CPPLapack_Tutorial-Japanese
cpptest培训手册parasoft c test traing presentation [template]共106页
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▪ Assertions
▪ Application Monitoring
▪ Coverage
▪ Runtime Error Detection
Parasoft Test – Overview
▪ Workflow is natural ▪ Work is task-based and focused on results
▪ THE KEY is uniform implementation in the group
▪ If implemented as a group practice, the result is huge productivity improvements
4 Cross reference 2 Check into Source
3 Execute nightly
Togetherness
▪ Single, uniform interface
▪ Same interface for all languages
▪ Support for all technologies needed in development
▪ Best Practices
Agenda
▪ Part 3
▪ Part 4
▪ Unit Test
▪ Concepts and Strategies ▪ Generating Unit Tests ▪ Stubs ▪ File Scope and Projects on Unit Test
project management)
Lapack中文帮助手册手册
PACKLAPACK简介1.1概述LAPACK API支持两种形式：一是标准的ANSI C；另一种是标准的FORTRAN77。
LAPACK 是开源的，官方网站是：/lapack/。
每个LAPACK例程都有四个形式，具体描述如下：精度例程前缀REAL精度SREAL DOUBLE精度DCOMPLEX单精度CCOMPLEX双精度Z下面例程是经过优化的。
DGETRF对一般矩阵进行LU分解。
DGETRS线性方程组求解。
DGETRI用LU分解求解一般矩阵的逆矩阵。
DGEQRF对一般矩阵进行QR分解。
DGELQF对一般矩阵进行LQ分解。
DPOTRF对对称正定矩阵进行Cholesky分解。
DPOTRS对线性方程组（对称正定）求解。
1.2函数的命名规则：LAPACK里的每个函数名已经说明了该函数的使用规则。
所有函数都是以XYYZZZ的形式命名，对于某些函数，没有第六个字符，只是XYYZZ的形式。
第一个字母X代表以下的数据类型：S REAL，单精度实数D DOUBLE PRECISION，双精度实数C COMPLEX，单精度复数Z COMPLEX*16或DOUBLE COMPLEX注：在新版LAPACK中含有使用重复迭代法的函数DSGESV和ZCDESV。
头2个字母表示使用的精度：DS输入数据是double双精度，算法使用单精度ZC输入数据是complex*16，算法使用complex单精度复数接下面两个字母YY代表数组的类型。
BD bidiagonal，双对角矩阵DI diagonal，对角矩阵GB general band，一般带状矩阵GE general(i.e.,unsymmetric,in some cases rectangular)，一般情形（即非对称，在有些情形下为矩形）GG general matrices,generalized problem(i.e.,a pair of general matrices)，一般矩阵，广义问题（即一对一般矩阵）GT general tridiagonal，一般三对角矩阵HB(complex)Hermitian band，（复数）厄尔米特带状阵HE(complex)Hermitian，（复数）厄尔米特矩阵HG upper Hessenberg matrix,generalized problem (i.e a Hessenberg and atriangular matrix)，上海森伯格矩阵，广义问题（即一个海森伯格矩阵和一个三角矩阵）HP (complex)Hermitian,packed storage ，（复数）压缩储存的厄尔米特矩阵HS upper Hessenberg ，上海森博格矩阵OP (real)orthogonal,packed storage ，（实数）压缩储存的正交阵OR (real)orthogonal ，（实数）正交阵PBsymmetric or Hermitian positive definite band ，对称或厄尔米特正定带状矩阵PO symmetric or Hermitian positive definite ，对称或厄尔米特正定矩阵PPsymmetric or Hermitian positive definite,packed storage ，压缩储存的对称或厄尔米特正定矩阵PTsymmetric or Hermitian positive definite tridiagonal ，对称或厄尔米特正定三对角阵SB (real)symmetric band ，（实数）对称带状阵SP symmetric,packed storage ，压缩储存的对称阵ST (real)symmetric tridiagonal ，（实数）对称三对角阵SY symmetric ，对称阵TB triangular band ，三角形带状矩阵TGtriangular matrices,generalized problem (i.e.,a pair of triangular matrices)，三角形矩阵，广义问题（即一对三角形阵）TP triangular,packed storage ，压缩储存的三角形阵TRtriangular (or in some cases quasi-triangular)，三角形阵（在某些情形下为类三角形阵）TZ trapezoidal ，梯形阵UN (complex)unitary ，（复数）酉矩阵UP (complex)unitary,packed storage ，（复数）压缩储存的酉矩阵最后三个字母ZZZ 代表计算方法。
Rcpp包使用指南说明书
Writing a package that uses Rcpp Dirk Eddelbuettel a and Romain François ba ;b https://romain.rbind.io/This version was compiled on July3,2023This document provides a short overview of how to use Rcpp(Ed-delbuettel et al.,2023a;Eddelbuettel and François,2011;Eddelbuettel, 2013)when writing an R package.It shows how usage of the function Rcpp.package.skeleton which creates a complete and self-sufficient ex-ample package using Rcpp.All components of the directory tree created by Rcpp.package.skeleton are discussed in detail.This document thereby complements the Writing R Extensions manual(R Core Team,2021)which is the authoritative source on how to extend R in general.Rcpp|package|R|C++1.IntroductionRcpp(Eddelbuettel et al.,2023a;Eddelbuettel and François,2011; Eddelbuettel,2013)is an extension package for R which offers an easy-to-use yet featureful interface between C++and R.However,it is somewhat different from a traditional R package because its key component is a C++library.A client package that wants to make use of the Rcpp features must link against the library provided by Rcpp.It should be noted that R has only limited support for C(++)-level dependencies between packages(R Core Team,2021).The LinkingTo declaration in the package DESCRIPTIONfile allows the client package to retrieve the headers of the target package(here Rcpp),but support for linking against a library is not provided by R and has to be added manually.This document follows the steps of the Rcpp.package.skeleton function to illustrate a recommended way of using Rcpp from a client package.We illustrate this using a simple C++function which will be called by an R function.We strongly encourage the reader to become familiar with the material in the Writing R Extensions manual(R Core Team,2021), as well as with other documents on R package creation such as Leisch(2008).Given a basic understanding of how to create R package,the present document aims to provide the additional information on how to use Rcpp in such add-on packages.ing Rcpp.package.skeleton2.1.Overview.Rcpp provides a function Rcpp.package.skeleton,modeled after the base R func-tion package.skeleton,which facilitates creation of a skeleton package using Rcpp.Rcpp.package.skeleton has a number of arguments documented on its help page(and similar to those of package.skeleton).The main argument is thefirst one which provides the name of the package one aims to create by invoking the function.An illustration of a call using an argument mypackage is providedbelow.Using Rcpp.package.skeleton is by far the simplest approach as it fulfills two roles.It creates the complete set offiles needed for a package,and it also includes the different components needed for using Rcpp that we discuss in the following sections.2.2.C++code.If the attributes argument is set to TRUE1,the following C++file is included in the src/directory:Thefile defines the simple rcpp_hello_world function that uses a few Rcpp classes and returns a List.This function is preceded by the Rcpp::export attribute to automatically handle argument conversion because R has to be taught how to e.g.handle the List class.Rcpp.package.skeleton then invokes compileAttributes on the package,which generates the RcppExports.cppfile(where we indented thefirst two lines for the more compact display here): 1Setting attributes to TRUE is the default.This document does not cover the behavior of Rcpp.package.skeleton when attributes is set to FALSE as we try to encourage package de-veloppers to use attributes.https:///package=Rcpp Rcpp Vignette|July3,2023|1–4Thisfile defines a function with the appropriate calling con-vention,suitable for.Call.It needs to be regenerated each time functions exposed by attributes are modified.This is the task of the compileAttributes function.A discussion on attributes is be-yond the scope of this document and more information is available in the attributes vignette(Allaire et al.,2023).2.3.R code.The compileAttributes also generates R code that uses the C++function.This is also a generatedfile so it should not be modified manually, rather regenerated as needed by compileAttributes.2.4.DESCRIPTION.The skeleton generates an appropriate DESCRIPTIONfile,using both Imports:and LinkingTo for Rcpp:Rcpp.package.skeleton adds the three last lines to the DESCRIPTIONfile generated by package.skeleton.The Imports declaration indicates R-level dependency between the client package and Rcpp;code from the latter is being imported into the package described here.The LinkingTo declaration indi-cates that the client package needs to use headerfiles exposed by Rcpp.2.5.Now optional:Makevars and Makevars.win.This behaviour changed with Rcpp release0.11.0.Thesefiles used to be manda-tory,now they are merely optional.We will describe the old settingfirst as it was in use for a few years.The new standard,however,is much easier and is described below.2.6.Releases up until0.10.6.Unfortunately,the LinkingTo decla-ration in itself was not enough to link to the user C++library of Rcpp.Until more explicit support for libraries is added to R,ones needes to manually add the Rcpp library to the PKG_LIBS variable in the Makevars and Makevars.winfiles.