Disadvantage

  • Steps are very expensive to compute. The complexity of one step of the basic QR-method = O(n^3).
  • Usually, many steps (much more than n) are required to converge. In fact, the basic QR-method can be arbitrarily slow if the eigenvalues are close to each other.


QR Decomposition in Machine learning

QR decomposition is a way of expressing a matrix as the product of two matrices: Q (an orthogonal matrix) and R (an upper triangular matrix). In this article, I will explain decomposition in Linear Algebra, particularly QR decomposition among many decompositions.

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What is QR Decomposition?

Decomposition or Factorization is dividing the original single entity into multiple entities for easiness. Decomposition has various applications in numerical linear algebra, optimization, solving systems of linear equations, etc. QR decomposition is a versatile tool in numerical linear algebra that finds applications in solving linear systems, least squares problems, eigenvalue computations, etc. Its numerical stability and efficiency make it a valuable technique in a range of applications....

Compute QR decomposition:

Gram-Schmidt Orthogonalization...

QR Decomposition using Python

Python3 import numpy as np # Create a numpy arrayarr = np.array([[1, 2, 4], [0, 0, 5],                [0, 3, 6]]) print(arr) # Find the QR factor of arrayq, r = np.linalg.qr(arr)print('\nQ:\n', q)print('\nR:\n', r)print(np.allclose(arr, np.dot(q, r)))  # to check result is correct or not...

Applications:

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Advantages

It has many applications in algebra and machine learning whether it is for least square method, linear regression, PCA, eigenvalue problem or regularization of model in machine learning. Few of them are written below....

Disadvantage:

It allows for a numerically stable and efficient solution of system of equation.Compared to LU decomposition, this method does not require that the decomposition be carried out on a square matrix....