Ordinary Algorithm Research Articles

On the worst case of three algorithms for computing the Jacobi symbol

We study the worst-case behavior of three iterative algorithms for computing the Jacobi symbol ( v u ) . Each algorithm is similar in format to the Euclidean algorithm for computing gcd( u, v ). Eisenstein's algorithm chooses an even quotient at each step. It is shown that the worst case occurs when u = 2 n + 1, v = 2 n − 1. Lebesgue's algorithm is essentially the least-remainder Euclidean algorithm with powers of 2 removed at each step. Its worst case occurs when u = 2 L n − L n − 1 , v = L n , where L 0 = 1, L 1 = 1, and L n = 2 L n − 1 + L n − 2 for n ≥ 2. The “ordinary” Jacobi symbol algorithm is essentially the ordinary Euclidean algorithm with powers of 2 removed at each step. It is the most interesting mathematically of the three. We prove that if the ordinary algorithm on input ( u , v ) performs n division steps, with u > v >0 and u + v as small as possible, then u = A n and v = A n − 1 , where A 0 = 1, A 1 = 3, A 2 n = A 2 n − 1 + 2 A 2 n − 2 for n ≥ 1, and A 2 n +1 = 2 A 2 n + A 2 n − 1 for n ≥ 1. We also discuss the worst-case inputs to the ordinary algorithm under the lexicographic and reverse lexicographic orderings.

On Latent Distance Analysis and the MLE Algorithm

Latent distance analysis proposed by Lazarsfeld (1968) is discussed and the method of the maximum likelihood estimation (MLE) of the latent parameters is studied. For this problem, it is well-known that the ordinary algorithm of MLE gives frequently improper solutions. In the present paper, an algorithm is proposed, which does not give any improper solution, after transforming the latent response parameters to the logistic form and applying the new MLE procedure. The algorithm depends on the EM algorithm. The actual data analysis is also given.

Eigenfactor solution of the matrix Riccati equation--A continuous square root algorithm

This paper introduces a new algorithm for solving the matrix Riccati equation. Differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix. The solution of these equations yields, then, the time behavior of the eigenvalues and eigenvectors of the solution matrix. A reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation. This algorithm serves two purposes. First, being a square root solution, it entails all the advantages of square root algorithms such as nonnegative definiteness and accuracy. Secondly, it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors. The algorithm which handles cases of distinct as well as multiple eigenvalues is tested on several examples. Through these examples it is seen that the algorithm is indeed more accurate than the ordinary one. Moreover, it is seen that the algorithm works in cases where the ordinary algorithm fails and even in cases where the closed-form solution cannot be computed as a result of numerical difficulties.

Restarting techniques in nonlinear programming algorithms applied to the optimal control of chemical plants

A convenient way for solving many optimal control problems in chemical engineering is to cast them into the form of nonlinear programming. While this transcription is formally straightforward, its implementation can give rise to some peculiar problems, most of which are connected with the high dimensionality resulting from this transformation. One of the crucial problems is how to continue the minimization procedure when the current step does not coincide with a descent direction. As a matter of fact this drawback is common to most minimization problems, but its frequency and importance increase with increasing dimensionality. Hence its significance in optimal control. Consequently robust methods are expected to be able to deal with these situations in which ordinary algorithms often get stuck. Restarting techniques are meant to overcome this difficulty with the choice of a new direction and/or of a new step length. Typically steps are computed according to the expression Δx_n=x_n−x_n-1=−αHn-1-1gn-1 where gn is the gradient vector and Hn a convenient approximation to the hessian matrix of the objective function f at the point x_n-1 and α an opportune real number (0,1] such that dfxn-1+αHn-1-1gn-1dα=0 We define Hn-1-1gn-1 an ascent direction if there is no such α or (which is computationally the same) it is zero to machine precision. In this case restarting techniques serve the purpose of modifying Δx_n either by improving (or even eliminating) approximations or by rejecting false assumptions. The following cases have been considered A) The expression for the computation of the steps is adequate, but the approximation to the hessian matrix is poor B) The objective function is far from being locally quadratic and descent directions lie in a narrow region along the gradient direction C) The objective function is "numerically" discontinuous, i.e. derivatives are very large and highly oscillating or are approximated badly by means of finite differences. The difficulty that arises in case A) can sometimes be removed by reduction of efficiency in favour of robustness (such as a more frequent updating in quasi-Newton algorithms). More often a different approximation has to be used (such as from Gauss-Newton to variable metric methods). The latter case is discussed in this paper. Case B) is generally more serious. It is frequently tackled by means of strategies that combine Newton-like and steepest descent steps , such as the dog-1eg algorithm. The Marquardt procedure and the pseudo-inverse method can also be regarded as belonging to this class, since the Marquardt parameter and the number of eigenvalues retained in the pseudo inverse method indicate the weight attached to each of the two components of the global step. A point that is discussed thorougly in this paper is whether the algorithm should vary the parameter that determines the contribution of each component continuously from iteration to iteration or whether this parameter should be a local property.Furthermore the convenience of adopting pure strategies (i.e. one always uses Newton-like directions and takes a few steepest descent steps only to overcome regions where Newton-like steps do not correspond to descent directions)is examined. A direct search has to be started in a situation like the one described in the case C) . In fact the gradient vector is either ill computed or does not provide much information. Sometimes a minimum has already been well located in spite of large values of derivatives, which are to be regarded as noise. Deterministic and random methods are compared as to the effectiveness of this procedure. A general algorithm, which can cope with the difficulties mentioned in A), B) and C) is proposed in this paper and applied to the optimal control of a chemical reactor.

