History
The QR algorithm was preceded by the LR algorithm, which uses the LU decomposition instead of the QR decomposition. The QR algorithm is more stable, so the LR algorithm is rarely used nowadays. However, it represents an important step in the development of the QR algorithm.
The LR algorithm was developed in the early 1950s by Heinz Rutishauer, who worked at that time as a research assistant of Eduard Stiefel at ETH Zurich. Stiefel suggested that Rutishauer use the sequence of moments y0T Ak x0, k = 0, 1, … (where x0 and y0 are arbitrary vectors) to find the eigenvalues of A. Rutishauer took an algorithm of Alexander Aitken for this task and developed it into the quotient–difference algorithm or qd algorithm. After arranging the computation in a suitable shape, he discovered that the qd algorithm is in fact the iteration Ak = LkUk (LU decomposition), Ak+1 = UkLk, applied on a tridiagonal matrix, from which the LR algorithm follows.
Read more about this topic: QR Algorithm
Famous quotes containing the word history:
“One classic American landscape haunts all of American literature. It is a picture of Eden, perceived at the instant of history when corruption has just begun to set in. The serpent has shown his scaly head in the undergrowth. The apple gleams on the tree. The old drama of the Fall is ready to start all over again.”
—Jonathan Raban (b. 1942)
“All objects, all phases of culture are alive. They have voices. They speak of their history and interrelatedness. And they are all talking at once!”
—Camille Paglia (b. 1947)
“To care for the quarrels of the past, to identify oneself passionately with a cause that became, politically speaking, a losing cause with the birth of the modern world, is to experience a kind of straining against reality, a rebellious nonconformity that, again, is rare in America, where children are instructed in the virtues of the system they live under, as though history had achieved a happy ending in American civics.”
—Mary McCarthy (1912–1989)