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Let P 4 (x) be a rational quartic polynomial which is not the square of a quadratic. Both Campbell and Ulas considered the problem of finding an rational arithmetic progression x 1 , x 2 ,. .. , x n , with P 4 (x i) a rational square for 1 ≤ i ≤ n. They found examples with n = 10 and n = 12. By simplifying Ulas' approach, we can derive more general… (More)

- Allan J Macleod
- 2005

SUMMARY Nine different algorithms for the acceleration of vector sequences are considered. All have the property that, in one dimension, they reduce to the standard Aitken A* process. Different rules for deciding when to accelerate are also discussed. Theoretical and practical results show a wide range of performance, but it is shown that certain methods… (More)

- Allan J Macleod
- 1993

We consider the approximation of the modified Strove functions Lo and L[ , and the related functions /o-I-o and Ix-Lx , where /o , Ix are modified Bessel functions. Chebyshev expansions are derived to an accuracy of 20D for these functions. By using generalized bilinear and biquadratic maps we optimize the number of coefficients for 20D accuracy.

There has recently been an upsurge of work on the (by now) classical problem of the stability of Gaussian elimination e.g., the papers of Trefethen &amp; Schreiber [5], and Higham and Higham [2]. This short note describes some experimental results in this area derived by the author. They are described more fully in a technical report [4].

- Allan J Macleod
- 2008

We consider the problem of determining a rational point (x, y), of infinite order, on the elliptic curve y 2 = x 3 + ax 2 + bx. The well-known method of 4-descent has proven very effective, and we give explicit formulae for the various stages. For points of large height, however, there can still be a significant search to be done. We describe a very simple… (More)

Low breeding success has been associated with declines in population and range of the re-introduced Capercaillie Tetrao urogallus in Scotland. Annual breeding success from 26 Scottish forests surveyed between 1991 and 2009 averaged only 0.6 chicks per female, the lowest rate recorded in 16 previous studies. Reduced breeding success was due to proportionally… (More)

- Allan J Macleod
- 2009

We discuss the problem of finding distinct integer sets {x 1 , x 2 ,. .. , x n } where each sum x i + x j , i = j is a square, and n ≤ 7. We confirm minimal results of Lagrange and Nicolas for n = 5 and for the related problem with triples. We provide new solution sets for n = 6 to add to the single known set. This provides new information for problem D15… (More)

- Allan J Macleod
- 2006

We consider the problem of finding four different rational squares, such that the product of any two plus the sum of the same two always gives a square. We give some historical background to this problem and exhibit one such quadruple.

- Allan Macleod
- 2005

We consider the problem of finding integer-sided triangles with R/r an integer, where R and r are the radii of the circumcircle and incircle respectively. We show that such triangles are relatively rare.

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