Editing Romberg's method (section)

== 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.
[[File:One-piece Trapezoid Approximation.svg|alt=One-piece approximation|thumb|One-piece. Note since it starts and ends at zero, this approximation yields zero area.]]
[[File:Two-piece Trapezoid Approximation.svg|alt=Two-piece approximation|thumb|Two-piece]]
[[File:Four-piece Trapezoid Approximation.svg|alt=Four-piece approximation|thumb|Four-piece]]
[[File:Eight-piece Trapezoid Approximation.svg|alt=Eight-piece approximation|thumb|Eight-piece]]
After trapezoid rule estimates are obtained, [[Richardson extrapolation]] is applied.

*For the first iteration the two piece and one piece estimates are used in the formula {{math|{{sfrac|4 × (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 {{math|{{sfrac|16 × (more accurate) − (less accurate)|15}}}}
*The third iteration uses the next power of 4: {{math|{{sfrac|64 × (more accurate) − (less accurate)|63}}}} on the values derived by the second iteration.
*The pattern is continued until there is one estimate.
{{Table alignment}}
{| class="wikitable col1right col2right"
 |-
 ! Number of pieces || Trapezoid estimates ||  First iteration || Second iteration || Third iteration
 |-
 |                  ||                     || {{math|1={{sfrac|4 {{abbr|MA|more accurate}} − {{abbr|LA|less accurate}}|3}}}} || {{math|1={{sfrac|16 MA − LA|15}}}} || {{math|1={{sfrac|64 MA − LA|63}}}}
 |-   
 |1||0  || {{math|1={{sfrac|4×16 − 0|3}} = 21.333...|size=90%}} || {{math|1={{sfrac|16×34.667 − 21.333|15}} = 35.556...|size=90%}} || {{math|1={{sfrac|64×42.489 − 35.556|63}} = 42.599...|size=90%}}
 |-
 |2||16 || {{math|1={{sfrac|4×30 − 16|3}} = 34.666...|size=90%}} || {{math|1={{sfrac|16×42 − 34.667|15}} = 42.489...|size=90%}} ||
 |-
 |4||30 || {{math|1={{sfrac|4×39 − 30|3}} = 42|size=90%}} ||   ||
 |-
 |8||39 ||   ||   ||
 |-
 |-
 |}