Longman method

A method for the approximate calculation of a definite integral

$$ I = \int\limits _ { a } ^ { b } f ( x) dx , $$

where $ f $ has exactly $ n $ roots $ x _ {i} $ inside the interval $ [ a, b] $,

$$ x _ {0} = a < x _ {1} < \dots < x _ {n} < b = x _ {n+1} , $$

and satisfies the conditions stated below. Let

$$ v _ {i} = ( - 1 ) ^ {i} \int\limits _ {x _ {i} } ^ { {x _ i+1} } f( x) dx ,\ \ i = 0 \dots n , $$

then $ I = S $, where

$$ S = \sum_{j=0}^ { n } ( - 1 ) ^ {j} v _ {j} . $$

It is assumed that $ f $ preserves its sign on the interval $ [ x _ {i} , x _ {i+1} ] $, has different signs on adjacent intervals, and $ v _ {i} \neq 0 $, $ i = 0 \dots n $. Such a function $ f $ is said to be oscillatory. The calculation of $ I $ by means of a quadrature formula for large $ n $ is difficult, since a good approximation of an oscillatory function on the whole interval $ [ a , b ] $ is impossible in practice. The use of the equality $ I = S $ leads to the need to calculate all integrals $ v _ {j} $, which is also inadvisable in the case of large $ n $.

The approximate calculation of $ I $ in Longman's method is based on the equality ( $ n \geq p $)

$$ \tag{1 } S = \sum_{k=0}^ { p-1} ( - 1 ) ^ {k} 2 ^ {- k - 1 } \Delta ^ {k} v _ {0} + $$

$$ + ( - 1) ^ {n} \sum_{k=0}^ { p-1} 2 ^ {- k - 1 } \Delta ^ {k} v _ {n-k+2} ^ {-p} ( - 1 ) ^ {p} \sum_{k=0}^{n-p} ( - 1 ) ^ {k} \Delta ^ {p} v _ {k} . $$

In (1) the finite differences of $ v _ {j} $ as functions of the discrete argument $ j $ occur:

$$ \Delta v _ {j} = v _ {j+1} - v _ {j} ,\ j= 0 \dots n- 1, $$

$$ \Delta ^ {r+1} v _ {j} = \Delta ^ {r} v _ {j+1} - \Delta ^ {r} v _ {j} , $$

$$ r = 1 \dots p- 1; \ j = 0 \dots n- r- 1 . $$

If $ v _ {j} $ is such that on the right-hand side of (1) one can neglect terms containing finite differences of order $ p $, then the approximate equality

$$ \tag{2 } S \cong \sum_{k=0}^{p-1} ( - 1 ) ^ {k} 2 ^ {- k - 1 } \Delta ^ {k} v _ {0} + ( - 1 ) ^ {n} \sum _ {k=0}^ { p-1} 2 ^ {- k - 1 } \Delta ^ {k} v _ {n-k} $$

can be used to calculate $ S $. To calculate the right-hand side of (2) it is sufficient to know the first $ p $ values $ v _ {j} $, that is, the values $ v _ {0} \dots v _ {p-1} $, and the last $ p $ values $ v _ {n} \dots v _ {n-p+1} $. Longman's method consists in the use of (2) for an approximate calculation of the sum $ S $.

If in the integral $ I $ the upper limit of integration $ b = + \infty $ and

$$ I = \sum_{i=0}^ \infty ( - 1 ) ^ {i} v _ {i} , $$

then instead of (1) one must use the equality

$$ \sum_{i=0}^ \infty ( - 1 ) ^ {i} v _ {i} = \sum_{i=0}^ \infty (-1) ^ {i} 2^{-i-1} \Delta ^ {i} v _ {0} $$

(the Euler transform) and replace the series on the right-hand side by a partial sum.

The method was proposed by I.M. Longman [1].

References

[1]	I.M. Longman, "A method for the numerical evaluation of finite integrals of oscillatory functions" Math. Comput. , 14 : 69 (1960) pp. 53–59
[2]	P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)