ExpressionTemplates

Expression templates are a C++ programming technique that allows for the creation of high-performance libraries, particularly in the domain of linear algebra and numerical computation. The core idea is to avoid intermediate object creation that often occurs with traditional operator overloading. Instead of evaluating an expression immediately and storing the result in a temporary object, expression templates defer the evaluation until the final result is needed. This is achieved by representing operations not as actual computations, but as expression trees.

These expression trees are composed of objects that represent operands and operators. When an expression is formed, say `A + B * C`, where A, B, and C are matrix objects, the result is not a new matrix object containing the sum. Instead, it's an object that describes the operation: "add A to the result of multiplying B and C." This description is passed around as a template argument.

The actual computation happens only when a function, such as an assignment or an output operation, is invoked. This function then traverses the expression tree, performing the necessary arithmetic operations directly into the final destination object. The benefits include eliminating temporary objects, which can be a significant performance bottleneck, and enabling the compiler to optimize the entire expression as a whole. This optimization can lead to code that is as fast as hand-written C loops, but with the much higher-level abstraction of mathematical notation. Libraries like Eigen extensively utilize expression templates for their performance.