Raphson method, Newton

This method is probably the most popular convergence method. It is somewhat more complicated since it requires the evaluation of a derivative. It also can lead to stability problems if the initial guess is poor and if the function is highly 

Newton-Raphson amounts to using the slope of the function curve to extrapolate to the correct value. Using the bubblepoint problem as a specific example, let us define the function f Ty 

We want to find the value of T that makes f T) equal to zero i.e., we want to find the root of f T). We guess a value of temperature 7J,. Then we evaluate the function at Tq, /(7-n,. Next we evaluate the slope of the function at Tq, f[jo) = 

Ti in Eq. (4.9) is the new guess of temperature. If the curve f T) were a straight line, we would converge to the correct solution in just one iteration 

The technique requires the evaluation offthe derivative of the function jjj., with respect to temperature. In our bubblepoint example this can be 

This is a numerical method to find the root of the function, i. e., the value of the independent variable that makes the dependent function zero. Using the first order Taylor expansion, the function value at Xj+i can be expressed as 

Sufficiently close to the root x, the second order Taylor expansion gives 

the new error is proportional to the square of the old error. If the old error is 0.1, then the new error would be of the order 0.01. The next error would be of the order 10 , and so on iteratively. The value Xj i is said to converge quadratically to the root Xr. 

The basic Simpson s 1 /3 Rule rule provides the value of the integral 

Note that each segment in the integration interval [xq, x f] has two subintervals. Thus, the number of subintervals, N, is even and the number of x s being (A - -1) is odd. 

This concept benefits from the Taylor expression. Similar to the direct method given in Plate 3.1 and tabulated below, the Newton-Raphson as stated earlier can be viewed at for comparison A reference is made to Plate 3.2. Additional solution procedures 

Direct integration method (Plate 3.1) Newton-Raphson method (Fig. 3.7) 

The idea is again local linear approximation, but now we use the tangent line at a current estimate x of the root. The tangent line will cross the zero at the abscissa 

While all the previous methods use two points, the correction (2.10) is based exclusively on the local behavior of the function as shown on Fig. 2.4. 

Therefore the method has excellent convergence properties near the root (with order of convergence p = 2), but may result in meaningless estimates otherwise. In addition, the number of equivalent function evaluations is usually larger than in the secant method, which does not require the derivative but has almost the same convergence rate. Neither the Newton-Raphson, nor the secant method are recommended if the function f has an extremum near the root. You can easily construct pathological cases to understand this rule. 

In the following module if the return value of the statue flag is ER = 2, the derivative f (x) vanishes in one of the iterations, and by (2.10) the procedure breaks down. 

Example 2.1.5 Molar volume by Newton-fRaphson method 

Clearly, either the equations obtained by taking T 7 [Eq. (4.19)] or the general quartic equation obtained from Eq. (4.13) are nonlinear and multivariable. Such equations can be represented in matrix form (by defining fa), as the rs, aft element of the t column vector) as 

An alternative to the above described perturbative procedure is the multi-variable Newton Raphson method. Such methods were used in the first molecular CC calculations (Paldus et n/., 1972). Here, one attempts to choose t such that the vector f(t) defined as 

The step lengths (corrections to to) can be obtained by solving the above set of linear equations and then used to update the t amplitudes 

These values of f can then be used as a new Iq vector for the next application of Eq. (4.24). This multidimensional Newton-Raphson procedure, which involves the solution of a large number of coupled linear equations, is then repeated until the At values are sufficiently small (convergence). Given the set of f J amplitudes, Eq. (4.16) can then be used to compute E. Although the first applications of the coupled cluster method to quantum chemistry did employ this Newton Raphson scheme, the numerical problems involved 

Show that the CC equations may be iterated to yield cluster amplitudes that, when used in the energy expression, give the MBPT third-order energy expression [see discussions following Eqs. (4.20) and (4.21)]. The third-order MBPT energy expression is given in Problem 3.2. 

If both functions are continuous and differentiable, the following system can be developed [3]  

This system is the heart of the Newton-Raphson procedure. Here the partial derivatives are evaluated at x, and y,. The quantities h and k are the unknowns, and are defined as 

The system given by Equation 9.10 is very straightforward to solve, but requires the Jacobian determinant, J (see Appendix A) 

For example, if raie were to solve the system y = cosjc x = siny 

in order for the Jacobian to vanish, either sinx or cosy must be negative that is 

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An alternative, and closely related, approach is the augmented Hessian method [25]. The basic idea is to interpolate between the steepest descent method far from the minimum, and the Newton-Raphson method close to the minimum. This is done by adding to the Hessian a constant shift matrix which depends on the magnitude of the gradient. Far from the solution the gradient is large and, consequently, so is the shift d. One... 

