- 2071 Bhadra · 8 marks
Quadratic least squares fit
Short answer
Normal equations give a=1.0357, b=−0.1929, c=0.2429. Best-fit quadratic: y=1.0357−0.1929x+0.2429x².
Find the best fit curve in the form of y = a + bx + cx² using least square approximation from the following discrete data.
| x | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |
|---|---|---|---|---|---|---|---|
| y | 1.1 | 1.3 | 1.6 | 2.0 | 2.7 | 3.4 | 4.1 |
For y = a + bx + cx² the least-squares normal equations are
Σy = n·a + bΣx + cΣx²
Σxy = aΣx + bΣx² + cΣx³
Σx²y = aΣx² + bΣx³ + cΣx⁴
| x | y | x² | x³ | x⁴ | xy | x²y |
|---|---|---|---|---|---|---|
| 1 | 1.1 | 1.0000 | 1.0000 | 1.0000 | 1.1000 | 1.1000 |
| 1.5 | 1.3 | 2.2500 | 3.3750 | 5.0625 | 1.9500 | 2.9250 |
| 2 | 1.6 | 4.0000 | 8.0000 | 16.0000 | 3.2000 | 6.4000 |
| 2.5 | 2 | 6.2500 | 15.6250 | 39.0625 | 5.0000 | 12.5000 |
| 3 | 2.7 | 9.0000 | 27.0000 | 81.0000 | 8.1000 | 24.3000 |
| 3.5 | 3.4 | 12.2500 | 42.8750 | 150.0625 | 11.9000 | 41.6500 |
| 4 | 4.1 | 16.0000 | 64.0000 | 256.0000 | 16.4000 | 65.6000 |
| Σx = 17.5 | 16.2 | 50.75 | 161.875 | 548.1875 | 47.650 | 154.475 |
With n = 7:
7a + 17.50b + 50.75c = 16.20
17.50a + 50.75b + 161.875c = 47.650
50.75a + 161.875b + 548.1875c = 154.475
Solving by Gauss elimination:
a = 1.0357, b = -0.1929, c = 0.2429
Answer: y = 1.0357 − 0.1929x + 0.2429x² (check: x = 4 gives 4.15 against 4.1).
✓ Answer checked by Claude Opus 5.5 (high effort)
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