Optimization of Linear Algebra Core Function Framework on Multicore Processors

The multi-core processor (FPGI for short) is a new type of high-performance computing system developed by PMC Corporation in the U.S. Its most important feature is that it can achieve high accuracy, high speed and low power consumption, and its core functions mainly include data set, state space and data compression[1]. In this chip design, researchers will conduct a comparative analysis under different parameters. Firstly, the relationship between the indicators will be understood; secondly, the related factors such as the mutual influence and force between the performance indicators will be understood; finally, the optimal control variables will be obtained by comparison, and the best optimization scheme will be developed according to these results so as to achieve the expected target requirements, and the optimization scheme of the multi-core processor will be finally determined[2].

Multi-core type design is a computer-based bionics, which uses computers for data processing to operate instead of the human brain, so as to accomplish tasks such as analysis, modeling and optimization of complex systems. At present, many companies at home and abroad have started to study and apply multi-channel development technology[3,4]. It has a strong data acquisition capability and high-speed computing speed, and it has been widely used in the military field. Microcontroller can run stably in various types of media environment, which makes it become one of the current research hotspots; at the same time, as the technology continues to develop and deepen, people's requirements for multi-core processor performance are getting higher and more difficult gradually. Therefore, how to optimize the design to meet the high performance and low power consumption is the current problem to be solved, and the core function of the multi-core processor as an important parameter, in the design and optimization process need to consider its stability, high reliability and low complexity[5].

Multi-core processor mapping refers to the operation on input variables with certain mathematical step functions, instead of directly mapping the input objects containing characteristic variables such as unknowns and value domains to the corresponding objects[6]. The core functions based on multi-core processors are nonlinear integers, which are stochastic in nature and require consideration of the complexity of the relationship between objects during the solution process. It is commonly used in solving the problem: first find a parameter (i.e., solution space); then calculate the optimal feasible solution according to the required obtained value. However, it can be optimized by using different types of multicore processors to achieve the best results due to different practical application environments and strict requirements[7,8]. The function is a vector structure composed of real and complex numbers, so the optimization of the function is to find the optimal solution after considering these parameters together[9].

1) Vector operations: multiplication, division and square root operations can be expressed as a single general operation as shown in Equation (1): (1) VEC=(xi(*)yip(+)zi),i∈{0,1,2,3},(*)∈{*,/},(+)∈{+,−},p∈{0,5,1}. \matrix{{VEC = \left({{x_i}\left(* \right)y_i^p\left(+ \right){z_i}} \right),i \in \left\{{0,1,2,3} \right\},} \hfill \cr {\left(* \right) \in \left\{{*,/} \right\},\left(+ \right) \in \left\{{+, -} \right\},p \in \left\{{0,5,1} \right\}.} \hfill \cr}

The expression can be converted as shown in Equation (2): (2) VEC= (2log2 xi(+)((log2 yi>>q))(+)zi),i∈{0,1,2,3},(+)∈{+,−},q∈{0,1} \matrix{{VEC = \,\left({{2^{{{\log}_2}\,{x_i}\left(+ \right)\left({\left({{{\log}_{2\,}}{y_i} > > q} \right)} \right)}}\left(+ \right){z_i}} \right),} \hfill \cr {i \in \left\{{0,1,2,3} \right\},\left(+ \right) \in \left\{{+, -} \right\},q \in \left\{{0,1} \right\}} \hfill \cr}

2) Trigonometric operations: According to the structure of VEC above, trigonometric functions, hyperbolas and their corresponding inverse trigonometric functions and inverse hyperbolas can be realized[10]. Here only the operations of sine, cosine and tangent functions are performed, which are calculated by Taylor series expansion of trigonometric functions, as shown in Equation (3): (3) TRG=c0xk0(+)c1xk1(k)c2xk2,(+)c3xk3(+)c4xk4,(+)∈{+,−}, \matrix{{TRG = {c_0}{x^{{k_0}}}\left(+ \right){c_1}{x^{{k_1}}}\left(k \right){c_2}{x^{{k_2}}},} \hfill \cr {\left(+ \right){c_3}{x^{{k_3}}}\left(+ \right){c_4}{x^{{k_4}}},\left(+ \right) \in \left\{{+, -} \right\},} \hfill \cr}

The expressions can be converted as shown in Equation (4): (4) VEC= (2log2 xi(+)((log2 yi>>q))(+)zi),i∈{0,1,2,3},(+)∈{+,−},q∈{1,0} \matrix{{VEC = \,\left({{2^{{{\log}_2}\,{x_i}\left(+ \right)\left({\left({{{\log}_{2\,}}{y_i} > > q} \right)} \right)}}\left(+ \right){z_i}} \right),} \hfill \cr {i \in \left\{{0,1,2,3} \right\},\left(+ \right) \in \left\{{+, -} \right\},q \in \left\{{1,0} \right\}} \hfill \cr}

