Logarithmic Properties

The logarithm is the inverse function to exponentiation. It means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.

eg.,

log_b x = y which is same to b^y = x

log₂ 8 = 3 same as 2³ = 8

The following are the properties of logarithm.

log_b xy = log_b x + log_b y

log_b (x/y) = log_b x - log_b y

log_b xⁿ = n log_b x

log_b a = log_c a / log_c b

log_b a = 1 / log_a b

Some basic values of logarithm

log_b1 = 0

log_bb = 1

log_bb² = 2

Exercise 2 - Amdahl's Law

A programmer has parallelized 99% of a program, but there is no value in increasing the problem size, i.e., the program will always be run with the same problem size regardless of the number of processors or cores used. What is the expected speedup on 20 processors? Solution As per Amdahl's law, the speedup, N - No of processors = 20 f - % of parallel operation = 99% = 1 / (1 - 0.99) + (0.99 / 20) = 1 / 0.01 + (0.99 / 20) = 16.807 The expected speedup on 20 processors is 16.807

Exercise 1 - Amdahl's Law

A programmer is given the job to write a program on a computer with processor having speedup factor 3.8 on 4 processors. He makes it 95% parallel and goes home dreaming of a big pay raise. Using Amdahl’s law, and assuming the problem size is the same as the serial version, and ignoring communication costs, what is the speedup factor that the programmer will get? Solution Speedup formula as per Amdahl's Law, N - no of processor = 4 f - % of parallel operation = 95% Speedup = 1 / (1 - 0.95) + (0.95/4) = 1 / 0.5 + (0.95/4) Speedup = 3.478 The programmer gets 3.478 as t he speedup factor.

Minor, Cofactor, Determinant, Adjoint & Inverse of a Matrix

Consider a matrix Minor of a Matrix I n the above matrix A, the minor of first element a 11 shall be Cofactor The Cofactor C ij of an element a ij shall be When the sum of row number and column number is even, then Cofactor shall be positive, and for odd, Cofactor shall be negative. The determinant of an n x n matrix can be defined as the sum of multiplication of the first row element and their respective cofactors. Example, For a 2 x 2 matrix Cofactor C 11 = m 11 = | a 22 | = a 22 = 2 Determinant The determinant of A is |A| = (3 x 2) - (1 x 1) = 5 Adjoint or Adjucate The Adjoint matrix of A , adjA is the transpose of its cofactor matrix. Inverse Matrix A matrix should be square matrix to have an inverse matrix and also its determinant should not be zero. The multiplication of matrix and its inverse shall be Identity matrix. The square matrix has no inverse is called Singular. Inv A = adjA / |A| [ adjoint A / determ...

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