(This has now changed with release0.11.0;see below).Rcpp provides the unexported function Rcpp:::LdFlags()to ease theprocess:The Makevars.win is the equivalent,targetingwindows.2.7.Releases since0.11.0.As of release0.11.0,this is no longer needed as client packages obtain the required code from Rcpp via explicit function registration.The user does not have to do anything.This means that PKG_LIBS can now be empty—unless some client libraries are needed.For example,RcppCNPy needs com-pression support and hence uses PKG_LIBS=-lz.Similarly,whena third-party library is required,it can and should be set here.SPACE.The Rcpp.package.skeleton function also cre-ates afile NAMESPACE.Thisfile serves three purposes.First,it ensure that the dynamic library contained in the package we are creating via Rcpp.package.skeleton will be loaded and thereby made avail-able to the newly created R package.Second,it declares which functions should be globally visible from the namespace of this package.As a reasonable default,we export all functions.Third,it instructs R to import a symbol from Rcpp.This sets up the import of all registered function and,together with the2|https:///package=Rcpp Eddelbuettel and FrançoisImports:statement in DESCRIPTION,provides what is needed for client packages to access Rcpp functionality.2.9.Helpfiles.Also created is a directory man containing two help files.One is for the package itself,the other for the(single)R function being provided and exported.The Writing R Extensions manual(R Core Team,2021)provides the complete documentation on how to create suitable content for helpfiles.2.10.mypackage-package.Rd.The helpfile mypackage-package.Rd can be used to describe the new package(and we once again indented somelines):2.11.rcpp_hello_world.Rd.The helpfile rcpp_hello_world.Rd serves as documentation for the ex-ample Ring modulesThis document does not cover the use of the module argument of Rcpp.package.skeleton.It is covered in the modules vignette (Eddelbuettel and François,2023).4.Further examplesThe canonical example of a package that uses Rcpp is the RcppEx-amples(Eddelbuettel and François,2019)package.RcppExam-ples contains various examples of using Rcpp.Hence,the RcppEx-amples package is provided as a template for employing Rcpp in packages.Other CRAN packages using the Rcpp package are RcppAr-madillo(Eddelbuettel et al.,2023b),and minqa(Bates et al., 2014).Several other packages follow older(but still supported and appropriate)instructions.They can serve examples on how to get data to and from C++routines,but should not be considered templates for how to connect to Rcpp.The full list of packages using Rcpp can be found at the CRAN page of Rcpp.5.Other compilersLess experienced R users on the Windows platform frequently ask about using Rcpp with the Visual Studio toolchain.That is simply not possible as R is built with the gcc compiler.Different compilers have different linking conventions.These conventions are particularly hairy when it comes to using C++.In short,it is not possible to simply drop sources(or headerfiles)from Rcpp into a C++project built with Visual Studio,and this note makes no attempt at claiming otherwise.Rcpp is fully usable on Windows provided the standard Win-dows toolchain for R is used.See the Writing R Extensions manual (R Core Team,2021)for details.6.SummaryThis document described how to use the Rcpp package for R and C++integration when writing an R extension package.The use of the Rcpp.package.skeleton was shown in detail,and references to further examples were provided.ReferencesAllaire JJ,Eddelbuettel D,François R(2023).Rcpp Attributes.Vignette included in R package Rcpp,URL https:///package=Rcpp.Eddelbuettel and François Rcpp Vignette|July3,2023|3Bates D,Mullen KM,Nash JC,Varadhan R(2014).minqa:Derivative-freeoptimization algorithms by quadratic approximation.R package version1.2.4,URL https:///package=minqa.Eddelbuettel D(2013).Seamless R and C++Integration with e R!Springer,New Y ork.ISBN978-1-4614-6867-7.Eddelbuettel D,François R(2011).“Rcpp:Seamless R and C++Integration.”Journal of Statistical Software,40(8),1–18.doi:10.18637/jss.v040.i08.URL https:///10.18637/jss.v040.i08.Eddelbuettel D,François R(2019).RcppExamples:Examples using Rcpp tointerface R and C++.R package version0.1.9,URL https://CRAN.R-Project.org/package=RcppExamples.Eddelbuettel D,François R(2023).Exposing C++functions and classes withRcpp modules.Vignette included in R package Rcpp,URL https://CRAN./package=Rcpp.Eddelbuettel D,François R,Allaire J,Ushey K,Kou Q,Russel N,Chambers J,Bates D(2023a).Rcpp:Seamless R and C++Integration.R package version1.0.11,URL https:///package=Rcpp.Eddelbuettel D,François R,Bates D,Ni B,Sanderson C(2023b).RcppArmadillo:Rcpp integration for Armadillo templated linear algebra library.R packageversion0.12.4.1.0,URL https:///package=RcppArmadillo.Leisch F(2008).“T utorial on Creating R Packages.”In P Brito(ed.),COMPSTA T2008–Proceedings in Computational Statistics.Physica Verlag,Heidelberg.URL https:///doc/contrib/Leisch-CreatingPackages.pdf.R Core Team(2021).Writing R extensions.R Foundation for Statistical Comput-ing,Vienna,Austria.URL https:///doc/manuals/R-exts.html.4|https:///package=Rcpp Eddelbuettel and François。
cpptest培训手册parasoft c test traing presentation [template]-PPT精品文档106页
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THE KEY is uniform implementation in the group
If implemented as a group practice, the result is huge productivity improvements
Generating a BDF
Prerequisite
The project is buildable on the machine
cpptesttrace
Monitors the build and puts all commands into the BDF The project needs to be buildable Options
Process
Testing
Suppressions Violations
Best Practices
Agenda
Part 3
Part 4
Unit Test
Concepts and Strategies Generating Unit Tests Stubs File Scope and Project
Overview Pre-commit vs. Post-commit
BDF
Static Analysis
Terminology Workflow
Authors, Reviewers, Observers Workflows Auditing the Code Review
What is: Data Flow Analysis
qbreakpad 编译
qbreakpad 编译qbreakpad是一个Google开源的C++崩溃处理库，可以在Windows，Mac和Linux上捕获应用程序崩溃时的信息并生成崩溃报告。
如果你要在项目中使用qbreakpad，你需要先将它编译成库文件。
本文将介绍qbreakpad的编译方法。
编译qbreakpad的前提条件是你已经安装了Google Breakpad，并将其加入了系统的环境变量中。
如果你还没有安装Google Breakpad，可以前往Google Breakpad的官网下载安装包并按照官方指南进行安装。
接下来，按照以下步骤编译qbreakpad：1. 下载qbreakpad的源代码，可以从GitHub上下载最新的代码。
2. 在下载的源代码目录下创建一个build文件夹。
3. 打开终端（Windows下为命令提示符），切换到build文件夹的目录下。
4. 输入以下命令进行CMake配置：cmake ..-DBREAKPAD_INCLUDE_DIRS=/path/to/breakpad/include-DBREAKPAD_LIBRARIES=/path/to/breakpad/lib其中，/path/to/breakpad/include和/path/to/breakpad/lib分别为Google Breakpad的头文件和库文件所在的路径。
5. 输入以下命令进行编译：make这个过程可能需要一些时间，取决于你的机器性能和代码的规模。
6. 编译完成后，在build文件夹中会生成libqbreakpad.a和qbreakpad_test可执行文件。
7. 将libqbreakpad.a添加到你的项目库文件中，并将qbreakpad_test用于测试qbreakpad的功能。
至此，你已经成功地编译了qbreakpad库。
如果你在使用过程中遇到了问题，可以查看官方文档或通过Google Breakpad官方社区进行咨询。
Rcpp快速参考指南说明书
return sum/n; // Obtain and return the Mean }
// Place dependent functions above call or // declare the function definition with: double muRcpp(NumericVector x);
double y0 = yy["foo"]; double y1 = yy["bar"];
// Matrix of 4 rows & 5 columns (filled with 0) NumericMatrix xx(4, 5);
// Fill with value int xsize = xx.nrow() * xx.ncol(); for (int i = 0; i < xsize; i++) {
for(int i = 0; i < n; i++){ sum += pow(x[i] - mean, 2.0); // Square
// [[Rcpp::export]]
2 | https:///package=Rcpp
Eddelbuettel and François
double varRcpp(NumericVector x, bool bias = true){ // Calculate the mean using C++ function double mean = muRcpp(x); double sum = 0; int n = x.size();
俄罗斯方块使用SDL库和C语言开发的小游戏
俄罗斯方块使用SDL库和C语言开发的小游戏俄罗斯方块是一款经典的益智游戏，游戏的目标是通过控制不同形状的积木，使其在游戏区域内形成完整的水平线，以便消除并获得分数。
为了实现这一目标，开发人员通常会使用各种编程语言和库来构建游戏。
本文将介绍使用SDL库和C语言开发俄罗斯方块小游戏的过程。
一、SDL库和C语言简介SDL（Simple DirectMedia Layer）是一个跨平台的多媒体库，可以提供对图形、声音、输入和网络等方面的底层访问。
它广泛应用于游戏开发，由于其易用性和高效性，成为许多开发人员的首选。
C语言是一种通用的高级编程语言，也是俄罗斯方块游戏开发中常用的语言之一。
二、游戏开发环境的搭建在开始开发俄罗斯方块小游戏之前，我们需要先搭建游戏开发环境。
首先，下载并安装SDL库的开发包，该开发包提供了一系列的头文件和库文件，方便我们在C语言中使用SDL库的功能。
