Extra Function Evaluations Research Articles

Small-Sample Statistical Condition Estimates for General Matrix Functions

A new condition estimation procedure for general matrix functions is presented that accurately gauges sensitivity by measuring the effect of random perturbations at the point of evaluation. In this procedure the number of extra function evaluations used to evaluate the condition estimate determines the order of the estimate. That is, the probability that the estimate is off by a given factor is inversely proportional to the factor raised to the order of the method. The “transpose-free” nature of this new method allows it to be applied to a broad range of problems in which the function maps between spaces of different dimensions. This is in sharp contrast to the more common power method condition estimation procedure that is limited, in the usual case where the Frechet derivative is known only implicitly, to maps between spaces of equal dimension. A group of examples illustrates the flexibility of the new estimation procedure in handling a variety of problems and types of sensitivity estimates, such as mix...

Polar Decomposition and Matrix Sign Function Condition Estimates

This paper presents reliable condition estimation procedures, based on Fréchet derivatives, for polar decomposition and the matrix sign function. For polar decomposition, the condition number for complex matrices is equal to the reciprocal of the smallest singular value, and rather surprisingly, for real matrices it is equal to the reciprocal of the average of the two smallest singular values. By using inverse power methods, both of these condition numbers can be evaluated at a fraction of the cost of finding the polar decomposition. Except for special cases, such as for normal matrices, the condition number of the matrix sign function does not have such a precise characterization. However, accurate condition estimates can be obtained by using explicit forms of the Fréchet derivative, or its finite-difference approximation, with a matricial inverse power method. These methods typically require two extra sign function evaluations, and it is an open problem whether accurate estimates can be obtained for a fraction of a function evaluation, as is the case for the polar decomposition. Related results for the stable Lyapunov equation and Newton’s method for the matrix square root problem are discussed.

A new 4th order runge-kutta method for initial value problems with error control

A new 4th order Runge-Kutta method for solving initial value problems is derived by replacing the arithmetic means in the formula where etc., by their geometric means i.e. etc. to yield initially a low order accuracy formula. However by re-comparing the Taylor series expansions of k 1 k 2 k 3 and k 4 in terms of the functional derivatives and the α ij parameters, a fourth order accuracy formula is obtained which is confirmed by numerical experiments. Then, a new fourth order Runge-Kutta method for solving linear initial value problems of the form y′ = Ay is derived which provides an estimate of the truncation error without any extra function evaluations. The idea follows from the fact that two numerical solutions of similar order can be obtained by using the arithmetic mean (AM) and the geometric mean (GM) averaging of the functional values. The numerical results given confirm that this new method is suitable to be used as an error control strategy.

Order barriers for continuous explicit Runge-Kutta methods

In this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations, the need for which occurs frequently in applications. Whereas most of the commonly used multi-step methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations, this is not always the case for one-step methods. We consider the class of explicit Runge-Kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given uniform order p. These bounds are similar to the Butcher barriers known for the discrete case, and are derived up to order p = 5 p = 5 . As far as we know, the examples we present of 8-stage continuous Runge-Kutta methods of uniform order 5 are the first of their kind.

Runge-Kutta interpolants based on values from two successive integration steps

New interpolants of the explicit Runge-Kutta method for the initial value problem are proposed. These interpolants are based on values of the solution and its derivative from two successive integration steps. In this paper, three interpolants withO(h6) local error (l.e.), for the fifth order solution, of the methods Fehlberg 4(5) (RKF 4(5)), Dormand and Prince 5(4) (RKDP 5(4)) and Verner 5(6) (RKV 5(6)) without extra cost are derived. An interpolant withO(h7) (l.e.) for the sixth order solution of the Verner's method with only one extra function evaluation per integration step is also constructed. The above advantages are obtained without any cost in the magnitude of the error.

