Sparse linear algebra solvers in SciPy - Mini Project: Build & Apply

Solving Sparse Linear Systems with SciPy

📖 Scenario: You work as a data scientist helping engineers solve large systems of equations that come from real-world problems like electrical circuits or network flows. These systems are often sparse, meaning most values are zero. Using special sparse matrix solvers saves time and memory.

🎯 Goal: Build a Python program that creates a sparse matrix and a vector, configures solver options, solves the sparse linear system using SciPy's sparse solver, and prints the solution.

📋 What You'll Learn

Create a sparse matrix using SciPy's csr_matrix with exact values

Create a vector b with exact values

Set a solver tolerance variable tol

Use scipy.sparse.linalg.spsolve with the sparse matrix, vector

Print the solution vector

💡 Why This Matters

🌍 Real World

Sparse linear solvers are used in engineering, physics, and computer science to efficiently solve large systems where most values are zero, saving memory and computation time.

💼 Career

Data scientists and engineers often need to solve sparse systems when working with network analysis, simulations, or optimization problems.

1

Create the sparse matrix and vector

Create a sparse matrix called A using scipy.sparse.csr_matrix with the exact data: values [10, 3, 1, 7, 2, 8], row indices [0, 0, 1, 2, 2, 2], and column indices [0, 2, 2, 0, 1, 2]. Also create a vector b as a NumPy array with values [19, 3, 35].

SciPy

import numpy as np
from scipy.sparse import csr_matrix
# Create sparse matrix A and vector b
# Your code here

Need a hint?

Use csr_matrix((data, (row_indices, col_indices)), shape=(3,3)) to create the sparse matrix.

2

Set the solver tolerance

Create a variable called tol and set it to 1e-5 to configure the solver tolerance.

SciPy

import numpy as np
from scipy.sparse import csr_matrix

A = csr_matrix((np.array([10, 3, 1, 7, 2, 8]), (np.array([0, 0, 1, 2, 2, 2]), np.array([0, 2, 2, 0, 1, 2]))), shape=(3, 3))
b = np.array([19, 3, 35])
# Set solver tolerance tol
# Your code here

Need a hint?

Just assign 1e-5 to the variable tol.

3

Solve the sparse linear system

Import spsolve from scipy.sparse.linalg. Use spsolve with the sparse matrix A, vector b to solve the system. Store the result in a variable called x.

SciPy

import numpy as np
from scipy.sparse import csr_matrix

A = csr_matrix((np.array([10, 3, 1, 7, 2, 8]), (np.array([0, 0, 1, 2, 2, 2]), np.array([0, 2, 2, 0, 1, 2]))), shape=(3, 3))
b = np.array([19, 3, 35])
tol = 1e-5
# Import spsolve and solve Ax = b
# Your code here

Need a hint?

Use x = spsolve(A, b) to solve the system.

4

Print the solution vector

Write a print statement to display the solution vector x.

SciPy

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

A = csr_matrix((np.array([10, 3, 1, 7, 2, 8]), (np.array([0, 0, 1, 2, 2, 2]), np.array([0, 2, 2, 0, 1, 2]))), shape=(3, 3))
b = np.array([19, 3, 35])
tol = 1e-5

x = spsolve(A, b)
# Print the solution vector x
# Your code here

Need a hint?

Use print(x) to show the solution.