UNIT 1 • FINDING ROOTS OF EQUATIONS

False Position Method
Regula Falsi Strategy

The False Position Method (Regula Falsi) improves upon the Bisection method by connecting boundary points $(a, f(a))$ and $(b, f(b))$ with a straight secant line. Instead of cutting blindly in the middle, it places the next guess right at the x-intercept of this secant line!

HOW REGULA FALSI WORKSSECANT INTERCEPT

C = \frac{a \cdot f(b) - b \cdot f(a)}{f(b) - f(a)}

1. Pick Starting Bracket [a, b]Choose points where

f(a)

and

f(b)

have opposite signs.

2. Draw Secant Line (C)Find x-intercept

C

where secant line crosses

y=0

3. Update Bound & RepeatReplace

a

b

with

C

keeping root bracketed.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the real root of $f(x) = x^3 - 2x - 5 = 0$ in the interval $[a, b] = [2, 3]$ .

STEP 01 • INITIAL BRACKET EVALUATION

Evaluate $f(x)$ at initial points $a = 2$ and $b = 3$ :

f(2) = (2)^3 - 2(2) - 5 = 8 - 4 - 5 = -1 \quad (\text{Negative})

f(3) = (3)^3 - 2(3) - 5 = 27 - 6 - 5 = +16 \quad (\text{Positive})

Since $f(2) < 0$ and $f(3) > 0$ , a root exists in $[2, 3]$ .

STEP 02 • COMPUTE FIRST INTERCEPT (C₁)

Substitute values into the Regula Falsi secant formula:

C_1 = \frac{a \cdot f(b) - b \cdot f(a)}{f(b) - f(a)} = \frac{2(16) - 3(-1)}{16 - (-1)} = \frac{32 + 3}{17} = \frac{35}{17} \approx 2.0588

f(2.0588) = (2.0588)^3 - 2(2.0588) - 5 = -0.3869 \quad (\text{Negative})

STEP 03 • UPDATE INTERVAL BOUNDS

Since $f(2.0588) < 0$ and $f(3) > 0$ , replace the old left bound $a = 2$ with $a = 2.0588$ .

New Search Bracket:

[a, b] = [2.0588, 3.0000]

STEP 04 • SUBSEQUENT ITERATIONS WALKTHROUGH

Iteration 2:

Secant Formula:

C_2 = \frac{2.0588(16) - 3(-0.3869)}{16 - (-0.3869)} = \frac{34.1019}{16.3869} \approx 2.0813

Function Evaluation:

f(2.0813) = -0.1425

(Negative)

Update: Set

a = 2.0813

. New bracket:

[2.0813, 3.0000]

Iteration 3:

Secant Formula:

C_3 = \frac{2.0813(16) - 3(-0.1425)}{16 - (-0.1425)} = \frac{33.728}{16.1425} \approx 2.0895

Function Evaluation:

f(2.0895) = -0.0516

(Negative)

Update: Set

a = 2.0895

. New bracket:

[2.0895, 3.0000]

Iteration 4:

Secant Formula:

C_4 \approx 2.0924

f(2.0924) = -0.0185

Update: Set

a = 2.0924

. New bracket:

[2.0924, 3.0000]

STEP 05 • ITERATION SUMMARY TABLE

Step (k)	Left Bound (a)	Right Bound (b)	Intercept (C)	Value f(C)	Sign of f(C)	Next Bracket
1	2.0000	3.0000	2.0588	-0.3869	f(c) < 0	[2.0588, 3.0000]
2	2.0588	3.0000	2.0813	-0.1425	f(c) < 0	[2.0813, 3.0000]
3	2.0813	3.0000	2.0895	-0.0516	f(c) < 0	[2.0895, 3.0000]
4	2.0895	3.0000	2.0924	-0.0185	f(c) < 0	[2.0924, 3.0000]
5	2.0924	3.0000	2.0935	-0.0066	f(c) < 0	[2.0935, 3.0000]

FINAL CONCLUSION

Approximated Root: x ≈ 2.0945

\text{Exact Root } \alpha \approx 2.09455, \quad f(2.09455) = 0

Iterating with the secant line intercept formula rapidly pulls the estimate toward the true root 2.09455 faster than standard bisection!

LABORATORY

Interactive Live Calculator & Visualizer

REGULA-FALSI LABORATORY

False Position Interactive Engine

PreviousBisection Method

Unit 1•2 of 28

NextIteration Method

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 1 • FINDING ROOTS OF EQUATIONS

False Position Method
Regula Falsi Strategy

HOW REGULA FALSI WORKSSECANT INTERCEPT

C = \frac{a \cdot f(b) - b \cdot f(a)}{f(b) - f(a)}

1. Pick Starting Bracket [a, b]Choose points where

f(a)

and

f(b)

have opposite signs.

