Continuity and Intermediate Value Theorem

For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point.

A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.

A function f(x) is continuous at a point a if and only if the following three conditions are satisfied:

f(a) is defined
lim_x→af(x) exists
lim_x→af(x) = f(a)

A function is discontinuous at a point a if it fails to be continuous at a. Discontinuities may be classified as removable, jump, or infinite.

Let f and g be continuous functions. Then

f + g is a continuous function.
fg is a continuous function.
f/g is a continuous function, when g(x) ≠ 0

The Intermediate Value Theorem

Suppose f(x) is a continuous function on the interval [a,b] with f(a) ≠ f(b). If N is a number between f(a) and f(b), then there is a point c between a and b such that f(c) = N.

The Intermediate Value Theorem guarantees that if a function is continuous over a closed interval, then the function takes on every value between the values at its endpoints.

We can locate root using Intermediate value theorem. The theorem fails for discontinuous functions.

Gaussian Elimination - Row reduction Algorithm

Gaussian elimination is a method for solving matrix equations of the form, Ax=b. This method is also known as the row reduction algorithm. Back Substitution Solving the last equation for the variable and then work backward into the first equation to solve it. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Pivot row The row that is used to perform elimination of a variable from other rows is called the pivot row. Example: Solving a linear equation The augmented matrix for the above equation shall be The equation shall be solved using back substitution. The eliminating the first variable (x1) in the first row (Pivot row) by carrying out the row operation. As the second row become zero, the row will be shifted to bottom by carrying out partial pivoting. Now, the second variable (x2) shall be eliminated by carrying out the row operation again. ...

Matrix Eigenvalues & Eigenvectors

The process of finding an unknown scalar, λ and a nonzero vector, x for a given non zero square matrix, A of dimension n x n is called matrix eigenvalue or eigenvalue. Ax = λ x The λ and x which satisfies the above equation is called eigen value and eigenvector. Ax should be proportional to x. The multiplication will produce a new vector that will have the same or opposite direction as the original vector. The set of all the eigenvalues of A is called the spectrum of A. The largest of the absolute values of the eigenvalues of A is called the spectral radius of A. To determine eigenvalue and eigenvector, the equation can be written in matrix notation, (A - λI)x = 0 By Cramer's theorem, the homogeneous linear system of equations has a nontrivial solution if and only if the corresponding determinant of the coefficients is zero. A - λI is called characteristic matrix and D( λ) is characteristic determinant of A. The above equation is called characteristic ...

Cache Memory

The supplementary memory system that temporarily stores frequently used instructions and data for quicker processing by CPU of a computer. Main memory consists of up to 2n addressable words, with each word having a unique n-bit address. there are M = 2n/K fixed length of blocks in main memory. where K - words The cache consists of m blocks called lines. The length of a line, not including tag and control bits, is the line size. As there are more blocks in main memory than lines in cache, an individual line cannot be uniquely and permanently dedicated to a particular block. Thus, each line includes a tag that identifies which particular block is currently being stored. The cache connects to the processor via data, control, and address lines. When a cache hit occurs, the data and address buffers are disabled and communication is only between processor and cache, with no system bus traffic. When a cache miss occurs, the desired address is loaded onto the system bus and the data are r...

The Data Ilm

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