C++ for Scientists - Technische Universität Dresden

56 CHAPTER 2. C++ BASICS
A function computing the inverse must return the same value and also pass the test agains
identity:
Matrix A inverse(inverse(A));
cout ≪ ”inverse(A) is \n” ≪ A inverse ≪ ”A ∗ AI is\n” ≪ Matrix(A inverse ∗ A);
assert(one norm(Matrix(A inverse ∗ A − I)) < 0.1);
After establishing tests for all components of our calculation we start with their implementations.
The first function we program is the inversion of an upper triangular matrix. This function
takes a dense matrix as argument and returns another matrix:
dense2D inline inverse upper(dense2D const& A) {
}
Since we do not need another copy of the input matrix we pass it as reference. The argument
shall not be changed so we can pass it as const. The constancy has several advantages:
• We improve the reliability of our program. Arguments passed as const are guaranteed
not to change, if we accidentally modify them the compiler will tell us and abort the
compilation. There is a way to remove the constancy but this should only be used as
last resort, e.g. for interfacing obsolete libraries written by others. Everything you write
yourself can be realized without eliminating the constancy of arguments.
• Compilers can optimize better when the objects are guaranteed not to alter.
• In case of references, the function can be called with expressions. Non-const references
require to store the expression into a variable and pass the variable to the function.
Another comment, people might tell you that it is too expensive to return containers as results
and it is more efficient to use references. This is true — in principle. For the moment we accept
this extra cost and pay more attention to clarity and convenience. Later in this book we will
introduce techniques how to minimize the cost of returning containers from functions.
So much for the function signature, let us now turn our attention to the function body. The
first thing we do is verifying that our argument is valid. Obviously the matrix must be square:
const unsigned n= num rows(A);
assert(num cols(A) == n); // Matrix must be square
The number of rows is needed several times in this function and is therefore stored in a variable,
well constant. Another prerequisite is that the matrix has no zero entries in the diagonal. We
leave this test to the triangular solver.
Speaking of which, we can get our inverse triangular matrix with a triangular solver of a linear
system, which we find in MTL4, more precisely the k-th vector of U −1 is the solution of
Ux = ek
where ek is the k-th unit vector. First we define a temporary variable for the result.
dense2D Inv(n, n);
Then we iterate over the columns of Inv: