What do you understand by numerical calculations?

In the mid-20th century, the availability and the growth in power of digital computers led to the increase in the mathematical model in the field of engineering and science. The analysis of these detailed models needs to be increased.

These analyses take a complex problem and break it into many smaller and simpler problems. Typically, in the present day, computers calculate these simple problems.

One must also define analytic calculations (or methods). In this case, a combination of algebra and calculus helps to arrive at an answer.

Numerical calculation is one of the ways we do science now. It isn't cheating; it's a legitimate method. If the results of the numerical calculation agree with actual data, then it works. It's that simple.

Building a numerical model

Let's say I have some object with a force on it. I don't want to use any fancy algebra or calculus to find the motion of this object. Instead, I want to use a numerical model. Here's what I will do. First, I will break the motion into tiny time steps. Let's say each step is 0.01 seconds. During each step, we have to do the following:

Calculate the force on the object as well as the acceleration. Use the acceleration and the velocity in the previous time step to find the new velocity. Use this new velocity and the old position to find the new position. Update the time. Repeat the steps for as long as I want.

Common perspectives in numerical analysis

The field of numerical analysis is concerned with all possible outcomes of the numerical problem, the numerical calculation methods for analysis are efficient in a similar manner to that of a computer program.

• When the numerical methods are introduced with a problem, they cannot get solved directly, they rather replace it with a problem that is easy to solve. The few examples of such problems are the use of the interpolation in numerical integration and root finding methods.
• There is quite a large use of the language and the results of the linear algebra and functional and real analysis (with its simplifying notation of norms, vector spaces, and operators).
• There are the lot of concerns with the size and the analytical form of the error. It is necessary to understand the nture of the error in the listed solution. The understanding the of the form of the error allows to improve the convergence behaviour of the numerical problem.
• The analytics are having some concerns with the stability of the numerical problems, any concept that refers to the stability or the sensitivity of the problem to make the changes at the small parameters of the problem.
• The numerical methods that are used for solving the problems must not be sensitive to make the changes in the data than the original problem that needs to be solved. The original problem that needs to be solved must be created in a manner such that the problem is stable and well conditioned.
• Numerical analysts are very interested in the effects of using finite precision computer arithmetic, which is especially important in numerical linear algebra, as large problems contain many rounding errors.
• The efficiency of the algorithm is the major field of interest for the the analyts.

Modern applications and computer software

The concept of numerical analysis and mathematical modeling is found to be essential in many areas of the modern world. The numerical analysis software is packed with various software and allows evaluation of the more detailed and complex models when the user is unaware of the mathematics utilized in the problem. Attaining the level of transparency the users want requires reliable, efficient, and accurate numerical analysis software. Problem Solving Environment is required in which the relatively easy problem is provided. The PSE's are based on the models that have excellence in the theory of the mathematical models and make them available for the users with the help of a convenient user interface.

Uses of numerical calculations

In the field of engineering, computer-aided engineering is an important subject for which various PSE's have been created. There is a wide range of numerical methods that are involved in providing the solution for such mathematical models. The basic Newtonian law of mechanics is followed by these models, but there can be many possible models and research that are constantly working on their design. Modeling the dynamics of a mechanical system is one of the important topics of CAE that involves both ordinary differential equations and algebraic equations. The analysis of the mixed systems is quite difficult but at the same time, it is quite necessary for the modeling of the moving mechanical systems. Some of the real-life examples of such type of problems are the creation of simulators for cars, planes, and other vehicles that requires solving the differential-algebraic equations.

Common mistakes

Students often encounter two errors in the solutions, which can be controlled and contained within some ordinary tolerance local.

1. Round-off error: It occurs due to limited storage space available inside the computer for storing mantissa part of a floating-point number, due to which these numbers are either chopped off or rounded.
2. Truncation error: It occurs due to the usage of a fixed or limited number of terms of an infinite series to approximate certain functions.

Context and Applications

This topic is significant in the professional exams for both graduate and postgraduate courses, especially for:

• Bachelor of Technology in Mechanical Engineering
• Bachelor of Science in Mathematics
• Master of Technology in Mechanical Engineering
• Master of Science in Mathematics
• Doctor of Philosophy in Mathematics

Related Concepts

• Partial differential equations
• Non-linear equations
• Vector algebra
• Types of mathematical model

Practice Problems

Q1. The Newton–Raphson method is also called _______.

1. Tangent method
2. Secant method
3. Chord method
4. Diameter method

Correct Option: (a)

Explanation: Newton–Raphson's method is also expressed as the tangent method. It is carried out by drawing a tangent to the curve at the point of initial guess.

Q2. The points where the Newton–Raphson method fail are called _________.

1. floating
2. continuous
3. non-stationary
4. stationary

Correct option: (d)

Explanation: The points where the function F(x) approaches infinity are called stationary points.

Q3. The convergence of which of the following methods depends on the initial assumed value?

1. False position
2. Gauss–Seidel method
3. Newton–Raphson method
4. Euler method

Correct option: (c)

Explanation: Newton-Raphson method does relies on the initial considered value for calculation.

Q4. Which error occurs due to usage of a fixed or limited number of terms of an infinite series to approximate a certain function?

1. Round-off error
2. Absolute error
3. Truncation error
4. Relative error

Correct Option: (c)

Explanation: Round-off Error: It occurs due to limited storage space available inside the computer for storing mantissa part of a floating-point number, due to which these numbers are either chopped off or rounded.

Truncation Error: It occurs due to the usage of a fixed or limited number of terms of an infinite series to approximate certain functions.

Absolute Error = |Exact value - Approximate value|

Relative Error = |(Exact value - Approximate value)/Exact error|

Q5. What is the full-form of CAE?

1. Computer-aided engineering
2. Computer approximated errors
3. Computer-aided errors
4. Computer approximated engineering

Correct Option: (a)

Explanation: Computer-aided engineering (CAE) is represented as the crucial topic for enhancing the designing and stimulation concept.