorithms also applicable to multiple roots of initially known and unknown multiplicity. Efficient methods are presented in this note for the evaluation of the multiplicity index of the root being sought. Also reviewed here are super-linear and super-cubic methods that converge contrarily or alternatingly, enabling us, not only to approach the root briskly and confidently but also to actually bound and bracket it as we progress.
文摘Considered in this note are numerical tracking algorithms for the accurate following of implicit curves. We start with a fixed point on the curve, and then systematically place on it additional points, one after the other. In this note we first go over the basic procedure of moving forward tangentially from an already placed point then orthogonally returning to the curve. Next, we further consider higher order forward stepping procedures for greater accuracy. We note, however, that higher order methods, desirable for greater accuracy, may harbor latent instabilities. This note suggests ways of holding such instabilities in check, to have stable and highly accurate tracing methods. The note has several supporting numerical examples, including the rounding of a dynamical “snap-through” point.
