- Asked 2/13
- 2071 Magh · 8 marks
False position: cos x = x e^x
Short answer
Use regula falsi with f(x)=cos x - x e^x on (0,1), updating the endpoint that keeps opposite signs. The successive c values stabilize at 0.5178, so the root is approximately 0.5178.
Find a root of the equation cos x = x e^x using the regula-falsi method correct up to four decimal places.
False Position (Regula Falsi) formula: for f(a)·f(b) < 0, c = (a·f(b) − b·f(a)) / (f(b) − f(a)). If f(a)·f(c) < 0 set b = c, otherwise set a = c.
Step 1: Bracket the root
f(x) = cos x − x·eˣ (radians) f(0) = 1 > 0 and f(1) = 0.5403 − 2.7183 = −2.1780 < 0, so a root lies in (0, 1). Take a = 0, b = 1.
Step 2: Iterations
Because the end b = 1 never moves, the method converges slowly here, so six decimals are shown for a, f(a), c and f(c).
| n | a | b | f(a) | f(b) | c | f(c) |
|---|---|---|---|---|---|---|
| 1 | 0.000000 | 1.0000 | 1.000000 | -2.1780 | 0.314665 | 0.519871 |
| 2 | 0.314665 | 1.0000 | 0.519871 | -2.1780 | 0.446728 | 0.203545 |
| 3 | 0.446728 | 1.0000 | 0.203545 | -2.1780 | 0.494015 | 0.070802 |
| 4 | 0.494015 | 1.0000 | 0.070802 | -2.1780 | 0.509946 | 0.023608 |
| 5 | 0.509946 | 1.0000 | 0.023608 | -2.1780 | 0.515201 | 0.007760 |
| 6 | 0.515201 | 1.0000 | 0.007760 | -2.1780 | 0.516922 | 0.002539 |
| 7 | 0.516922 | 1.0000 | 0.002539 | -2.1780 | 0.517485 | 0.000829 |
| 8 | 0.517485 | 1.0000 | 0.000829 | -2.1780 | 0.517668 | 0.000271 |
| 9 | 0.517668 | 1.0000 | 0.000271 | -2.1780 | 0.517728 | 0.000088 |
| 10 | 0.517728 | 1.0000 | 0.000088 | -2.1780 | 0.517748 | 0.000029 |
| 11 | 0.517748 | 1.0000 | 0.000029 | -2.1780 | 0.517754 | 0.000009 |
| 12 | 0.517754 | 1.0000 | 0.000009 | -2.1780 | 0.517756 | 0.000003 |
| 13 | 0.517756 | 1.0000 | 0.000003 | -2.1780 | 0.517757 | 0.000001 |
The last three values of c (0.517754, 0.517756, 0.517757) all round to 0.5178, and f(c) is practically zero, so the root is fixed to four decimals.
Answer: root ≈ 0.5178
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Secant: x e^x - cos x · Numerical Method, chapter 2 · asked 3×
- False position: e^cos x - sin x - 1 · Numerical Method, chapter 2
- False position pseudo-code · Numerical Method, chapter 2 · asked 2×
- Secant algorithm, versus false position · Numerical Method, chapter 2
- Secant: cos x + e^x - 5 · Numerical Method, chapter 2
- N-R: sin x = e^-x with limitations · Numerical Method, chapter 2
- Step-up cycloconverter, four times frequency · Power Electronics, chapter 6
- Position analysis of a four-bar mechanism · Theory of Mechanism and Machine I, chapter 2
- Pedal equation of r^m = a^m cos mθ · Mathematics I, chapter 1 · asked 2×
- Bernoulli's equation and four applications · Fluid Mechanics, chapter 5