 
Summary: A Class of Numerical Integration Rules With First Order
Derivatives
Mohamad Adnan AIAlaoui"
Abstract
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 twosegment rule, is
obtained by interpolating the corrected trapezoidal rule and the Simpson onethird rule. The third
member, a threesegment rule, is obtained by interpolating the corrected trapezoidal rule and the
Simpson threeeights rule. The fourth member, a foursegment rule is obtained by interpolating
the twosegment 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 NewtonCotes
rules to cancel Out an additional term in the EulerMacLaurin 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 roundoff 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 NewtonCotes rules.
Members of the proposed family are shown to be viable alternatives to Gaussian quadrature.
Key words: Numerical integration. Interpolation. Roundoff error. Truncation error. Simpson's
rule. Trapezoidal rule. Boole's rule. NewtonCotes rules. Gaussian quadrature. Romberg
