Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions

Ali Seron Jason

2 months ago

https://www.gyanodhan.com/video/7A2.%20Computer%20Science/Numerical%20Methods/681.%20Difference%20bw%20algebraic%20and%20transcendental%20equationtranscendental%20functions.mp4

Contents

0.1 Difference Between Algebraic and Transcendental Equations & Functions
- 0.1.1 1. Algebraic Equation
- 0.1.2 2. Transcendental Equation
0.2 3. Difference Between Algebraic & Transcendental Functions
0.3 4. Numerical Methods to Solve Transcendental Equations
0.4 Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions
0.5 UNIT 2 SOLUTION OF ALGEBRAIC AND …

1 1. Algebraic Equations
2 2. Transcendental Equations
3 3. Transcendental Functions
- 3.1 Definition:
- 3.2 Common Transcendental Functions:
4 Summary Table
5 In Numerical Methods (Computer Science Use)

Difference Between Algebraic and Transcendental Equations & Functions

1. Algebraic Equation

An algebraic equation is an equation that involves only polynomials with constants and variables, combined using addition, subtraction, multiplication, and division.

General Form:

$anxn+an−1xn−1+⋯+a1x+a0=0a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0$

where $an,an−1,…,a0a_n, a_{n-1}, \dots, a_0$ are constants, and $x$ is the variable.

Examples:

$x^2 – 4x + 3 = 0$
$3×5−2×3+x−7=03x^5 – 2x^3 + x – 7 = 0$
$2×4+5×2−9=02x^4 + 5x^2 – 9 = 0$

2. Transcendental Equation

A transcendental equation contains transcendental functions, such as exponential, logarithmic, or trigonometric functions, which cannot be expressed as a finite polynomial.

Examples:

$e^x – 3x = 0$
$sin⁡x−x=0\sin x – x = 0$
$\log x = 5$

These equations do not have a closed-form solution and often require numerical methods like Bisection Method, Newton-Raphson Method, or Secant Method for solving.

3. Difference Between Algebraic & Transcendental Functions

Aspect	Algebraic Function	Transcendental Function
Definition	A function involving polynomials only.	A function that goes beyond polynomials (contains exponentials, logarithms, or trigonometry).
Form	$x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$	$f(x) = e^x, \sin x, \log x, etc.$
Example	$x^2 + 3x + 5$ , $4×3−2x+74x^3 – 2x + 7$	$e^x, \log x, \sin x, \tan x$
Solvability	Can be solved algebraically using factorization, quadratic formula, etc.	Requires numerical methods for solving.
Graph Nature	Generally smooth and polynomial-like.	More complex curves, often periodic or asymptotic.

4. Numerical Methods to Solve Transcendental Equations

Since transcendental equations cannot be solved directly, we use iterative methods:

Bisection Method – Divides the interval and checks for root signs.
Newton-Raphson Method – Uses derivatives for fast convergence.
Regula Falsi (False Position) Method – A modified bisection method.
Secant Method – Uses two initial approximations instead of derivatives.

Would you like a step-by-step example of solving a transcendental equation using numerical methods?

Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions

UNIT 2 SOLUTION OF ALGEBRAIC AND …

Here’s a clear and concise explanation of the difference between algebraic and transcendental equations, along with an understanding of transcendental functions in the context of Computer Science and Numerical Methods:

1. Algebraic Equations

Definition:

An algebraic equation is any equation formed using a finite number of algebraic operations (addition, subtraction, multiplication, division, and roots) on polynomials.

Examples:

$x^2 + 3x – 5 = 0$
$x^3 – 7 = 0$
$x+2x=4\sqrt{x} + 2x = 4$

Nature:

Variables appear in polynomial form.
Can be solved exactly using algebraic methods (e.g., factorization, quadratic formula).

2. Transcendental Equations

Definition:

A transcendental equation is one that contains at least one transcendental function of the variable — i.e., a function that cannot be expressed as a finite combination of algebraic operations.