A new embedded pair of Runge-Kutta formulas of orders 5 and 6

A new pair of embedded Runge-Kutta (RK) formulas of orders 5 and 6 is presented. It is derived from a family of RK methods depending on eight parameters by using certain measures of accuracy and stability. Numerical tests comparing its efficiency to other formulas of the same order in current use are presented. With an extra function evaluation per step, a C 1-continuous interpolant of order 5 can be obtained.

Condition Estimates for Matrix Functions

A sensitivity theory based on Fréchet derivatives is presented that has both theoretical and computational advantages. Theoretical results such as a generalization of Van Loan’s work on the matrix exponential are easily obtained: matrix functions are least sensitive at normal matrices. Computationally, the central problem is to estimate the norm of the Fréchet derivative, since this is equal to the function’s condition number. Two norm-estimation procedures are given; the first is based on a finite-difference approximation of the Fréchet derivative and costs only two extra function evaluations. The second method was developed specifically for the exponential and logarithmic functions; it is based on a trapezoidal approximation scheme suggested by the chain rule for the identity $e^X = ( e^{X/2^n } )^{2^n } $. This results in an infinite sequence of coupled Sylvester equations that, when truncated, is uniquely suited to the “scaling and squaring” procedure for $e^X $ or the “inverse scaling and squaring” procedure for log X. Both the trapezoid approximation method and the more general finite-difference approach yield excellent condition estimates for a large class of problems taken from the literature. The problems in this set illustrate that condition estimates based on the Fréchet derivative have the virtue of reliability and general applicability.

New interpolants for runge-kutta algorithms using second derivatives

In this paper a general interpolant for the Explicit Runge-Kutta methods is proposed. These interpolants are based on second derivatives on mesh-points of the integration interval, and first derivatives on interior points of each step. These first derivatives can be produced using lower order interpolants. Here an interpolant with 0(h 6) local truncation error for the fifth order solution used in RKF4(5) method is presented, with a cost of “about” one extra function evaluation per integration step.

A new fourth order Runge-Kutta formula for y′ = Ay with stepsize control

A new fourth order Runge-Kutta method for solving linear initial value problems of the form y′ = Ay is derived which provides as estimate of the truncation error without any extra function evaluations. The idea follows from the fact that two numerical solutions of similar order can be obtained by using the arithmetic mean (AM) and the geometric mean (GM) averaging of the functional values. The numerical results given confirm that this new method is suitable to be used as an error control strategy.

Interpolants for Runge-Kutta formulas

A general procedure for the construction of interpolants for Runge-Kutta (RK) formulas is presented. As illustrations, this approach is used to develop interpolants for three explicit RK formulas, including those employed in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are required to obtain an interpolant with O ( h 5 ) local truncation error for the fifth-order RK formula used in RKF45; two extra function evaluations per step are required to obtain an interpolant with O ( h 6 ) local truncation error for this RK formula.

Optimal Load Flow Solution Using the Hessian Matrix

The rapid convergence that Newton's method possesses, by use of the Jacobian matrix, has led to an investigation of using a higher order matrix, the Hessian, for an even faster convergence. It turns out that this approach unifies the fields of nonlinear programming methods and Newton based methods. The load flow problem can be defined as the solution of a system of simultaneous equations f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</sub> (x)= O, i= l, ..., n. It can be shown the Newton's method proceeds in a direction that minimizes F=∑f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</sub> (x) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . The Hessian load flow also minimizes F by assuming that it is a quadratic function, such that the linearizations become HΔx=-g, where the Hessian H is the matrix of the second partials of F and the vector g is the gradient of F. The optimal load flow problem can be formulated by including some additional terms in F so that a single algorithm, based on the Hessian, essentially solves both the normal and the optimal load flow problems. An interesting aspect of the method is that an existing Newton's program can be updated to a Hessian program quite simply. The H matrix is somewhat less sparse than the corresponding Jacobian but enough so that sparse techniques should be used. Furthermore, the Hessian can be completely obtained from the Jacobian, thus avoiding extra explicit function evaluations in the program. The paper presents enough details of the method for an implementation of a computer program. Numerical examples are given and compared with Newton's method results.