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:
“To the questions of the officiously meddling police Falter replied absently and tersely; but, when he finally grew tired of this pestering, he pointed out that, having accidentally solved the riddle of the universe, he had yielded to artful exhortation and shared that solution with his inquisitive interlocutor, whereupon the latter had died of astonishment.”
—Vladimir Nabokov (18991977)
“The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
—Aldous Huxley (18941963)
“Men decide far more problems by hate, love, lust, rage, sorrow, joy, hope, fear, illusion, or some other inward emotion than by reality, authority, any legal standard, judicial precedent, or statute.”
—Marcus Tullius Cicero (10643 B.C.)