Background
On a computer, one can often avoid explicitly transposing a matrix in memory by simply accessing the same data in a different order. For example, software libraries for linear algebra, such as BLAS, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement.
However, there remain a number of circumstances in which it is necessary or desirable to physically reorder a matrix in memory to its transposed ordering. For example, with a matrix stored in row-major order, the rows of the matrix are contiguous in memory and the columns are discontiguous. If repeated operations need to be performed on the columns, for example in a fast Fourier transform algorithm (e.g. Frigo & Johnson, 2005), transposing the matrix in memory (to make the columns contiguous) may improve performance by increasing memory locality. Since these situations normally coincide with the case of very large matrices (which exceed the cache size), performing the transposition in-place with minimal additional storage becomes desirable.
Also, as a purely mathematical problem, in-place transposition involves a number of interesting number theory puzzles that have been worked out over the course of several decades.
Read more about this topic: In-place Matrix Transposition
Famous quotes containing the word background:
“In the true sense one’s native land, with its background of tradition, early impressions, reminiscences and other things dear to one, is not enough to make sensitive human beings feel at home.”
—Emma Goldman (1869–1940)
“Pilate with his question “What is truth?” is gladly trotted out these days as an advocate of Christ, so as to arouse the suspicion that everything known and knowable is an illusion and to erect the cross upon that gruesome background of the impossibility of knowledge.”
—Friedrich Nietzsche (1844–1900)
“I had many problems in my conduct of the office being contrasted with President Kennedy’s conduct in the office, with my manner of dealing with things and his manner, with my accent and his accent, with my background and his background. He was a great public hero, and anything I did that someone didn’t approve of, they would always feel that President Kennedy wouldn’t have done that.”
—Lyndon Baines Johnson (1908–1973)