Central differences

Subroutine FUNDR. This subroutine calculates the required derivatives for REGRES by central difference, using EVAL to calculate the objective functions. 

Third card FORMAT(8F10.2), size of increments to be used in central difference formula for calculating derivatives with respect to the independent variables. 

TRANSFER VECTOR FOR PARAMETERS VECTOR OF CENTRAL DIFFERENCE INCREMENTS FOP CALCULATING DERIVATIVES WRT THE PARAMETERS VECTOR OF CENTRAL DIFFERENCE INCREMENTS FOR... 

If we write the time-dependent Sclnodinger equation as 5i i/(T = -(i/li)f7i i, then, after replacing the time derivative by a central difference, we obtain... 

The leap-frog algorithm uses the simplest central difference I or-m n la for a derivative... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

A central difference expression may be derived by combining the equations for forward and backward differences. 

The first derivative of f at x, may then be given in terms of the central difference expression as... 

Coefficients of central difference expressions for derivatives up to order four of O(h ) are given in Figure 1-56 and of O(h ) in Figure 1-57. 

Given a data table with evenly spaced values of x, and rescaling x so that h = one unit, forward differences are usually used to find f(x) at x near the top of the table and backward differences at x near the bottom. Interpolation near the center of the set is best accomplished with central differences. 

To use central differences, the origin of x must be shifted to a base line (shaded area in Table 1-13) and x rescaled so one full (two half) line spacing = 1 unit. Sterling s formula (full lines as base) is defined as... 

Central Difference Table with Base Line... 

If 3 u/3x is represented by a central difference expression and du/dy by a backward difference expression an implicit solution may be obtained where... 

The tube radius is divided into a number of equally sized increments, Ar = R/I, where / is an integer. For reasons of convergence, we prefer to use a second-order, central difference approximation for the first partial derivative ... 

Setting A = 0 gives a second-order, central difference. ... 

The second derivative is constant (independent of a) for this second-order approximation. We consider it to be a central difference. ... 

It is apparent that the central difference approximations converge O(Ax ). The forward and backward approximations to the hrst derivative converge 0(Ax). This is because they are really approximating the derivatives at the points x = Ax rather than at x = 0. 

Apply finite difference approximations to Equation (9.15) using a backwards difference for da/d and a central difference for d a/d. The result is... 

The best solution to such numerical difficulties is to change methods. Integration in the reverse direction eliminates most of the difficulty. Go back to Equation (9.15). Continue to use a second-order, central difference approximation for d a/d, but now use a first-order, forward... 

The marching equation for reverse shooting. Equation (9.24), was developed using a first-order, backward difference approximation for dajdz, even though a second-order approximation was necessary for (faldz. Since the locations j—l, j, and j+ are involved an5rway, would it not be better to use a second-order, central difference approximation for dajdz ... 

Thus, the right and left difference derivatives generate approximations of order 1 to Lu = u, while the central difference derivative approximates to the second order the same. 

Since the central difference derivative in t approximates to the second order in r and Au = -f 0 h ), scheme (53) approximates... 

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

Central difference as extensively used in this chapter... 

Central differences were used in Equation (5.8), but forward differences or any other difference scheme would suffice as long as the step size h is selected to match the difference formula and the computer (machine) precision with which the calculations are to be executed. The main disadvantage is the error introduced by the finite differencing. 

Finite difference Newton method. Application of Equation (5.8) to/(jc) = x2 - x is illustrated here. However, we use a forward difference formula for f x) and a three-point central difference formula for/"(jc)... 

When the user, whether working on stand-alone software or through a spreadsheet, supplies only the values of the problem functions at a proposed point, the NLP code computes the first partial derivatives by finite differences. Each function is evaluated at a base point and then at a perturbed point. The difference between the function values is then divided by the perturbation distance to obtain an approximation of the first derivative at the base point. If the perturbation is in the positive direction from the base point, we call the resulting approximation a forward difference approximation. For highly nonlinear functions, accuracy in the values of derivatives may be improved by using central differences here, the base point is perturbed both forward and backward, and the derivative approximation is formed from the difference of the function values at those points. The price for this increased accuracy is that central differences require twice as many function evaluations of forward differences. If the functions are inexpensive to evaluate, the additional effort may be modest, but for large problems with complex functions, the use of central differences may dramatically increase solution times. Most NLP codes possess options that enable the user to specify the use of central differences. Some codes attempt to assess derivative accuracy as the solution progresses and switch to central differences automatically if the switch seems warranted. 

As a consequence, the gradient of the objective function and the Jacobian matrix of the constraints in the nonlinear programming problem cannot be determined analytically. Finite difference substitutes as discussed in Section 8.10 had to be used. To be conservative, substitutes for derivatives were computed as suggested by Curtis and Reid (1974). They estimated the ratio /x of the truncation error to the roundoff error in the central difference formula... 

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