Solving least Square Problem using QR DECOMPOSITION

Orthogonal bases-orthonormal vectors-Gram Schmidt Process-QR Decomposition-Solving inconsistent system of matrices

#linearalgebra

The least squares problem is a method for finding the best-fitting solution to a system of linear equations that has more equations than unknowns. The goal is to find a solution that minimizes the sum of the squares of the residuals, which are the differences between the observed values and the predicted values.

One way to solve the least squares problem is using QR decomposition, which involves decomposing a matrix into two matrices: an orthogonal matrix (Q) and an upper triangular matrix (R). The orthogonal matrix has orthonormal columns, and the upper triangular matrix has the same rank as the original matrix.

The QR decomposition can be used to solve the least squares problem by transforming the linear system into a triangular system, which can be solved more efficiently. The steps are as follows:

Decompose the coefficient matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R: A = QR.

Multiply the vector of observations b by the orthogonal matrix Q: b' = Q^T b.

Solve the triangular system Rx = b' for the unknowns x using back substitution.

The solution x to the least squares problem is obtained by solving for x in the original equation Ax = b.

The QR decomposition method is a fast and efficient way to solve the least squares problem, especially for large and sparse matrices. It is widely used in many areas of science and engineering, including signal processing, control systems, and numerical analysis.