اجتہاد کی شرعی حیثیت اور عصر حاضر ميں اجتہاد کی عملی صورتیں

Ijtihad is an invaluable secondary source of the Muslim jurisprudence. In Islamic law, ijtihad refers to the independent interpretation of problems not precisely covered by the sacred scripture of Islam, Qur’an, and Prophetic traditions. The savants who undertake ijtihad must be firmly rooted in knowledge and savvy what the demands of the contemporary era are. Since new challenges call for innovative solutions, the faithful cannot genuinely live by Islam without their jurists deriving the laws of sharia from its sources.

Some Novel Multistep Iterative Methods for Solving Nonlinear Equations and Their Dynamics

Researchers have been contributing a lot to develop root …nding methods for solving nonlinear equations and system of nonlinear equations from many decades. The research started growing since the publication of the books by Traub in 1964 and by Ortega and Rheinboldt in 1970. Finding root of these equations have remained a very important problem in mechanical, electrical and aeronautical engineering. Some complicated techniques exist for solving cubic or quartic equations but higher nonlinear equations are rarely of a form that allows the roots to be determined exactly. So, numerical techniques must be used to solve complex nonlinear equations. Many numerical techniques have been developed earlier in literature to …nd the zero of a nonlinear equation to a speci…ed accuracy. These methods start with an initial approximation of the exact root and iteratively improve this approximation until the required accuracy is obtained. There are several contributors to this problem; Newton, Laguerre, Grae¤e, Baristow, Mueller, Traub and many others. The methods developed by all these researchers are single step. Among these techniques, Newton’s method [15, 104] is most popular method for …nding roots of the nonlinear equations. Newton’s method is quadratically convergent but it may not converge to real root if the initial guess does not lie in the vicinity of root or f 0 is zero in the neighborhood of the real root. Multipoint iterative methods allow us not to discard information that had already been computed. These methods require evaluations of the nonlinear function and derivatives of nonlinear function at several values of the independent variable [104]. The root …nding methods that use only information from the current iteration are called methods without-memory and the root …nding methods that use information from the current and previous iteration are termed as methods with-memory. Ostrowski [77] de…ned the e¢ ciency index of an iterative method as q 1 nf ; where q is the convergence order of the method and nf is the number of function evaluations required per iteration. Kung and Traub [56] conjectured that a without-memory multipoint method requiring n + 1 function evaluations per iteration have optimal order at most 2n and it can attain the e¢ ciency index at most 2 n n+1 : The methods satisfying above hypothesis of Kung and Traub are known as optimal. The main aim of this thesis is to investigate and develop some new optimal and computationally e¢ cient iterative schemes to …nd simple and multiple roots of nonlinear equations as well as for …nding roots of systems of nonlinear equations using various techniques. We have developed some novel multistep with and without-memory iterative methods for solving nonlinear equations by using the weight function approach, with-memorization, rational and inverse interpolation techniques. The basins of attractions and stability analysis of the methods have also been investigated for deep study. A large number of real world applications are reduced to solve systems of nonlinear equations numerically. Solution of such systems has been one of the most challenging problems in numerical analysis. Newton’s method is a basic method for this problem which is also extended for solving systems of nonlinear equations. Several iterative methods for solving systems of nonlinear equations are brought forward. One of the main advantages of these schemes was to achieve high order of convergence with few Jacobian and function evaluations. We have established in this thesis, a new family of optimal fourth order Jarratt type methods for solving nonlinear equations and have extended it to solve systems of nonlinear equations. Convergence analysis for both cases shows that the order of convergence of the new methods is at least four. Cost of computations, numerical tests and basins of attraction are presented which show that the new methods are better alternates to existing methods of similar kind. In addition stability analysis shows the stable behavior of new methods. We have also given applications of the proposed methods to well known Burger’s equation and global positioning system (GPS). In this thesis, we have developed two new classes of optimal eighth order without-memory methods for …nding simple roots of nonlinear equations using weight function approach and four parameters. These methods are extendable to with-memory scheme as well. We have also developed general classes of optimal derivative-free npoint iterative methods based on inverse and rational interpolations that satisfy Kung–Tarub’s Hypothesis [56]. The proposed schemes require n + 1 function evaluations to acquire the convergence order 2n and e¢ - ciency index 2 n n+1. Some dynamical aspects and basins of attraction are studied for the presented methods. Moreover, we have studied the stability analysis of the proposed methods by using the polynomial p(z) = z21. With-memory multi-step iterative methods that use information from the current and previous iterations, increase the convergence order and computational e¢ ciency of the multi-step iterative methods without-memory without any additional function evaluations. The increase in the order of convergence is based on one or more accelerator parameters which appear in the error equations of the without-memory methods. For this reason, several multi-step withand without-memory iterative methods have been developed in recent years. For a background study regarding the acceleration of convergence order via withmemorization, one may see e.g. [78,79]. In this work, we have presented two new e¢ cient with-memory iterative methods for simple roots of nonlinear equations based on newly developed optimal eighth order derivative-free without-memory methods involving four parameters. iiipoint iterative methods based on inverse and rational interpolations that satisfy Kung–Tarub’s Hypothesis [56]. The proposed schemes require n + 1 function evaluations to acquire the convergence order 2n and e¢ - ciency index 2 n n+1. Some dynamical aspects and basins of attraction are studied for the