Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions
Computer Science/Numerical Methods/. Difference b/w algebraic and transcendental equation transcendental functions
Contents [hide]
- 0.1 Difference Between Algebraic and Transcendental Equations & Functions
- 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 - 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 xx is the variable.
Examples:
- x2−4x+3=0x^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:
- ex−3x=0e^x – 3x = 0
- sinx−x=0\sin x – x = 0
- xlogx=5x \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 | f(x)=xn+an−1xn−1+⋯+a1x+a0f(x) = x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 | f(x)=ex,sinx,logx,etc.f(x) = e^x, \sin x, \log x, etc. |
| Example | x2+3x+5x^2 + 3x + 5, 4×3−2x+74x^3 – 2x + 7 | ex,logx,sinx,tanxe^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:
-
x2+3x−5=0x^2 + 3x – 5 = 0
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x3−7=0x^3 – 7 = 0
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x+2x=4\sqrt{x} + 2x = 4
Nature:
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Variables appear in polynomial form.
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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:
-
ex=3e^x = 3
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sin(x)=x/2\sin(x) = x/2
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x⋅ln(x)=1x \cdot \ln(x) = 1
Nature:
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Cannot be solved analytically in most cases.
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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: exe^x
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Logarithmic functions: ln(x),log10(x)\ln(x), \log_{10}(x)
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Trigonometric functions: sin(x),cos(x),tan(x)\sin(x), \cos(x), \tan(x)
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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 | x2+2x−3=0x^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)
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Algebraic Equations: Often solved with direct or iterative methods.
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Transcendental Equations: Require root-finding algorithms like:
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Newton-Raphson Method
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Bisection Method
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Secant Method
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These are essential in simulations, control systems, machine learning models, etc.
Would you like visual examples or Python code for solving such equations numerically?