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Qr Householder Method

2021年4月23日

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This article will discuss QR Decomposition in Python. In previous articles we have looked at LU Decomposition in Python and Cholesky Decomposition in Python as two alternative matrix decomposition methods. QR Decomposition is widely used in quantitative finance as the basis for the solution of the linear least squares problem, which itself is used for statistical regression analysis.
One of the key benefits of using QR Decomposition over other methods for solving linear least squares is that it is more numerically stable, albeit at the expense of being slower to execute. Hence if you are performing a large quantity of regressions as part of a trading backtest, for instance, you will need to consider very extensively whether QR Decomposition is the best fit (excuse the pun).
For a square matrix $A$ the QR Decomposition converts $A$ into the product of an orthogonal matrix $Q$ (i.e. $Q^TQ=I$) and an upper triangular matrix $R$. Hence: begin{eqnarray*} A = QR end{eqnarray*}
5,6 is reorthogonalization. Unfortunately this method has not become very popular because at about the same time a new way to compute the decompo-sition 1 was proposed 4 using elementary orthogonal Householder-matrices. This algorithm is stable and the only objection is that it does not yield an explicit representation of Q. The householderDecomposition method is a member of the cMatrix object. The cMatrix object is a rather simple object to perform a few matrix operations. It has a few operator overloads and the ability to transpose a matrix. Download the project at the end of this post to see the details. Here is the Householder decomposition. Determine the computational complexity for QR decomposition using Gram-Schmidt Modified Gram-Schmidt Householder reflections Givens rotations Compare the complexity of Householder vs Givens for a sparse matrix Implement QR decomposition using Householder reflections, (input matrix A of full column rank and output Q,R).
There are a few different algorithms for calculating the matrices $Q$ and $R$. We will outline the method of Householder Reflections, which is known to be more numerically stable the the alternative Gramm-Schmidt method. I’ve outlined the Householder Reflections method below.
Note, the following explanation is an expansion of the extremely detailed article on QR Decomposition using Householder Reflections over at Wikipedia.
A Householder Reflection is a linear transformation that enables a vector to be reflected through a plane or hyperplane. Essentially, we use this method because we want to create an upper triangular matrix, $R$. The householder reflection is able to carry out this vector reflection such that all but one of the coordinates disappears. The matrix $Q$ will be built up as a sequence of matrix multiplications that eliminate each coordinate in turn, up to the rank of the matrix $A$.
The first step is to create the vector $mathbb{x}$, which is the $k$-th column of the matrix $A$, for step $k$. We define $alpha = -sgn(mathbb{x}_k)(||mathbb{x}||)$. The norm $||cdot||$ used here is the Euclidean norm. Given the first column vector of the identity matrix, $I$ of equal size to $A$, $mathbb{e}_1 = (1,0,...,0)^T$, we create the vector $mathbb{u}$: begin{eqnarray*} mathbb{u} = mathbb{x} + alpha mathbb{e}_1 end{eqnarray*}
Once we have the vector $mathbb{u}$, we need to convert it to a unit vector, which we denote as $mathbb{v}$: begin{eqnarray*} mathbb{v} = mathbb{u}/||mathbb{u}|| end{eqnarray*}
Now we form the matrix $Q$ out of the identity matrix $I$ and the vector multiplication of $mathbb{v}$: begin{eqnarray*} Q = I - 2 mathbb{v} mathbb{v}^T end{eqnarray*}
$Q$ is now an $mtimes m$ Householder matrix, with $Qmathbb{x} = left( alpha, 0, ..., 0right)^T$. We will use $Q$ to transform $A$ to upper triangular form, giving us the matrix $R$. We denote $Q$ as $Q_k$ and, since $k=1$ in this first step, we have $Q_1$ as our first Householder matrix. We multiply this with $A$ to give us: begin{eqnarray*} Q_1A = begin{bmatrix} alpha_1&star&dots&star 0 & & & vdots & & A’ & 0 & & & end{bmatrix} end{eqnarray*}
The whole process is now repeated for the minor matrix $A’$, which will give a second Householder matrix $Q’_2$. Now we have to ’pad out’ this minor matrix with elements from the identity matrix such that we can consistently multiply the Householder matrices together. Hence, we define $Q_k$ as the block matrix: begin{eqnarray*} Q_k = begin{pmatrix} I_{k-1} & 0 0 & Q_k’end{pmatrix} end{eqnarray*}
Once we have carried out $t$ iterations of this process we have $R$ as an upper triangular matrix: begin{eqnarray*} R = Q_t ... Q_2 Q_1 A end{eqnarray*}
$Q$ is then fully defined as the multiplication of the transposes of each $Q_k$: begin{eqnarray*} Q = Q^T_1 Q^T_2 ... Q^T_t end{eqnarray*}
This gives $A=QR$, the QR Decomposition of $A$.
