Solve Least Squares Problems by the QR Decomposition

Least Square Problem

In many real life problem solving, when a solution \( x \) to a system of equations of the form \[ A x = B \] cannot be found (i.e. the system is inconsistent), an approximate solution \( \hat x \) to the given system \( A x = B \) may be enough.
Solve the least squares problems is to find an approximate solution \( \hat x \) such that the distance between the vectors \( Ax \) and \( B \) given by \( || A\hat x - B || \) is the smallest.
It is shown in Linear Algebra and its Applications that the approximate solution \( \hat x \) is given by the equation \[ R \hat x = Q^T B \] where \( R \) and \( Q \) are such that \( A = Q R \) is the \( QR \) decomposition of matrix \( A \); \( Q^T\) is the transpose of matrix \( Q \) and \( R \) is an upper triangualar matrix.
Note that this method



Examples with Solutions

Example 1
Use the \( QR \) decomposition to solve the least square problem related to the inconsistent system \( A x = B \) with \( A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \\ 1 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} \).


Solution to Example 1
Given
\( A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \\ 1 & 2 \end{bmatrix} \)
Use the Gram-Schmidt process to find the orthogonal matrix \( Q \) and decompose matrix \( A \) as \( A = QR \).
\( Q = \begin{bmatrix} \dfrac{2\sqrt{5}}{5}&\dfrac{-4\sqrt{105}}{105}\\ 0&\dfrac{\sqrt{105}}{21}\\ \dfrac{\sqrt{5}}{5}&\dfrac{8\sqrt{105}}{105} \end{bmatrix} \)
We now calculate matrix \( R \).
Multiply both sides of \( A = QR \) by \( Q^T\) where \( Q^T \) is the transpose of \( Q \).
\( Q^T A = Q^T Q R \)
One of the properties of orthogonal matrices \( Q \) is that \( Q^T Q = I\), hence the above simplifies to
\( R = Q^T A \)
\( Q^T = \begin{bmatrix} \dfrac{2\sqrt{5}}{5} & 0 & \dfrac{\sqrt{5}}{5}\\ \dfrac{-4\sqrt{105}}{105} & \dfrac{\sqrt{105}}{21} & \dfrac{8\sqrt{105}}{105} \end{bmatrix} \)
Calculate \( R \)
\( R = Q^T A = \begin{bmatrix} \dfrac{2\sqrt{5}}{5} & 0 & \dfrac{\sqrt{5}}{5}\\ \dfrac{-4\sqrt{105}}{105} & \dfrac{\sqrt{105}}{21} & \dfrac{8\sqrt{105}}{105} \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 0 & 1 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} \sqrt{5}&\frac{2}{\sqrt{5}}\\ 0 & \sqrt{\frac{21}{5}} \end{bmatrix} \)
Calculate \( Q^T B \)
\( Q^T B = \begin{bmatrix} \dfrac{2\sqrt{5}}{5} & 0 & \dfrac{\sqrt{5}}{5}\\ \dfrac{-4\sqrt{105}}{105} & \dfrac{\sqrt{105}}{21} & \dfrac{8\sqrt{105}}{105} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} = \begin{bmatrix} \sqrt{5}\\ \dfrac{4\sqrt{5}}{\sqrt{21}} \end{bmatrix} \)

We now substitute \( R \) and \( Q^T B \) by their numerical values in the equation \( R \hat x = Q^T B \) and write the system
\( \begin{bmatrix} \sqrt{5} & \dfrac{2}{\sqrt{5}}\\ 0 & \sqrt{\dfrac{21}{5}} \end{bmatrix} \cdot \hat x = \begin{bmatrix} \sqrt{5}\\ \dfrac{4\sqrt{5}}{\sqrt{21}} \end{bmatrix} \)
Solve the above using any method to obtain
\( \hat x = \begin{bmatrix} \dfrac{13}{21}\\ \dfrac{20}{21} \end{bmatrix} \)
Note that since matrix \( R \) is an upper triangular matrix, using the method of back substitution to solve the system \( R \hat x = Q^T B \) is the most efficient.



Example 2

Use
the \( QR \) decomposition to solve the least square problem related to the inconsistent system \( A x = B \) with \( A = \begin{bmatrix} 1 & 0 & 1\\ 2 & 0 & 0 \\ 0 & -1 & 1\\ 0 & -2 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 \\ 3 \\ 4 \\ 7 \end{bmatrix} \).


