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4,101 | Replacing each dot product with a unique scalar gives |
4,102 | and these equations can be expressed as the single matrix equation |
4,103 | This matrix equation relates the scalar components of a in the n basis with those in the e basis . Each matrix element cjk is the direction cosine relating nj to ek. The term direction cosine refers to the cosine of the angle between two unit vectors, which is also equal to their dot product. Therefore, |
4,104 | By referring collectively to e1, e2, e3 as the e basis and to n1, n2, n3 as the n basis, the matrix containing all the cjk is known as the "transformation matrix from e to n", or the "rotation matrix from e to n" , or the "direction cosine matrix from e to n" . The properties of a rotation matrix are such that its inv... |
4,105 | The properties of a direction cosine matrix, C are: |
4,106 | The advantage of this method is that a direction cosine matrix can usually be obtained independently by using Euler angles or a quaternion to relate the two vector bases, so the basis conversions can be performed directly, without having to work out all the dot products described above. |
4,107 | By applying several matrix multiplications in succession, any vector can be expressed in any basis so long as the set of direction cosines is known relating the successive bases. |
4,108 | With the exception of the cross and triple products, the above formulae generalise to two dimensions and higher dimensions. For example, addition generalises to two dimensions as |
4,109 | The cross product does not readily generalise to other dimensions, though the closely related exterior product does, whose result is a bivector. In two dimensions this is simply a pseudoscalar |
4,110 | A seven-dimensional cross product is similar to the cross product in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products. |
4,111 | Vectors have many uses in physics and other sciences. |
4,112 | In abstract vector spaces, the length of the arrow depends on a dimensionless scale. If it represents, for example, a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 new... |
4,113 | Often in areas of physics and mathematics, a vector evolves in time, meaning that it depends on a time parameter t. For instance, if r represents the position vector of a particle, then r gives a parametric representation of the trajectory of the particle. Vector-valued functions can be differentiated and integrated by... |
4,114 | The position of a point x = in three-dimensional space can be represented as a position vector whose base point is the origin |
4,115 | Given two points x = , y = their displacement is a vector |
4,116 | The velocity v of a point or particle is a vector, its length gives the speed. For constant velocity the position at time t will be |
4,117 | Acceleration a of a point is vector which is the time derivative of velocity. Its dimensions are length/time2. |
4,118 | Force is a vector with dimensions of mass×length/time2 and Newton's second law is the scalar multiplication |
4,119 | Work is the dot product of force and displacement |
4,120 | In the language of differential geometry, the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a contravariant vector to be a tensor of contravariant rank one. Alternatively, a contravariant vector is defined to be a tangent vector... |
4,121 | Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip and gain a minus sign. A transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change the orientation of space. A vector which gains a minus sign whe... |
4,122 | One example of a pseudovector is angular velocity. Driving in a car, and looking forward, each of the wheels has an angular velocity vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the reflection of this angular velocity vector points to the right, b... |
4,123 | This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties. See parity . |
4,124 | MIMD architectures may be used in a number of application areas such as computer-aided design/computer-aided manufacturing, simulation, modeling, and as communication switches. MIMD machines can be of either shared memory or distributed memory categories. These classifications are based on how MIMD processors access me... |
4,125 | An example of MIMD system is Intel Xeon Phi, descended from Larrabee microarchitecture. These processors have multiple processing cores that can execute different instructions on different data. |
4,126 | Most parallel computers, as of 2013, are MIMD systems. |
4,127 | In shared memory model the processors are all connected to a "globally available" memory, via either software or hardware means. The operating system usually maintains its memory coherence. |
4,128 | From a programmer's point of view, this memory model is better understood than the distributed memory model. Another advantage is that memory coherence is managed by the operating system and not the written program. Two known disadvantages are: scalability beyond thirty-two processors is difficult, and the shared mem... |
4,129 | There are many examples of shared memory : UMA , COMA . |
4,130 | MIMD machines with shared memory have processors which share a common, central memory. In the simplest form, all processors are attached to a bus which connects them to memory. This means that every machine with shared memory shares a specific CM, common bus system for all the clients. |
