COSC342: Lab 02
Matrices and Vectors in C++March 14th 20161 IntroductionIn this lab you will practice using a simple Matrix library in C++. Onlinedocumentation for this library is available at http://www.cs.otago.ac.nz/cosc342/docs/matrices/index.html. The library provides two classes, Matrixand Vector, and Vector is a subclass of Matrix.Each class has a header file, Class.h, and an implementation, Class.cpp.There is also a sample file, matrixTest.cpp illustrating some basic operations,and a Makefile which builds the project.You can get the source code from/home/cshome/coursework/342/pickup/labs/lab02-MatricesIt is easiest to work from the command line for this lab. Start by makinga directory to store your COSC342 work, if you have not already. Open aterminal, and enter the following commandscd ∼/Documentsmkdir cosc342cd cosc342If you have already made a directory to store your COSC342 work, change tothat directory. Next we’ll get a copy of the lab filescp -r /home/cshome/coursework /342/ pickup/labs/lab02 -Matrices .The -r option copies files recursively, and the dot, ., at the end means copyto here.You should now be able to build and run the program:cd lab02 -Matricesmake./ matrixYou should see the following output:The Matrix A is:1 0 00 0.5 -12 2 01The Vector v is:123u = A*V is:1-262 Working with the LibraryThis may be your first time working with C++, but the sample file should giveyou enough information to get started with the Matrix and Vector classes.We’ll go through some basics, do some exercises, then you can go back to lookat the classes and their implementations in more detail.Matrix and Vector objects are most commonly made with constructors thattake the size as parameters:Matrix A(3,2); // A 3x2 matrixVector v(3); // A 3-vectorIf no arguments are given then a 1× 1 Matrix or Vector is made:Matrix A; // A 1x1 matrixVector v; // A 1-vectorYou can access the elements of a Matrix or Vector with a function-stylesyntax. Following C/C++ convention, indices start from 0 rather than 1.Matrix A(3,3); // A 3x3 matrixA(0,0) = 1; // Set top left element to 1A(1,2) = 2; // Set 3rd element of 2nd row to 2A(2,2) = 3; // Set bottom right element to 3C++ uses streams for output, and the standard output stream, std::cout,is defined in the iostream header. The << operator is used to send informationto the stream, so “Hello World” in C++ looks like this:#includeint main () { std::cout << "Hello , World" << std::endl; return 0; } The special constant std::endl ends a line and flushes the stream. You can also use << with Matrix and Vector objects: 2 std::cout << A << std::endl; Vectors are a single column, so it is often more convenient to output their transpose: std::cout << v.transpose () << std::endl; C++ supports operator overloading, which we’ll discuss in detail later. For now all you need to know is that this means you can write fairly natural arith- metic expressions using Matrix and Vector objects: Matrix A(3,3); Vector v(3), u(3); u = A*v; 3 Exercises These exercises are a subset of last week’s tutorial, so you should be able to check your answers against them. For each exercise you should write code to evaluate the required expressions, and display the results in the console with the std::cout stream. 3.1 Vector Arithmetic Let u = (1,−3, 2) and v = (3, 2, 0). Evaluate the following: 1. 2u 2. u+ v 3. u− v 4. 2u− 3v 3.2 Dot and Cross Products Vector objects have methods dot and cross which take another Vector as a parameter and return the appropriate product. Evaluate the following dot products: 1. (1, 2,−2) · (1, 2,−2) 2. (1, 2,−2) · (−2, 2, 1) 3. (1, 2,−2) · (2, 3, 1) Evaluate the following cross products: 1. (1, 2,−2)× (−2, 2, 1) 2. (1, 2,−2)× (−2,−4, 4) 3. (1, 2,−2) · ((1, 2,−2)× (−2, 2, 1)) 3 3.3 Matrix Arithmetic Evaluate the following: 1. 2 [ 1 −2 3 4 ] − [ 3 2 4 1 ] 2. [ 2 1 2 2 ] [ 1 −2 −4 1 ] 3. [ 1 −2 −4 1 ] [ 2 1 2 2 ] 3.4 Matrices and Vectors These following exercises make use of the matrix A = [ 1 1 0 0 1 2 ] , and the vectors u = (1, 2,−2) and v = (−2, 2, 1). Compute the following: 1. Au 2. uuT 3. uTv 4 Inside the Library This may be your first time working with C++, but many of the concepts should be familiar to you from your C and Java experience. For the most part you should just be able to follow along for now, but here are some notes that might help you make sense of things. I’d recommend skimming over these at first, having a go at writing some code, and then coming back for a closer look. There are also plenty of comments in the code which should help you get started, as well as the online documentation. Classes in C++ are defined using a similar syntax to Java (technically Java borrowed a lot of syntax from C++), and each class typically has two files asso- ciated with it – a header file (e.g.: Matrix.h) and an implementation file (e.g.: Matrix.cpp). If you open Matrix.h in a text editor, you will see the definitions (or signatures) of many methods, but no implementation details. These details are all in Matrix.cpp, and the header file just gives the interface. Some of the syntax may be unfamiliar to you, in particular operator over- loading. A good example is the multiplication method for Matrices: friend Matrix operator *( const Matrix& lhs , const Matrix& rhs); The short version of what this does, is allow us to write code like 4 Matrix A(3,3), B(3,4), C(3,4); // Code to fill A and B with values C = A*B; This code creates a 3×3 matrix called A, a 3×4 