presented methods. Moreover, we have studied the stability analysis of the proposed methods by using the polynomial p(z) = z21. With-memory multi-step iterative methods that use information from the current and previous iterations, increase the convergence order and computational e¢ ciency of the multi-step iterative methods without-memory without any additional function evaluations. The increase in the order of convergence is based on one or more accelerator parameters which appear in the error equations of the without-memory methods. For this reason, several multi-step withand without-memory iterative methods have been developed in recent years. For a background study regarding the acceleration of convergence order via withmemorization, one may see e.g. [78,79]. In this work, we have presented two new e¢ cient with-memory iterative methods for simple roots of nonlinear equations based on newly developed optimal eighth order derivative-free without-memory methods involving four parameters. For this, we approximate the involved parameters with the help of Newton’s interpolating polynomials passing through best saved iterative points to construct highly e¢ cient with-memory methods. This is a novel idea since there are a few with-memory iterative methods in the literature involving four accelerators. The R-order of convergence [73] of the new with-memory methods raises from 8 to 15:5156 without additional function evaluations and e¢ ciency index is signi…cantly improved from 81=4 1:68179 to 15:515601=4 1:9847. We have also presented a general class of with-memory methods as an extension of newly developed derivative-free family of npoint without-memory optimal methods employing a self-accelerating parameter. At each iterative step, we use a suitable variation of the free parameter. The convergence order of the existing family is improved from 2n to 2n + 2n1 without additional function evaluations. An extensive comparison of our with-memory methods is done with the existing withand without-memory methods in terms of e¢ ciency index, residual error and computational order of convergence using some nonlinear equations. In this thesis, we have also established some new families of methods to …nd multiple roots of univariate nonlinear equations. Two families are of sixth order convergent methods and the other two are of optimal eighth order convergent methods. These families are based on modi…ed Newton’s method and weight function approach. An extensive convergence analysis is presented for each of the presented schemes with the help of symbolic computations on programming package Mathematica 8. In addition, we have also demonstrated the applicability of the presented schemes on some real-life problems and illustrated that the proposed methods are more e¢ cient among the available multiple root …nding techniques. The numerical tests of all the problems considered in this thesis have been carried out by using the programming package Maple 16 based on highprecision calculations on few initial estimations. Comparison of the performance of proposed and existing methods has also been carried out by drawing dynamical phase portraits of the stability behavior of the methods on the complex plane, that allows us to know how wide is the set of initial guesses that leads us to the required roots. Both of the comparisons give us complementary information that ivpoint without-memory optimal methods employing a self-accelerating parameter. At each iterative step, we use a suitable variation of the free parameter. The convergence order of the existing family is improved from 2n to 2n + 2n1 without additional function evaluations. An extensive comparison of our with-memory methods is done with the existing withand without-memory methods in terms of e¢ ciency index, residual error and computational order of convergence using some nonlinear equations. In this thesis, we have also established some new families of methods to …nd multiple roots of univariate nonlinear equations. Two families are of sixth order convergent methods and the other two are of optimal eighth order convergent methods. These families are based on modi…ed Newton’s method and weight function approach. An extensive convergence analysis is presented for each of the presented schemes with the help of symbolic computations on programming package Mathematica 8. In addition, we have also demonstrated the applicability of the presented schemes on some real-life problems and illustrated that the proposed methods are more e¢ cient among the available multiple root …nding techniques. The numerical tests of all the problems considered in this thesis have been carried out by using the programming package Maple 16 based on highprecision calculations on few initial estimations. Comparison of the performance of proposed and existing methods has also been carried out by drawing dynamical phase portraits of the stability behavior of the methods on the complex plane, that allows us to know how wide is the set of initial guesses that leads us to the required roots. Both of the comparisons give us complementary information that iv1 without additional function evaluations. An extensive comparison of our with-memory methods is done with the existing withand without-memory methods in terms of e¢ ciency index, residual error and computational order of convergence using some nonlinear equations. In this thesis, we have also established some new families of methods to …nd multiple roots of univariate nonlinear equations. Two families are of sixth order convergent methods and the other two are of optimal eighth order convergent methods. These families are based on modi…ed Newton’s method and weight function approach. An extensive convergence analysis is presented for each of the presented schemes with the help of symbolic computations on programming package Mathematica 8. In addition, we have also demonstrated the applicability of the presented schemes on some real-life problems and illustrated that the proposed methods are more e¢ cient among the available multiple root …nding techniques. The numerical tests of all the problems considered in this thesis have been carried out by using the programming package Maple 16 based on highprecision calculations on few initial estimations. Comparison of the performance of proposed and existing methods has also been carried out by drawing dynamical phase portraits of the stability behavior of the methods on the complex plane, that allows us to know how wide is the set of initial guesses that leads us to the required roots. Both of the comparisons give us complementary information that helps us to fully understand the numerical performance of the iterative schemes and to establish the conclusions." xml:lang="en_US