Sparse linear algebra solvers in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of sparse linear system solver using SciPy

What is the output of the following code snippet that solves a sparse linear system using SciPy's conjugate gradient solver?

SciPy

import numpy as np
from scipy.sparse import diags
from scipy.sparse.linalg import cg

# Create a sparse diagonal matrix A
diagonals = [np.ones(3)*2]
A = diags(diagonals, offsets=[0], shape=(3, 3))

b = np.array([1, 2, 3])

x, info = cg(A, b)
print(np.round(x, 2))

A [ 1. 2. 3.]

B [ 0.33 0.67 1. ]

C [ 0.5 1. 1.5]

D [ 0.25 0.5 0.75]

💡 Hint

Think about how the conjugate gradient solver solves Ax = b for sparse A.

The conjugate gradient solver finds x such that Ax = b. Given the sparse matrix A and vector b, the solution x is approximately [0.5, 1.0, 1.5].

❓ data_output

intermediate

Number of non-zero elements after sparse matrix multiplication

Given two sparse matrices A and B, what is the number of non-zero elements in the product matrix C = A * B?

SciPy

import numpy as np
from scipy.sparse import csr_matrix

A = csr_matrix([[0, 1, 0], [2, 0, 0], [0, 0, 3]])
B = csr_matrix([[1, 0, 0], [0, 0, 4], [5, 0, 0]])

C = A.dot(B)
print(C.nnz)

A 6

B 5

C 4

D 3

💡 Hint

Count the non-zero elements in the resulting sparse matrix after multiplication.

Multiplying A and B results in a sparse matrix with 3 non-zero elements.

🔧 Debug

advanced

Identify the error in sparse linear solver usage

What error does the following code raise when trying to solve a sparse linear system?

SciPy

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

A = csc_matrix([[0, 1], [1, 0]])
b = np.array([1, 2])
x = spsolve(A, b)
print(x)

A TypeError: unsupported operand type(s)

B No error, prints [2. 1.]

C ValueError: dimension mismatch

D LinAlgError: Singular matrix

💡 Hint

Check if the matrix A is invertible before solving.

No error occurs because matrix A is invertible (determinant = -1). The solution is x = [2. 1.].

❓ visualization

advanced

Visualizing sparsity pattern of a matrix

Which option produces a plot showing the sparsity pattern of a sparse matrix A?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.sparse import random

A = random(10, 10, density=0.2, format='csr', random_state=42)

# Choose the correct code to visualize sparsity pattern

A

plt.spy(A)
plt.show()

B

plt.imshow(A)
plt.show()

C

plt.plot(A)
plt.show()

D

plt.scatter(A)
plt.show()

💡 Hint

Use the function designed to visualize sparse matrix structure.

plt.spy() is the correct function to visualize the sparsity pattern of a matrix.

🚀 Application

expert

Choosing the best solver for a large sparse symmetric positive definite matrix

You have a large sparse symmetric positive definite matrix A and vector b. Which solver is the best choice to efficiently solve Ax = b?

A Use scipy.sparse.linalg.cg (Conjugate Gradient solver)

B Use numpy.linalg.solve with dense matrix conversion

C Use scipy.sparse.linalg.lsqr (Least squares solver)

D Use scipy.sparse.linalg.bicgstab (BiConjugate Gradient Stabilized solver)

💡 Hint

Consider matrix properties and solver efficiency for symmetric positive definite matrices.

The conjugate gradient solver (cg) is optimized for large sparse symmetric positive definite matrices and is more efficient than other solvers.