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Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given. Directions: List three sequences of transformations that take pre-image ABCD to image ABCD. For important special cases, extensions of the general results are presented. Understand that a sequence of rigid transformations including translations, reflections, and. Common properties and results on convergence acceleration and stability are given. rigid transformations is congruent to the original figure. As illustration, two new sequence transformations are derived. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. Here, we review known Levin-type sequence transformations and put them in a common theoretical framework. Special cases of such Levin-type transformations belong to the most powerful currently known extrapolation methods for scalar sequences and series. Then, nonlinear sequence transformations are obtained. As shown first by Levin, it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the n are simple functions of a few sequence elements sn.
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Such remainder estimates provide an easy-to-use possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations. Transformations that depend not only on the sequence elements or partial sums sn but also on an auxiliary sequence of the so-called remainder estimates n are of Levin-type if they are linear in the sn, and nonlinear in the n. The basic idea is to construct from a given sequence.
SEQUENCE OF TRANSFORMATIONS SERIES
Translate 5 units in the positive Y directionĬf(bx+a)+d = Translate by a units in the negative X direction, then scale by a factor of 1/b parallel to the X-axis, then scale by a factor of c parallel to the Y-axis, then translate by d units in the positive Y direction.Ĭ+d = Scale by a factor of 1/a parallel to the X-axis, then translate by b units in the negative X direction, then scale by a factor of c parallel to the Y axis, then translate by d units in the positive Y direction.Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. Scale by a factor of 3 parallel to the Y axis Scale by a factor of 1/2 parallel to the X axis Translate 4 units in the positive X direction So scale parallel to the X axis by a factor of 1/2, then move left by 2 units. Hence, the original point becomes x= (8/2)-2 = 2ĭescribe the transformation of 3f(2x-4) + 5. If we want to do scaling first, we need to factorise into f 2(x+2). Hence, the original point becomes x= (8-4)/2 = 2 For f(2x+4), we do translation first, then scaling. Let’s look at this example to illustrate the difference: Example 1. Knowing whether to scale or translate first is crucial to getting the correct transformation. Move left by 4 units, then scale parallel to the X axis by a factor of 1/2. In the transformation of graphs, knowing the order of transformation is important. Let’s look at this example to illustrate the difference:įor f(2x+4), we do translation first, then scaling.
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In the transformation of graphs, knowing the order of transformation is important.
SEQUENCE OF TRANSFORMATIONS HOW TO
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