False Position Method Formula and Solved Example
The concept of False Position Method is essential in mathematics and helps in solving real-world and exam-level problems efficiently. This root-finding technique is frequently tested in board exams, competitive tests, and engineering courses due to its practical and conceptual importance.
Understanding False Position Method
The False Position Method, also called Regula Falsi Method, is a numerical method to solve equations of the form \( f(x) = 0 \). It is most useful when algebraic solutions are difficult or impossible. This method is widely used in numerical analysis, engineering mathematics, and various scientific computations. It is particularly popular for quickly finding the real root of nonlinear equations.
Formula Used in False Position Method
The standard formula is: \( c = b - \frac{f(b)(b-a)}{f(b)-f(a)} \)
Here’s a helpful table to understand False Position Method calculations more clearly:
False Position Method Formula Table
| Step | Expression | Description |
|---|---|---|
| 1 | \( c = b - \frac{f(b)(b-a)}{f(b)-f(a)} \) | Finds the next approximation (c) of the root |
| 2 | Update a or b based on sign of \( f(c) \) | Interval adjustment for next iteration |
This table shows how the pattern of the False Position Method formula is applied in each iteration to approach the actual root.
Worked Example – Solving a Problem
Let’s solve for the root of \( f(x) = 2e^x \sin x - 3 \) correct to three decimal places using the False Position Method.
1. Choose two initial guesses:
Let \( a = 0 \), \( b = 1 \)
2. Calculate \( f(a) \) and \( f(b) \):
\( f(0) = 2e^0 \sin 0 - 3 = -3 \)
\( f(1) = 2e^1 \sin 1 - 3 \approx 1.5747 \)
3. Since \( f(a) \) and \( f(b) \) have opposite signs, a root exists between 0 and 1.
4. Compute first approximation (\( x_1 \)):
\( x_1 = \frac{a f(b) - b f(a)}{f(b) - f(a)} \)
\( x_1 = \frac{0 \times 1.5747 - 1 \times (-3)}{1.5747 - (-3)} = \frac{3}{4.5747} \approx 0.6557 \)
5. Evaluate \( f(x_1) \):
\( f(0.6557) = 2e^{0.6557} \sin(0.6557) - 3 \approx -0.6507 \)
6. The sign of \( f(x_1) \) is negative, so update \( a = 0.6557 \) (since \( f(b) > 0 \)).
7. Compute second approximation (\( x_2 \)):
\( x_2 = \frac{a f(b) - b f(a)}{f(b) - f(a)} \)
\( x_2 = \frac{0.6557 \times 1.5747 - 1 \times (-0.6507)}{1.5747 - (-0.6507)} = \frac{1.0325 + 0.6507}{2.2254} \approx 0.7563 \)
8. Evaluate \( f(x_2) \):
\( f(0.7563) = 2e^{0.7563} \sin(0.7563) - 3 \approx -0.0761 \)
9. Since \( f(x_2) < 0 \), update \( a = 0.7563 \).
10. Compute third approximation (\( x_3 \)):
\( x_3 = \frac{a f(b) - b f(a)}{f(b) - f(a)} \)
\( x_3 = \frac{0.7563 \times 1.5747 - 1 \times (-0.0761)}{1.5747 - (-0.0761)} = \frac{1.1909 + 0.0761}{1.6508} \approx 0.7675 \)
11. Thus, after three iterations, the best approximation to the root is 0.768 (to three decimal places).
False Position Method vs Other Methods
A common student confusion is how the False Position Method differs from other root-finding techniques.
| Method | Process | Convergence | Updates |
|---|---|---|---|
| False Position | Uses secant line for root estimate, interval always brackets root | Faster than bisection, may slow if function skewed | One end of interval is updated, other may “stick” |
| Bisection | Halves interval each time | Guaranteed, but slow | Both ends can change, always shrink |
| Secant | Like False Position, but doesn’t always bracket root | Usualy faster, not guaranteed | No guaranteed bracketing, both points move |
These distinctions help students answer common board and competitive exam questions.
Important Tips and Common Mistakes
- Don’t forget: initial guesses must bracket the root (opposite signs for \( f(a) \) and \( f(b) \)).
