Sparse direct solvers (spsolve) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of sparse linear system solution

What is the output of the following code that solves a sparse linear system using spsolve?

SciPy

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

A = csc_matrix([[3, 0, 0], [0, 4, 0], [0, 0, 5]])
b = np.array([3, 8, 10])
x = spsolve(A, b)
print(x)

A [0.33333333 0.5 0.2]

B [9. 32. 50.]

C [1. 2. 2.]

D Raises a ValueError

💡 Hint

Remember that spsolve solves Ax = b for x.

The matrix A is diagonal with entries 3, 4, and 5. Solving Ax = b means dividing each element of b by the corresponding diagonal element of A, resulting in [1, 2, 2].

🧠 Conceptual

intermediate

Understanding spsolve input requirements

Which of the following statements about the inputs to spsolve is TRUE?

A The matrix A must be a sparse matrix format supported by SciPy, like CSC or CSR.

B The vector b must be a sparse matrix.

C The vector b can be a list or a NumPy array.

D The matrix A must be a dense NumPy array.

💡 Hint

Check the documentation for the expected matrix type for A.

spsolve requires A to be a sparse matrix in formats like CSC or CSR. The vector b can be a dense array or list, but A must be sparse.

🔧 Debug

advanced

Identify the error in sparse solver usage

What error will the following code raise when executed?

SciPy

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

A = csr_matrix([[1, 2], [3, 4], [5, 6]])
b = np.array([7, 8])
x = spsolve(A, b)
print(x)

A ValueError: dimension mismatch between A and b

B TypeError: unsupported operand type(s)

C No error, outputs solution vector

D LinAlgError: singular matrix

💡 Hint

Check the shapes of A and b before solving.

Matrix A has shape (3, 2) but vector b has length 2, so dimensions do not align for solving Ax = b. This causes a ValueError.

❓ data_output

advanced

Resulting vector from sparse system with zero entries

Given the sparse matrix and vector below, what is the output vector after solving with spsolve?

SciPy

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

A = csc_matrix([[10, 0, 0], [0, 0, 0], [0, 0, 5]])
b = np.array([20, 0, 15])
x = spsolve(A, b)
print(x)

A [20.0, 0.0, 15.0]

B [2.0, 0.0, 3.0]

C [2.0, nan, 3.0]

D Raises a LinAlgError due to singular matrix

💡 Hint

Check if the matrix A is invertible.

The second row of A is all zeros, making A singular (not invertible). spsolve raises a LinAlgError in this case.

🚀 Application

expert

Using spsolve for a large sparse system

You have a large sparse matrix A of size 10000x10000 with mostly zeros and a vector b of length 10000. Which approach is MOST efficient to solve Ax = b using SciPy?

A Convert A to a dense NumPy array and use numpy.linalg.solve

B Use scipy.sparse.linalg.spsolve directly on the sparse matrix A and vector b

C Convert b to a sparse matrix and use spsolve

D Use a for-loop to solve each equation separately

💡 Hint

Consider memory and computation time for large sparse matrices.

spsolve is optimized for sparse matrices and large systems. Converting to dense wastes memory and time. Using spsolve directly is efficient.