其次，选择一个适合的集成开发环境（IDE），比如Code::Blocks或者Visual Studio等，以便我们方便地编写和调试代码。
三、游戏的基本框架在开始编写游戏代码之前，我们需要先了解游戏的基本框架。
俄罗斯方块游戏通常由游戏区域、积木、分数和游戏状态等组成。
游戏区域是一个矩形区域，用来放置不同形状的积木。
积木由四个小方块组成，可以旋转和移动。
分数用来记录玩家的得分情况。
游戏状态用来判断游戏是进行中还是已结束。
四、游戏的初始化在游戏开始之前，我们需要先进行一些初始化的工作。
首先，我们需要初始化SDL库，包括初始化视频子系统、音频子系统和定时器等。
其次，我们需要创建游戏窗口，并设置窗口的标题和大小等属性。
最后，我们需要加载游戏的资源，比如积木的纹理、音效和背景音乐等。
五、游戏的主循环游戏的主循环是游戏的核心部分，它不断地更新游戏的状态，并根据用户的输入进行相应的处理。
在每一帧的更新过程中，我们需要先处理用户的输入，比如检测用户是否按下了方向键或者空格键等。
CLAPACK在vc++6.0中成功调用
CLAPACK在vc++6.0中成功调用Kevin Wong2010-3-24序：CLAPACK是LAPACK的C语言接口。
LAPACK的全称是Linear Algebra Package，是非常著名的线性代数库。
原版的LAPACK是用Fortran写的，为了方便C/C++程序的使用，就有了LAPACK的C接口CLAPACK。
网上虽然给出了一些关于CLAPACK在windows下的调用讲解, 但用起来总会有这样那样的问题. 对于Visual Studio C++ 6.0环境如何实现对CLAPACK的调用,基本上就没有相关资料了.本人青睐于vc++6.0的运行速度, 研究了一下再此环境中调用CLAPACK的攻略. 终于功夫不负苦心人, 在此将基本步骤梳理一下,希望对大家有所帮助.一、你需要准备什么?(1) MS vc++6.0. (有些费话了,呵呵)(2)下载clapack-3.2.1-CMAKE.tgz并解压缩.(3)下载CMAKE并安装.有了(2)和(3)的准备可以提供vc各个版本需要的项目和工程文件了.二、编译库文件1) 使用CMAKE∙∙∙∙∙ 打开CMAKE-gui∙∙∙∙∙ 将Source目录指向你的CLAPACK-3.2.1-CMAKE文件夹;∙∙∙∙∙ 将Bulid目录指向你准备存放生成文件的文件夹(最好不同于Source文件夹)∙∙∙∙∙ 点击configure, 选择vc++6.0作为你的编译器.(如果你的电脑上没有其他编译器,默认就可以了)∙∙∙∙∙ 你需要再次点击configure直到所有选项都变成白色.∙∙∙∙∙ 点击generate, 将会生成Visual Studio 需要的项目和工程文件, 这就完成了.∙∙∙∙∙ 关闭CMAKE2)查看"build"指向的文件夹,打开CLAPACK.dsw.3)激活"ALL_BUILD"工程,就可以组建(bulid)出所需要的所有的库文件了.编译完成后，找到libf2c.lib blas.lib lapack.lib tmglib.lib。
CLAPACK的安装与使用
CLAPACK的安装与简单使用ysmcleverysm@CLAPACK是LAPACK的C语言接口。
LAPACK的全称是Linear Algebra PACKage，是非常著名的线性代数库。
LAPACK是用Fortran写的，为了方便C/C++程序的使用，就有了LAPACK的C接口库CLAPACK。
LAPACK的主页是/lapack/，CLAPACK则在/clapack/。
安装CLAPACK首先自从其主页上下载CLAPACK包/clapack/clapack.tgz，解压。
在其目录下的INSTALL目录中有make.inc的范本文件，比如在linux下就是make.inc.LINUX，将这个文件拷到CLAPACK目录下并改名为make.inc，并修改此文件中的相应参数，如果需要的话。
CLAPACK需要F2CLIBS的libI77.a和libF77.a，一个tmglib和blas，这几个库都包含在了CLAPACK的安装包中，但是blas可以选用其他优化过的版本以得到更好的性能。
在此我们使用CLAPACK中的blas。
首先编译F2CLIBS，用于将fortran转换为c语言，在CLAPACK目录下，make f2clib，在CLAPACK/F2CLIBS下就会生成libI77.a和libF77.a。
编译tmglib，在CLAPACK目录下，make tmglib，会生成CLAPACK/tmglib_LINUX.a。
编译blas，在CLAPACK目录下，make blaslib，会生成CLAPACK/blas_LINUX.a。
最后是编译CLAPACK，make，会生成CLAPACK/lapack_LINUX.aCLAPACK下的clapack.h就是所需要的头文件，除此之外还需要的一个头文件是F2CLIBS/f2c.h。
现在就通过使用CLAPACK中的一个函数sgesv_解线性方程组来学习一下使用的方法。
GLPKAPI R包用户指南说明书
Introduction to glpkAPILouis Luangkesorn∗November10,20221IntroductionThis document introduces the use of the glpkAPI package1for R.The GNU Linear Programming Package(GLPK)is intended for solving linear program-ming(LP)and mixed integer programming(MIP)and other related problems. In addition,it includes facilities for converting problem information between the GNU MathProg language(a subset of the AMPL mathematical programming language),free andfixed MPS,and the CPLEX LP formats.2The GLPK pack-age is an interface into the C Application Programming Interface(API)to the GLPK solver.This document will introduce the use of the GLPK package through the use of the cannery problem from Dantzig3which is used in the GNU MathProg documentation.4The modelfile describing the cannery problem can be found in Appendix A.2Entering the modelTo use glpk,first load the package.>library(glpkAPI)Next read in the model and data.There are several ways of entering the model.glpk can read the model and data in a GNU MathProg Language(GMPL)modelfile.Alternatively,the model and data can be entered using the GLPK API.∗*******************.Thanks to Leo Lopes for his comments and suggestions.1Package glpkAPI maintained by Gabriel Gelius-Dietrich2GNU Linear Programming Kit:Reference Manual,Version4.54Draft,March2014.3The demand data here is from the GLPK documentation,which differs slightly from Dantzig,Linear Programming and Extensions,Princeton University Press,Princeton,NJ, 1963.The documentation demand values are used here for consistancy.4GNU Linear Programming Kit:Modeling Language GNU MathProg,Version4.50Draft, May2013.12.1Reading a GNU MathProg Language modelTo use a GNU MathProg model requires several steps.1.Allocating the workspace using initProbGLPK().The problem can thenbe given an name using setProbNameGLPK().2.Reading model section using mplAllocWkspGLPK()and mplReadModelGLPK().3.Reading data section(s)using mplReadDataGLPK().4.Generating the model using mplGenerateGLPK().5.Building the problem object using result<-mplBuildProbGLPK().6.Solving the problem using solveSimplexGLPK().7.Postsolving the model using mplPostsolveGLPK().8.Freeing the workspace using mplFreeWkspGLPK()and delProbGLPK()>mip<-initProbGLPK()>setProbNameGLPK(mip,"transport")>trans<-mplAllocWkspGLPK()>result<-mplReadModelGLPK(trans,+system.file("extdata","transport.mod",package="glpkAPI"),skip=0) >result<-mplGenerateGLPK(trans)>result<-mplBuildProbGLPK(trans,mip)If the data was in a separatefile,it would need to be read in usingmplReadDataGLPK(trans,"transport.mod")Then examine the problem size within R.The rows represent the objective function as well as the supply and demandconstraints.>numrows<-getNumRowsGLPK(mip)>numrows[1]6>for(i in1:numrows){+print(getRowNameGLPK(mip,i))+}[1]"cost"[1]"supply[Seattle]"[1]"supply[San-Diego]"[1]"demand[New-York]"[1]"demand[Chicago]"[1]"demand[Topeka]"2The columns represent the decision variables,which are the units sent over the cannary-market links.>numcols<-getNumColsGLPK(mip)>numcols[1]6>for(j in1:numcols){+print(getColNameGLPK(mip,j))+}[1]"x[Seattle,New-York]"[1]"x[Seattle,Chicago]"[1]"x[Seattle,Topeka]"[1]"x[San-Diego,New-York]"[1]"x[San-Diego,Chicago]"[1]"x[San-Diego,Topeka]">print(getNumNnzGLPK(mip))[1]18After the model and data are entered,the model can then be solved using any one of many algorithms and the output would go to the specified output file.For the Simplex method,the solveSimplexGLPK()takes the problem name and solves it using the Simplex method.>return<-solveSimplexGLPK(mip)>return<-mplPostsolveGLPK(trans,mip,GLP_MIP);We can then look at the solution in terms of the objective and constraints >for(i in1:numrows){+print(getRowNameGLPK(mip,i))+print(getRowPrimGLPK(mip,i))+}[1]"cost"[1]153.675[1]"supply[Seattle]"[1]350[1]"supply[San-Diego]"[1]550[1]"demand[New-York]"[1]325[1]"demand[Chicago]"[1]300[1]"demand[Topeka]"[1]2753as well as the decision variables.>for(j in1:numcols){+print(getColNameGLPK(mip,j))+print(getColPrimGLPK(mip,j))+}[1]"x[Seattle,New-York]"[1]50[1]"x[Seattle,Chicago]"[1]300[1]"x[Seattle,Topeka]"[1]0[1]"x[San-Diego,New-York]"[1]275[1]"x[San-Diego,Chicago]"[1]0[1]"x[San-Diego,Topeka]"[1]275Finally,clean up the workspace.>mplFreeWkspGLPK(trans)>delProbGLPK(mip)2.2Using the APIIf the problem data already in R,such as pulled from a database or the result of previous analysis,the model and the data can be specified using the API.First create R data objects to hold the various model parameters.>print("USING API")[1]"USING API">canneries<-c("Seattle","San-Diego")>capacity<-c(350,600)>markets<-c("New-York","Chicago","Topeka")>demand<-c(325,300,275)>distance<-c(2.5,2.5,1.7,1.8,1.8,1.4)>dim(distance)<-c(2,3)>freight<-90To use the API,define a problem instance and indicate that the objective is to minimize cost.>lpi<-initProbGLPK()>setProbNameGLPK(lpi,"cannery API")>setObjNameGLPK(lpi,"Total Cost")>setObjDirGLPK(lpi,GLP_MIN)4There are6columns,corresponding to the six potential cannery-market pairswhose transport the model solving for,each of which has a lower bound of zero.>numlinks<-length(distance)>nummarkets<-length(markets)>numcanneries<-length(canneries)>addColsGLPK(lpi,numlinks)[1]1>for(i in1:numcanneries){+cannerystartrow<-(i-1)*nummarkets+for(j in1:nummarkets){+colname<-toString(c(canneries[i],markets[j]))+transcost<-distance[i,j]*freight/1000+setColNameGLPK(lpi,cannerystartrow+j,colname)+setColBndGLPK(lpi,cannerystartrow+j,GLP_LO,0.0,0.0)+setObjCoefsGLPK(lpi,cannerystartrow+j,transcost)+}+}Next,we will add constraints.There are5constraints,two supply con-straints relating to the canneries and three demand constraints relating to themarkets.In addition,we will make thefirst row correspond to the objectivefunction.The objective row will be free,and does not have upper or lowerbounds.