- Mixing up formulae for False Position, Secant, and Bisection Method is common—always write them out and double-check which update to use.
- False Position can converge slowly if one end of the interval is “sticky” and doesn’t update—watch out for this in practical use.
Real-World Applications
The False Position Method is applied in engineering, physics, and financial computations. For example, it is used to find the break-even point in cost equations, chemical engineering reactions, and error correction in computational models. Vedantu helps students see how maths applies beyond the classroom.
Practice Problems
- Use the False Position Method to solve \( x^3 + x - 1 = 0 \) for a root between 0 and 1 up to three decimal places.
- Apply the Regula Falsi formula to \( \cos x = x \) in the interval \( [0,1] \).
- Compare the number of steps needed by False Position and Bisection Method for \( x^2 - 4 = 0 \) between 0 and 3.
Summary
We explored the idea of False Position Method, how to apply its formula, carry out step-by-step calculations, and understand its real-life relevance. Practice more with Vedantu to build confidence in these numerical methods and ace your exams!
Related Maths Topics
FAQs on False Position Method (Regula Falsi) in Numerical Methods
1. What is the False Position Method?
The False Position Method, also called Regula Falsi Method, is an iterative numerical technique to find the roots of nonlinear equations of the form f(x) = 0. It uses linear interpolation between two points where the function values have opposite signs to successively approximate the root.
2. What is the formula for the Falsi method?
The key Regula Falsi formula to find the next approximation c is:c = b - [f(b) * (b - a)] / [f(b) - f(a)]
where a and b are two initial points with f(a) and f(b) having opposite signs.
3. What is the difference between bisection and false position?
The bisection method divides the interval into two equal halves regardless of function values, whereas false position method uses the values of the function at the endpoints to interpolate the root, often resulting in faster convergence. Both require that the function changes sign in the interval but differ in step calculation.
4. How accurate is the false position method?
The false position method generally has an order of convergence greater than 1, making it faster than bisection but potentially slower than the secant method. Its accuracy depends on the function's behavior and initial guesses. Sometimes, convergence can be slow if one endpoint remains fixed during iteration.
5. What is the secant method and how does it relate?
The secant method is another iterative root-finding algorithm that approximates the root by using two recent approximations but unlike false position, it does not require the function values at points to have opposite signs. It generally converges faster but lacks the guaranteed bracketing property of the false position method.
6. Is there a calculator or code for the false position method?
Yes, there are many online false position method calculators. Additionally, code implementations in Python and Matlab are widely available and helpful for assignments and competitive exam practice. These automate iterations and provide stepwise results efficiently.
7. Why does the false position method sometimes converge slowly?
Slow convergence occurs because the method keeps one endpoint fixed if the function values at one side do not change sign consistently, causing the interval to shrink asymmetrically. This is common when the root is closer to one end and the method approaches it mostly from one side.
8. Why is it called “Regula Falsi”?
The name Regula Falsi, Latin for "false rule," refers to the method's use of a linear interpolation (a "false position") to estimate the root by connecting points of known function values rather than directly solving the equation.
9. Can the false position method fail to find a root?
While false position method generally converges if the initial interval brackets a root, it can fail or converge very slowly if the function is not well-behaved, or if the root is repeated or multiple roots exist within the interval. Proper initial guesses and function analysis help avoid failure.
10. What makes students confuse the secant and false position methods?
Students often confuse these because both use two points to approximate roots, but the secant method does not require bracketing (opposite sign) and uses the last two approximations, while false position method requires the function to have opposite signs at interval ends. Understanding the key difference in sign requirements clarifies this confusion.
11. How can I apply false position to real-world engineering problems?
The false position method is used in engineering for solving nonlinear equations arising in circuit analysis, structural engineering, fluid mechanics, and control systems. It provides accurate root approximations vital for design and analysis tasks where analytical solutions are difficult.
12. What is the order of convergence of the false position method?
The order of convergence of the false position method is superlinear, typically between linear and quadratic (approximately 1.5). This makes it faster than the bisection method but generally slower than Newton-Raphson and secant methods.