Fuzzy solution in fuzzy linear programming problems

Fuzzy linear programming problems are discussed in which both constraints and objective functions are assumed to be fuzzy inequalities. The problem is to obtain a fuzzy solution such that the fuzzy inequalities hold. It is shown how to reduce this problem to the conventional linear programming problem so that it can be easily solved by the ordinary algorithms of linear programming.

A hybrid algorithm for finding a global minimum

One of the most important problems in non-linear programming is to find out the global minimum of a given objective function. In this paper, a new hybrid algorithm which combines the random optimization method of Matyas (1965) and one of the well-known ordinary descent algorithms having an effective convergence property (for example, the Fletcher-Reeves conjugate gradient method, the Davidon-Fletcher-Powell quasi Newton method, etc.) is proposed in order to find out a global minimum in as small a number of steps as possible. Several computational experiments on multimodal objective functions are carried out in order to test the efficiency of the proposed hybrid algorithm. The results obtained imply that the proposed hybrid algorithm is useful for finding out a global minimum in a small number of steps. A theorem that predicts convergence to a global minimum is also given.

Load models for short-term load forecasting in multinode power systems

The problem of on-line short-term load demand forecasting at all the major loading nodes of an interconnected power system is studied. A multivariable state space model is proposed. The innovations representation is utilized, so that an ordinary recursive least-squares algorithm can be used to give unbiased estimates of the model parameters. State estimation is carried out by using adaptive estimation techniques. The parameter identification and state estimation algorithms are combined in a bootstrap manner. A scheme for detecting abnormal load behaviour is proposed. A weather sensitive load model is also presented for longer load time forecasts. Actual data provided by the Ontario Hydro for four loading buses are used for five-minute, ten-minute and hourly forecasts. The results show that the proposed approach is very attractive.

A numerical study of augmented penalty function algorithms for terminally constrained optimal control problems

Three augmented penalty function algorithms are tested and compared with an ordinary penalty function algorithm for two demonstration optimal control problems. Although the augmented penalty function is quite helpful in solving control problems with terminal state constraints, the convergence can be improved significantly by providing systematic increases in the penalty constant.

Modified Merson's integration algorithm which saves two evaluations at each step

This paper discusses a modified version of Merson's integration algorithm which can save two evaluations of the system at each step. For a complex system the saving in computing time may come to 40%. If you now use the ordinary Merson algorithm, you will find it easy to switch to the modified algorithm. The flow chart appended to this paper shows that switching to the modified algorithm involves adding on four blocks to Merson's method.

Proving simultaneous positivity of linear forms

In this paper we study the computational effort involved in establishing that m linear forms ll(x),..., lm(x), where x ----(x 1 ,..., x,,), are simultaneously non-negative for a substitution y e R n, i.e. that 0 ~< li(y), 1 <~ i ~ m. We develop the notion of a complete proof for simultaneous positivity and show that under certain natural conditions, proofs for the simultaneous positivity ofm forms will always requirem steps. Appropriate theorems and counterexamples show that if the conditions in the main theorem do not hold, then the conclusion need not hold. A number of applications of the main theorem are given. The present discussion brings out ideas and concepts which may be of interest and applicability beyond the present context. We shall devote some time to present these ideas here. Let P be a computational problem. We can distinguish between three levels of treatment of the problem. The highest level is finding a solution S to the problem. Depending on P, the solution S may be a number, a permutation, a truth value, etc. I f the algorithm for arriving at the solution S is a valid one, then the fact that S was produced, is in itself also a proof that S is indeed a solution of the problem P. The next level is that of checking a proposed solution. Given a possible solution S for the problem P, we wish to check whether S is indeed a solution. As an illustration of the difference between finding a solution and checking a proposed solution, consider the problem of finding a proper divisor of an integer n known to be composite. The ordinary algorithm for finding a divisor of n may take up to about ~/n steps. However, if given a supposed divisor k of n, then to check it we just have to divide n by k.