4,131 | For example, if we consider a bus with clients A, B, C connected on one side and P, Q, R connected on the opposite side,
any one of the clients will communicate with the other by means of the bus interface between them. |
4,132 | MIMD machines with hierarchical shared memory use a hierarchy of buses to give processors access to each other's memory. Processors on different boards may communicate through inter-nodal buses. Buses support communication between boards. With this type of architecture, the machine may support over nine thousand proce... |
4,133 | In distributed memory MIMD machines, each processor has its own individual memory location. Each processor has no direct knowledge about other processor's memory. For data to be shared, it must be passed from one processor to another as a message. Since there is no shared memory, contention is not as great a problem w... |
4,134 | Examples of distributed memory include MPP , COW and NUMA . The former is complex and expensive: Many super-computers coupled by broad-band networks. Examples include hypercube and mesh interconnections. COW is the "home-made" version for a fraction of the price. |
4,135 | In an MIMD distributed memory machine with a hypercube system interconnection network containing four processors, a processor and a memory module are placed at each vertex of a square. The diameter of the system is the minimum number of steps it takes for one processor to send a message to the processor that is the far... |
4,136 | In an MIMD distributed memory machine with a mesh interconnection network, processors are placed in a two-dimensional grid. Each processor is connected to its four immediate neighbors. Wrap around connections may be provided at the edges of the mesh. One advantage of the mesh interconnection network over the hypercube ... |
4,137 | Such machines exploit data level parallelism, but not concurrency: there are simultaneous computations, but each unit performs the exact same instruction at any given moment . SIMD is particularly applicable to common tasks such as adjusting the contrast in a digital image or adjusting the volume of digital audio. Mos... |
4,138 | Modern graphics processing units are often wide SIMD implementations, capable of branches, loads, and stores on 128 or 256 bits at a time. |
4,139 | The first use of SIMD instructions was in the ILLIAC IV, which was completed in 1966. |
4,140 | SIMD was the basis for vector supercomputers of the early 1970s such as the CDC Star-100 and the Texas Instruments ASC, which could operate on a "vector" of data with a single instruction. Vector processing was especially popularized by Cray in the 1970s and 1980s. Vector processing architectures are now considered sep... |
4,141 | The first era of modern SIMD computers was characterized by massively parallel processing-style supercomputers such as the Thinking Machines CM-1 and CM-2. These computers had many limited-functionality processors that would work in parallel. For example, each of 65,536 single-bit processors in a Thinking Machines CM-2... |
4,142 | The current era of SIMD processors grew out of the desktop-computer market rather than the supercomputer market. As desktop processors became powerful enough to support real-time gaming and audio/video processing during the 1990s, demand grew for this particular type of computing power, and microprocessor vendors turne... |
4,143 | The first widely deployed desktop SIMD was with Intel's MMX extensions to the x86 architecture in 1996. This sparked the introduction of the much more powerful AltiVec system in the Motorola PowerPC and IBM's POWER systems. Intel responded in 1999 by introducing the all-new SSE system. Since then, there have been sever... |
4,144 | All of these developments have been oriented toward support for real-time graphics, and are therefore oriented toward processing in two, three, or four dimensions, usually with vector lengths of between two and sixteen words, depending on data type and architecture. When new SIMD architectures need to be distinguished ... |
4,145 | An application that may take advantage of SIMD is one where the same value is being added to a large number of data points, a common operation in many multimedia applications. One example would be changing the brightness of an image. Each pixel of an image consists of three values for the brightness of the red , green... |
4,146 | With a SIMD processor there are two improvements to this process. For one the data is understood to be in blocks, and a number of values can be loaded all at once. Instead of a series of instructions saying "retrieve this pixel, now retrieve the next pixel", a SIMD processor will have a single instruction that effectiv... |
4,147 | Another advantage is that the instruction operates on all loaded data in a single operation. In other words, if the SIMD system works by loading up eight data points at once, the add operation being applied to the data will happen to all eight values at the same time. This parallelism is separate from the parallelism p... |
