- 2071 Bhadra · 8 marks
Lagrange: y at x = 3.0
Short answer
Using Lagrange interpolation, y(3.0)=20.2672. Sum of products yᵢ·Lᵢ(3.0) with coefficients L₀=0.664773, L₁=0.361634, L₂=−0.004295, L₃=−0.030545, L₄=0.008433.
Use Lagrange's Interpolation formula to find the value of y when x = 3.0, from the following table.
| x | 3.2 | 2.7 | 1.0 | 4.8 | 5.6 |
|---|---|---|---|---|---|
| y | 22.0 | 17.8 | 14.2 | 38.3 | 51.7 |
The x values are unequally spaced and unordered, so Lagrange's formula is the right tool:
y(x) = Σ yᵢ·Lᵢ(x), Lᵢ(x) = Π (x − xⱼ)/(xᵢ − xⱼ) (j ≠ i)
Data: x₀ = 3.2, x₁ = 2.7, x₂ = 1.0, x₃ = 4.8, x₄ = 5.6 with y = 22.0, 17.8, 14.2, 38.3, 51.7. Required x = 3.0.
Lagrange coefficients at x = 3.0
For example L₀(3) = [(3−2.7)(3−1.0)(3−4.8)(3−5.6)] / [(3.2−2.7)(3.2−1.0)(3.2−4.8)(3.2−5.6)] = (0.3)(2.0)(−1.8)(−2.6) / (0.5)(2.2)(−1.6)(−2.4) = 2.808/4.224 = 0.664773.
| Term | Numerator Π(3 − xⱼ) | Denominator Π(xᵢ − xⱼ) | Value |
|---|---|---|---|
| L0(3) | (0.3) × (2.0) × (-1.8) × (-2.6) | (0.5) × (2.2) × (-1.6) × (-2.4) | 0.664773 |
| L1(3) | (-0.2) × (2.0) × (-1.8) × (-2.6) | (-0.5) × (1.7) × (-2.1) × (-2.9) | 0.361634 |
| L2(3) | (-0.2) × (0.3) × (-1.8) × (-2.6) | (-2.2) × (-1.7) × (-3.8) × (-4.6) | -0.004295 |
| L3(3) | (-0.2) × (0.3) × (2.0) × (-2.6) | (1.6) × (2.1) × (3.8) × (-0.8) | -0.030545 |
| L4(3) | (-0.2) × (0.3) × (2.0) × (-1.8) | (2.4) × (2.9) × (4.6) × (0.8) | 0.008433 |
Sum
| i | yᵢ | Lᵢ(3.0) | yᵢ·Lᵢ(3.0) |
|---|---|---|---|
| 0 | 22.0 | 0.664773 | 14.6250 |
| 1 | 17.8 | 0.361634 | 6.4371 |
| 2 | 14.2 | -0.004295 | -0.0610 |
| 3 | 38.3 | -0.030545 | -1.1699 |
| 4 | 51.7 | 0.008433 | 0.4360 |
| Sum | 20.2672 |
Check: ΣLᵢ = 1.000000 (must equal 1).
Answer: y(3.0) ≈ 20.2672
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Lagrange interpolation pseudo-code · Numerical Method, chapter 4 · asked 2×
- Newton interpolation: steam table · Numerical Method, chapter 4
- Lagrange: f(27.5) · Numerical Method, chapter 4
- Newton interpolation: y(3), y(7) · Numerical Method, chapter 4
- Stirling formula: y(35) · Numerical Method, chapter 4
- State and prove Lagrange's mean value theorem · Mathematics I, chapter 1 · asked 6×
- Calorific value of fuel by Dulong's formula · Internal Combustion Engines, chapter 3
- Truth value of a compound formula (a) · Discrete Structure, chapter 1
- Truth value of a compound formula (b) · Discrete Structure, chapter 1