matrix called B, then multiplies them together, storing the result in C. In more detail: • friend Tells us that even though this function is not exactly a method of the Matrix class it can access the internal elements of Matrix objects as if it was. • Matrix Tells us that the result of multiplying two Matrix objects together is another Matrix object. • operator* Tells us that this function provides an implementation for the multiplication operator. Basically, you can use A*B as a shorthand for Matrix::operator*(A,B). • Finally the parameters are Matrix objects. These are passed by reference (&) to avoid copying complex objects, and the const keyword provides a guarantee that if you call A*B, A and B will remain unchanged. • The parameter names, lhs and rhs, are supposed to remind you that they are on the left hand side and right hand side of the * operator respectively. Operator overloading is a very powerful tool for providing concise and intu- itive interfaces to mathematical objects like Matrices and Vectors. It also lets us provide a few other useful things, such as overloading the assignment operator, which is also used in the line C = A*B;, above. Another useful pair of overloads are the stream operators, << and >>, which are used in C++ to provide a pow- erful syntax for input and output through streams. The classic Hello World program in C++ is #include int main () { std::cout << "Hello World" << std::endl; return 0; } The iostream library provides input and output streams, one of which is std::cout – this is an output stream which goes to the console (unless redi- rected by the operating system). The std:: part just says that the stream cout exists in the standard namespace, which prevents cluttering the global names- pace with lots of stuff. In this context the << operator provides the ability to send information to the stream. In this case, we are sending a string, and this gets printed in the console. 5 Overloading << means that we can just send Matrix operators to these streams: std::cout << "Matrix A is:" << std::endl; std::cout << A << std::endl; One final way that operator overloading is used is to access elements of Matrix objects. This is done with the function call operator, operator(). There are two versions of this, and here are their signatures: double& operator ()( size_t row , size_t col); const double& operator ()( size_t row , size_t col) const; These let us write code like Matrix A(3,3); double x = A(1,2); On the first line, the brackets tell us that the parameters to send to the con- structor are 3 and 3. On the second line, however, the brackets are calls to the function call operator, which returns the corresponding element of the matrix (indexed from 0, of course, since this is C++). So the second line gets the value from the second row and third column of A, and stores it in x. Note that the function call operators return a reference, this means that you can also assign values to Matrix elements like this: A(2,1) = 4.5; This is also why there are two versions of the function call operator. C++ makes a lot of use of the idea of const correctness, which is using the keyword const to tell the compiler when things should not be altered. If A is const, then we want to allow read access to its elements, but don’t want to be able to write to them. The second version of the function call operator tells us that if we call operator() on a const Matrix then we get back a const double&, and so can’t assign a new value to it. Have a look at the source code for the Matrix and Vector classes. Notice that the output of the operators on Vector and Matrix objects might be a Vector, a Matrix or a double (a scalar). This helps to catch errors at com- pile time, but other errors (such as multiplying Matrix objects whose interior dimensions don’t agree) are caught at run time with asserts. Have a look through the code and see how it works. Ask us if you have any questions. 5 More Advanced Matrix Libraries This Matrix code is designed to be easy to use and reasonably easy to un- derstand. More advanced C++ programming methods allow for very efficient implementations, but they are much harder to understand. One such library is Eigen, which can be found at http://eigen.tuxfamily.org. 6 Eigen uses a feature C++ called templates, which are a bit like Java generics (again, Java stole this from C++) but more powerful. Matrices in Eigen are declared by giving their size and type as template parameters: // A 3x4 matrix of doubles Eigen :: Matrix M<3, 4, double > The use of templates allows the compiler to do a lot of reasoning about the size and type of matrices. It means that the compiler can determine automatically that the result of multiplying a 3× 4 matrix by a 4× 2 matrix is a 3× 2 matrix. If you try assigning this to an incorrectly sized matrix, then you get an error at compile time rather than runtime. Templates also allow for a crazy C++ technique called template metaprogram- ming. It turns out that the template structure of C++ is Turing complete, so you can do all sorts of things with it. This lets the compiler realise when it is dealing with small matrices and unroll loops etc. to make faster code. Other advanced features of Eigen like lazy evaluation, avoidance of temporary objects, and vectorisation make the code very fast, while keeping a fairly simple interface. For our purposes, however, code that you can understand is better than code which is very fast.