## Abstract

Filon developed a method for evaluating intcgrals of the form

He uscs a second order curve fitted to the middlc and end points of a

panel. The formula rcduces to Simpson’s rulc whcn p approachcs zero.

In rcf. 1, Flinn describes a modification of Filon’s method. He fits a

fifth order curve to the values of the function and of its first derivative, at

the same abscissas as Filon. This formula rcduces to Simpson’s rule with

end correction when p approaches zero, and to Filon’s formula if the values

of the dcrivatives are expressed in terms of the function values.

Both formulas are rclatcd to the Newton-Cotes type formulas in the sense

that they are based on a Stirling approximation and require the function

values in cquidistant points.

In this sense, the proposed method is of the Gaussian type as it is based

on an expansion into orthogonal polynomials and demands function valucs

in abscissas which are not cqually spaced, while no values of the derivative

are needed.

Two sets of orthogonal polynomials were tried out: Chcbychev’s and

Legendre’s. As could be expected, the precision of the latter is slightly better.

Therefore constants will be given only for the Lcgendre formula.

He uscs a second order curve fitted to the middlc and end points of a

panel. The formula rcduces to Simpson’s rulc whcn p approachcs zero.

In rcf. 1, Flinn describes a modification of Filon’s method. He fits a

fifth order curve to the values of the function and of its first derivative, at

the same abscissas as Filon. This formula rcduces to Simpson’s rule with

end correction when p approaches zero, and to Filon’s formula if the values

of the dcrivatives are expressed in terms of the function values.

Both formulas are rclatcd to the Newton-Cotes type formulas in the sense

that they are based on a Stirling approximation and require the function

values in cquidistant points.

In this sense, the proposed method is of the Gaussian type as it is based

on an expansion into orthogonal polynomials and demands function valucs

in abscissas which are not cqually spaced, while no values of the derivative

are needed.

Two sets of orthogonal polynomials were tried out: Chcbychev’s and

Legendre’s. As could be expected, the precision of the latter is slightly better.

Therefore constants will be given only for the Lcgendre formula.

Original language | English |
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Publisher | SCK CEN |

Number of pages | 9 |

State | Published - Oct 1964 |

Externally published | Yes |

### Publication series

Name | SCK•CEN Reports |
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Publisher | Studiecentrum voor Kernenergie |

No. | BLG-333 |