3) Power exponential and logarithmic operations: Power operations can also be converted into multiplication operations in the logarithmic region as shown in Equation (5): (5) POW=xy=2(y*log2 x) POW = {x^y} = {2^{\left({y*{{\log}_2}\,x} \right)}}

The logarithm based on an arbitrary base can be calculated by converting Equation (6) into a multiplication method: (6) LOG= logb x=k*log2 x,k=1/log2 b. \matrix{{LOG = \,{{\log}_b}\,x = k*{{\log}_2}\,x,} \hfill \cr {k = 1/{{\log}_2}\,b.} \hfill \cr}

4) Logarithmic conversion: Any fixed-point number α can be approximated by log2X, so that the operation of any two numbers can be converted into a logarithmic operation, which reduces the complexity of the calculation[11]. Any 32-bit binary fixed-point logarithm x can be expressed as shown in Equation (7): (7) x=zkzk−1⋯z2z1z0, z−1 z−2 z−3⋯zj,x=∑i=jk2izi,zi=0/1 \matrix{{x = {z_k}{z_{k - 1}} \cdots {z_2}{z_1}{z_0},\,{z_{- 1}}\,{z_{- 2}}\,{z_{- 3}} \cdots {z_j},} \hfill \cr {x = \sum\nolimits_{i = j}^k {{2^i}{z_i},{z_i} = 0/1}} \hfill \cr}

If the highest bit Z_k is 1, x can be expressed as shown in Equation (8): (8) x=2k(1+f),f=∑i=jk−12i−kzi,0≤f≤1 \matrix{{x = {2^k}\left({1 + f} \right),} \hfill \cr {f = \sum\nolimits_{i = j}^{k - 1} {{2^{i - k}}{z_i},}} \hfill \cr {0 \le f \le 1} \hfill \cr}

Its logarithm can then be expressed as shown in Equation (9): (9) log2x=k+log2(1+f) {\log _2}x = k + {\log _2}\left({1 + f} \right) k is the integer part of the logarithm and log₂(1+f) is its trailing part in the range [0,1). In the method of segmented linear approximation, the nonlinear group log₂(1+f) can be approximated as shown in Equation (10). (10) log2(1+f)≅ai*f+bi {\log _2}\left({1 + f} \right) \cong {a_i}*f + {b_i}

Based on the area and power consumption considerations, an optimal coefficient is chosen and Equation (10) can be written as (11): (11) ai*f+bi≅f+[invi0(f)>>pi]+[invi1(f)>>qi]+[invi1(f)>>ri]+biΔ__log2(1+f). \matrix{{{a_i}*f + {b_i} \cong f + \left[{inv_i^0\left(f \right) > > {p_i}} \right] +} \hfill \cr {\left[{inv_i^1\left(f \right) > > {q_i}} \right] + \left[{inv_i^1\left(f \right) > > {r_i}} \right] +} \hfill \cr {{b_i} \buildrel \Delta \over = {{\log}_2}\left({1 + f} \right).} \hfill \cr} a_i is calculated from the sum of the shift results, invij inv_i^j is a bit-flip function, and p_i, q_i and r_iv are the number of shifts defined for each approximation region i. Therefore, log₂X can be expressed as shown in Equation (12). (12) log2 x=k+log2(1+f) {\log _2}\,x = k + {\log _2}\left({1 + f} \right)

5) Opposite number conversion: Opposite number conversion converts the result of the logarithmic part of the calculation into a fixed number of principle as follows:

x is a logarithmic value, and x can be expressed as shown in equation (13). (13) x=k+f, 0≤f≤1. x = k + f,\,0 \le f \le 1. k and f denote the integer and trailing digits of x, respectively. Therefore, the inverse number conversion of x can be expressed as shown in Equation (14). (14) 2x=2k*2f {2^x} = {2^k}*{2^f}

In the above equation, it is just a shift operation, and the nonlinearity needs to be approximated linearly by segments, as shown in equation (15). (15) 2f≅ai*f+bi {2^f} \cong {a_i}*f + {b_i}

In the above equation, both a_i and b_i are approximation coefficients defined inside each approximation region i, 0 ≤ f ≤ 1. Since the error of the inverse number conversion basically spans the whole region, f is divided into 8 approximation regions, and Equation (15) can be expressed as Equation (16): (16) ai*f+bi≅f+[invi0(f)>>pi]+[invi1(f)>>qi]+bΔ__2f \matrix{{{a_i}*f + {b_i} \cong f + \left[{inv_i^0\left(f \right) > > {p_i}} \right] +} \hfill \cr {\left[{inv_i^1\left(f \right) > > {q_i}} \right] + b \buildrel \Delta \over = {2^f}} \hfill \cr}

SMT-PAAG consists of 16 processing units (Processing El-ement, PE) interconnected to form a 4 × 4 two-dimensional array, also includes a front-end processor, four co-processors, two schedulers and two storage management, the overall structure of the system is shown in Figure 1[12,13].