>numcanneries<-length(canneries)>nummarkets<-length(markets)>addRowsGLPK(lpi,numcanneries+nummarkets+1)[1]1>setRowsNamesGLPK(lpi,1,getObjNameGLPK(lpi))>for(i in1:numcanneries){+setRowsNamesGLPK(lpi,i+1,toString(c("Supply",canneries[i])))+setRowBndGLPK(lpi,i+1,GLP_UP,0,capacity[i])+}>for(j in1:nummarkets){+setRowsNamesGLPK(lpi,numcanneries+j+1,toString(c("Demand",markets[j]))) +setRowBndGLPK(lpi,numcanneries+j+1,GLP_LO,demand[j],0)+}Now,load the constraint matrix which represents the objective function andthe constraints.The non-zero values of the matrix are entered as three vectors,each with one element for each non-zero value.A vector to indicate the row,avector to indicate the column,and a vector which contains the matrix elementst,we call loadMatrixGLPK(lpi)tofinish.5>#create variables to hold the constraint information>ia<-numeric()>ja<-numeric()>ar<-numeric()>#add in objective coefficients>>for(i in1:numcols){+ia[i]<-1+ja[i]<-i+ar[i]<-getObjCoefGLPK(lpi,i)+}>for(i in1:numcanneries){+#supply constraints+cannerysupplyrow=numcols+(i-1)*nummarkets+for(j in1:nummarkets){+ia[cannerysupplyrow+j]<-(i+1)+ja[cannerysupplyrow+j]<-(i-1)+numcanneries*(j-1)+1 +ar[cannerysupplyrow+j]<-1+}+#demand constraints+marketdemandrow=numcols+numcanneries*nummarkets+for(j in1:nummarkets){+colnum<-(i-1)*nummarkets+j+ia[marketdemandrow+colnum]<-numcanneries+j+1+ja[marketdemandrow+colnum]<-colnum+ar[marketdemandrow+colnum]<-1+}+}>loadMatrixGLPK(lpi,length(ia),ia,ja,ar)Then,examine the problem entered in the API.>numrows<-getNumRowsGLPK(lpi)>numrows[1]6>numcols<-getNumColsGLPK(lpi)>numcols[1]6>for(i in1:numrows){+print(getRowNameGLPK(lpi,i))+}[1]"Total Cost"[1]"Supply,Seattle"6[1]"Supply,San-Diego"[1]"Demand,New-York"[1]"Demand,Chicago"[1]"Demand,Topeka">for(j in1:numcols){+print(getColNameGLPK(lpi,j))+}[1]"Seattle,New-York"[1]"Seattle,Chicago"[1]"Seattle,Topeka"[1]"San-Diego,New-York"[1]"San-Diego,Chicago"[1]"San-Diego,Topeka">print(getNumNnzGLPK(lpi))[1]18Finally solve using the simplex method and look at the solution. >solveSimplexGLPK(lpi)[1]0>for(i in1:numrows){+print(getRowNameGLPK(lpi,i))+print(getRowPrimGLPK(lpi,i))+}[1]"Total Cost"[1]153.675[1]"Supply,Seattle"[1]325[1]"Supply,San-Diego"[1]575[1]"Demand,New-York"[1]325[1]"Demand,Chicago"[1]300[1]"Demand,Topeka"[1]275>for(j in1:numcols){+print(getColNameGLPK(lpi,j))+print(getColPrimGLPK(lpi,j))+}7[1]"Seattle,New-York"[1]325[1]"Seattle,Chicago"[1]300[1]"Seattle,Topeka"[1]0[1]"San-Diego,New-York"[1]0[1]"San-Diego,Chicago"[1]0[1]"San-Diego,Topeka"[1]275And save the results to afile.>printSolGLPK(lpi,"transout.api")[1]02.3Using API to modify the modelNow,we will solve the version of the problem that is found in Dantzig.The demand at New York and Topeka are both300instead of325and275.This next section will use the API to modify the problem as read through the MathProg file.In order to examine an individual row,we need to index the rows and columns.This is done through the use of createIndexGLPK().Then we can use the findRowGLPK()and findColGLPK()>cindex<-createIndexGLPK(lpi)>new_york_row=findRowGLPK(lpi,"Demand,New-York")>topeka_row=findRowGLPK(lpi,"Demand,Topeka")>new_york_row[1]4>topeka_row[1]6>setRowBndGLPK(lpi,new_york_row,GLP_LO,300,0)>setRowBndGLPK(lpi,topeka_row,GLP_LO,300,0)We can solve this modified problem and look at the results.>solveSimplexGLPK(lpi)[1]08>for(i in1:numrows){+print(getRowNameGLPK(lpi,i))+print(getRowPrimGLPK(lpi,i))+print(getRowDualGLPK(lpi,i)) +}[1]"Total Cost"[1]151.2[1]0[1]"Supply,Seattle"[1]300[1]0[1]"Supply,San-Diego"[1]600[1]0[1]"Demand,New-York"[1]300[1]0.225[1]"Demand,Chicago"[1]300[1]0.153[1]"Demand,Topeka"[1]300[1]0.126>for(j in1:numcols){+print(getColNameGLPK(lpi,j))+print(getColPrimGLPK(lpi,j))+print(getColDualGLPK(lpi,j))+print(getObjCoefGLPK(lpi,j)) +}[1]"Seattle,New-York"[1]300[1]0[1]0.225[1]"Seattle,Chicago"[1]300[1]0[1]0.153[1]"Seattle,Topeka"[1]0[1]0.036[1]0.162[1]"San-Diego,New-York"[1]0[1]09[1]0.225[1]"San-Diego,Chicago"[1]0[1]0.009[1]0.162[1]"San-Diego,Topeka"[1]300[1]0[1]0.126Finally,clean up the workspace.>delProbGLPK(lpi)A ModelfileTRANSPORT.MOD#A TRANSPORTATION PROBLEM##This problem finds a least cost shipping schedule that meets#requirements at markets and supplies at factories.##References:#Dantzig,G B.,Linear Programming and Extensions#Princeton University Press,Princeton,New Jersey,1963, #Chapter3-3.set I;/*canning plants*/set J;/*markets*/param a{i in I};/*capacity of plant i in cases*/param b{j in J};/*demand at market j in cases*/param d{i in I,j in J};/*distance in thousands of miles*/param f;/*freight in dollars per case per thousand miles*/10param c{i in I,j in J}:=f*d[i,j]/1000;/*transport cost in thousands of dollars per case*/var x{i in I,j in J}>=0;/*shipment quantities in cases*/minimize cost:sum{i in I,j in J}c[i,j]*x[i,j];/*total transportation costs in thousands of dollars*/s.t.supply{i in I}:sum{j in J}x[i,j]<=a[i];/*observe supply limit at plant i*/s.t.demand{j in J}:sum{i in I}x[i,j]>=b[j];/*satisfy demand at market j*/data;set I:=Seattle San-Diego;set J:=New-York Chicago Topeka;param a:=Seattle350San-Diego600;param b:=New-York325Chicago300Topeka275;param d:New-York Chicago Topeka:=Seattle 2.5 1.7 1.8San-Diego 2.5 1.8 1.4;param f:=90;end;B OutputThe following is the output of the command:printSolGLPK(lpi,"transout.api") Problem:cannery APIRows:6Columns:6Non-zeros:18Status:OPTIMALObjective:Total Cost=153.675(MINimum)11No.Row name St Activity Lower bound Upper bound Marginal------------------------------------------------------------------------1Total Cost B153.6752Supply,SeattleB325350 3Supply,San-DiegoB575600 4Demand,New-YorkNL3253250.225 5Demand,ChicagoNL3003000.153 6Demand,TopekaNL2752750.126 No.Column name St Activity Lower bound Upper bound Marginal------------------------------------------------------------------------1Seattle,New-YorkB32502Seattle,ChicagoB30003Seattle,TopekaNL000.036 4San-Diego,New-YorkNL00<eps 5San-Diego,ChicagoNL000.009 6San-Diego,TopekaB2750Karush-Kuhn-Tucker optimality conditions:KKT.PE:max.abs.err=2.84e-14on row1max.rel.err=9.22e-17on row1High qualityKKT.PB:max.abs.err=0.00e+00on row0max.rel.err=0.00e+00on row0High qualityKKT.DE:max.abs.err=0.00e+00on column0max.rel.err=0.00e+00on column0High qualityKKT.DB:max.abs.err=0.00e+00on row0max.rel.err=0.00e+00on row012High qualityEnd of output13。
C++基础学习中常见英语单词汇总及简介
一、基础语法单词1、include:包含,文件包含,指C+中包含头文件命令,用于将指定头文件嵌入源文件中。
2、iostream:输入输出流,包括输入输出操作,简单理解in、out的首字母与stream 结合。
3、using:使用,使用,释放一个或多个对象。
4、namespace:命名空间,指标识符的各种可见范围。
5、std:标准,C++标准程序库中的,所有标识符都被定义于一个名为std的nameapace中。
6、main:主函数,main()函数称之为主函数,一个C程序总是从main（）函数开始执行的。
7、cout:字符输出,用于在计算机屏幕上显示信息,是C++中iostream类型的对象。
8、cin:字符输入,是一个C++标准的输入流对象,从设备键盘取得数据,送到输入流对象ci中,然后送到内存。
9、endl:换行,一行输出结束,然后输出下一行。
10、return:返回,返回到主调函数继续执行。
二、数据类型单词1、char:字符、字符型,字符型变量可表示单个字符,字符型变量只占用1个字节2、int:integer,整型,整型变量表示整数类型的数据。
3、short:短,短整型。
4、long:长,长整型。
5、f0at:浮点数,单精度浮点型。
浮点型变量表示小数类型的数据,4个字节。
6、double:双,双精度浮点型,8个字节7、bool:布尔,布尔型,布尔数据类型表示真或假的值。
8、Void:空白,无类型。
9、size:大小,函数,求字符串长度。
10、sizeof:运算符,大小,sizeof关键字可统计数据类型所占用的内存大小。
11、true:真,成立时为真。
12、false:真,不成立时为假。
13、fixed:固定的,表示用一般的方式输出浮点数。
14、const:常量,常数,constant(不变的)的缩写。
三、控制流单词l、if：如果,在布尔表达式为真时执行。
2、else:否则,与if语句搭配,在布尔表达式为假时执行。
LAPACK线性代数函数手册说明书
