### 数学代写|优化算法代写optimization algorithms代考| Algebraic Computing Complexity

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|优化算法代写optimization algorithms代考|Formal Computational Models

Turing machines (TM) [3] is a class of the most well-known formal models for the analysis of the problem of limited complexity. The problem is considered to be algorithmically solved if its solution can be built using the corresponding TM. It

should be noted that the class of problems that can be solved with TM is left to be solved moving from TM to another formal model [3,241, 271]. All problems of algebraic complexity are divided into two classes (the class $\mathrm{P}$ is a problem that can be solved with polynomial complexity on deterministic Turing machines (DTM), and the NP class is the class where the problems can be solved with polynomial complexity on nondeterministic Turing machines (NTM) [3]). As the characteristics of the computational complexity, computing time (number of steps that are necessary to use the solving problem of the algorithm) and memory (the amount of operating domain that is used by the algorithm) are used.

Here are some known relations between the time complexity $(T(n), n$ is the amount of input data) and by the amount of memory $(S(n))$ [249] that are obtained for TM.

Let DTIME $(T(n))$ (DSPACE $(S(n)))$ be a class of problem that suppose DTM per hour $T(n)$ (with a memory $(S(n)$ ). The classes of problems are determined likewise NTIME $(\cdot)$, NSPACE $(\cdot)$ for HTM. Then
$$\begin{gathered} \operatorname{DTIME}(T(n)) \subseteq \operatorname{NTIME}(T(n)) \ \operatorname{NTIME}(T(n)) \subseteq \operatorname{DTIME}\left(2^{O(T(n))}\right) \ \operatorname{DSPACE}(S(n)) \subseteq \operatorname{NSPACE}(S(n)) \ \operatorname{NSPACE}(S(n)) \subseteq \operatorname{DSPACE}\left(S^{2}(n)\right) \ \operatorname{NTIME}(T(n)) \subseteq \operatorname{DSPACE}(T(n)) \ \operatorname{DTIME}(T(n)) \subseteq \operatorname{DSPACE}\left(T(n) / \log _{2}(T(n))\right) \ \operatorname{NSPACE}(T(n)) \subseteq \operatorname{DTIME}\left(2^{O(S(n))}\right) \end{gathered}$$
An important example of complex problems is NP-complete problems. The problem $f$ is considered to be NP-completed if it belongs to the NP class and each NP problem can be polynomial complexity that is reduced to $f$. The central point in the theory of NP-completeness is whether or not the classes $\mathrm{P}$ and NP are congruent, in other words if the problem (from the class NP) is provided by practice that is related to problems (of class P) that can be solved. There are reasons to assume that the solution of the most complex problems of the NP class (NP-complete problems) requires (as it can be seen from the estimates) the deterministic exponential time; in other words, the classes P and NP are different. The NP-completeness of many problems is proved $[3,48]$. The difficulty is to prove that each NP problem can be polynomially transformed to this problem.

It should be noted that the definition of the NP class and the proof of the polynomial complexity of many “reset” problems had great practical importance. Together with practical valuation, it destroyed some illusions regarding the practical constructing of solving a problem that has a solution; it has been found that the existence of only one algorithm for solving a certain mass issue is not enough for

practice. On the other hand, the algorithms for which acceptable polynomial upper estimated were proved and found some practical use.

The basic possibility of classification by complexity is provided by the so-called theorems on the hierarchy. The hierarchy theorem for a given complexity (by time or memory) determines which decrease in the upper complexity estimate leads to the narrowing of the class of functions that can be computed with this complexity.

## 数学代写|优化算法代写optimization algorithms代考|Asymptotic Qualities of “Fast” Algorithms

The purpose of a lower complexity estimate construction is to prove that none of the algorithms in this computational model has less complexity of computation than the given function $\varphi(t)$. Unfortunately, the well-known “high” (nontrivial) lower estimates are perhaps the exception, not the rule.

The scheme of upper estimates of complexity constructing is as following. Based on some methods of solving problem, CA is built in a particular computational model, and it is proved that the computational complexity does not exceed some function from input data in the class. This function is called the upper estimate of the computational complexity of solving problem constructing.

