Sparse iterative solvers (gmres, cg) in SciPy - Mini Project: Build & Apply

Solving Sparse Linear Systems with GMRES and CG

📖 Scenario: You work as a data scientist helping engineers solve large systems of equations that come from real-world problems like network flows or physical simulations. These systems are often very large but have mostly zero values, called sparse systems.To solve these efficiently, you will use special methods called iterative solvers: GMRES and CG. These methods find approximate solutions quickly without using too much memory.

🎯 Goal: You will create a sparse matrix and a vector, then use the GMRES and CG solvers from scipy.sparse.linalg to find solutions. Finally, you will print the results to see how well the solvers worked.

📋 What You'll Learn

Create a sparse matrix using scipy.sparse

Create a vector with exact values

Set a tolerance level for the solver

Use gmres and cg solvers from scipy.sparse.linalg

Print the solution vectors

💡 Why This Matters

🌍 Real World

Sparse iterative solvers are used in engineering, physics, and computer science to solve large systems efficiently, such as in simulations, network analysis, and optimization.

💼 Career

Knowing how to use sparse solvers helps data scientists and engineers handle big data problems and scientific computations where memory and speed are critical.

1

Create a sparse matrix and vector

Create a sparse matrix called A using scipy.sparse.csr_matrix with these exact values: [[4, 1, 0], [1, 3, 1], [0, 1, 2]]. Also create a vector called b as a NumPy array with values [1, 2, 3].

SciPy

# Create sparse matrix A and vector b
# Your code here

Need a hint?

Use csr_matrix to create the sparse matrix A with the exact 3x3 values. Use np.array to create vector b.

2

Set the tolerance for the solvers

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

SciPy

import numpy as np
from scipy.sparse import csr_matrix

A = csr_matrix([[4, 1, 0], [1, 3, 1], [0, 1, 2]])
b = np.array([1, 2, 3])
# Set tolerance tol to 1e-5
# Your code here

Need a hint?

Simply assign 1e-5 to a variable named tol.

3

Solve the system using GMRES and CG

Import gmres and cg from scipy.sparse.linalg. Use gmres to solve Ax = b with tolerance tol and store the solution in x_gmres. Use cg to solve Ax = b with tolerance tol and store the solution in x_cg. Ignore the second returned value from both solvers.

SciPy

import numpy as np
from scipy.sparse import csr_matrix

A = csr_matrix([[4, 1, 0], [1, 3, 1], [0, 1, 2]])
b = np.array([1, 2, 3])
tol = 1e-5
# Import gmres and cg and solve Ax=b
# Your code here

Need a hint?

Use gmres(A, b, tol=tol) and cg(A, b, tol=tol). Assign the first returned value to x_gmres and x_cg respectively, and ignore the second value with _.

4

Print the solutions

Print the solution vectors x_gmres and x_cg each on a separate line.

SciPy

import numpy as np
from scipy.sparse import csr_matrix
from scipy.sparse.linalg import gmres, cg

A = csr_matrix([[4, 1, 0], [1, 3, 1], [0, 1, 2]])
b = np.array([1, 2, 3])
tol = 1e-5

x_gmres, _ = gmres(A, b, tol=tol)
x_cg, _ = cg(A, b, tol=tol)
# Print x_gmres and x_cg
# Your code here

Need a hint?

Use print(x_gmres) and print(x_cg) to show the solution vectors.