应用数学的数值方法代写|NUMERICAL METHODS FOR APPLIED MATHEMATICS MATH266代写

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While working exactly is often a good starting point, it is often not enough to solve the problem completely. In many cases, we encounter integrals that cannot be evaluated analytically, or equations that are too complex to solve by hand. This is where numerical methods come in, as they allow us to approximate solutions to these problems using iterative processes that are well-suited to computers.

One of the advantages of numerical methods is that they allow us to handle real-world data that do not fit neatly into simple mathematical models. For example, if we have data from an experiment that we want to analyze, we may need to use numerical methods to fit a curve to the data or to approximate an integral that describes some aspect of the system being studied.

When using numerical methods, it is important to be aware of how errors propagate through computations. Even small errors in initial data or in the numerical algorithms can lead to significant errors in the final result. Therefore, it is important to understand the sources of errors and to develop techniques for controlling and minimizing them.

Some of the most important numerical methods that are commonly used in applied mathematics include methods for finding roots of equations, approximating integrals, and interpolating data. In each case, there are many different methods to choose from, each with its own advantages and disadvantages in terms of accuracy and efficiency. It is important to choose the right method for the problem at hand, taking into account factors such as the size of the problem, the desired level of accuracy, and the available computational resources.

Biological signaling and regulation networks often involve cycles in which a protein backbone is transformed through a collection of modified states with different numbers of phosphate groups attached. A basic cycle might be described by the reaction network:
$$\mathrm{A} \stackrel{k_1}{\rightarrow} \mathrm{B} \stackrel{k_2}{\rightarrow} \mathrm{C} \stackrel{k_3}{\rightarrow} \mathrm{A}$$
where A, B and $\mathrm{C}$ have the same protein backbone with different numbers of phosphate groups. Of course, some kind of energy input is required to maintain a cycle, which is not represented above.

Write down the stoichiometry matrix $\mathbf{S}$ for this reaction network.

$\mathbf{S}=\left(\begin{array}{ccc}-1 & 0 & 1 \ 1 & -1 & 0 \ 0 & 1 & -1\end{array}\right)$

Characterize the null space of $\mathbf{S}$ in terms of a dimension and a basis. What does this tell you about the fluxes (reaction rates) in the network at steady state? What physical interpretation can you provide for this?

The null space has dimension 1 and a basis: $(1,1,1)$. This indicates that the fluxes are equal at steady state.

Characterize the left null space of $\mathbf{S}$ in terms of a dimension and a basis. What does this tell you about the time evolution of the protein concentrations? What physical interpretation can you provide for this?

The left null space has dimension 1 and a basis: $(1,1,1)$. This indicates that the sum of the protein concentrations is constant in time.

应用数学的数值方法代写|NUMERICAL METHODS FOR APPLIED MATHEMATICS MATH226 University of Liverpool Assignment

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Assignment-daixieTM为您提供利物浦大学University of Liverpool NUMERICAL METHODS FOR APPLIED MATHEMATICS MATH226应用数学的数值方法代写代考辅导服务！

Instructions:

Applied mathematics often deals with complex problems that cannot be solved exactly using analytical methods. In such cases, numerical methods provide an alternative approach to solve these problems using computers. These methods involve approximating solutions by breaking down the problem into smaller parts and performing iterative calculations until a desired level of accuracy is achieved.

Numerical methods are commonly used in a variety of fields, such as physics, engineering, finance, and biology, to solve problems that involve real-world data. For example, in finance, numerical methods are used to calculate the value of complex financial derivatives, while in biology, these methods are used to simulate the behavior of large-scale biochemical networks.

One of the key advantages of numerical methods is their ability to handle complex problems that would be impractical or impossible to solve by hand. These methods allow for the efficient solution of problems that involve large amounts of data, complex equations, or nonlinear relationships.

However, numerical methods are not without their drawbacks. One major issue is the propagation of errors, where small errors in the initial data or calculations can lead to significant errors in the final results. To mitigate this issue, it is essential to carefully analyze the sources of error and design algorithms that minimize their impact.

In this module, you will learn about various numerical methods, including methods for finding roots, approximating integrals, and interpolating data. You will examine the advantages and disadvantages of different approaches in terms of accuracy and efficiency, and you will learn how to write computer programs to handle calculations automatically. By the end of the module, you will have a solid understanding of numerical methods and be able to apply them to solve complex problems in various fields.

Create a real $4 \times 4$ matrix with rank 2 . Propose a vector $\mathbf{b} \in \mathbb{R}^4$ such that the system of equations $\mathbf{A x}=\mathbf{b}$ has a family solutions. What is that family of solutions? Propose a vector $\mathbf{b} \in \mathbb{R}^4$ such that the system of equations $\mathbf{A x}=\mathbf{b}$ has no solutions. Explain why this is the case.

There are many possible answers. A diagonal matrix with rank 2 would be:
$$\left(\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right)$$
A value of $\mathbf{b}=(1,0,0,0)$ would give a family of solutions $\mathbf{x}=(1,0, a, b)$ where $a$ and $b$ are arbitrary. A value of $\mathbf{b}=(0,0,1,0)$ would give no solution since this vector is not in the range space of the matrix.

Write a MATLAB ${ }^6$ function that takes advantage of this sparsity pattern to compute the product of $\mathbf{S}$ with a vector. Your function should take as an input the fluxes associated with each reaction in the cycle and return the rate of change for the concentration of each species in the cycle. Be sure that your function does not compute the stoichiometry matrix explicitly.

function $y=$ sparse_mult $(x)$
$$y(1)=-x(1)+x(\text { end })$$
for $i=2:$ length $(x)$
$$y(i)=x(i-1)-x(i) ;$$
end;

Develop an expression for the characteristic polynomial of the $\mathrm{N}$ component stoichiometry matrix, S. The roots represent the eigenvalues of $\mathbf{S}$.

$$\operatorname{det}(\mathbf{S}-\lambda \mathbf{I})=(-1-\lambda) M_{11}+(-1)^{N-1} M_{1 N}=0$$
Because the minors are determinants of diagonal/upper triangular matrices, these are simple to calculate: $M_{11}=(-1-\lambda)^{N-1}, M_{1 N}=1$. Therefore the secular polynomial satisfies $(-1-\lambda)^N+(-1)^{N-1}=0$.