2. Draw Secant Line (C)Find x-intercept

C

where secant line crosses

y=0

3. Update Bound & RepeatReplace

a

b

with

C

keeping root bracketed.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the real root of $f(x) = x^3 - 2x - 5 = 0$ in the interval $[a, b] = [2, 3]$ .

STEP 01 • INITIAL BRACKET EVALUATION

Evaluate $f(x)$ at initial points $a = 2$ and $b = 3$ :

f(2) = (2)^3 - 2(2) - 5 = 8 - 4 - 5 = -1 \quad (\text{Negative})

f(3) = (3)^3 - 2(3) - 5 = 27 - 6 - 5 = +16 \quad (\text{Positive})

Since $f(2) < 0$ and $f(3) > 0$ , a root exists in $[2, 3]$ .

STEP 02 • COMPUTE FIRST INTERCEPT (C₁)

Substitute values into the Regula Falsi secant formula:

C_1 = \frac{a \cdot f(b) - b \cdot f(a)}{f(b) - f(a)} = \frac{2(16) - 3(-1)}{16 - (-1)} = \frac{32 + 3}{17} = \frac{35}{17} \approx 2.0588

f(2.0588) = (2.0588)^3 - 2(2.0588) - 5 = -0.3869 \quad (\text{Negative})

STEP 03 • UPDATE INTERVAL BOUNDS

Since $f(2.0588) < 0$ and $f(3) > 0$ , replace the old left bound $a = 2$ with $a = 2.0588$ .

New Search Bracket:

[a, b] = [2.0588, 3.0000]

STEP 04 • SUBSEQUENT ITERATIONS WALKTHROUGH

Iteration 2:

Secant Formula:

C_2 = \frac{2.0588(16) - 3(-0.3869)}{16 - (-0.3869)} = \frac{34.1019}{16.3869} \approx 2.0813

Function Evaluation:

f(2.0813) = -0.1425

(Negative)

Update: Set

a = 2.0813

. New bracket:

[2.0813, 3.0000]

Iteration 3:

Secant Formula:

C_3 = \frac{2.0813(16) - 3(-0.1425)}{16 - (-0.1425)} = \frac{33.728}{16.1425} \approx 2.0895

Function Evaluation:

f(2.0895) = -0.0516

(Negative)

Update: Set

a = 2.0895

. New bracket:

[2.0895, 3.0000]

Iteration 4:

Secant Formula:

C_4 \approx 2.0924

f(2.0924) = -0.0185

Update: Set

a = 2.0924

. New bracket:

[2.0924, 3.0000]

STEP 05 • ITERATION SUMMARY TABLE

Step (k)	Left Bound (a)	Right Bound (b)	Intercept (C)	Value f(C)	Sign of f(C)	Next Bracket
1	2.0000	3.0000	2.0588	-0.3869	f(c) < 0	[2.0588, 3.0000]
2	2.0588	3.0000	2.0813	-0.1425	f(c) < 0	[2.0813, 3.0000]
3	2.0813	3.0000	2.0895	-0.0516	f(c) < 0	[2.0895, 3.0000]
4	2.0895	3.0000	2.0924	-0.0185	f(c) < 0	[2.0924, 3.0000]
5	2.0924	3.0000	2.0935	-0.0066	f(c) < 0	[2.0935, 3.0000]

FINAL CONCLUSION

Approximated Root: x ≈ 2.0945

\text{Exact Root } \alpha \approx 2.09455, \quad f(2.09455) = 0

Iterating with the secant line intercept formula rapidly pulls the estimate toward the true root 2.09455 faster than standard bisection!

LABORATORY

Interactive Live Calculator & Visualizer

REGULA-FALSI LABORATORY

False Position Interactive Engine

PreviousBisection Method

Unit 1•2 of 28

NextIteration Method

False Position Method Regula Falsi Strategy

Step-by-Step Worked Example

Approximated Root: x ≈ 2.0945

Interactive Live Calculator & Visualizer

False Position Interactive Engine

NETZ Engine

False Position Method Regula Falsi Strategy

Step-by-Step Worked Example

Approximated Root: x ≈ 2.0945

Interactive Live Calculator & Visualizer

False Position Interactive Engine

False Position Method
Regula Falsi Strategy

False Position Method
Regula Falsi Strategy