Dr. Brajesh Kumar Jha

Approximate Matrix Multiplication using Random Sampling Background: Approximate Matrix Multiplication (AMM) Matrix multiplication is a fundamental operation in scientific computing, machine learning, data mining, and numerical linear algebra. Given two matrices \( A \in \mathbb{R}^{n \times d} \) and \( B \in \mathbb{R}^{d \times m} \), the exact computation of \( AB \) requires \( O(ndm) \) arithmetic operations, which becomes computationally expensive when the inner dimension \( d \) is large. Approximate Matrix Multiplication (AMM) addresses this challenge by computing an approximation \( \widetilde{C} \approx AB \) that is significantly faster to obtain while maintaining provable accuracy guarantees. The key idea is to reduce the dimensionality of the problem using randomized techniques, thereby lowering computational cost and memory usage. Random Sampling of Columns and Rows (Sketching) One of the simplest and most intuitive approaches to AMM is ra...

Quiz: Big Data Matrix Instructions: Enter your details, answer all questions, and click Submit Quiz . Name: Roll Number: 1. What is a Big Data Matrix? A. A small table stored in Excel B. A large-scale matrix representation of big data C. A graphical visualization tool D. A database index 2. In an Object–Attribute Matrix, rows represent ______. A. Attributes B. Columns C. Objects or entities D. Algorithms 3. Which matrix is commonly used in recommendation systems? A. Document–Term Matrix B. Object–Attribute Matrix C. User–Item Matrix D. Time-Series Matrix 4. A Sparse Matrix is characterized by: A. All values are non-zero B. Mostly negative values C. Mostly zero values D. Equal rows and columns 5. Document–Term Matrix is mainly used in: A. Image processing B. Network routing C. Natural Language Processing D. Sorting algorithms 6. Which matrix represents relationships between nodes? A. Time-Series Matrix...

Limits of Functions of Two Variables Limits of Functions of Two Variables Definition Let f(x,y) be defined in a neighborhood of the point (a,b) , except possibly at the point itself. We say the limit of f(x,y) as (x,y) → (a,b) is L, and write: lim (x,y)→(a,b) f(x,y) = L if for every ε>0, there exists δ>0 such that: 0 < √((x-a)² + (y-b)²) < δ ⇒ |f(x,y)-L| < ε Intuitive Explanation The value of f(x,y) gets closer to L as (x,y) approaches (a,b) from any direction. The limit must be the same along all paths approaching (a,b). Examples Example 1 Function: f(x,y) = x + y Find: lim (x,y)→(1,2) f(x,y) Solution: f(1,2) = 1 + 2 = 3 Answer: 3 Example 2 Function: f(x,y) = xy / (x² + y²) Find: lim (x,y)→(0,0) f(x,y) Solution: Along y=0: f(x,0)=0 Along x=0: f(0,y)=0 Along y=x: f(x,x)=1/2 Answer: Limit does not exist (different values along different paths) Example 3 Function: f(x,y) = x² + y² Find: lim (x,y)→(1,1) f(x,y) ...

  Function of Two variable: Let u be a symbol which has a definite value for every pair of values of x and y, then u is called a function of two independent variable x and y and is written as u=f(x,y)   Function of Two Variables – Graphical Representation A function of two variables is written as: z = f(x, y) 1. General 3D Graph z | | • | • | • | • |•____________ y / / x Explanation: This shows a surface in 3D space where z depends on x and y. 2. Plane Surface (z = x + y) z | | / | / | / |/________ y / / x Application: Cost, temperature variation. 3. Paraboloid (z = x² + y²) z | __|__ / | \ / | \ /_______|_______\ y | x Application: Heat distribution, potential energy. 4....

 Calculus of Several Variables Table of Content                 1 Functions of two variables               2 Limits of function of two variables              3 Continuity of function of two variables               4 Partial derivatives               5 Partial derivatives              6 Partial derivatives               7 Total derivatives              8 Total derivatives              9 Maxima and minima .        ...

Unit -1   Big Data Matrix What is a Big Data Matrix? A Big Data Matrix is a very large data structure arranged in rows and columns, where: Rows represent entities (users, sensors, documents) Columns represent features or attributes Data volume is too large for traditional systems Characteristics of Big Data Matrix Large scale (millions or billions of rows) High dimensionality Sparse data (many zero values) Stored and processed in distributed systems Why Big Data Matrices are Important Used in Machine Learning and Artificial Intelligence Helps in data analytics and decision making Supports large-scale scientific research Enables real-time data processing Types of Data Matrix Numerical Data Matrix Binary Data Matrix Categorical Data Matrix Sparse Data Matrix Distributed Data Matrix 1. Numerical Data Matrix Definition A Numerical Data Matrix is a matrix in which all elements are numerical val...

📥 Download Partial Fractions PDF Integration by Partial Fractions — Examples & Solutions Integration by Partial Fractions Types and worked examples (converted from the uploaded notes). Type 1: Distinct linear factors When the denominator is expressible as a product of distinct linear factors. Example 1 Evaluate \( \displaystyle \int \frac{x-1}{(x+1)(x-2)} \, dx \). Write \( \dfrac{x-1}{(x+1)(x-2)} = \dfrac{A}{x+1} + \dfrac{B}{x-2} \) Then \( x-1 = A(x-2) + B(x+1) \) Putting \(x=2\): \(1 = 3B \Rightarrow B=\tfrac{1}{3}\). Putting \(x=-1\): \(-2 = -3A \Rightarrow A=\tfrac{2}{3}\). Thus \( \dfrac{x-1}{(x+1)(x-2)} = \dfrac{2/3}{x+1} + \dfrac{1/3}{x-2} \) Integrate: \( \int \frac{x-1}{(x+1)(x-2)} dx = \frac{2}{3}\ln|x+1| + \frac{1}{3}\ln|x-2| + C \) Example 2 Evaluate \( ...