If the whole problem was parallelizable, we would, of course, expect the speed up to double also.Therefore, throwing in more hardware is not necessarily the optimal approach.…

Algorithms vary significantly in how parallelizable they are, ranging from easily parallelizable to completely unparallelizable.Further, a given problem may accommodate different algorithms, which may be more or less parallelizable.Some problems are easy to divide up into pieces in this way - these are called embarrassingly parallel problems.…

For example when M is a circle, or more generally a torus, such frames exist; but not when M is a 2-sphere.A manifold that does have a global moving frame is called parallelizable.Note for example how the unit directions of latitude and longitude on the Earth's surface break down as a moving frame at the north and south poles.…

This still requires the developer to know that the loop is parallelizable, but all the other work is done by the library.MSDN describes the Parallel Patterns Library as an "imperative programming model that promotes scalability and ease-of-use for developing concurrent applications."…

In an n-dimensional Riemannian manifold or pseudo-Riemannian manifold, a frame field is a set of orthonormal vector fields which forms a basis for the tangent space at each point in the manifold.This is possible globally in a continuous fashion if and only if the manifold is parallelizable.As before, frames can be specified in terms of a given coordinate basis, and in a non-flat region, some of their pairwise Lie brackets will fail to vanish.…

Switching between multiple coordinate charts is necessary because, except in trivial cases, it is not possible for a single coordinate chart to cover the entire manifold.Changing to and between general tetrads is much similar and equally necessary (except for parallelizable manifolds).…

Thus, Amdahl's law is based on the assumption of a fixed problem size: it assumes the overall workload of a program does not change with respect to machine size (i.e., the number of processors).Both laws assume the parallelizable part is evenly distributed over P processors.The impact of Gustafson's law was to shift research goals to select or reformulate problems so that solving a larger problem in the same amount of time would be possible.…

Its Euler characteristic is 0, by the product property.More generally, any parallelizable manifold, including any Lie group, has Euler characteristic 0.The Euler characteristic of any closed odd-dimensional manifold is also 0.…

Parallel transport can always be defined along curves on a surface using only the metric on the surface.Thus tangent planes along a curve can be identified using the intrinsic geometry, even when the surface itself is not parallelizable.Parallel transport along geodesics, the "straight lines" of the surface, is easy to define.…

For example a differential manifold M has a principal bundle of frames associated to its tangent bundle.A global section will exist (by definition) only when M is parallelizable, which implies strong topological restrictions.In number theory there is a (superficially different) reason to consider principal homogeneous spaces, for elliptic curves E defined over a field K (and more general abelian varieties).…

Tangent bundles are not, in general, trivial bundles: for example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem.In general, a manifold is said to be parallelizable if and only if its tangent bundle is trivial.Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles.…

It states that a small portion of the program which cannot be parallelized will limit the overall speed-up available from parallelization.A program solving a large mathematical or engineering problem will typically consist of several parallelizable parts and several non-parallelizable (sequential) parts.…

A smooth manifold is parallelizable if it admits a smooth global frame.This is equivalent to the tangent bundle being trivial.…

Reyes renders curved surfaces, such as those represented by parametric patches, by dividing them into micropolygons, small quadrilaterals each less than one pixel in size.Although many micropolygons are necessary to approximate curved surfaces accurately, they can be processed with simple, parallelizable operations.A Reyes renderer tessellates high-level primitives into micropolygons on demand, dividing each primitive only as finely as necessary to appear smooth in the final image.…

Algorithms vary significantly in how parallelizable they are, ranging from easily parallelizable to completely unparallelizable.Further, a given problem may accommodate different algorithms, which may be more or less parallelizable.…

That puts the sequential problem in P. Then, it will be in NC if and only if it is parallelizable.Many other problems have been proved to be P-complete, and therefore are widely believed to be inherently sequential.…

There is the additional topological obstruction to the existence of this structure.In particular, a noncompact world manifold must be parallelizable.…

For relatively small values of k, the Viterbi algorithm is universally used as it provides maximum likelihood performance and is highly parallelizable.Viterbi decoders are thus easy to implement in VLSI hardware and in software on CPUs with SIMD instruction sets.…

Also note: an n-dimensional manifold admits n vector fields that are linearly independent at each point if and only if its frame bundle admits a global section.In this case, the manifold is called parallelizable.…

Although this process might sound slow, it is very cache-local and highly parallelizable due to the lack of register dependencies and therefore in fact has excellent performance on modern out-of-order execution CPUs.A red-black tree for example performs much better on paper, but is highly cache-unfriendly and causes multiple pipeline and TLB stalls on modern CPUs which makes that algorithm bound by memory latency rather than CPU speed.…