Title lapack()—LAPACK linear-algebra functionsSyntax Description Remarks and examples Reference Also seeSyntaxvoid flopin(numeric matrix A)void lapack function(...)void flopout(numeric matrix A)where lapack function may beLA DGBMV()LA DGEBAK()LA ZGEBAK()LA DGEBAL()LA ZGEBAL()LA DGEES()LA ZGEES()LA DGEEV()LA ZGEEV()LA DGEHRD()LA ZGEHRD()LA DGGBAK()LA ZGGBAK()LA DGGBAL()LA ZGGBAL()LA DGGHRD()LA ZGGHRD()LA DHGEQZ()LA ZHGEQZ()LA DHSEIN()LA ZHSEIN()LA DHSEQR()LA ZHSEQR()LA DLAMCH()LA DORGHR()LA DSYEVX()LA DTGSEN()LA ZTGSEN()LA DTGEVC()LA ZTGEVC()LA DTREVC()LA ZTREVC()LA DTRSEN()LA ZTRSEN()LA ZUNGHR()DescriptionLA DGBMV(),LA DGEBAK(),LA ZGEBAK(),LA DGEBAL(),LA ZGEBAL(),...are LAPACK func-tions in original,as-is form;see[M-1]LAPACK.These functions form the basis for many of Mata’s linear-algebra capabilities.Mata functions such as cholesky(),svd(),and eigensystem()are implemented using these functions;see[M-4]matrix.Those functions are easier to use.The LA*() functions provide more capability.flopin()and flopout()convert matrices to and from the form required by the LA*() functions.12lapack()—LAPACK linear-algebra functionsRemarks and examples LAPACK stands for Linear Algebra PACK age and is a freely available set of Fortran90routines for solving systems of simultaneous equations,eigenvalue problems,and singular-value problems.The original Fortran routines have six-letter names like DGEHRD,DORGHR,and so on.The Mata functions LA DGEHRD(),LA DORGHR(),etc.,are a subset of the LAPACK double-precision real and complex routine.All LAPACK double-precision functions will eventually be made available.Documentation for the LAPACK routines can be found at /lapack/,although we recommend obtaining LAPACK Users’Guide by Anderson et al.(1999).Remarks are presented under the following headings:Mapping calling sequence from Fortran to MataFlopping:Preparing matrices for LAPACKWarning on the use of rows()and cols()afterflopin()Warning:It is your responsibility to check infoExampleMapping calling sequence from Fortran to MataLAPACK functions are named withfirst letter S,D,C,or Z.S means single-precision real,D means double-precision real,C means single-precision complex,and Z means double-precision complex.Mata provides the D*and Z*functions.The LAPACK documentation is in terms of S*and C*.Thus,tofind the documentation for LA DGEHRD,you must look up SGEHRD in the original documentation.The documentation(Anderson et al.1999,227)reads,in part,SUBROUTINE SGEHRD(N,ILO,IHI,A,LDA,TAU,WORK,LWORK,INFO)INTEGER IHI,ILO,INFO,LDA,LWORK,NREAL A(LDA,*),TAU(*),WORK(LWORK)and the documentation states that SGEHDR reduces a real,general matrix,A,to upper Hessenberg form,H,by an orthogonal similarity transformation:Q ×A×Q=H.The corresponding Mata function,LA DGEHRD(),has the same arguments.In Mata,arguments ihi, ilo,info,lda,lwork,and n are real scalars.Argument A is a real matrix,and arguments tau and work are real vectors.You can read the rest of the original documentation tofind out what is to be placed(or returned) in each argument.It turns out that A is assumed to be dimensioned LDA×something and that the routine works on A(1,1)(using Fortran notation)through A(N,N).The routine also needs work space,which you are to supply in vector WORK.In the standard LAPACK way,LAPACK offers you a choice:you can preallocate WORK,in which case you have to choose a fairly large dimension for it, or you can do a query tofind out how large the dimension needs to be for this particular problem.If you preallocate,the documentation reveals that the WORK must be of size N,and you set LWORK equal to N.If you wish to query,then you make WORK of size1and set LWORK equal to−1.The LAPACK routine will then return in thefirst element of WORK the optimal size.Then you call the function again with WORK allocated to be the optimal size and LWORK set to equal the optimal size.Concerning Mata,the above works.You can follow the LAPACK documentation to the e J()to allocate matrices or vectors.Alternatively,you can specify all sizes as missing value(.),and Mata willfill in the appropriate value based on the assumption that you are using the entire matrix.lapack()—LAPACK linear-algebra functions3 Thus,in LA DGEHRD(),you could specify lda as missing,and the function would run as if you had specified lda equal to cols(A).You could specify n as missing,and the function would run as if you had specified n as rows(A).Work areas,however,are treated differently.You can follow the standard LAPACK convention outlined above;or you can specify the sizes of work areas(lwork)and specify the work areas themselves (work)as missing values,and Mata will allocate the work areas for you.The allocation will be as you specified.One feature provided by some LAPACK functions is not supported by the Mata implementation.If a function allows a function pointer,you may not avail yourself of that option.Flopping:Preparing matrices for LAPACKThe LAPACK functions provided in Mata are the original LAPACK functions.Mata,which is C based, stores matrices PACK,which is Fortran based,stores matrices columnwise.Mata and Fortran also disagree on how complex matrices are to be organized.Functions flopin()and flopout()handle these issues.Coding flopin(A)changes matrixA from the Mata convention to the LAPACK convention.Coding flopout(A)changes A from theLAPACK convention to the Mata convention.The LA*()functions do not do this for you because LAPACK often takes two or three LAPACK functions run in sequence to achieve the desired result,and it would be a waste of computer time to switch conventions between calls.Warning on the use of rows()and cols()afterflopin()Be careful using the rows()and cols()functions.rows()of aflopped matrix returns the logical number of columns and cols()of aflopped matrix returns the logical number of rows!The danger of confusion is especially great when using J()to allocate work areas.If a LAPACK function requires a work area of r×c,you code,LA function(...,J(c,r,.),...)Warning:It is your responsibility to check infoThe LAPACK functions do not abort with error on failure.They instead store0in info(usually the last argument)if successful and store an error code if not successful.The error code is usually negative and indicates the argument that is a problem.ExampleThe following example uses the LAPACK function DGEHRD to obtain the Hessenberg form of matrixA.We will begin with1234112342456737891048910114lapack()—LAPACK linear-algebra functionsThefirst step is to use flopin()to put A in LAPACK order::_flopin(A)Next we make a work-space query to get the optimal size of the work area.:LA_DGEHRD(.,1,4,A,.,tau=.,work=.,lwork=-1,info=0):lwork=work[1,1]:lwork128After putting the work-space size in lwork,we can call LA DGEHRD()again to perform the Hessenberg decomposition::LA_DGEHRD(.,1,4,A,.,tau=.,work=.,lwork,info=0)LAPACK function DGEHRD saves the result in the upper triangle and thefirst subdiagonal of A.We must use flopout()to change that back to Mata order,andfinally,we extract the result: :_flopout(A):A=A-sublowertriangle(A,2):A123411-5.370750529.0345341258.39223227032-11.3578166925.18604651-4.40577178-.656148389930-1.660145888-.1860465116.1760901813400-8.32667e-16-5.27356e-16ReferenceAnderson,E.,Z.Bai,C.Bischof,S.Blackford,J.Demmel,J.J.Dongarra,J.Du Croz,A.Greenbaum,S.Hammarling,A.McKenney,and PACK Users’Guide.3rd ed.Philadelphia:Society for Industrial andApplied Mathematics.Also see[M-1]LAPACK—The LAPACK linear-algebra routines[R]copyright lapack—LAPACK copyright notification[M-4]matrix—Matrix functions。
LAPACK安装和调试(包含C语言借口的安装)
Linux环境下Lapack软件包的编译和使用自行下载编译过程1. 从 netlab 下载最新版本的源代码2. 将源代码解压缩3. 编辑 make.inc 文件，进行一些系统相关的编译参数设置一些编译参数，PLAT 表示使用的平台，会接在生成lib库的名字中，原本是 _LINUX。
FORTRAN 表示你使用的fortran编译器，根据自己安装的编译器来设置，我用的是 gfo rtran。
OPTS 表示编译时的优化程度，设置为 -O2即可。
LOADER 设置成和FORTRAN 基本一样就可以了。
确定生成目标如下，BLASLIB = /libblas$(PLAT).aLAPACKLIB = liblapack$(PLAT).aTMGLIB = libtmglib$(PLAT).aEIGSRCLIB = libeigsrc$(PLAT).aLINSRCLIB = liblinsrc$(PLAT).a原来的这些库文件的名字前面没有lib的，这里加上。
因为linux上库文件一般有前缀lib的，在链接的时候-l后面带的是库的名字，那个时候是不加lib的。
如果已经编译生成了这些库，可以重命名，而不用重新编译。
PLAT ：设置生成的库函数的后缀，比如 SUN， LINUX之类的，当然也可以不设置；FORTRAN ：设置编译器，比如 g77， gfortran， ifort， g95 等等；（我设置为FORTRAN = ifort -g）OPT：设置编译选项，根据具体的编译器和优化要求进行设置；LOADER ：设置成和FORTRAN 一样就可以了；4. 编辑 Makefile找到 lib：选项。
然后设置需要编译的库函数如下#lib： lapacklib tmgliblib： blaslib lapacklib tmglib默认的情况是不编译 blas 库的。
（除非事先编译好了blas库，否则应一起编译。
）5. 编译和测试 Lapack.在源代码的根目录下输入 make 回车。
fortran lapack例子
