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 herewith commission you to carry out all preparations with regard to ... a total solution of the Jewish question in those territories of Europe which are under German influence.... I furthermore charge you to submit to me as soon as possible a draft showing the ... measures already taken for the execution of the intended final solution of the Jewish question.”
—Hermann Goering (1893–1946)
“Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.”
—Ralph Waldo Emerson (1803–1882)
“Those great ideas which come to you in your sleep just before you awake in morning, those solutions to the world’s problems which, in the light of day, turn out to be duds of the puniest order, couldn’t they be put to some use, after all?”
—Robert Benchley (1889–1945)