统计代写|数值分析和优化代写numerical analysis and optimazation代考|Error Testing and Order of Convergence

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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

统计代写|数值分析和优化代写numerical analysis and optimazation代考|Error Testing and Order of Convergence

Often an algorithm first generates an approximation to the solution and then improves this approximation again and again. This is called an iterative numerical process. Often the calculations in each iteration are the same. However, sometimes the calculations are adjusted to reach the solution faster. If the process is successful, the approximate solutions will converge to a solution. Note that it is a solution, not the solution. We will see this beautifully illustrated when considering the fractals generated by Newton’s and Halley’s methods.

More precisely, convergence of a sequence is defined as follows. Let $x_{0}, x_{1}$, $x_{2}, \ldots$ be a sequence (of approximations) and let $x$ be the true solution. We define the absolute error in the $n^{\text {th }}$ iteration as
$$\epsilon_{\mathrm{n}}=x_{n}-x$$
The sequence converges to the limit $x$ of the sequence if
$$\lim {n \rightarrow \infty} \epsilon{n}=0$$

Note that convergence of a sequence is defined in terms of absolute error.
There are two forms of error testing, one using a target absolute accuracy $\epsilon_{t}$, the other using a target relative error $\delta_{t}$. In the first case the calculation is terminated when
In the second case the calculation is terminated when
Both methods are flawed under certain circumstances. If $x$ is large, say $10^{20}$, and $u=10^{-16}$, then $\epsilon_{n}$ is never likely to be much less than $10^{4}$, so condition $(1.5)$ is unlikely to be satisfied if $\epsilon_{t}$ is chosen too small even when the process converges. On the other hand, if $\left|x_{n}\right|$ is very small, then $\delta_{t}\left|x_{n}\right|$ may underflow and test (1.6) may never be satisfied (unless the error becomes exactly zero). As (1.5) is useful when $(1.6)$ is not, and vice versa, they are combined into a mixed error test. A target error $\eta_{t}$ is prescribed and the calculation is terminated when
$$\left|\epsilon_{n}\right| \leq \eta_{t}\left(1+\left|x_{n}\right|\right)$$
If $\left|x_{n}\right|$ is small, $\eta_{t}$ is regarded as target absolute error, or if $\left|x_{n}\right|$ is large $\eta_{t}$ is regarded as target relative error.

Tests such as $(1.7)$ are used in modern numerical software, but we have not addressed the problem of estimating $\epsilon_{n}$, since the true value $x$ is unknown. The simplest formula is
$$\epsilon_{n} \approx x_{n}-x_{n-1}$$
However, theoretical research has shown that in a wide class of numerical methods, cases arise where adjacent values in an approximation sequence have the same value, but are both the incorrect answer. Test (1.8) will cause the algorithm to terminate too early with an incorrect solution.
A safer estimate is
$$\epsilon_{n} \approx\left|x_{n}-x_{n-1}\right|+\left|x_{n-1}-x_{n-2}\right|$$
but again research has shown that even the approximations of three consecutive iterations can all be the same for certain methods, so (1.9) might not work either. However, in many problems convergence can be tested independently, for example when the inverse of a function can be easily calculated (calculating the $k^{\text {th }}$ power as compared to taking the $k^{\text {th }}$ root). Error and convergence testing should always be fitted to the underlying problem.

统计代写|数值分析和优化代写numerical analysis and optimazation代考|Computational Complexity

A well-designed algorithm should not only be robust, and have a fast rate of convergence, but should also have a reasonable computational complexity. That is, the computation time shall not increase prohibitingly with the size of the problem, because the algorithm is then too slow to be used for large problems.

Suppose that some operation, call it $\odot$, is the most expensive in a particular algorithm. Let $n$ be the size of the problem. If the number of operations of the algorithm can be expressed as $O[f(n)]$ operations of type $\odot$, then we say that the computational complexity is $f(n)$. In other words, we neglect the less expensive operations. However, less expensive operations cannot be neglected, if a large number of them need to be performed for each expensive operation.
For example, in matrix calculations the most expensive operations are multiplications of array elements and array references. Thus in this case the

operation $\odot$ may be defined to be a combination of one multiplication and one or more array references. Let’s consider the multiplication of $n \times n$ matrices $A=\left(A_{i j}\right)$ and $B=\left(B_{i j}\right)$ to form a product $C=\left(C_{i j}\right)$. For each element in $C$, we have to calculate
$$C_{i j}=\sum_{k=1}^{n} A_{i k} B_{k j}$$
which requires $n$ multiplications (plus two array references per multiplication). Since there are $n^{2}$ elements in $C$, the computational complexity is $n^{2} \times n=n^{3}$.
Note that processes of lower complexity are absorbed into higher complexity ones and do not change the overall computational complexity of an algorithm. This is the case, unless the processes of lower complexity are performed a large number of times.

For example, if an $n^{2}$ process is performed each time an $n^{3}$ process is performed then, because of
$$O\left(n^{2}\right)+O\left(n^{3}\right)=O\left(n^{3}\right)$$
the overall computational complexity is still $n^{3}$. If, however, the $n^{2}$ process was performed $n^{2}$ times each time the $n^{3}$ process was performed, then the computational complexity would be $n^{2} \times n^{2}=n^{4}$.

统计代写|数值分析和优化代写numerical analysis and optimazation代考|Condition

The condition of a problem is inherent to the problem whichever algorithm is used to solve it. The condition number of a numerical problem measures the asymptotically worst case of how much the outcome can change in proportion to small perturbations in the input data. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. The condition number is a property of the problem and not of the different algorithms that can be used to solve the problem.

As an example consider the problem where a graph crosses the line $x=0$. Naively one could draw the graph and measure the coordinates of the crossover points. Figure $1.1$ illustrates two cases. In the left-hand problem it would be easier to measure the crossover points, while in the right-hand problem the crossover points lie in a region of candidates. A better (or worse) algorithm would be to use a higher (or lower) resolution. In the chapter on non-linear systems we will encounter many methods to find the roots of a function that is the points where the graph of a function crosses the line $x=0$.

统计代写|数值分析和优化代写numerical analysis and optimazation代考|Error Testing and Order of Convergence

εn=Xn−X

|εn|≤这吨(1+|Xn|)

εn≈Xn−Xn−1

εn≈|Xn−Xn−1|+|Xn−1−Xn−2|

统计代写|数值分析和优化代写numerical analysis and optimazation代考|Computational Complexity

C一世j=∑ķ=1n一种一世ķ乙ķj

广义线性模型代考

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MATLAB代写

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