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# 数学代写|数值分析代写Numerical analysis代考|STAT360 From Gaussian elimination to LU factorization

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## 数学代写|数值分析代写Numerical analysis代考|From Gaussian elimination to LU factorization

In this section, we show that the standard GE without row swapping can be done through elementary matrix operations, which naturally leads to an LU factorization without pivoting.

Let us briefly review GE. Given a system of linear equations $\mathbf{A x}=\mathbf{b}$, where $\mathbf{A} \epsilon$ $\mathbb{R}^{n \times n}$ is nonsingular, and $\mathbf{b} \in \mathbb{R}^n$, GE updates the coefficient matrix and the right-hand side by elementary row operations, one column at a time, eliminating all entries below the main diagonal. This produces a new linear system $\mathbf{U x}=\mathbf{c}$, where $\mathbf{U}$ is an upper triangular matrix. Then back substitution is used to solve $\mathbf{U x}=\mathbf{c}$ for $\mathbf{x}$. Note that this procedure does not change the solution to $\mathbf{A x}=\mathbf{b}$, that is, it is the same $\mathbf{x}$ in $\mathbf{A x}=\mathbf{b}$ and $\mathbf{U x}=\mathbf{c}$.

After reviewing this process below, we will show how it can be “saved” in an LU factorization $\mathbf{A}=\mathbf{L U}$, with $\mathbf{L}$ lower triangular and $\mathbf{U}$ upper triangular. The $\mathbf{U}$ is the upper triangular matrix that results from the GE procedure. The $\mathbf{L}$ is a matrix with ones on the diagonal, and whose entries are easily determined from the GE procedure (although the mathematical reasons why it works are a little more complicated). In your linear algebra course, the GE procedure could be performed in many different ways, so long as the Gauss rules are not violated. However, for a computer algorithm, it is necessary to avoid ambiguity and to have a more precise algorithm: we will always move left to right, and use the diagonal entry to eliminate entries below it.

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

GE breaks down at Step $i$ if the $i$ th diagonal entry of the current (modified) coefficient matrix, referred to as the pivot, is zero (or close to 0 ), since there is no way to eliminate a nonzero entry using a zero pivot. A zero pivot may arise at any step of GE, making the algorithm fail, even if $\mathbf{A}$ is nonsingular and a unique solution to $\mathbf{A x}=\mathbf{b}$ exists. Consider, for example, the linear equation
$$\left(\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array}\right)\left(\begin{array}{l} x_1 \ x_2 \end{array}\right)=\left(\begin{array}{l} 0 \ 2 \end{array}\right) .$$
There is a unique solution $\left(x_1=2\right.$ and $\left.x_2=0\right)$, but GE fails at the first step.

## 数学代写|数值分析代写NUMERICAL ANALYSIS代 考|PIVOTING

GE 在 Step 崩溃 $i$ 如果 $i$ 当前的第 th 个对角线条目 modified系数矩阵，称为主元，为零orcloseto0，因为无法使用零主元诮除非零项。零主元可能 出现在 $G E$ 的任何一步，使算法失败，即使 $\mathbf{A}$ 是非奇异的并且是唯一的解决方案 $\mathbf{A x}=\mathbf{b}$ 存在。例如，考虑线性方程
$$\left(\begin{array}{lll} 0 & 11 & 0 \end{array}\right)\left(x_1 x_2\right)=\left(\begin{array}{ll} 02 \end{array}\right) .$$

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