Solution to Example 2
Given
\( A = \begin{bmatrix} 1 & 0 & 1\\ 2 & 0 & 0 \\ 0 & -1 & 1\\ 0 & -2 & 0 \end{bmatrix} \)
Use
the Gram-Schmidt process to find the orthogonal matrix \( Q \) and decompose matrix \( A \) as \( A = QR \).
\( Q = \begin{bmatrix} \dfrac{\sqrt{5}}{5} & 0 & \dfrac{\sqrt{10}}{5}\\ \dfrac{2\sqrt{5}}{5} & 0 & -\dfrac{\sqrt{10}}{10}\\ 0&-\dfrac{\sqrt{5}}{5} & \dfrac{\sqrt{10}}{5}\\ 0&-\dfrac{2\sqrt{5}}{5} & -\dfrac{\sqrt{10}}{10} \end{bmatrix} \)
We now calculate matrix \( R \). Multiply both sides of \( A = QR \) by \( Q^T\) where \( Q^T \) is the transpose of \( Q \).
\( Q^T A = Q^T Q R \)
One of the properties of orthogonal matrices \( Q \) is that \( Q^T Q = I\), hence the above simplifies to
\( R = Q^T A \)
\( Q^T = \begin{bmatrix} \dfrac{\sqrt{5}}{5}&\dfrac{2\sqrt{5}}{5}&0&0\\ 0&0&-\dfrac{\sqrt{5}}{5}&-\dfrac{2\sqrt{5}}{5}\\ \dfrac{\sqrt{10}}{5}&-\dfrac{\sqrt{10}}{10}&\dfrac{\sqrt{10}}{5}&-\dfrac{\sqrt{10}}{10} \end{bmatrix} \)
Calculate \( R \)
\( R = Q^T A = \begin{bmatrix} \dfrac{\sqrt{5}}{5}&\dfrac{2\sqrt{5}}{5}&0&0\\ 0&0&-\dfrac{\sqrt{5}}{5}&-\dfrac{2\sqrt{5}}{5}\\ \dfrac{\sqrt{10}}{5}&-\dfrac{\sqrt{10}}{10}&\dfrac{\sqrt{10}}{5}&-\dfrac{\sqrt{10}}{10} \end{bmatrix} \begin{bmatrix} 1 & 0 & 1\\ 2 & 0 & 0 \\ 0 & -1 & 1\\ 0 & -2 & 0 \end{bmatrix} = \begin{bmatrix} \sqrt{5} & 0 & \dfrac{1}{\sqrt{5}}\\ 0 & \sqrt{5} & -\dfrac{1}{\sqrt{5}}\\ 0 & 0 & \dfrac{2 \sqrt 2}{\sqrt{5}} \end{bmatrix} \)
Calculate \( Q^T B \)
\( Q^T B = \begin{bmatrix} \dfrac{\sqrt{5}}{5}&\dfrac{2\sqrt{5}}{5} & 0 & 0\\ 0 & 0 & -\dfrac{\sqrt{5}}{5} & -\dfrac{2\sqrt{5}}{5}\\ 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \\ 4 \\ 7 \end{bmatrix} = \begin{bmatrix} \frac{8}{\sqrt{5}}\\ -\frac{18}{\sqrt{5}}\\ \sqrt{\frac{2}{5}} \end{bmatrix} \)

We now substitute \( R \) and \( Q^T B \) by their numerical values in the equation \( R \hat x = Q^T B \) and write the system
\( \begin{bmatrix} \begin{bmatrix} \sqrt{5} & 0 & \dfrac{1}{\sqrt{5}}\\ 0 & \sqrt{5} & -\dfrac{1}{\sqrt{5}}\\ 0 & 0 & \dfrac{2}{\sqrt{5}} \end{bmatrix} \end{bmatrix} \cdot \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} = \begin{bmatrix} \frac{8}{\sqrt{5}}\\ -\frac{18}{\sqrt{5}}\\ \sqrt{\frac{2}{5}} \end{bmatrix} \)
Solve to obtain
\( x_1 = \dfrac{3}{2} \) , \( x_2 = \dfrac{-7}{2} \) , \( x_3 = \dfrac{1}{2} \)
\( \hat x = \begin{bmatrix} \dfrac{3}{2} \\ \dfrac{-7}{2}\\ \dfrac{1}{2} \end{bmatrix} \)



More References and links

  1. Vector Spaces - Questions with Solutions
  2. Solve Least Squares Problems by the Normal Equations.
  3. Linear Algebra and its Applications - 5 th Edition - David C. Lay , Steven R. Lay , Judi J. McDonald
  4. Elementary Linear Algebra - 7 th Edition - Howard Anton and Chris Rorres

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