fortran lapack例子Fortran LAPACK (Linear Algebra Package) is a widely used library for performing linear algebra operations, such as solving systems of linear equations, eigenvalue problems, and singular value decomposition. In this article, we will explore some real-world examples of using LAPACK in Fortran and explain each step in detail.1. Introduction to LAPACK and Fortran:LAPACK is a collection of Fortran subroutines that provideshigh-performance implementations of numerical linear algebra routines. It is designed to be portable and efficient, making it an ideal choice for scientific and engineering applications. Fortran, specifically Fortran 90 and later versions, is a programming language commonly used in scientific and technical computing, known for its efficiency and numerical capabilities.2. Setting up the Environment:Before we can use LAPACK in Fortran, we need to set up the environment correctly. This involves installing a Fortran compiler, such as GNU Fortran (gfortran), and linking the LAPACK library. We can do this by downloading the LAPACK library from Netlib and following the installation instructions. Once the environment is set up, we can proceedto use LAPACK in our Fortran programs.3. Solving a System of Linear Equations:One of the most common tasks in linear algebra is solving a system of linear equations. Let's consider the following example:Ax = bwhere A is a square matrix of size n×n, x is a vector of size n, and b is a vector of size n. To solve this system using LAPACK, we can use the DGESV subroutine. This subroutine takes the matrix A, the right-hand side vector b, and returns the solution vector x.We need to initialize the values of A and b before calling the DGESV subroutine. Once the solution is obtained, we can print the values of x to verify the result. Additionally, we should check the return value of the subroutine to ensure that the solution was successfully obtained.4. Computing Eigenvalues and Eigenvectors:Another important task in linear algebra is computing the eigenvalues and eigenvectors of a matrix. LAPACK provides several subroutines for this purpose, such as DSYEV and DGEEV. Let's consider the DSYEVsubroutine, which computes all the eigenvalues and, optionally, the eigenvectors of a real symmetric matrix.To use DSYEV, we need to initialize the matrix A and choose the appropriate input parameters. We can then call the DSYEV subroutine, which returns the eigenvalues in ascending order and optionally, the eigenvectors. We can print the eigenvalues and eigenvectors to display the results.5. Performing Singular Value Decomposition:Singular Value Decomposition (SVD) is another important technique in linear algebra, which decomposes a matrix into three separate matrices: U, Σ, and V. LAPACK provides the DGESVD subroutine to perform SVD.To use DGESVD, we need to initialize the matrix A. We can then call the DGESVD subroutine, which returns the matrices U, Σ, and V. We can print these matrices to verify the correctness of the SVD operation.6. Error Handling and Optimization:When using LAPACK in Fortran, it is essential to handle errors properly to ensure the correct execution of the program. LAPACK subroutines often return error codes, which we can check to detect any issues duringcomputation. We can use error handling routines, such as IF statements, to handle these errors and continue the execution of the program.To optimize the performance of LAPACK in Fortran, we can make use of various compiler flags and optimization techniques. For example, we can enable vectorization, loop unrolling, and compiler-specific optimizations to improve the execution speed of our programs.In conclusion, LAPACK provides a powerful set of subroutines for solving linear algebra problems efficiently. By using LAPACK in Fortran, we can perform tasks such as solving systems of linear equations, computing eigenvalues and eigenvectors, and performing singular value decomposition. By carefully following the steps outlined in this article, we can effectively utilize LAPACK to solve real-world problems in scientific and engineering domains.。
Lapack中文帮助手册手册
Lapack中文帮册手册Lapack中文帮册手册第一章概述1.1 Lapack简介Lapack（Linear Algebra Package）是一套用于高效解决线性代数问题的软件包。
它提供了各种线性代数操作的实现，包括线性方程组求解、特征值和特征向量计算、矩阵分解等。
1.2 Lapack的特点Lapack具有以下特点：- 高效性：Lapack使用高效的算法实现线性代数操作，能够处理大规模矩阵和稠密矩阵。
- 跨平台性：Lapack可以在各种操作系统和计算机体系结构上运行，并与许多编程语言兼容。
- 可扩展性：Lapack提供了很多可调用的子程序和函数，可以通过组合和调用这些子程序和函数来解决不同的线性代数问题。
1.3 Lapack的安装和使用详细的安装和使用指南可以参考附件A中的文档。
第二章线性代数基础2.1 矩阵和向量的表示2.2 矩阵的运算2.3 线性方程组的求解方法2.4 特征值和特征向量的计算方法2.5 矩阵分解方法第三章 Lapack子程序和函数3.1 基本线性代数子程序(BLAS) 3.2 Lapack线性代数子程序第四章常用的线性代数操作4.1 线性方程组的求解4.2 特征值和特征向量的计算4.3 矩阵分解第五章高级应用5.1 稀疏矩阵的处理5.2 并行计算第六章附录附录A Lapack安装和使用指南附录B 术语表附录A Lapack安装和使用指南安装和使用Lapack的详细步骤，请参考附件A中的文档。
附录B 术语表术语1：线性代数定义：线性代数是数学中研究向量空间和线性映射的分支。
术语2：特征值定义：对于一个n阶方阵A，如果存在标量λ和非零向量v，使得Av=λv成立，则称λ为矩阵A的特征值，v为对应的特征向量。
术语3：矩阵分解定义：将一个矩阵表示为若干个简单矩阵的乘积的过程被称为矩阵分解。
1、本文档涉及附件：附件A Lapack安装和使用指南2、本文所涉及的法律名词及注释：无。
lapack矩阵乘法c语言
lapack矩阵乘法c语言摘要：1.引言pack矩阵乘法简介3.C语言实现lapack矩阵乘法4.总结正文：【引言】LAPACK（Linear Algebra Package）是一个线性代数软件包，提供了丰富的线性代数操作功能，如矩阵乘法等。
在C语言中，我们可以利用LAPACK 提供的库函数实现矩阵乘法。
本文将详细介绍LAPACK矩阵乘法在C语言中的实现方法。
【lapack矩阵乘法简介】LAPACK矩阵乘法是利用LAPACK库函数进行矩阵乘法运算。
它提供了多种矩阵乘法算法，如普通矩阵乘法、复数矩阵乘法等。
LAPACK库函数采用了高效的算法和数据结构，能够有效地提高矩阵乘法运算的效率。
【C语言实现lapack矩阵乘法】要在C语言中实现LAPACK矩阵乘法，首先需要安装LAPACK库，并配置好相应的编译环境。
然后，通过引入头文件和调用LAPACK库函数，可以实现矩阵乘法运算。
以下是一个简单的C语言实现LAPACK矩阵乘法的示例：```c#include <stdio.h>#include <stdlib.h>#include <math.h>#include <lapacke.h>int main() {int n, lda, ldb, ldc;double *A, *B, *C;printf("请输入矩阵A的大小：");scanf("%d", &n);lda = n;ldb = n;ldc = n;A = (double *)malloc(n * n * sizeof(double));B = (double *)malloc(n * n * sizeof(double));C = (double *)malloc(n * n * sizeof(double));printf("请输入矩阵A的元素：");for (int i = 0; i < n * n; i++) {scanf("%lf", &A[i]);}printf("请输入矩阵B的元素：");for (int i = 0; i < n * n; i++) {scanf("%lf", &B[i]);}dgemm_(&transa, &transb, &n, &n, &n, &one, A, &lda, B, &ldb, &one, C, &ldc);printf("矩阵乘法结果为：");for (int i = 0; i < n * n; i++) {printf("%.2lf ", C[i]);}free(A);free(B);free(C);return 0;}```【总结】本文首先介绍了LAPACK矩阵乘法的基本概念，然后通过一个简单的C语言示例，详细阐述了如何在C语言中实现LAPACK矩阵乘法。
malloclab-handout实验步骤
malloclab-handout实验步骤以下是CSAPP实验中Malloclab-handout的实验步骤：1. 解压文件：使用`tar xvf malloclab-handout.tar`命令解压文件。
2. 修改文件：需要修改的唯一文件是`mm.c`，包含如下几个需要实现的函数：- `mm_init()`:在调用`mm_malloc()`、`mm_realloc()`或`mm_free()`之前，调用`mm_init()`进行初始化，正确返回0。
- `mm_malloc()`:在堆区域分配指定大小的块，分配的空间，返回的指针应该是8字节对齐的。
- `mm_free()`:释放指针指向的block。
- `mm_realloc()`:返回指向一个大小为`size`的区域指针，满足以下条件：- 如果`ptr`为空，调用等同于`mm_malloc(size)`；- 如果`size`为0，调用等同于`mm_free(ptr)`；- 如果`ptr`不为空：先按照`size`指定的大小分配空间，将原有数据从头到尾拷贝到新分配的内存区域，而后释放原来`ptr`所指内存区域。
3. 实验验证：`mdriver.c`负责测试`mm.c`的正确性、空间利用率和吞吐量。
- `-f <tracefile>`：在`-f`后面添加一些`tracefile`来测试实现的函数。
- `-V`：打印出诊断信息。
- `./mdriver -V -f short1-bal.rep`：运行程序。
编程规则：1. 不能改变`mm.c`中函数接口。
2. 不能直接调用任何内存管理的库函数和系统函数`malloc()`、`calloc()`、`free()`、`realloc()`、`sbrk()`、`brk()`。
3. 不能定义任何全局或者静态复合数据结构，如arrays、structs、trees。
4. 允许使用整数、浮点数和指针等简单数据类型。
puar原理
puar原理
PUAR原理指的是通过使用ptrace系统调用，实现对运行中的进程进行追踪。
具体来说，ptrace可以追踪进程的内存数据、寄存器的值、接收的信号以及指定进程的运行方式等。
调用ptrace追踪进程时（gdb attach -p $pid），被追踪进程会发生以下事情：
- 追踪进程会变为被追踪进程的父进程。
- 进程状态会进入TASK_TRACED，表示当前进程正在被追踪，此时进程会暂停下来，等待追踪进程的操作。
- 发送给被追踪进程的信号会被转发给父进程，除了SIGKILL，子进程则会被阻塞。
- 父进程收到信号之后可以对子进程进行修改，来让子进程继续运行。
通过使用PUAR原理，可以实现对进程的监控和调试，在操作系统和应用程序的开发和调试中具有重要的作用。
LAPACK线性代数包（LAPACK）例程说明书
