Introduction to numerical analysis
Numerical Analysis deals with the study of Methods, Techniques, or Algorithms for obtaining approximations for solutions to Mathematical problems. It has played a tremendous role in the advancement of science and technology.
Numerical Analysis as a Science and As an Art:
Numerical Analysis is a two-edged knife and serves as a science as well as an art. As a science numerical Analysis is concerned with the methods (processes) for obtaining a solution to complicated mathematical problems by means of arithmetical and logical operations.
Sometimes these methods involve the development of an algorithm for the solution of problems. As an art, it deals with the development of an algorithm and applying it to a particular problem. As a matter of fact, numerical analysis is a different discipline, than it was thirty years ago.
With the invention of the digital computer, high-speed computation has revolutionized it · as an art and has given an enormous impetus to it as a science.
The Role of Numerical Analysis in Science:
In science, we are mainly concerned with some particular aspect of the physical world and thus we investigate by using mathematical models. The use of the model serves two purposes.
- It enables us to isolate the relevant aspects of a complex physical situation and it also enables us to specify with Complete precision the problem to be, solved.
- When the model has been established, the next step is to write down equations expressing the constraints and physical Laws that apply. These equations may be simple algebraic equations or differential or integral equations.
The difficulty with conventional mathematical analysis lies in solving the equations. As everybody knows it is easier to write down equations than to solve them. In the case of a differential equation, it may be possible to obtain a useful solution whereas it may be quite impossible to do so in the case of another equation.
Numerical Analysis is much more general in its application and usually, when solutions exist, they can be computed. The great advantage of Numerical Analysis is that it enables more realistic models to be treated.
It is unfortunately not true that if results are required to a slow degree of precision, the calculations can ‘be done throughout to the same low degree of precision. Sometimes it is necessary to work with quite a high accuracy in order to get an answer which is accurate to 95 %.
Why Numerical Techniques?
In your Mathematics courses, you might have concentrated mainly on Analytical techniques. Therefore, it is likely that you know how to calculate and also how to solve a differential equation.
You are also familiar with the determinant and matrix techniques for solving a system of simultaneous linear equations. Therefore, your first reaction to encountering a book such as this may be – Why Numerical methods? or what are Numerical techniques?
While studying Integration, you have learned many techniques for integrating a variety of functions, such as integration by substitution, by parts, by partial fractions, etc. But how to integrate a function when the values are given in the tabular form?
data is given as under for time t sec, the velocity is v feet/ sec2.
How to find the distance traveled in 50 Secs i.e. How to evaluate Also consider the solution of Simultaneous Linear equations, and the use of Cramer’s Rule, or inversion of the Matrix, these methods do not present much trouble when solving a system of three equations in three unknowns.
But what happens if you have to solve a system of fifty equations in fifty unknowns, which can occur when dealing with space frames which are used in roof trusses, bridge trusses, pylons, etc?
For that purpose, you need an application and great advantage of numerical technique and a digital computer. There are certainly more problems that require numerical treatment for their solutions.
- The equation occurs in physics. What value of a satisfies this equation?
- The equation Sin wt = e-at occurs in Astronomy wh t is the smallest positive value of t for given values of α and ω that satisfies the equation.
- The Integral occurs when obtaining the heat capacity of a solid i.e.
by a method based on the vibrational frequencies of the crystal. What is the value of this integral for a certain value of a?
Sources of errors
Numerical answers to problems generally contain errors that arise in two areas namely,
- Errors inherent in the mathematical formulation of the problem.
- errors incurred when the mathematical statement of a problem is only an approximation to the physical situation, and we desire to solve it numerically Such errors are often
There are three main sources of computational error.
- gross error or blunder, which is familiar to all users. Digital computers reduced the probability of such errors enormously.
- The other two types of errors in which we are mainly interested are
- The error was caused by solving the problem not as formulated but rather using some approximations. This is usually caused by the replacement of an infinite (i.e. summation or integration) or infinitesimal (i. e. differentiation) process by a finite approximation, examples are:
- Calculation of an elementary function says
neglecting the contribution of the rest of the terms.
- Approximation of the Integral; of a function by a finite summation of functional values as in the trapezoidal or Simpson’s rules (we shall discuss them later.
- A solution of a system of Linear equations by Jacobi or Gauss-Seidel methods or a solution of Non-linear equations by the Newton-Raphson method is to be discussed.
Later, this type of error is usually called the ‘Truncation’ error because we limit the iterations to a certain number whereas these can go to infinity and the contribution of the remaining terms or iterations is not taken into account.
The other source of error is that caused by the fact. that arithmetic calculations can almost never be carried out with complete accuracy, most numbers have infinite decimal representation which must be rounded. This kind of error is called ’roundoff error.