Sparse matrix factorizations in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of sparse LU factorization

What is the shape of the L and U factors after performing LU factorization on a 4x4 sparse matrix?

SciPy

import numpy as np
from scipy.sparse import csc_matrix
from scipy.sparse.linalg import splu

A = csc_matrix(np.array([[4, 0, 0, 0], [3, 5, 0, 0], [0, 1, 7, 0], [0, 0, 2, 6]]))
lu = splu(A)
L_shape = lu.L.shape
U_shape = lu.U.shape
print(L_shape, U_shape)

A (4, 4) (4, 4)

B (3, 3) (3, 3)

C (4, 3) (3, 4)

D (5, 5) (5, 5)

💡 Hint

LU factorization of an n x n matrix produces L and U matrices of the same shape as the original matrix.

LU factorization decomposes a square matrix A into L (lower triangular) and U (upper triangular) matrices, both having the same shape as A.

❓ data_output

intermediate

Number of non-zero elements after Cholesky factorization

Given a 3x3 sparse positive definite matrix, what is the number of non-zero elements in the Cholesky factor?

SciPy

import numpy as np
from scipy.sparse import csc_matrix
from scipy.sparse.linalg import cholesky

A = csc_matrix(np.array([[4, 1, 0], [1, 3, 1], [0, 1, 2]]))
ch = cholesky(A)
L = ch.L()
nnz = L.nnz
print(nnz)

A 6

B 5

C 4

D 7

💡 Hint

Count the non-zero elements in the lower triangular Cholesky factor matrix.

The Cholesky factor L is lower triangular and sparse. Counting its non-zero elements gives 5 in this case.

🔧 Debug

advanced

Identify the error in sparse QR factorization code

What error will this code raise when trying to perform QR factorization on a sparse matrix using scipy?

SciPy

import numpy as np
from scipy.sparse import csc_matrix
from scipy.sparse.linalg import qr

A = csc_matrix(np.array([[1, 2], [3, 4]]))
Q, R = qr(A)
print(Q.toarray())

A No error, prints Q matrix

B TypeError: cannot unpack non-iterable NoneType object

C ValueError: too many values to unpack (expected 2)

D AttributeError: module 'scipy.sparse.linalg' has no attribute 'qr'

💡 Hint

Check if scipy.sparse.linalg has a qr function.

scipy.sparse.linalg does not provide a qr function. Attempting to call qr from this module raises AttributeError.

🚀 Application

advanced

Choosing factorization for solving sparse linear systems

You have a large sparse symmetric positive definite matrix. Which factorization method is best suited for solving linear systems efficiently?

A LU factorization

B Singular Value Decomposition (SVD)

C Cholesky factorization

D QR factorization

💡 Hint

Consider matrix properties and factorization efficiency.

Cholesky factorization is optimized for symmetric positive definite matrices and is more efficient than LU or QR for this case.

🧠 Conceptual

expert

Effect of fill-in during sparse LU factorization

What does 'fill-in' mean in the context of sparse LU factorization, and why is it important?

A Fill-in refers to the creation of new non-zero elements in L and U that were zero in the original matrix, increasing memory and computation cost.

B Fill-in is the process of filling missing data in sparse matrices before factorization.

C Fill-in means removing zero elements from the matrix to make it denser for better factorization.

D Fill-in describes the compression of sparse matrices to reduce storage space.

💡 Hint

Think about how factorization changes the sparsity pattern.

Fill-in occurs when zeros in the original matrix become non-zero in the factors, which can increase memory use and slow down computations.