Algorithm 380: in-situ transposition of a rectangular matrix [F1]

Correctness proof of an in-place permutation

“…end end end A special case of the permutation problem arises in the transposition of a rectangular matrix without using extra storage [2,3]. In case the matrix A [i,j], i = l(1)m and j = l(1)n is columnwise mapped onto a vector VEC [k], k=l(1)m.n, G=m.n, the function f is defined as follows in ALGOL-60:…”

In-Place Transposition of Rectangular Matrices

“…We now describe the overall features of Algorithm MIPT by further contrasting it to the three earlier algorithms [6,8,9].…”

“…Both Brenner [8] and later Cate and Twigg [9] improved Laflin and Brebner's algorithm [6] by combining the dual and self dual cases into a single processing case. Brenner went further when he applied some elementary Abelian group theory: the natural numbers 0 < i < mn partition into n d Abelian groups where n d is the number of divisors d of φ(q); φ is Euler's phi function.…”

“…However, sometimes the cycle minima search is drastically reduced; in those cases the overall processing time of the new double inner loop is greatly reduced. All three algorithms [6,8,9] combine a variant of our bit vector algorithm with Gower's algorithm. By variant we mean they all use an integer sized array of size IWORK to just hold a bit.…”

“…References [6,4,[8][9][10]13] all emphasize the importance of the fundamental mapping P (k) = mk mod q. Here 0 < k < q, i = k/n , j = k − ni and P (k) = i + mj is the location of A ij assuming Fortran storage order and 0-origin subscripting of A ij .…”