Title LAPACK—Linear algebra package(LAPACK)routinesDescription Syntax Option for set lapack mklRemarks and examples Acknowledgments ReferencesAlso seeDescriptionLAPACK stands for Linear Algebra PACK age and is a freely available set of Fortran90routines for solving systems of simultaneous equations,eigenvalue problems,and singular value problems.Many of the LAPACK routines are based on older EISPACK and LINPACK routines,and the more modern LAPACK does much of its computation by using Basic Linear Algebra Subprograms(BLAS).Stata contains two sets of LAPACK and BLAS libraries;one is from Netlib,and the other is from the Intel Math Kernel Library(MKL).set lapack mkl sets which LAPACK library will be used.set lapack mkl cnr sets the conditional numerical reproducibility mode for the Intel MKL LAPACK routines.SyntaxSet whether Intel MKL LAPACK routines will be usedset lapack mklon|off,permanentlySet the conditional numerical reproducibility mode for the Intel MKL LAPACK routinesset lapack mkl cnrdefault|auto|compatible|offOption for set lapack mklpermanently specifies that,in addition to making the change right now,the setting be remembered and become the default setting when you invoke Stata in the future.Remarks and examples Remarks are presented under the following headings:LAPACK in Mataset lapack mklIntel MKL conditional numerical reproducibilityset lapack mkl cnr12LAPACK—Linear algebra package(LAPACK)routinesLAPACK in MataThe LAPACK and BLAS routines form the basis for many of Mata’s linear algebra capabilities.Individual functions of Mata that use LAPACK routines always make note of that fact.Stata on all platforms,except ARM-based Mac(Apple Silicon),contains two sets of LAPACK and BLAS libraries.One is based on the source code from Netlib’s LAPACK and has been used in Stata since Stata9.Stata,since Stata17,also contains LAPACK and BLAS libraries from the Intel MKL.Because the Intel MKL does not support ARM-based Mac,ARM-based Mac supports only Netlib’s LAPACK library.For up-to-date information on LAPACK,see /lapack/.For up-to-date information on Intel MKL,see https:///mkl.Advanced programmers can directly access the LAPACK functions;see[M-5]lapack().set lapack mklFor platforms that support both LAPACK libraries,the default is set lapack mkl on,meaning the Intel MKL LAPACK routines are used.To instead use Netlib’s LAPACK library,you can set lapack mkl off.To determine which library is being used,check the contents of c(lapack mkl).On ARM-based Mac,lapack mkl is not settable and is always off.Note that set lapack mkl should be specified in Stata,not in Mata.Intel MKL conditional numerical reproducibilityRegardless of which LAPACK routine you use,you may encounter slight numeric differences when you run the same code in the same version and platform of Stata on a different OS.This is normal and can be caused by several reasons.For example,Stata is compiled with different compilers on different operating systems,and different compilers could produce slightly different numeric results.You might also get different results on the same operating system if you use the Intel MKL routines.Rosenquist(2011)explains why:“To get the best performance wherever a program is run,Intel MKL will check on the processor type at run time and can dispatch processor-specific code accordingly.Ifa particular instruction set or cache of a certain size is available a specialized code path may exploitit.These code paths are different enough that they will again cause a different order of operations and will cause slightly different results on different processors.”This means that there could be slight numeric differences when running the same code on two machines with the sameflavor of Stata, the same operating system,but with different CPU s.That said,you can obtain different levels of numerical reproducibility with set lapack mkl cnr,discussed below.set lapack mkl cnrIntel offers a couple of different conditional numerical reproducibility(CNR)functions for obtaining different levels of reproducibility,and the type of reproducibility desired can be specified with the lapack mkl cnr setting.The lapack mkl cnr setting should be specified in Stata,not Mata, and it will take effect the next time you launch Stata.The possible values are default,auto, compatible,or off.set lapack mkl cnr default sets the reproducibility level to SSE42on Intel CPUs.On non-Intel CPU s,set lapack mkl cnr default is the same as set lapack mkl cnr auto.LAPACK—Linear algebra package(LAPACK)routines3 set lapack mkl cnr auto allows the Intel MKL to automatically determine the code path based on the different features the CPU supports.The setting takes advantage of the specific features of the CPU,hence the improved performance.The cost is the lower level of numerical reproducibility;the same LAPACK function may return different results on different CPU s,assuming you are using the function on the same operating system andflavor of Stata.Note that the automatically determined code path will not necessarily be the most efficient.For example,we noticed that the A VX instruction set caused excessively high CPU usage on Intel CPU s when Stata was idle.set lapack mkl cnr compatible provides a higher level of numerical reproducibility compared with set lapack mkl cnr auto.In this setting,the LAPACK functions try to produce the same results by using a set of the most common features supported by different CPU s.This setting is also the slowest in terms of performance.set lapack mkl cnr off turns off the code that tries to maintain conditional numerical repro-ducibility,which results in greater performance.With this setting,you may get different results across different runs of the same LAPACK function on the same copy of Stata and on the same machine.For a detailed introduction to the Intel MKL conditional numerical reproducibility,see Rosenquist(2011). AcknowledgmentsWe thank the authors of LAPACK for their excellent work:E.Anderson,Z.Bai,C.Bischof,S.Blackford,J.Demmel,J.Dongarra,J.Du Croz,A.Greenbaum,S.Hammarling,A.McKenney,and D.Sorensen.ReferencesAnderson,E.,Z.Bai,C.Bischof,S.Blackford,J.Demmel,J.J.Dongarra,J.Du Croz,A.Greenbaum,S.Hammarling,A.McKenney,and PACK Users’Guide.3rd ed.Philadelphia:Society for Industrial andApplied Mathematics.Rosenquist,T.2011.Getting reproducible results with Intel MKL.https:///content/www/us/en/develop/articles/getting-reproducible-results-with-intel-mkl.html. Also see[M-5]lapack()—Linear algebra package(LAPACK)functions[R]Copyright LAPACK—LAPACK copyright notification[M-1]Intro—Introduction and adviceStata,Stata Press,and Mata are registered trademarks of StataCorp LLC.Stata andStata Press are registered trademarks with the World Intellectual Property Organization®of the United Nations.Other brand and product names are registered trademarks ortrademarks of their respective companies.Copyright c 1985–2023StataCorp LLC,College Station,TX,USA.All rights reserved.。
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CPPLapack Tutoriale-mail:ueshima@a.mei.titech.ac.jpCPPLapack CP-PLapack CPPLapack1CPPLapack1.1CPPLapack linux1.2linuxapt-get install blaslapackapt-get install lapackBLAS(yum)LAPACK BLAS CPPLapack•mkdir~/local mkdir~local/lib~/local/lib•cd~/local/lib•/projects/cpplapack/CPPLapack cpplapack-XXXX_XX_XX.tar.gz•tar xzf cpplapack-XXXX_XX_XX.tar.gz CPPLapack•lu-s cpplapack-XXXX_XX_XX cpplapackCygwin VC++CPPLapack1.3cpplapack.h CPPLapackiostream cmathA= 175346#include‘‘cpplapack.h’’void main(void){CPPL::dgematrix A(2,3);A(0,0)=1;A(0,1)=7;A(0,2)=5;A(1,0)=3;A(1,1)=4;A(1,2)=6;std::cout<<A<<std::endl;}main.cpp1.4˜/local/lib/cpplapack/makefiles Makefile.g++main.cpp>mv Makefile.g++MakefileMakefileOBJECTS=main.o>makeA.OUT>makeg++-c main.cpp-O2-Wall-Wno-unknown-pragmas-I/home/hoge/local/lib/cpplapack/includeg++main.o-O2-Wall-Wno-unknown-pragmas-L/usr/local/lib/LAPACK-L/usr/local/lib/BLAS-llapack-lblas-lg2c-lm-o A.OUT1.5A.OUT>./A.OUT>./A.OUT1753462()2.1CPPLapackdgematrix.dgesv dgbmatrix.dgbsv dsymatrix.dsysv zgematrix.zgesv zgbmatrix.zgbsv zhematrix.zhesv2.2doubleAx=yxyA.dgesv(y);2.3•LU A•AL ,U(L)yx2.411−2−3213−12x 0x 1x 2 = 123//====================================================================[include]#include‘‘cpplapack.h’’//=======================================================================[main]/*!main*/int main(int argc,char**argv){////make dgematrix A////CPPL::dgematrix A(3,3);A(0,0)=1;A(0,1)=1;A(0,2)=-2;A(1,0)=-3;A(1,1)=2;A(1,2)=1;A(2,0)=3;A(2,1)=-1;A(2,2)=2;////make dcovector y////CPPL::dcovector y(3);y(0)=1;y(1)=2;y(2)=3;////solve Ax=y////A.dgesv(y);////print////std::cout<<‘‘x=\n’’<<y<<std::endl;return0;}x=121]3()3.1CPPLapackdgematrix.dgels dgematrix.dgelss zgematrix.zgels zgematrix.zgelss3.2doubleAx=yAMNxyA.dgels(y,r);A.dgelss(y,S,RANK,RCOND);dgelsm ≥nm ≤nrrdgelssSdcovectorRANK AdoubleRCONDAdouble-1.0()3.3•AdgelsQR,LQdgelssyx•dgelsAdgelss3.441351−1220−315x 