Examples:

$e^x = 3$
$sin⁡(x)=x/2\sin(x) = x/2$
$\cdot \ln(x) = 1$

Nature:

Cannot be solved analytically in most cases.
Requires numerical methods (e.g., Bisection, Newton-Raphson, Secant Method) for approximation.

3. Transcendental Functions

Definition:

These are functions that go beyond algebraic — they cannot be expressed as roots of polynomial equations with rational coefficients.

Common Transcendental Functions:

Exponential functions: $e^x$
Logarithmic functions: $ln(x), \log_{10}(x)$
Trigonometric functions: $sin⁡(x),cos⁡(x),tan⁡(x)\sin(x), \cos(x), \tan(x)$
Inverse trigonometric functions: $arcsin⁡(x),arccos⁡(x)\arcsin(x), \arccos(x)$

Summary Table

Feature	Algebraic Equation	Transcendental Equation
Involves	Polynomials only	Transcendental functions (sin, log, exp, etc.)
Solvability	Often solved exactly	Usually solved approximately (numerical methods)
Example	$x^2 + 2x – 3 = 0$	$sin⁡(x)−x/2=0\sin(x) – x/2 = 0$
Application	Found in basic modeling	Found in complex physical, engineering simulations

In Numerical Methods (Computer Science Use)

Algebraic Equations: Often solved with direct or iterative methods.
Transcendental Equations: Require root-finding algorithms like:
- Newton-Raphson Method
- Bisection Method
- Secant Method

These are essential in simulations, control systems, machine learning models, etc.

Would you like visual examples or Python code for solving such equations numerically?

Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions

Difference Between Algebraic and Transcendental Equations & Functions

1. Algebraic Equation

2. Transcendental Equation

3. Difference Between Algebraic & Transcendental Functions

4. Numerical Methods to Solve Transcendental Equations

Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions

UNIT 2 SOLUTION OF ALGEBRAIC AND …

1. Algebraic Equations

Definition:

Examples:

Nature:

2. Transcendental Equations

Definition:

Examples:

Nature:

3. Transcendental Functions

Definition:

Common Transcendental Functions:

Summary Table

In Numerical Methods (Computer Science Use)

Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions

Algebraic and Transcendental Equation and It’s Applications

Numerical Analysis by Dr. Anita Pal Assistant Professor …

Semester–V L T P C 15PHU505A NUMERICAL METHODS 5

Difference Between Algebraic and Transcendental Equations & Functions

1. Algebraic Equation

2. Transcendental Equation

3. Difference Between Algebraic & Transcendental Functions

4. Numerical Methods to Solve Transcendental Equations

Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions

UNIT 2 SOLUTION OF ALGEBRAIC AND …

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f9ee.png" alt="🧮" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> 1. Algebraic Equations

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Definition:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Examples:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Nature:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f310.png" alt="🌐" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> 2. Transcendental Equations

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Definition:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Examples:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Nature:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f4c8.png" alt="📈" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> 3. Transcendental Functions

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Definition:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f539.png" alt="🔹" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Common Transcendental Functions:

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f9e0.png" alt="🧠" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> Summary Table

<img src="https://s.w.org/images/core/emoji/13.0.1/72x72/1f4bb.png" alt="💻" class="wp-smiley amp-wp-843f19c" width="72" height="72" data-amp-original-style="height: 1em; max-height: 1em;"> In Numerical Methods (Computer Science Use)

Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions

Algebraic and Transcendental Equation and It’s Applications

Numerical Analysis by Dr. Anita Pal Assistant Professor …

Semester–V L T P C 15PHU505A NUMERICAL METHODS 5

1. Algebraic Equations

Definition:

Examples:

Nature:

2. Transcendental Equations

Definition:

Examples:

Nature:

3. Transcendental Functions

Definition:

Common Transcendental Functions:

Summary Table

In Numerical Methods (Computer Science Use)