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# 数学代写|数值分析代写Numerical analysis代考|MAT12004 Solution of systems of equations

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## 数学代写数值分析代写Numerical analysis代考|Solution of systems of equations

Consider the linear system $A \boldsymbol{x}=\boldsymbol{b}$ where $A \in \mathbb{R}^{n \times n}$ and $\boldsymbol{x}$ and $\boldsymbol{b}$ are column vectors of size $n$. According to Theorem $2.3$ there exist a permutation matrix $P \in \mathbb{R}^{n \times n}$, a unit lower triangular matrix $L \in \mathbb{R}^{n \times n}$ and an upper triangular matrix $U \in \mathbb{R}^{n \times n}$ such that $P A=L U$. Having obtained the LU factorisation of the matrix $P A$, the solution of the system of linear equations $A \boldsymbol{x}=\boldsymbol{b}$ is straightforward: multiplying both sides of $A \boldsymbol{x}=\boldsymbol{b}$ on the left by the permutation matrix $P$, we obtain that
$$P A \boldsymbol{x}=P \boldsymbol{b} ;$$
equivalently, $L U \boldsymbol{x}=P \boldsymbol{b}$. On defining $\boldsymbol{y}=U \boldsymbol{x}$ we can rewrite (2.28) as the following coupled set of linear equations:
$$L \boldsymbol{y}=P \boldsymbol{b}, \quad U \boldsymbol{x}=\boldsymbol{y}$$

## 数学代写|数值分析代写Numerical analysis代考|Computational work

In this section we shall show that the work involved in factorising an $n \times n$ matrix in the form $A=L U$ is proportional to $n^3$. An estimate of the amount of computational work of this kind is important in deciding in advance how long a calculation would take for a very large matrix, and is also useful in comparing different methods for the solution of a given problem. For example, in the next chapter we shall derive a method for solving a system of equations with a symmetric positive definite matrix; that method requires only half the amount of work involved in the standard LU factorisation algorithm which takes no account of symmetry.

Accurate estimates of the time taken by a computation are very complicated and require some detailed knowledge of the computer being used. The estimates which we shall give are simple but crude; they are normally good enough for the types of comparisons we have just mentioned.

## 数学代写数值分析代写Numerical analysis代考|Solution of systems of equations

$$P A \boldsymbol{x}=P \boldsymbol{b} ;$$

$$L \boldsymbol{y}=P \boldsymbol{b}, \quad U \boldsymbol{x}=\boldsymbol{y}$$

## MATLAB代写

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