0x 1x 2=8205//====================================================================[include]#include‘‘cpplapack.h’’//=======================================================================[main]/*!main*/int main(int argc,char**argv){////make dgematrix A////CPPL::dgematrix A(4,3);A(0,0)=4;A(0,1)=1;A(0,2)=3;A(1,0)=5;A(1,1)=1;A(1,2)=-1;A(2,0)=2;A(2,1)=2;A(2,2)=0;A(3,0)=-3;A(3,1)=1;A(3,2)=5;////make dcovector y////CPPL::dcovector y(4);y(0)=8;y(1)=2;y(2)=0;y(3)=5;////solve Ax=y////A.dgels(y);////print////std::cout<<‘‘x=\n’’<<y<<std::endl;return0;}x=0.928934-0.924221.740034()4.1CPPLapackdgematrix.dgeevdgematrix.dggevdsymatrix.dsyevzgematrix.zgeevzgematrix.zggevzsymatrix.zsyev4.2doubleAA.dgeev(wr,wi);wr,wi doubledcovector vrr,vriA.dgeev(wr,wi,vrr,vri);drovector vlr,vliA.dgeev(wr,wi,vli,vri);dggev4.3A4.4A= 321200421//====================================================================[include]#include‘‘cpplapack.h’’//=======================================================================[main]/*!main*/int main(int argc,char**argv){////make dgematrix A////CPPL::dgematrix A(3,3);A(0,0)=3;A(0,1)=2;A(0,2)=1;A(1,0)=2;A(1,1)=0;A(1,2)=0;A(2,0)=4;A(2,1)=2;A(2,2)=1;////make wr wi vr////std::vector<double>wr,wi;std::vector<CPPL::dcovector>vrr,vri;////dgeev////A.dgeev(wr,wi,vrr,vri);////print////for(int i=0;i<3;i++){std::cout<<‘‘####‘‘<<i<<‘‘th eigen####’’<<std::endl;std::cout<<‘‘wr=’’<<wr[i]<<std::endl;std::cout<<‘‘wi=’’<<wi[i]<<std::endl;std::cout<<‘‘vrr=\n’’<<vrr[i]<<std::endl;std::cout<<‘‘vri=\n’’<<vri[i]<<std::endl;}}####0th eigen####wr=5wi=0vrr=0.6201740.2480690.744208vri=####1th eigen####wr=-1wi=0vrr=0.447214-0.8944271.23361e-16vri=####2th eigen####wr=1.36434e-16wi=0vrr=-5.22952e-17-0.4472140.894427vri=5()5.1CPPLapackdgematrix.dgesvdzgematrix.zgesvd5.2doubleA A M NA.dgesvd(S,U,VT);S dcovector U VT dgematrix5.3A5.4A=413 51−1220−315//====================================================================[include]#include‘‘cpplapack.h’’//=======================================================================[main]/*!main*/int main(int argc,char**argv){////make dgematrix A////CPPL::dgematrix A(4,3);A(0,0)=4;A(0,1)=1;A(0,2)=3;A(1,0)=5;A(1,1)=1;A(1,2)=-1;A(2,0)=2;A(2,1)=2;A(2,2)=0;A(3,0)=-3;A(3,1)=1;A(3,2)=5;////make S,U and VT////CPPL::dcovector S;CPPL::dgematrix U,VT;////SVD A////A.dgesvd(S,U,VT);////print S,U,and VT////std::cout<<‘‘S=\n’’<<S<<std::endl;std::cout<<‘‘U=\n’’<<U<<std::endl;std::cout<<‘‘VT=\n’’<<VT<<std::endl;return0;}S=7.613055.945521.64083U=-0.404476-0.6724230.445651-0.430862-0.678796-0.0902972-0.04937610.727079-0.287353-0.181365-0.872916-0.3500750.541353-0.711899-0.1922890.403933VT=-0.947144-0.1466730.285317-0.230125-0.309031-0.9227910.22352-0.9396740.2589446()6.1CPPLapack6.2m(x 1,y 1)(x n ,y n )m 1x 1x 21···x n 11x 2x 22···x n 2............1x m x 2m ···x n nk 0k 1...k n=y 1y 2...y mAx=yk 0k n6.34(0,-4.04),(1,-1.98),(2,2.02),(3,13.86)3//====================================================================[include]#include‘‘cpplapack.h’’//======================================================================[solve]void solve(int N,CPPL::dcovector&x,CPPL::dcovector&y){////make dgematrix A////CPPL::dgematrix A(x.l,N+1);for(int i=0;i<A.m;i++){for(int j=0;j<A.n;j++){A(i,j)=std::pow(x(i),(double)j);}}////solve Ak=y////A.dgels(y);}//=======================================================================[main]/*!main*/int main(int argc,char**argv){////make dcovector x,y////CPPL::dcovector x(4),y(4);x(0)=0;y(0)=-4.04;x(1)=1;y(1)=-1.98;x(2)=2;y(2)=2.02;x(3)=3;y(3)=13.86;////solve////solve(3,x,y);////print////for(int i=0;i<3;i++){std::cout<<‘‘k’’<<i<<‘‘=’’<<y(i)<<std::endl;}return0;}k0=-4.04k1=3.05667k2=-1.987() 7.17.2X k=N−1n=0x n e−j2πknNW0W0W0···W0W0W1W2···W nW0W2W4 (2)............W0W N−1W(N−1)2···W(N−1)nx0x1x2...x n=X0X1X2...X N−1Ax=y n7.3data.txt10dft.txt//====================================================================[include]#include‘‘cpplapack.h’’//========================================================================[DFT]void DFT(CPPL::zcovector&X,int n){////make zgematrix A////CPPL::zgematrix A(X.l,n+1);for(int i=0;i<A.m;i++){for(int j=0;j<A.n;j++){A(i,j)=std::complex<double>(std::cos((i*j)*2*MPI/(X.l)),-std::sin((i*j)*2*MPI/(X.l)));}}////solve Ax=X////A.zgels(X);}//=======================================================================[main]/*!main*/int main(int argc,char**argv){////make zcovector X////CPPL::zcovector X;X.read(‘‘data.txt’’);////solve////DFT(X,10);////write////X.write(‘‘dft.txt’’);return0;}data.txtzcovector100(1,3.97901e-10)(0.998027,-0.0627905)(0.992115,-0.125333)(0.982287,-0.187381)(0.968583,-0.24869)(0.951057,-0.309017)(0.929776,-0.368125)(0.904827,-0.425779)(0.876307,-0.481754)(0.844328,-0.535827)...dft.txtzcovector11(2.07977e-07,6.53992e-09)(1,-6.4705e-07)(-2.07977e-07,6.54019e-09)(-1.03885e-07,6.54021e-09)(-6.91412e-08,6.5401e-09)(-2.80132e-08,6.53977e-09)(-4.12676e-08,6.54009e-09)(-3.42625e-08,6.54017e-09)(-2.92385e-08,6.54009e-09)(-3.86341e-08,6.53992e-09)(-2.24983e-08,6.54008e-09)]8()8.18.21d2φ∆x2=ρj φ1=0φN=0−211−21···1−21···1−211−2φ2φ3...φj...φN−2φN−1=∆x2ρ2ρ3...ρj...ρN−2ρN−1Ax=yφ8.3d2φ//====================================================================[include]#include‘‘cpplapack.h’’//=======================================================================[main]/*!main*/int main(int argc,char**argv){////declare objects////const int N(201);const double dx(1/(double)(N-1));const double k(2*MPI);CPPL::dgbmatrix A(N-2,N-2,1,1);CPPL::dcovector rho(N-2);////make dgbmatrix A////A.identity();A*=-2;for(int i=0;i<A.n-1;i++){A(i,i+1)=1;A(i+1,i)=1;}////make dcovector rho////for(int i=0;i<rho.l;i++){rho(i)=-k*k*sin((i+1)*k*dx);}rho*=dx*dx;////solve A*phi=dx^2*rho////A.dgbsv(rho);////print////std::cout<<‘‘x phi(x)Ans’’<<std::endl;std::cout<<‘‘000’’<<std::endl;for(int i=0;i<rho.l;i++){std::cout<<(i+1)*dx<<‘‘‘‘<<rho(i)<<‘‘‘‘<<sin((i+1)*k*dx)<<std::endl;}std::cout<<‘‘100’’<<std::endl;}x phi(x)0000.0050.03141330.03141080.010.06279570.06279050.0150.09411610.09410830.020.1253440.1253330.0250.1564470.1564340.030.1873970.1873810.0350.2181610.2181430.040.248710.248690.0450.2790140.2789910.050.3090420.3090170.0550.3387660.3387380.060.3681550.3681240.0650.3971810.3971480.070.4258140.4257790.0750.4540280.453990.080.4817930.4817540.0850.5090830.5090410.090.5358710.5358270.0950.562130.5620830.10.5878340.5877850.1050.6129570.6129070.110.6374760.6374240.1150.6613660.6613120.120.6846030.684547...9()9.1(PointJacobi)9.2n n n+11φn+1 j =1201101···101···10110φn2φn3...φn j...φn N−2φn N−1−∆x2ρ2ρ3...ρj...ρN−2ρN−1φ0 φn+1−φn ≤ φ9.3section8//====================================================================[include]#include‘‘cpplapack.h’’//=======================================================================[main]/*!main*/int main(int argc,char**argv){////declare objects////const int N(201);const double dx(1/(double)(N-1));const double pai(3.141592);const double k(2*pai);const double eps(1.0e-6);CPPL::dgbmatrix A(N-2,N-2,1,1);CPPL::dcovector rho(N-2);CPPL::dcovector phi(N-2),phi_old(N-2);////make dgbmatrix A////A.zero();for(int i=0;i<A.n-1;i++){A(i,i+1)=1;A(i+1,i)=1;}////make dcovector rho////for(int i=0;i<rho.l;i++){rho(i)=-k*k*sin((i+1)*k*dx);}rho*=dx*dx;////solve////do{phi_old=phi;phi=A*phi_old-rho;phi*=0.5;}while(nrm2(phi-phi_old)>eps);////print////std::cout<<‘‘x phi(x)’’<<std::endl;std::cout<<‘‘000’’<<std::endl;for(int i=0;i<rho.l;i++){std::cout<<(i+1)*dx<<‘‘‘‘<<phi(i)<<‘‘‘‘<<sin((i+1)*k*dx)<<std::endl;}std::cout<<‘‘100’’<<std::endl;}x phi(x)0000.0050.0314070.03141080.010.0627830.06279050.0150.0940970.09410830.020.1253180.1253330.0250.1564160.1564340.030.1873590.1873810.0350.2181170.2181430.040.248660.248690.0450.2789580.2789910.050.308980.3090170.0550.3386970.3387380.060.368080.3681240.0650.39710.3971480.070.4257280.4257790.0750.4539360.453990.080.4816960.4817540.0850.508980.5090410.090.5357620.5358270.0950.5620160.5620830.10.5877150.5877850.1050.6128330.6129070.110.6373470.6374240.1150.6612320.6613120.120.6844650.684547...。