Convergence rate quadratic

I - FH) r on a quadratic surface. Here I is the unit matrix and denotes the nth cycle. The ultimate convergence rate is governed by the magnitude of the largest eigenvalue of the matrix (I - FH). This will be... 

In the above equation, the norm is usually the Euclidean norm. We have a linear convergence rate when 0 is equal to 1. Superlinear convergence rate refers to the case where 0=1 and the limit is equal to zero. When 0=2 the convergence rate is called quadratic. In general, the value of 0 depends on the algorithm while the value of the limit depends upon the function that is being minimized. 

If the quadratic model or one of its modifications are used, the local region presents no difficulties. All methods converge rapidly since they effectively reduce to Newton s method in the local region. In this section we briefly discuss local convergence rates and stopping criteria. 

Quasi-Newton methods may be used instead of our full Newton iteration. We have used the fast (quadratic) convergence rate of our Newton algorithm as a numerical check to discriminate between periodic and very slowly changing quasi-periodic trajectories the accurate computed elements of the Jacobian in a Newton iteration can be used in stability computations for the located periodic trajectories. There are deficiencies in the use of a full Newton algorithm, such as its sometimes small radius of convergence (Schwartz, 1983). Several other possibilities for continuation methods also exist (Doedel, 1986 Seydel and Hlavacek, 1986). The pseudo-arc length continuation was sufficient for our calculations. 

Steepest descent is simple to implement and requires modest storage, O(k) however, progress toward a minimum may be very slow, especially near a solution. The convergence rate of SD when applied to a convex quadratic function, as in Eq. [22], is only linear. The associated convergence ratio is no greater than [(k - 1)/(k + l)]4 where k, the condition number, is the ratio of largest to smallest eigenvalues of A ... 

Figure 12 Steepest descent and conjugate gradient quantities that affect the convergence rate for quadratic functions (see text for the distinct context of these functions).

Unfortunately, there is a disparity between this theoretical convergence result and the practical behavior of the method in general. Thus, modifications of the classic Newton iteration are essential for guaranteeing global convergence, with quadratic convergence rate near the solution. 

Perhaps a month later 1 decided to compute the r s in the transcendental case numerically. This problem was even slower to compute than the quadratic one. Again, it became apparent that the r s converged geometrically, and altogether amazingly, the convergence rate was the same 4.669 that I remembered by virtue of my efforts to fit it. 

The order of accuracy of the upwind scheme can be improved by using a higher-order accurate scheme such as QUICK (quadratic upwind interpolation for convective kinematics).The concentration at an interface is interpolated by means of a parabola instead of a straight line. The use of QUICK or similar methods may, however, complicate implementation of boundary conditions or lessen the convergence rate of the solution algorithm. 

Whenever Newton s method converges, its convergence rate is quadratic. 

For roots of multiplicity 1, Newton s method has quadratic convergence when Xn is close to the solution. If the multiplicity of a root is greater than 1, the convergence rate becomes linear. Knowing the multiplicity k of the root one can obtain quadratic convergence using... 

Newton s method is thus quadratically convergent A detailed analysis of the convergence rate of Newton s method in terms of error vectors is given in Section 11.5.2. 

The chief merit of this method is its rapid rate of convergence starting with a set of good initial guesses. Since the discussion of its quadratic convergence... 

As you can see from the third and fourth columns in the table the rate of convergence of Newton s method is superlinear (and in fact quadratic) for this function. 

Determine the relative rates of convergence for (1) Newton s method, (2) a finite difference Newton method, (3) quasi-Newton method, (4) quadratic interpolation, and (5) cubic interpolation, in minimizing the following functions ... 

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