Using For Solution To Linear Inverse Problems
Compared to the direct matrix inverse, inverse solutions using QR decomposition are more numerically stable as evidenced by their reduced condition numbers .
To solve the underdetermined linear problem where the matrix A has dimensions and rank, first find the QR factorization of the transpose of A:, where Q is an orthogonal matrix (i.e. ), and R has a special form: . Here is a square right triangular matrix, and the zero matrix has dimension . After some algebra, it can be shown that the solution to the inverse problem can be expressed as: 
where is found by Gaussian elimination.
To find a solution to the overdetermined problem which minimizes the norm, first find the QR factorization of A: . The solution can then be expressed as, where and are the same as before, but now is a projection matrix that maps a vector in into .
Read more about this topic: QR Decomposition
Famous quotes containing the words solution, inverse and/or problems:
“I can’t quite define my aversion to asking questions of strangers. From snatches of family battles which I have heard drifting up from railway stations and street corners, I gather that there are a great many men who share my dislike for it, as well as an equal number of women who ... believe it to be the solution to most of this world’s problems.”
—Robert Benchley (1889–1945)
“The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
—Aldous Huxley (1894–1963)
“It’s so easy during those first few months to think that the problems will never end. You feel as if your son will never sleep through the night, will always spit up food after eating, and will never learn to smile—even though you don’t know any adults or even older children who still act this way.”
—Lawrence Kutner (20th century)