Question

Scientific Computing Course - please explain in detail

PI has a value of 3.14159.... Round PI to 4 decimal places using chopping and then again using rounding.
What are the relative errors of using chopping and rounding?

Answer 1

1. Explanation for Scientific Computing:

Scientific computing is nothing but the scientific discovery of analysis and experiments. The problems themselves usually arise from other disciplines such as mathematics, engineering, biology, physics, chemistry and other natural sciences. For example some problems cannot be solved by olden experimental or theoretical means, such as predicting the climate change. But these kinds of problems can be now solved using computers or super computers to predict. Other such examples like characterization of toxic materials or the problem would be too expensive or time-consuming to try to solve by other avenues etc., Furthermore, computer simulations can be embedded in optimization algorithms for optimal designs, e.g. the optimal design of aircrafts in the computer instead of experience driven trial and error designs with the support of expensive wind tunnel experiments. Another important part of Scientific Computing is that it is a multidisciplinary activity. Generally, it involves experts in the application at hand, and also applied mathematicians and computer scientists that help to implement computational solution.

According to the definition:

“Scientific Computing is the collection of tools, techniques, and theories required to solve on a computer mathematical models of problems in Science and Engineering.”
The development of the electronic computer, however, signaled a new era in the approach to the solution of scientific problems. Considerations that where irrelevant or unimportant for hand calculation now became of utmost importance for the efficient and correct use of a large Computer System. Many of these considerations – programming languages, operating systems, management of large quantities of data, correctness of programs – were subsumed under the new discipline of Computer Science, on which scientific computing now depends heavily. In summary, then, scientific computing draws on mathematics and computer science to develop the best way to use computer systems to solve problems from science and engineering.

Scientific Computing Course:

Applications of mathematics in the analysis, modeling and solution of complex engineering problems often involve scientific computing—a combination of advanced mathematical techniques, high-performance computing, computer simulations and methods for optimization, data analysis and visualization. The development and use of these models and computational methods require a combination of modern mathematics and numerical analysis, as well as computer programming—collectively known as scientific computing. This focus area provides the student with a solid understanding of the underlying abstract and applied mathematical tools, as well as techniques that provide the necessary basis for the modeling of the problems and the design of efficient computer algorithms.

2. PI has a value of 3.14159.... Round PI to 4 decimal places using chopping and then again using rounding.
What are the relative errors of using chopping and rounding?

The number π has an infinite decimal expansion of the form

                       π = 3.14159

Written in normalized decimal form, we have

                       π = 0.314159 × 10.

Chopping: The floating-point form of π into four digit places using chopping is

                       f l(π) = 0.31415 × 10 = 3.1415.

Rounding: The sixth digit of the decimal expansion of π is a 9, so the floating-point form of π using five-digit rounding is

                       f l(π) = (0.31415 + 0.00001) × 10 = 3.1416.

Suppose that p∗ is an approximation to p then the relative error is |p − p∗| / |p| , provided that p ≠ 0.