QR Decomposition - Using For Solution To Linear Inverse Problems

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: 
x = Q
\begin{bmatrix} (R_1^T)^{-1}b \\ 0 \end{bmatrix}
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:

    Give a scientist a problem and he will probably provide a solution; historians and sociologists, by contrast, can offer only opinions. Ask a dozen chemists the composition of an organic compound such as methane, and within a short time all twelve will have come up with the same solution of CH4. Ask, however, a dozen economists or sociologists to provide policies to reduce unemployment or the level of crime and twelve widely differing opinions are likely to be offered.
    Derek Gjertsen, British scientist, author. Science and Philosophy: Past and Present, ch. 3, Penguin (1989)

    Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.
    Ralph Waldo Emerson (1803–1882)

    As our disorderly, competitive technological society is piling up its victims and constantly developing new problems of maladjustment, we must use our scientific knowledge to determine the cause and prevention of suffering rather than putting all our emphasis on its alleviation ...
    Agnes E. Meyer (1887–1970)