Linear Algebra: Practice Programs in Python and C Languages
Computation of QR Decomposition by Using Python (Linear Algebra: Matrix Decomposition)
QR decomposition of a matrix in Python' is primarily achieved using the numpy.linalg.qr() function. This function provides a convenient and efficient way to compute the QR decomposition.
Linear Algebra
PROGRAMMES
IN PYTHON AND C LANGUAGES
UNIT − IV:
MATRIX DECOMPOSITION
COMPUTATION OF QR DECOMPOSITION BY USING PYTHON
QR
decomposition of a matrix in Python' is primarily achieved using the
numpy.linalg.qr() function. This function provides a convenient and efficient
way to compute the QR decomposition.
Here's
how to compute QR decomposition using Python:
Python
import numpy as np
# create a sample matrix
A = np.array([12,−51,4],
[6, 167, −68],
[−4, 24, −41]])
# Perorm QR decomposition
Q, R = np.linalg.qr (A)
# print the original matrix, Q and R
print ("Original Matrix A:")
print (A)
print ("\n orthogonal Matrix Q:")
print (Q)
print ("\n Upper Triangular Matrix R:")
print (R)
#verify the decomposition by multiplying Q and R
A_reconstructed = Q@R
print ("\nReconstructed Matrix (Q @ R):")
print (A_reconstructed)
Explanation
• Import numpy:
The numpy library is essential for numerical operations in Python, including
linear algebra.
• Create a matrix A:
Define the matrix for which you want to compute the QR decomposition.
• np.linalg.qr(A):
This function takes the matrix A as input and returns two matrices:
• Q:
An orthogonal matrix (its columns are orthonormal vectors). For real matrices,
QTQ=1, where I is the identity matrix.
• R:
An upper triangular matrix.
• Verification:
Multiplying and R should reconstruct the original matrix A, which serves as a
basic check of the decomposition's correctness.
This
method leverages optimized underyling implementations, making it the
recommended approach for practical applications. While it's possible to
implement QR decomposition using methods like Gram−Schmidt or Householder
reflections in pure Python, numpy, linalg.qr() offers significantly better
performance and numerical stability.
Linear Algebra: Practice Programs in Python and C Languages : Tag: maths, mathematics : - Computation of QR Decomposition by Using Python (Linear Algebra: Matrix Decomposition)
Linear Algebra: Practice Programs in Python and C Languages
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Linear Algebra
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