The following illustrations, taken from Introduction to Algortihms, Cormen et. Taking the dot product of vectors just means that you take the products of the individual components and then add up the results. In the product C = A X B we define the entry, the entry in the row and the column of A, as being the dot product of the row of A with the column of B. Let us start with two square matrices A and B which are both of size n by n. Cormen draws an analogy between the asymptotic comparison of two functions f and g and the comparison of two real numbers a and b as follows: By analogy, -notation (small omega) is to -notation (big omega) as o-notation is to O-notation. The definitions of O-notation and o-notation are similar, but the main difference is that in, the bound holds for some constant c > 0, but in, the bound holds for all constants c > 0. Small o-notation is used to denote an upper bound that is not asymptotically tight. Asymptotic notation is primarily used to describe the running times of algorithms, with each type of notation specifying a different bound (lower, upper, tight bound, etc.).Д«ig-Theta: -notation bounds a function to within constant factors.Д«ig-Oh: -notation gives an upper bound for a function to within a constant factor.Д«ig-Omega: -notation gives a lower bound for a function to within a constant factor. Asymptotic notationД«efore we start, let us briefly talk about asymptotic notation. Various algorithms have been devised for computing the matrix product, especially for large matrices. Since it is such a central operation in many applications, matrix multiplication is one of the most well-studied problems in numerical computing. The definition of matrix multiplication is motivated by linear equations and linear transformations on vectors, which have numerous applications in applied mathematics, physics, and engineering. It just comes up all the time in important applications. Tim Roughgarden states thatД¬omputers as long as theyвЂ™ve been in use from the time they were invented uptil today, a lot of their cycles is spent multiplying matrices. Matrix multiplication is a fundamental problem in computing. There will be illustrations along the way, which are taken from CormenвЂ™s textbook Introduction to Algorithms and Tim RoughgardenвЂ™s slides from his Algorithms specialization on Coursera. I will start with a brief introduction about how matrix multiplication is generally observed and implemented, apply different algorithms (such as Naive and Strassen) that are used in practice with both pseduocode and Python code, and then end with an analysis of their runtime complexities. In this post I will explore how the divide and conquer algorithm approach is applied to matrix multiplication.

Divide and conquer all settlements 4.5