Summary: A Class of Numerical Integration Rules With First Order
Mohamad Adnan AI-Alaoui"
A novel approach to deriving a family of quadrature formulae is presented. The first member
of the new family is the corrected trapezoidal rule. The second member, a two-segment rule, is
obtained by interpolating the corrected trapezoidal rule and the Simpson one-third rule. The third
member, a three-segment rule, is obtained by interpolating the corrected trapezoidal rule and the
Simpson three-eights rule. The fourth member, a four-segment rule is obtained by interpolating
the two-segment rule with the Boole rule. The process can be carried on to generate a whole class
of integration rules by interpolating the proposed rules appropriately with the Newton-Cotes
rules to cancel Out an additional term in the Euler-MacLaurin error formula. The resulting rules
integrate correctly polynomials of degrees less or equal to n+3 if n is even and n+2 if n is odd,
where n is the number of segments of the single application rules. The proposed rules have
excellent round-off properties, close to those of the trapezoidal rule. Members of the new family
obtain with two additional fianctional evaluations the same order of errors as those obtained by
doubling the number of segments in applying the Romberg integration to Newton-Cotes rules.
Members of the proposed family are shown to be viable alternatives to Gaussian quadrature.
Key words: Numerical integration. Interpolation. Round-off error. Truncation error. Simpson's
rule. Trapezoidal rule. Boole's rule. Newton-Cotes rules. Gaussian quadrature. Romberg