page_title: Romberg's method
section_title: A geometric example
hierarchical_section_title: Romberg's method / A geometric example
context_page_description: In numerical analysis, Romberg's method is used to estimate the definite integral
  by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule. The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points.
  The integrand must have continuous derivatives, though fairly good results
  may be obtained if only a few derivatives exist.
  If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate.
  The method is named after Werner Romberg, who published the method in 1955.
context_section_description: 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 extrapolation is applied.
  For the first iteration the two piece and one piece estimates are used in the formula (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 (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.
  
   
  
  MA stands for more accurate, LA stands for less accurate
caption: Four-piece