The divide part of the divide-and-conquer algorithm comes from the realization that a tridiagonal matrix is "almost" block diagonal.
The size of submatrix we will call, and then is . Note that the remark about being almost block diagonal is true regardless of how is chosen (i.e., there are many ways to so decompose the matrix). However, it makes sense, from an efficiency standpoint, to choose .
We write as a block diagonal matrix, plus a rank-1 correction:
The only difference between and is that the lower right entry in has been replaced with and similarly, in the top left entry has been replaced with .
The remainder of the divide step is to solve for the eigenvalues (and if desired the eigenvectors) of and, that is to find the diagonalizations and . This can be accomplished with recursive calls to the divide-and-conquer algorithm, although practical implementations often switch to the QR algorithm for small enough submatrices.
Read more about this topic: Divide-and-conquer Eigenvalue Algorithm
Famous quotes containing the word divide:
“We have to divide mother love with our brothers and sisters. Our parents can help us cope with the loss of our dream of absolute love. But they cannot make us believe that we haven’t lost it.”
—Judith Viorst (20th century)
“Grief can take care of itself, but to get the full value of a joy you must have somebody to divide it with.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“The king said, -Divide the living boy in two; then give half to the one, and half to the other. But the woman whose son was alive said to the king -because compassion for her son burned within her - -Please, my lord, give her the living boy; certainly do not kill him! The other said, -It shall be neither mine nor yours; divide it. Then the king responded: -Give the first woman the living boy; do not kill him. She is his mother.”
—Bible: Hebrew, 1 Kings. 3:25-37.
Solomon resolves a dispute between two women over a child. Solomon’s wisdom was proven by this story.