4,148 | To remedy problems 1 and 5, RISC-V's vector extension uses an alternative approach: instead of exposing the sub-register-level details to the programmer, the instruction set abstracts them out as a few "vector registers" that use the same interfaces across all CPUs with this instruction set. The hardware handles all al... |
4,149 | An order of magnitude increase in code size is not uncommon, when compared to equivalent scalar or equivalent vector code, and an order of magnitude or greater effectiveness is achievable with Vector ISAs. |
4,150 | ARM's Scalable Vector Extension takes another approach, known in Flynn's Taxonomy as "Associative Processing", more commonly known today as "Predicated" SIMD. This approach is not as compact as Vector processing but is still far better than non-predicated SIMD. Detailed comparative examples are given in the Vector pro... |
4,151 | Small-scale SIMD became popular on general-purpose CPUs in the early 1990s and continued through 1997 and later with Motion Video Instructions for Alpha. SIMD instructions can be found, to one degree or another, on most CPUs, including IBM's AltiVec and SPE for PowerPC, HP's PA-RISC Multimedia Acceleration eXtensions... |
4,152 | Intel's AVX-512 SIMD instructions process 512 bits of data at once. |
4,153 | SIMD instructions are widely used to process 3D graphics, although modern graphics cards with embedded SIMD have largely taken over this task from the CPU. Some systems also include permute functions that re-pack elements inside vectors, making them particularly useful for data processing and compression. They are also... |
4,154 | Adoption of SIMD systems in personal computer software was at first slow, due to a number of problems. One was that many of the early SIMD instruction sets tended to slow overall performance of the system due to the re-use of existing floating point registers. Other systems, like MMX and 3DNow!, offered support for dat... |
4,155 | SIMD on x86 had a slow start. The introduction of 3DNow! by AMD and SSE by Intel confused matters somewhat, but today the system seems to have settled down and newer compilers should result in more SIMD-enabled software. Intel and AMD now both provide optimized math libraries that use SIMD instructions, and open sourc... |
4,156 | Apple Computer had somewhat more success, even though they entered the SIMD market later than the rest. AltiVec offered a rich system and can be programmed using increasingly sophisticated compilers from Motorola, IBM and GNU, therefore assembly language programming is rarely needed. Additionally, many of the systems t... |
4,157 | SIMD within a register, or SWAR, is a range of techniques and tricks used for performing SIMD in general-purpose registers on hardware that does not provide any direct support for SIMD instructions. This can be used to exploit parallelism in certain algorithms even on hardware that does not support SIMD directly. |
4,158 | It is common for publishers of the SIMD instruction sets to make their own C/C++ language extensions with intrinsic functions or special datatypes guaranteeing the generation of vector code. Intel, AltiVec, and ARM NEON provide extensions widely adopted by the compilers targeting their CPUs. |
4,159 | The GNU C Compiler takes the extensions a step further by abstracting them into a universal interface that can be used on any platform by providing a way of defining SIMD datatypes. The LLVM Clang compiler also implements the feature, with an analogous interface defined in the IR. Rust's packed_simd crate uses this in... |
4,160 | C++ has an experimental interface std::experimental::simd that works similarly to the GCC extension. LLVM's libcxx seems to implement it. For GCC and libstdc++, a wrapper library that builds on top of the GCC extension is available. |
4,161 | Microsoft added SIMD to .NET in RyuJIT. The System.Numerics.Vector package, available on NuGet, implements SIMD datatypes. Java also has a new proposed API for SIMD instructions available in OpenJDK 17 in an incubator module. It also has a safe fallback mechanism on unsupported CPUs to simple loops. |
4,162 | Instead of providing an SIMD datatype, compilers can also be hinted to auto-vectorize some loops, potentially taking some assertions about the lack of data dependency. This is not as flexible as manipulating SIMD variables directly, but is easier to use. OpenMP 4.0+ has a #pragma omp simd hint. This OpenMP interface ha... |
4,163 | Consumer software is typically expected to work on a range of CPUs covering multiple generations, which could limit the programmer's ability to use new SIMD instructions to improve the computational performance of a program. The solution is to include multiple versions of the same code that uses either older or newer S... |
4,164 | FMV, manually coded in assembly language, is quite commonly used in a number of performance-critical libraries such as glibc and libjpeg-turbo. Intel C++ Compiler, GNU Compiler Collection since GCC 6, and Clang since clang 7 allow for a simplified approach, with the compiler taking care of function duplication and sele... |
4,165 | As using FMV requires code modification on GCC and Clang, vendors more commonly use library multi-versioning: this is easier to achieve as only compiler switches need to be changed. Glibc supports LMV and this functionality is adopted by the Intel-backed Clear Linux project. |