There are several types of CA (which these estimates are implemented on). They are optimal, order optimal, and asymptotically optimal. Optimal CA corresponds to the case when the upper and lower boundaries are congruent. Two other types of CA concem, respectively, the estimates with the “accuracy to the multiplicative constant” and “accuracy to additive constants.” The practical use of algorithms is based on estimates that have an explicit specificity.

Consider these questions briefly. Let $A(0, X) \neq \varnothing A$ consider the computer model of sequential computations. Then
$$T\left(I_{n}(f), X, Y\right)=T_{I}\left(I_{n}(f), Y\right)+T_{a}(X, Y),$$
where $T_{I}(\cdot)=\sum_{1}^{r} \alpha_{i} n_{i}(n), T_{a}(\cdot)=\sum_{1}^{r} \alpha_{i} m_{i}(n, a), \alpha_{i}$ is a price of the $i$-operation from the model $c ; n_{i}(n), m_{i}(n, a)$ is the number of operations of the $i$-type that are necessary for the computation of the set of functionals $I_{n}(f)$ and the solution of the problem $f$ by the algorithm $a \in A$, provided that the set $I_{n}(f)$ is known; and $n$ is a number of functionals in the set. The values $T_{l}, T_{a}$ are called, respectively, informational and combinatorial (computational) complexities (solving computation) [270].

Note that the value $T$ depends essentially on $n$ and the character of the dependence $\left{n_{i}, m_{i}\right}$ from $n$. For example, by solving a system of $n$ linear equations, $A x=b$ by Gaussian elimination (for given $A, b) n_{i}=0, m_{i}=O\left(n^{3}\right), i=1,2$ (there is about the operations of addition and multiplication of two numbers).

In the general case, there is a possibility to assume that $n_{i}=O(n)$ (the functional $I_{n}(f)$ has a limited complexity) and $m_{i}(n)$ can be functions of $n$, for example, polynomial or exponential (or higher) complexity. Then the question arises on the

possibility of a solution computation with less computational complexity (see, for example, the class of NP-complete problems).

Of course, the character of dependence $m_{i}$ from $n$ is not determinative in the practical acceptability of the algorithm for solving a specific problem. It must be also considered that the constants in the functional dependences $m_{i}(n)$ can be that sort of algorithms with a lower order of complexity increasing, and advantage will be only for infinite values $n$. For example, offered algorithms of solving systems of linear algebraic equations for which $m_{i}=O\left(n^{\beta}\right), \beta<3$, have advantages over the complexity of Gaussian elimination for infinite values $n$. In addition, it is needed to pay attention to the possible loss of numerical stability of the algorithm. The fast Fourier transform (FFT) algorithm is used to multiply two numbers, and it has the complexity $O(n \log n)$ where $n$ is the number of binary digit bits for the number notation. The practical advantage of a high speed next to the traditional way of multiplication $\left(O\left(n^{2}\right)\right)$ is achieved for $n>100$.

## 数学代写|优化算法代写optimization algorithms代考|Accuracy and Complexity of Computations

The theory of analytic complexity is closely related to the theory of errors in the approximate solving problem. The value of the processing time is often determined by the requirements to the accuracy of the approximate solution; the relation of the components of the global error; the dependence of the error on the type, structure, volume of input data and their accuracy, bit grid of computer, and rounding rules; the type of error estimates; and the method of estimates constructing from below and from above. Therefore, there is a good reason to consider advisably these two characteristics: the error of the approximate solution and the process time [297, 301$]$.
Considering that it is difficult to build high lower and lower upper estimates in the given model of computation (when $E$ is a global error), some idealized models are considered that to consider only individual components of the global error (more often the errors of the method) and the influence of the individual components of computational models on error and complexity. For such incomplete models, it is possible to conclude the impossibility of constructing $\varepsilon$-solution based on this information.

The dependence of the approximate solution accuracy and the complexity of the $\varepsilon$-solution computation from the various components of the computational model will be considered next.

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## MATLAB代写

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