The full Newton-Raphson method computes the full Hessian A of second derivatives and then computes a new guess at the 3X coordinate vector X, according to... 

Second Derivative Methods The Newton-Raphson Method... 

Xk) is the inverse Hessian matrix of second derivatives, which, in the Newton-Raphson method, must therefore be inverted. This cem be computationally demanding for systems u ith many atoms and can also require a significant amount of storage. The Newton-Uaphson method is thus more suited to small molecules (usually less than 100 atoms or so). For a purely quadratic function the Newton-Raphson method finds the rniriimum in one step from any point on the surface, as we will now show for our function f x,y) =x + 2/. 

The unknowns in this equation are the local coordinates of the foot (i.e. and 7]). After insertion of the global coordinates of the foot found at step 6 in the left-hand side, and the global coordinates of the nodal points in a given element in the right-hand side of this equation, it is solved using the Newton-Raphson method. If the foot is actually inside the selected element then for a quadrilateral element its local coordinates must be between -1 and +1 (a suitable criteria should be used in other types of elements). If the search is not successful then another element is selected and the procedure is repeated. 

Generalizing the Newton-Raphson method of optimization (Chapter 1) to a surface in many dimensions, the function to be optimized is expanded about the many-dimensional position vector of a point xq... 

Numerical Derivatives The results given above can be used to obtain numerical derivatives when solving problems on the computer, in particular for the Newton-Raphson method and homotopy methods. Suppose one has a program, subroutine, or other function evaluation device that will calculate/given x. One can estimate the value of the first derivative at Xq using... 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

This equation must be solved for y The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL(Ref. 224). 

The sets of equations can he solved using the Newton-Raphson method. The first form of the derivative gives a tridiagonal system of equations, and the standard routines for solving tridiagonal equations suffice. For the other two options, some manipulation is necessary to put them into a tridiagonal form (see Ref. 105). 

This represents a set of nonlinear algebraic equations that can he solved with the Newton-Raphson method. However, in this case, a viable iterative strategy is to evaluate the transport coefficients at the last value and then solve... 

The advantage of this approach is that it is easier to program than a full Newton-Raphson method. If the transport coefficients do not vary radically, then the method converges. If the method does not converge, then it maybe necessary to use the full Newton-Raphson method. 

Given values fori7(/c), solve equations h(x, 17) = 0 for x(/c). These will be m equations in m unknowns. If the equations are nonlinear, solving can be done using a variant of the Newton-Raphson method. 

Newton-Raphson method (or any or several variants to it) is used to solve the equations, the jacobian matrix and its LU fac tors are... 

Simultaneous solution by the Newton-Raphson method yields x = 0.9121, y = 0.6328. Accordingly, the fractional compositions are ... 

There are several reasons that Newton-Raphson minimization is rarely used in mac-romolecular studies. First, the highly nonquadratic macromolecular energy surface, which is characterized by a multitude of local minima, is unsuitable for the Newton-Raphson method. In such cases it is inefficient, at times even pathological, in behavior. It is, however, sometimes used to complete the minimization of a structure that was already minimized by another method. In such cases it is assumed that the starting point is close enough to the real minimum to justify the quadratic approximation. Second, the need to recalculate the Hessian matrix at every iteration makes this algorithm computationally expensive. Third, it is necessary to invert the second derivative matrix at every step, a difficult task for large systems. 

Equation 5-197 is a polynomial of the third degree, and by employing either a numerieal method or a spreadsheet paekage sueh as Mierosoft Exeel, the roots (C ) of the equation ean be determined. A developed eomputer program PROGS 1 using the Newton-Raphson method to determine was used. The Newton-Raphson method for the roots of Equation 5-197 is... 

Using the Newton-Raphson method for solving the nonlinear system of Equations 5-253 gives... 

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

Usually, modified Newton-Raphson methods with relaxation are applied. Additional iteration loops are necessary for the determination of the dynamic pressure losses in ducts and duct fittings. 

Pseudo-Newton-Raphson methods have traditionally been the preferred algorithms with ab initio wave function. The interpolation methods tend to have a somewhat poor convergence characteristic, requiring many function and gradient evaluations, and have consequently primarily been used in connection with semi-empirical and force field methods. 

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