Figure 1

The connection between the coprocessor (ACCL) and the surrounding processing units (PEs) is shown in Figure 1. There are 4 external PEs connected to the coprocessor, each of which sends an opcode, an operation valid signal and operands a, b. There is a preprocessing module in the coprocessor that sequentially adjusts the operations of the 4 PEs and sends them to the pipeline for calculation[14].

The linear algebraic model on a multicore processor is a time-varying iterative optimization method whose main purpose is to improve the algorithm solution accuracy by imposing constraints on the inputs, outputs and performance metrics[15]. The multicore processor is shown in Figure 2. In practice, we would like to find an optimal solution to achieve optimal control. However, the desired goal may not be ideal or even have errors or defects, which may lead to inefficient operation of the system.

Figure 2

The multicore processor can operate, control and analyze multiple discrete individual objects, and it is capable of automatic operation in different environments with its intelligent microcomputer with multiple functions. It uses advanced algorithm design methods to optimize the processing process. It is widely used in systems for complex data calculation and real-time monitoring due to the advantages of high accuracy and scalability of multi-channel functions; at the same time, it can use multiple single-point discretization solvers to transform complex problems into several simple algebraic equations or vector matrices, which simplifies the complexity and computation and improves the computation accuracy[16]. The internal structure of the multi-core processor is shown in Figure 3.

Figure 3

Multicore hybrid network data transmission refers to the use of different communication protocols to exchange information between multiple nodes, and its specific data transmission structure is shown in Figure 4.

Figure 4

There are four typical multi-core processors: Hydra processors, Piranha processors, IBM Power series multi-core processors and Cell processors.

Hydra processor: Hydra is a typical high-order inertial system, which consists of multiple processors. The processor has better fault tolerance, and it can be optimized to a greater extent when it performs special processing on some specific objects. Its main function is to generate error signals during motion to improve dynamic performance; at the same time, it can also provide a certain amount of information to help control strategy adjustment and optimize decision-making. The Hydra processor structure is shown in Figure 5.

Figure 5

Piranha processor: Piranhaov et al. (1996) proposed a classical solver-based optimization method that replaces the traditional linear mapping calculus by using coefficients with nonlinear terms and a finite order as an input variable. The basic idea is that the function can be approximately fitted to zero under certain conditions. The algorithm can effectively solve the local optimal solution problem; it is also able to overcome the disadvantages such as the limitation of the number of iterations and the non-convergence problem, and it has high global convergence. The Piranha processor structure is shown in Figure 6.

Figure 6

IBM Power Series Multicore Processor: IBMPower Series Multicore Processor (ARM) is a non-linear design language with virtualization. It is implemented in the form of graphics, functions, etc. on different interfaces to simulate the trajectory of things moving in the real world. The chip is capable of real-time monitoring and automatic tracking functions of objects, and realizes the control and processing capabilities of human-computer interaction interface by connecting with computer communication system; at the same time, it can complete various operation processes according to user instructions; it can also realize the processing and storage of different data information according to the actual requirements of users[17]. The processing flow chart of the data packet of the processor is shown in Figure 7.

Figure 7

Cell processor: Cell is a high-performance, highly integrated, high-speed and low-energy products, its core function mainly refers to the different functional structures and in a certain frequency range can work properly. Cell processor structure is shown in Figure 8.

Figure 8

In the process of optimizing the design of a multicore processor, the most important thing is how to translate the optimal objective function into something that is easy to understand and test. This requires specific analysis of different types of optimization objects. This places high demands on the multicore computing drive system, the key technologies involved in various integrated development environments, and other aspects. Therefore, it is very difficult to accomplish the above problems and achieve their optimal performance. At the same time, there are interactions and constraints among the units, which leads to some difficult challenges in the design process. Therefore, researchers must perform specific analysis for different types of optimization objects before determining the best solution.

In the long history of modern science and technology development, computer technology has become an indispensable part of the development of human society. With the continuous improvement of computer technology and manufacturing process level, people are conducting in-depth research on intelligent machines. The multi-core processor is a new type of high-speed precision computing system (FATS), which is one of the most cutting-edge, advanced and efficient and stable control algorithms in the past 20 years. It has the advantages of small size, high efficiency and multiple functions; it is widely used in military field and computer network field due to its powerful nonlinear planning ability and good global optimization performance, and it plays an increasingly important role in the military field. As a complex system, the performance of multicore processor is also uneven or unpredictable, so the optimization design method of fitting polynomial algorithm is very important. And in practical applications, the performance of a multicore processor as a nonlinear process is also affected by a variety of factors. Therefore, in this paper, we propose a simulation calculation method that combines an improved linear programming problem solver and a first-order single-loop model.