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Introduction

Makefile 定义了一系列的规则来指定，哪些文件需要先编译，哪些文件需要后编译，哪些文件需要重新编译，甚至进行更复杂的操作。

Makefile 带来的好处 —— “自动化编译”。写好后，只需要一个make命令，整个工程完全自动编译。 - make 是一个命令工具，一个解释 Makefile 中指令的命令工具 - Delphi： make - Visual C++: nmake - Linux: GNU make

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Finite Volumn Method

Introduction

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Chapter 3 Matrix Tree Theorem

title: Chapter 3 Matrix Tree Theorem icon: file order: 3 author: Krigo category: - MATH tag: - MATH - Graph Theory footer: Thank's myself hardwork. copyrigh: 无版权 date: 2024-05-28

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Chapter 4 Laplacian Eigenpairs

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Chapter 5 Algebraic Connectivity

Definition 5.1

IMPORTANT

The algebraic connectivity of $G$ is the second smallest eigenvalue, $\lambda_2(G)$ , of the Laplacian matrix $L(G)$ . We denote the algebraic connectivty of $G$ by $a(G)$ .

$\lambda_2(G) = a(G)$
$\lambda_2(G)$ is the second smallest eigenvalues.

Krigo大约 3 分钟

The implementation of parallel Computing in upwind scheme (OpenMP)

Version 0.1

Fortran Code

subroutine get_stream_tn_parallel
    use omp_lib
    
    implicit none
    integer :: i, thread_id,partial_Sum,total_Sum

    t1_vertex(0)%value = t0_vertex(0)%value - dt * (t0_vertex(0)%value - t0_vertex(nx)%value)/dx
    !$OMP PARALLEL PRIVATE(partial_Sum) SHARED(total_Sum)
    
    !$OMP DO
    do i = 1,nx + nghost
        t1_vertex(i)%value = t0_vertex(i)%value - dt * (t0_vertex(i)%value - t0_vertex(i-1)%value)/dx
        !thread_id = OMP_GET_THREAD_NUM()
        !print *, "Upwind process: ", thread_id
    enddo
    !$OMP end DO
    
    !$OMP DO
    do i = 1 - nghost, nx + nghost
        t0_vertex(i)%value = t1_vertex(i)%value
    enddo
    !$OMP end DO 
    !$OMP END PARALLEL
end subroutine get_stream_tn_parallel

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Baisc knowlodges of parallel computing

Two familers of parallel machines

Shared-memory architecture

These parallel machines are build up on a set of processors which have access to a common memory. Usually the name of SMP machines is used for computers based on this architecture, where SMP stands for Symmetric Multi Processing.

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CMU 15-418 Lecture 01

Course Main Theme

Designing and writing parallel programs —— scale! (规模)
- Parallel thing
  1. Decomposing(分解) the work into pieces that can safely be performed in parallel
  2. Assigning work to processors
  3. Managing communication/synchronization(同步) between the processors so that it does not limit speedup
Parallel computer hardware implemtation: how parallel computers work
- Mechanisms(机制) used to implement abstractions effeciently
  1. Performance(性能) characteristics of implementations
  2. Design trade-offs: performance vs. convenience vs. cost
Thinking about efficiency
- Fast $\neq$ Efficient
- Your program runs faster on a parallel computer, it deos not mean it is using the hardware efficiently.

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Numerical Method for Linear equations

重要

Consider the time-dependent Cauchy probelm of scalar equation,

\begin{equation} \left\{ \begin{array}{lr} u_t + au_x = 0, \qquad -\infty < x < \infty, t \geq 0 \\ u(x,0) = u_0(x) \end{array} \right. \end{equation}

Define:

$h = \Delta x$
$k = \Delta t$
$x_j = jh, j = \cdots,-1,0,1,\cdots$
$t_n = nk, n = 0,1,2,\cdots$
$x_{j+\frac{1}{2}} = x_j + h/2 = (j+\frac{1}{2})h$
$\lambda = \frac{k}{h}$

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The proof of Theorem 1.7

Theorem 1.7

The characterristic polynomial of $A(G)$ :

\begin{equation} p(A(G),\mu) = det(\mu I_{n} - A(G)) = \mu^{n} + c_1 \mu^{n-1} + c_2\mu^{n-2} + c_3\mu^{n-3} + \cdots + c_n \end{equation}

$c_1$ = 0;
$-c_2$ is the number of edges of $G$ ;
$-c_3$ is twice the number of triangles in $G$ ;
$c_n = (-1)^{n}\det(A(G))$ , 这一条性质只能用在 $c_n$ 为多项式的最后一项的时候，对于 $c_4 \thicksim c_{n-1}$ 请使用 Theorem 1.18

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