Error function, definition derivatives

From a fundamental point of view, integration is less demanding than differentiation, as far as the conditions imposed on the class of functions. As a consequence, numerical integration is a lot easier to carry out than numerical differentiation. If we seek explicit functional forms (sometimes referred to as closed forms) for the two operations of calculus, the situation is reversed. You can find a closed form for the derivative of almost any function. But even some simple functional forms cannot be integrated expliciUy, at least not in terms of elementary functions. For example, there are no simple formulas for the indefinite integrals J e dx or J dx. These can, however, be used for definite new functions, namely, the error function and the exponential integral, respectively. 

This discussion and the calculated relative errors have so far considered only the double sided derivative definition of Eq. (5.4). Figure 5.3 shows relative error for the single sided derivative of Eq. (5.3) when evaluated for the power function as used in Figure 5.2. As expected for the larger values off the relative error in the derivative now varies directly with the e value as seen by the slope of the dotted curve in the figure. For small values off the relative error increases with the same upper limit due to machine precision as for the double sided derivative. For this example of a power function, the minimum relative error in the derivative is around 10 and is seen to occur at an f value of about 10. A similar calculation with the exp() function as used in Figure 5.1 would show very similar data and trends. 

Computer code implementing Richardson s extrapolation for the derivative is shown in Listing 5.2. The function deriv() starts out with the basic central difference definition of the derivative on lines 3 through 5 with the result stored in a table (a[]). Then a Richardson improvement loop is coded from lines 7 to 16. Within this loop a test of the accuracy achieved is made on lines 13 through 15 and the loop is exited if sufficient accuracy is achieved. Note that the test is made on line 14 of the difference in value between two iterative values. The loop at lines 10-12 implements a multiple Richardson improvements based upon all previous data which is saved in a table. The deriv() function returns 3 values on line 17 the derivative value, the number of Richardson cycles used and the estimated relative error in the derivative. 

With these definitions, the mathematical derivation of the FCV family of clustering algorithms depends upon minimizing the generalized weighted sum-of-squared-error objective functional... 

Basic functional controls will include those in Table 11.19. Most of the values can be derived from a straightforward transfer function but several, particularly those determining maximum charge current, discharge current, and state-of-charge, will require careful scrutiny of the definition, calculation, and error analysis. 

In the present work Gallerkin s method of weighted residuals is used to derive the weak form of the equilibrium equations. Hence, the first step towards finite element discretisation of the governing equations is the definition of shape functions for the domain variables, i.e. displacement, pore water pressure and pore air pressure. Introducing these shape functions into equations 13, 14 and 15 the governing equations are approximated with a certain accuracy. The approximation errors, termed... 

In a single run, however, the estimated FES does not converge to a definite value but fluctuates around the exact value. The accuracy and efficiency of the MTD algorithm depend on the size and shape of the energy hills, that is the height w and the width of the Gaussians, and on the deposition interval t. The exact expression for the error as a function of these quantities has been provided in [12], A simpler, empirical expression was also derived [13], which reads... 

Listing 10.21 shows code to solve the set of equations for the circuit example above. The equation set is defined in the f() function on lines 8 through 13. The reader should be readily able to correlate the code formulation with the basic equation set of Eq.(10.61). The major differences are the use of the v[4] variable for the inductor current in the equation set and the notation vp[] for the various derivatives. A selected set of element values are defined on lines 5 and 6 to complete the equation definition. The set of equations is solved on line 15 by a call to the function odeivseQ which is the adaptive step size algorithm with a returned error estimate. Other choices for functions to generate the solution include the odeiv() routine with a specified range of times and number of solution time points... 

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