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. Then the variable (x3) can be determined now with the row reduced matrix, by which other variable (x1, x2) can be calculated. The value of variables are

x3 = 2 ; x2 = 4 ; x1 = 2

The same operation can be done with a single command, Row reduced echelon form (rref) in Matlab or Octave as follows.

The system is overdetermined, if the equations are more than unknowns; determined, if equations are equal to unknowns; underdetermined, if equations are lesser than unknowns.

The system is called consistent, if it has at least on solution and inconsistent, if it doesn't have any solution.

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.

BITS Work Integrated Learning Program - M.Tech Data Science & Engineering

BITS Pilani offers work integrated learning program (WILP) on M.Tech Data Science and Engineering which is UGC approved. The course is a four semester programme designed to help work professionals to build their skills required for data science engineering which enable them to become a Data Scientist. It is a 100% online course and lectures would be delivered by BITS Pilani faculty on weekends. Those who are working in software industry as Software Engineer, Programmer, Data Analyst, Business Analyst can apply for the course. Minimum eligibility criteria to apply for the course. Those who are employed holding B.E/B.Tech/MCA/M.Sc or Equivalent with 60% marks and minimum one year relevant work experience. The candidates should have basic programming and engineering mathematics knowledge. The following subjects shall be covered in the course. Semester Subjects Data Mining Mathematical Fundamentals for Data Science Data Structure and Algorithms Design Co...

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