A Geometric Example
To estimate the area under a curve the trapezoid rule is applied first to one-piece, then two, then four, and so on.
After trapezoid rule estimates are obtained Richardson's Extrapolation is applied
- For the first iteration the two piece and one piece estimates are used in the formula (4 X (more accurate) - (less accurate))/3 The same formula is then used to compare the four piece and the two piece estimate, and likewise for the higher estimates
- For the second iteration the values of the first iteration are used in the formula (16(more accurate)-less accurate)/15
- The third iteration uses the next power of 4: (64 (More accurate) - less accurate)/63 on the values derived by the second iteration.
- The pattern is continued until there is one estimate.
| Number of pieces | Trapezoid estimates | First iteration | second iteration | third iteration |
| (4MA-LA)/3* | (16MA-LA)/15 | (64MA-LA)/63 | ||
| 1 | 0 | (4*480-0)/3 = 640 | (16*880-640)/15 =896 | (64*1015.11-896)/63 = 1017.002 |
| 2 | 480 | (4*780-480)/3 = 880 | (16*1006.67-880)/15 = 1015.11.. | |
| 4 | 780 | (4*950-780)/3 =1006.666.. | ||
| 8 | 950 |
- MA stands for more accurate, LA stands for less accurate
Read more about this topic: Romberg's Method
Famous quotes containing the word geometric:
“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o’ th’ day
The clock doth strike, by algebra.”
—Samuel Butler (1612–1680)