# 数学代写|最优化作业代写optimization theory代考|MATH683

## 数学代写|最优化作业代写optimization theory代考|Exercise Set

Exercise 2.8.1 (Easy)
Calculate the following integral:
$$\int_{-2}^3 x^4 d x$$
first by the trapezoidal method, then Simpson’s method, and finally Gauss-Legendre quadrature with 3 points ( $n=2$ ), by using in the three cases the bounds of the integration interval as calculation interval. Comment.
Exercise 2.8.2 (Easy)
The error function $\operatorname{erf}(x)$ is defined by
$$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_0^x \exp \left(-t^2\right) d t=1-\frac{2}{\sqrt{\pi}} \int_x^{+\infty} \exp \left(-t^2\right) d t$$

1. Using Gauss-Legendre quadrature with 4 points, give an approximation of both integrals for $x=1$. For the calculation of the second integral, it will be useful to think about the choice of the bound to use to replace $+\infty$ and the influence of the value of the term $\exp \left(-t^2\right)$. It may be useful to make a few trials to better understand this influence. Discuss the results thus obtained. Deduce an approximation of $\operatorname{erf}(x)$.
2. Do the same estimation with the first integral by Simpson’s method with a step $h=0.1$. Compare the result thus obtained.

Remark: The error function is used in many problems of physics. Consider a solid plate where the one-dimensional heat transfer is ruled by Fourier’s law
$$\frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2}$$
subjected to a constant temperature at the interface $T(x=0, t)=T_s$. Let $T(x, t=0)=$ $T_0$ he the initial temperature. The temperature along time (Incropera and DeWitt 1996) is expressed as
$$\frac{T(x, t)-T_s}{T_0-T_s}=\operatorname{erf}\left(\frac{x}{2 \sqrt{\alpha t}}\right)$$ where α is the thermal diffusivity.

## 数学代写|最优化作业代写optimization theory代考|Equation Solving by Iterative Methods

The problem is to develop adequate methods to find the solutions of the general equation
$$f(x)=0$$
The roots will be noted $\alpha_i$. In a given number of cases, $f(x)$ will be supposed to be a polynomial of degree $n$
$$f(x)=x^n+a_1 x^{n-1}+\cdots+a_{n-1} x^1+a_n$$
but some methods are applicable to any type of function.
There exist four main classes of methods (Gritton et al. 2001) to find the roots of a nonlinear equation of the form
$$f(x)=0$$

1. Local methods that require an initial estimation of the root (e.g. successive substitutions, Newton). Frequently, local methods are designed to search for only one real root of the nonlinear equation, even if multiple roots exist. Nevertheless, they are often very robust and nearly always converge (e.g. Newton, quasi-Newton), but they present the drawback to need to provide an initial estimation sufficiently close to the root.
2. Global methods that find a root from an arbitrary initial value (e.g. homotopy). They are adapted to the search of multiple roots.
3. Interval methods that find all the roots in a specified domain of $x$ (e.g. dichotomy, regula falsi). They are robust but slow.
4. Graphical methods or spreadsheet that uses a graphical view of $f(x)$ in a specified domain of $x$.

Some of the methods presented below (Graeffe, Bernoulli, Bairstow) are more interesting from a mathematical point of view than for real applications, but they present a historical interest and their exposure may promote future ideas.

# 最优化代考

## 数学代写|最优化作业代写optimization theory代考|习题集

.

$$\int_{-2}^3 x^4 d x$$

$$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_0^x \exp \left(-t^2\right) d t=1-\frac{2}{\sqrt{\pi}} \int_x^{+\infty} \exp \left(-t^2\right) d t$$

$$\frac{\partial T}{\partial t}=\alpha \frac{\partial^2 T}{\partial x^2}$$

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