To calculate the QR Decomposition of a matrix $A$ with NumPy/SciPy, we can make use of the built-in linalg library via the linalg.qr function. This is significantly more efficient than using a pure Python implementation:
The output of the QR decomposition includes $A$, $Q$ and $R$. As a basic sanity check we can see that $R$ is in fact an upper triangular matrix:
You aren’t likely to ever need a pure Python implementation of QR Decomposition (homework notwithstanding), but I feel that it is helpful to gain an understanding of the Householder Reflections algorithm, so I have written my own implementation:
The output from the Householder method implemented in pure Python is given below:
You can see that we get the same answers as above in the SciPy implementation, albeit with a few more significant figures! One has to be extremely careful in numerical algorithms when dealing with floating point arithmetic, but that is a discussion for another day. Next:The Schur Decomposition and Up:ch1 Previous:Householder transformation
QR Decomposition
The QR decomposition (or factorization)is an algorithm that converts a given matrix into a product of an orthogonal matrix and a rightor upper triangular matrix with . In the following we consider two methods for the QR decomposition.
*The Householder transformation:
We first construct a Householder matrix based on the first column vector of , by which will be converted into a vector with its last elements equal to zero: We then construct second transformation matrix (47)where is an dimensional Householder matrix to eliminate the last elements of the first column of : This process is repeated with the general kth transformation matrix (49)where is an dimensional Householder matrix to eliminate the last elements of the first column of the sub-matrix of the same dimension obtained from the previous steps. After such iterations becomes an upper triangular matrix: Define and pre-multiply it on both sides we get the QR decomposition of : (51)where defined aobve is an orthogonal matrix:
To find the complexity of this algorithm, we note that the Householder transformation of a column vector is , which is needed for all columns of a submatrix in each of the iterations. Therefore the total complexity is . However, if is a Hessenberg matrix (or more specially a tridiagonal matrix), then the Householder transformation of each column vector is constant (for the only subdiagonal component), and the complexity becomes .
Note that this method cannot convert into a diagonal matrix, by performing Householder transformations on the rows (from the right) as well as on the columns (from the left), similar to the tridiagonalization of a symmetric matrix previously considered. This is because while post-multiplying a Householder matrix to convert a row, the column already converted by pre-multiplying a Householder matrix will be changed again.
*The Gram-Schmidt process:
The QR decomposition of can also be obtained by converting the column vectors in , assumed to be independent, into a set of orthonormal vectors , which form the columns of the orthogonal matrix .
We first find as the normalized version of the first column of , and then find each of the remaining orthogonal vector (), with all previous already normalized: (53)followed by normalization so that . Now can be represented as a linear combination of vectors as: We define and , and rewrite the above as (55)where is the ith column of , and are yet to be generated (but they are multiplied by zero and play no role here). Combining the equations above in matrix form, we get where is orthogonal and is upper triangular.
The complexity for computing and is , and the total complexity of this algorithm is .
The QR decomposition finds many applications, such as solving a linear system :(57)where can be obtained as:
and then can be obtained by solving with an upper triangular coefficient matrix by back-substitution.
More importantly, the QR decomposition is the essential part of the QR algorithm for solving the eigenvalue problem of a general matrix, to be considered in the following section.
The Matlab code listed below carries out the QR decomposition by both the Householder transformation and the Gram-Schmidt method:
Next:The Schur Decomposition and Up:Householder Matrixch1 Previous:Qr Householder Method MeaningHouseholder transformationRuye Wang2018-08-20
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