Numerical Analysis

Numerical Analysis deals with the process of getting the numerical solution to complex problems. The majority of mathematical problems in science and engineering are difficult to answer precisely, and in some cases it is impossible. To make a tough Mathematical problem easier to solve, an approximation is essential. Numerical approximation has become more popular as a result of tremendous advances in computational technology. As a result, a great deal of scientific software is being developed to solve more complex challenges quickly and easily. Let us go through the definition of numerical analysis as well as the various concepts included, such as errors, interpolation and so on in this article.

Introduction to Numerical Analysis

Numerical analysis is a discipline of mathematics concerned with the development of efficient methods for getting numerical solutions to complex mathematical problems. There are three sections to the numerical analysis. The first section of the subject deals with the creation of a problem-solving approach. The analysis of methods, which includes error analysis and efficiency analysis, is covered in the second section. The efficiency analysis shows us how fast we can compute the result, while the error analysis informs us how correct the result will be if we utilize the approach. The construction of an efficient algorithm to implement the approach as a computer code is the subject’s third part. All three elements must be familiar to have a thorough understanding of the numerical analysis.

Meanwhile, there are at least three reasons to learn the theoretical foundations of numerical methods:

1. Learning various numerical methods and analyzing them will familiarize a person with the process of inventing new numerical methods. When the existing approaches are insufficient or inefficient to handle a certain problem, this is critical.
2. In many cases, there are multiple solutions to a problem. As a result, using the right procedure is critical for getting a precise answer in less time.
3. With a solid foundation, one can effectively apply methods (especially when a technique has its own restrictions and/or drawbacks in certain instances) and, more significantly, analyze what went wrong when results did not meet expectations.

Let’s have a look at some of the key topics in numerical analysis.

Different Types of Errors

The disparity between the approximate representation of a real number and the actual value is termed an error.

Error = True Value – Approximate Value, is the formula for calculating the error in a computed amount.

The absolute error is defined as the absolute value of the error defined above.

Relative Error = Error / True Value is a measurement of the error in respect to the magnitude of the true value.

The relative error is multiplied by 100 to get the percentage error.

The phrase “truncation error” refers to the error that occurs when a smooth function is approximated by reducing its Taylor series representation to a limited number of terms.

Significant Digits

If x_A is an approximation to x, so we can conclude that x_A approximates x to r significant β-digits if |x − x_A| ≤ (½)β^s−r+1 with “s” the greatest integer such that β^s ≤ |x|.

As an example, the approximate value x_A = 0.333 includes three significant digits for x = ⅓, since |x − x_A| ≈ .00033 < 0.0005 = 0.5 × 10⁻³.

But 10−1 < 0.333 · · · = x.

Hence, in this case s = −1 and and therefore r = 3.

Propagation of Errors

When an error is committed, it has an impact on subsequent outcomes because it propagates through subsequent calculations. We’ll look at how utilizing approximate numbers rather than actual numbers affects the outcomes before moving on to function evaluation. We’ll now explore how error propagates in four basic arithmetic operations.

• In addition and subtraction, the total of the error bounds for the terms provides an error bound for the results .
• In multiplication and division, The sum of the bounds for the relative errors of the given integers gives a limitation for the relative error of the results.

Finite Difference Operators

Now, let us discuss the various finite difference operators in brief.

Forward Operator

Assume that “h” be the finite difference, then

Δf(x) = f(x+h) – f(x)

Δ² f(x) = f(x+2h)-2f(x+h) + f(x)

Δ³ f(x)= f(x+3h) – 3f(x+2h) + 2f(x+h) – f(x)

Then,

Shift Operator

Assume that h be the finite difference.

Then, E f(x) = f(x+h)

Eⁿ f(x) = f(x+nh)

Backward Difference

Suppose h be the finite difference.

Thus,

….

….

….

Central Difference Operator

Averaging Operator

Factorial Notation

Suppose h be the finite difference.

Hence,

Relation Between Different Finite Operators

Relationship Between Δ and E

E ≡ 1 + Δ and Δ ≡ E-1

Hence, Eⁿ ≡ (1+Δ)ⁿ and Δⁿ ≡ (E-1)ⁿ

Interpolation

Interpolation is the process of determining the approximate value of a function f(x) for an x between multiple x values x_0, x₁, …, x_n for which the value of f(x) is known.

I.e., f(x_i) = f_i (i = 0, 1, 2, …, n)

If the real-valued function f(x) has (n+1) different values, then x₀ x₁, ..x_n. A polynomial of degree n or less is P_n(x_i) = f(x). It indicates that there can only be one polynomial with a degree less than or equal to n that interpolates f(x) at (n+1) unique points x₀, x₁, x₂, …x_n.

Solved Example on Numerical Analysis

Example:

Show that μ₄ = μ₃ + Δμ₂ + Δ²μ₁ + Δ³μ₁

Solution:

As we know that

Δμ_x = μ_x+h – μ_x

Hence, μ₄ – μ₃ = Δμ₃

μ₃ – μ₂ = Δμ₂

μ₂ – μ₁ = Δμ₁

Therefore,

μ₄ = μ₃ + Δμ₃

μ₄ = μ₃ + Δ(μ₂ – Δμ₂) [Since, μ₃ – μ₂ = Δμ₂]

μ₄ = μ₃ + Δμ₂ – Δ²μ₂

μ₄ = μ₃ + Δμ₂ – Δ²(μ₁ + Δμ₁) [Since, Δμ₁ = μ₂ – μ₁]

μ₄ = μ₃ + Δμ₂ + Δ²μ₁ + Δ³μ₁

Hence, proved.

Stay tuned to BYJU’S – The Learning App and download the app all the Maths-related concepts easily by exploring more videos.