4,166 | In 2013 John McCutchan announced that he had created a high-performance interface to SIMD instruction sets for the Dart programming language, bringing the benefits of SIMD to web programs for the first time. The interface consists of two types: |
4,167 | Instances of these types are immutable and in optimized code are mapped directly to SIMD registers. Operations expressed in Dart typically are compiled into a single instruction without any overhead. This is similar to C and C++ intrinsics. Benchmarks for 4×4 matrix multiplication, 3D vertex transformation, and Mandelb... |
4,168 | McCutchan's work on Dart, now called SIMD.js, has been adopted by ECMAScript and Intel announced at IDF 2013 that they are implementing McCutchan's specification for both V8 and SpiderMonkey. However, by 2017, SIMD.js has been taken out of the ECMAScript standard queue in favor of pursuing a similar interface in WebAss... |
4,169 | Emscripten, Mozilla's C/C++-to-JavaScript compiler, with extensions can enable compilation of C++ programs that make use of SIMD intrinsics or GCC-style vector code to the SIMD API of JavaScript, resulting in equivalent speedups compared to scalar code. It also supports the WebAssembly 128-bit SIMD proposal. |
4,170 | It has generally proven difficult to find sustainable commercial applications for SIMD-only processors. |
4,171 | One that has had some measure of success is the GAPP, which was developed by Lockheed Martin and taken to the commercial sector by their spin-off Teranex. The GAPP's recent incarnations have become a powerful tool in real-time video processing applications like conversion between various video standards and frame rates... |
4,172 | A more ubiquitous application for SIMD is found in video games: nearly every modern video game console since 1998 has incorporated a SIMD processor somewhere in its architecture. The PlayStation 2 was unusual in that one of its vector-float units could function as an autonomous DSP executing its own instruction stream,... |
4,173 | A later processor that used vector processing is the Cell Processor used in the Playstation 3, which was developed by IBM in cooperation with Toshiba and Sony. It uses a number of SIMD processors and is geared towards the huge datasets required by 3D and video processing applications. It differs from traditional ISAs ... |
4,174 | Ziilabs produced an SIMD type processor for use on mobile devices, such as media players and mobile phones. |
4,175 | Larger scale commercial SIMD processors are available from ClearSpeed Technology, Ltd. and Stream Processors, Inc. ClearSpeed's CSX600 has 96 cores each with two double-precision floating point units while the CSX700 has 192. Stream Processors is headed by computer architect Bill Dally. Their Storm-1 processor conta... |
4,176 | Arcade system boards have used specialized graphics circuits since the 1970s. In early video game hardware, RAM for frame buffers was expensive, so video chips composited data together as the display was being scanned out on the monitor. |
4,177 | A specialized barrel shifter circuit helped the CPU animate the framebuffer graphics for various 1970s arcade video games from Midway and Taito, such as Gun Fight , Sea Wolf , and Space Invaders . The Namco Galaxian arcade system in 1979 used specialized graphics hardware that supported RGB color, multi-colored sprites... |
4,178 | The Atari 2600 in 1977 used a video shifter called the Television Interface Adaptor. Atari 8-bit computers had ANTIC, a video processor which interpreted instructions describing a "display list"—the way the scan lines map to specific bitmapped or character modes and where the memory is stored . 6502 machine code subro... |
4,179 | The NEC µPD7220 was the first implementation of a personal computer graphics display processor as a single large-scale integration integrated circuit chip. This enabled the design of low-cost, high-performance video graphics cards such as those from Number Nine Visual Technology. It became the best-known GPU until the... |
4,180 | In 1984, Hitachi released ARTC HD63484, the first major CMOS graphics processor for personal computers. The ARTC could display up to 4K resolution when in monochrome mode. It was used in a number of graphics cards and terminals during the late 1980s. In 1985, the Amiga was released with a custom graphics chip including... |
4,181 | In 1987, the IBM 8514 graphics system was released. It was one of the first video cards for IBM PC compatibles to implement fixed-function 2D primitives in electronic hardware. Sharp's X68000, released in 1987, used a custom graphics chipset with a 65,536 color palette and hardware support for sprites, scrolling, and m... |
4,182 | IBM introduced its proprietary Video Graphics Array display standard in 1987, with a maximum resolution of 640×480 pixels. In November 1988, NEC Home Electronics announced its creation of the Video Electronics Standards Association to develop and promote a Super VGA computer display standard as a successor to VGA. S... |
4,183 | In 1991, S3 Graphics introduced the S3 86C911, which its designers named after the Porsche 911 as an indication of the performance increase it promised. The 86C911 spawned a variety of imitators: by 1995, all major PC graphics chip makers had added 2D acceleration support to their chips. Fixed-function Windows accelera... |
