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1. Qr Householder Pokemon
2. Householder Qr Complexity
3. Householder Qr Decomposition

Section7.5QR decomposition by Householder reflectors

1. The LU factorization stores the work of Gaussian elimination, QR stores the Householder triangulation process (see below), and the Cholesky factorization stores Cholesky's algorithm.
2. Get the free 'QR-Zerlegung einer Matrix' widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Widget Gallery widgets in Wolfram Alpha.
3. This Householder QR decomposition is faster, but less numerically stable and less feature-full than FullPivHouseholderQR or ColPivHouseholderQR. This class supports the inplace decomposition mechanism.
4. Householder QR Householder transformations are simple orthogonal transformations corre- sponding to re ection through a plane.

A = QR: This factorization can be constructed by three methods: 1. Givens † Property 3.3 (Reduced QR) Suppose the rank of A 2 Rm£n is n for which A = QR is known. Then A = QR where Q and R are submatrices of Q and R given respectively by Q = Q = Q(1: m;1: n); R = R(1: n;1: n): Moreover Q has.

Another algorithm for determining the (QR) decomposition of a matrix which requires fewer operations and is less susceptible to rounding error is the method of Householder reflectors. The method depends upon the following theorem.

Given two such vectors (vec{x},vec{w}inR^ntext{,}) define (vec{v}=vec{w}-vec{x}text{,}) and consider the matrix

Qr Householder Pokemon

While it may be confusing to think that this is a matrix, recall: since (vec{v}) is an (ntimes 1) column matrix, (vec{v}vec{v}^T) is an (ntimes n) symmetric matrix. In fact,

and

A matrix satisfying (P^2=P) is called a projection matrix. The purpose of using (P) is that (vec{x}-Pvec{x}) is the projection of (vec{x}) onto (vec{w+x}text{,}) and therefore (vec{x}-2Pvec{x} = vec{w}text{!})

For brevity, I'll refer to the (QR) decomposition by Householder reflectors as HHQR. The process of HHQR for a matrix (A) iterates through the columns of (A) just like Gram-Schmidt, but with far less numerical instability. We'll explain the process without use of an example, as the process becomes extremely unwieldy in exact arithmetic.

Suppose we begin with a (mtimes n) matrix¹The discussion of the method contained here originally appears in Timothy Sauer's excellent Numerical Analysis, 2nd edition. (A = [a_{i,j}]text{,}) where (m>ntext{.}) We will choose as our first vector (vec{x}_1 = vc{a_{1,1},a_{2,1},ldots, a_{m,1}}text{,}) the first column of (Atext{.}) Since we desire to reflect it onto a coordinate axis vector of the same magnitude, we can choose (vec{w_1}=vc{pmabs{vec{x}_1},0,0,ldots,0}text{.}) Since subtracting nearly identical floating point numbers is problematic and leads to rounding error, we will choose the sign of the first coordinate of (vec{w}_1) to be the opposite of the sign of the first coordinate of (vec{x}_1text{.}) This allows us to determine the first Householder reflector (H_1) such that (H_1vec{x}_1 = vec{w}_1text{.}) Now, putting (R_1 = H_1A) gives us a matrix whose first column is (vec{w}_1text{,}) by construction; notably, this is the first iteration and in the first column (R_1) is the start of an upper triangular matrix. Specifically, if we write (R_1=[r_{i,j}]text{,}) we have (r_{i,1} = 0) whenever (i>1text{.})

Householder Qr Complexity

In order to continue this process, we will produce from (R_1) a new matrix (R_2=[r_{i,j}]) which has (r_{i,j}=0) whenever (i>j) and (jleq 2text{.}) We do so by ignoring the first row and column of (R_1text{;}) doing so gives us (R'_1text{,}) an ((m-1)times(n-1)) matrix. If we let (vec{x}_2) be the first column of (R'_1) and (vec{w}_2=vc{pmabs{vec{x}_2},0,0,ldots,0}text{,}) we can find the Householder reflector (hat{H}_2) such that (hat{H}_2vec{x}_2=vec{w}_2text{.}) Say then that (hat{H}_2 = [hat{h}_{i,j}]text{.}) We will create our full reflection matrix by inserting (hat{H}_2) into the bottom right corner of an identity matrix. Here is the formula for (h_{i,j}) along with the more intuitive picture diagram:

Then (R_2=H_2R_1=H_2H_1Atext{.})

We continue this process in the way established: at each step, we reflect the first column of a submatrix onto a coordinate-axis vector of the same length, flipping signs of the first coordinate, and after processing (n) columns in this way we have (R=H_nH_{n-1}cdots H_2H_1Atext{,}) which is an upper triangular matrix. Moreover, the matrices (H_j) are all orthogonal matrices, so (H_j^{-1}=H_jtext{.}) Hence we obtain

Householder Qr Decomposition