4,184 | Throughout the 1990s, 2D GUI acceleration evolved. As manufacturing capabilities improved, so did the level of integration of graphics chips. Additional application programming interfaces arrived for a variety of tasks, such as Microsoft's WinG graphics library for Windows 3.x, and their later DirectDraw interface for... |
4,185 | In the early- and mid-1990s, real-time 3D graphics became increasingly common in arcade, computer, and console games, which led to increasing public demand for hardware-accelerated 3D graphics. Early examples of mass-market 3D graphics hardware can be found in arcade system boards such as the Sega Model 1, Namco System... |
4,186 | The term "GPU" was coined by Sony in reference to the 32-bit Sony GPU in the PlayStation video game console, released in 1994. |
4,187 | In the PC world, notable failed attempts for low-cost 3D graphics chips included the S3 ViRGE, ATI Rage, and Matrox Mystique. These chips were essentially previous-generation 2D accelerators with 3D features bolted on. Many were pin-compatible with the earlier-generation chips for ease of implementation and minimal cos... |
4,188 | OpenGL appeared in the early '90s as a professional graphics API, but originally suffered from performance issues which allowed the Glide API to become a dominant force on the PC in the late '90s. These issues were quickly overcome and the Glide API fell by the wayside. Software implementations of OpenGL were common du... |
4,189 | Microsoft began to work more closely with hardware developers and started to target the releases of DirectX to coincide with those of the supporting graphics hardware. Direct3D 5.0 was the first version of the API to gain widespread adoption in the gaming market, and it competed directly with more-hardware-specific, of... |
4,190 | Nvidia was first to produce a chip capable of programmable shading: the GeForce 3. Each pixel could now be processed by a short program that could include additional image textures as inputs, and each geometric vertex could likewise be processed by a short program before it was projected onto the screen. Used in the Xb... |
4,191 | In October 2002, with the introduction of the ATI Radeon 9700 , the world's first Direct3D 9.0 accelerator, pixel and vertex shaders could implement looping and lengthy floating point math, and were quickly becoming as flexible as CPUs, yet orders of magnitude faster for image-array operations. Pixel shading is often u... |
4,192 | With the introduction of the Nvidia GeForce 8 series and new generic stream processing units, GPUs became more generalized computing devices. Parallel GPUs are making computational inroads against the CPU, and a subfield of research, dubbed GPU computing or GPGPU for general purpose computing on GPU, has found applicat... |
4,193 | Nvidia's CUDA platform, first introduced in 2007, was the earliest widely adopted programming model for GPU computing. OpenCL is an open standard defined by the Khronos Group that allows for the development of code for both GPUs and CPUs with an emphasis on portability. OpenCL solutions are supported by Intel, AMD, Nvi... |
4,194 | In 2010, Nvidia partnered with Audi to power their cars' dashboards, using the Tegra GPU to provide increased functionality to cars' navigation and entertainment systems. Advances in GPU technology in cars helped advance self-driving technology. AMD's Radeon HD 6000 series cards were released in 2010, and in 2011 AMD r... |
4,195 | The PS4 and Xbox One were released in 2013; they both use GPUs based on AMD's Radeon HD 7850 and 7790. Nvidia's Kepler line of GPUs was followed by the Maxwell line, manufactured on the same process. Nvidia's 28 nm chips were manufactured by TSMC in Taiwan using the 28 nm process. Compared to the 40 nm technology from ... |
4,196 | In 2018, Nvidia launched the RTX 20 series GPUs that added ray-tracing cores to GPUs, improving their performance on lighting effects. Polaris 11 and Polaris 10 GPUs from AMD are fabricated by a 14 nm process. Their release resulted in a substantial increase in the performance per watt of AMD video cards. AMD also rele... |
4,197 | In 2019, AMD released the successor to their Graphics Core Next microarchitecture/instruction set. Dubbed RDNA, the first product featuring it was the Radeon RX 5000 series of video cards. |
4,198 | The company announced that the successor to the RDNA microarchitecture would be incremental . AMD unveiled the Radeon RX 6000 series, its RDNA 2 graphics cards with support for hardware-accelerated ray tracing. The product series, launched in late 2020, consisted of the RX 6800, RX 6800 XT, and RX 6900 XT. The RX 6700 ... |
4,199 | The PlayStation 5 and Xbox Series X and Series S were released in 2020; they both use GPUs based on the RDNA 2 microarchitecture with incremental improvements and different GPU configurations in each system's implementation. |
4,200 | Intel first entered the GPU market in the late 1990s, but produced lackluster 3D accelerators compared to the competition at the time. Rather than attempting to compete with the high-end manufacturers Nvidia and ATI/AMD, they began integrating Intel Graphics Technology GPUs into motherboard chipsets, beginning with the... |
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