Sparse iterative solvers (gmres, cg) in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the GMRES solver from scipy.

SciPy

from scipy.sparse.linalg import [1]

Drag options to blanks, or click blank then click option'

Acg

Bgmres

Csolve

Dinv

Attempts:

3 left

💡 Hint

Common Mistakes

Importing 'cg' instead of 'gmres'.

Trying to import 'solve' which is not an iterative solver.

Using 'inv' which is for matrix inversion.

The GMRES solver is imported from scipy.sparse.linalg as gmres.

2fill in blank

medium

Complete the code to solve a linear system using the CG solver.

SciPy

x, info = [1](A, b)

Drag options to blanks, or click blank then click option'

Asolve

Binv

Cgmres

Dcg

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'solve' which is a direct solver, not iterative.

Using 'inv' which computes matrix inverse.

Using 'gmres' which is a different iterative solver.

The CG solver is called by cg(A, b) to solve the system Ax = b.

3fill in blank

hard

Fix the error in the code to correctly call GMRES with a restart parameter of 50.

SciPy

x, info = gmres(A, b, restart=[1])

Drag options to blanks, or click blank then click option'

A50

B'50'

Crestart

DNone

Attempts:

3 left

💡 Hint

Common Mistakes

Passing the restart value as a string.

Passing the parameter name as a value.

Passing None which disables restart.

The restart parameter must be an integer, so use 50 without quotes.

4fill in blank

hard

Fill both blanks to create a dictionary comprehension that maps each word to its length if the length is greater than 3.

SciPy

{word: [1] for word in words if [2]

Drag options to blanks, or click blank then click option'

Alen(word)

Blen(word) > 3

Cword.startswith('a')

Dword

Attempts:

3 left

💡 Hint

Common Mistakes

Using the word itself as the value instead of its length.

Using a wrong condition like startswith instead of length check.

Not using a condition at all.

The dictionary comprehension maps each word to its length using len(word) and filters words with length greater than 3 using len(word) > 3.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that maps uppercase keys to values where values are greater than zero.

SciPy

result = [1]: [2] for [3], [2] in data.items() if [2] > 0}

Drag options to blanks, or click blank then click option'

Ak.upper()

Bv

Ck

Dvalue

Attempts:

3 left

💡 Hint

Common Mistakes

Using the wrong variable names in the loop.

Not converting keys to uppercase.

Using incorrect variable names for values.

The comprehension uses k.upper() as keys, v as values, and iterates over k, v in data.items(). It filters values greater than zero.

Practice

1. Which of the following statements about scipy.sparse.linalg.cg is true?

easy

A. cg is slower than gmres for all matrices.

B. cg can solve any linear system regardless of matrix properties.

C. cg uses dense matrix methods internally.

D. cg requires the matrix to be symmetric and positive definite.

Solution

1. Step 1: Understand the requirements of cg

   The conjugate gradient method (cg) is designed for symmetric positive definite matrices only.

2. Step 2: Compare with other options

   cg cannot solve any matrix (B is wrong), it is often faster than gmres for suitable matrices (A is wrong), and it uses sparse methods, not dense (D is wrong).

3. Final Answer:

   cg requires the matrix to be symmetric and positive definite. -> Option D

4. Quick Check:

   cg needs symmetric positive definite matrix [OK]

Hint: Remember: CG needs symmetric positive definite matrices [OK]

Common Mistakes:

• Thinking CG works for any matrix
• Confusing CG with GMRES
• Assuming CG uses dense matrix methods

2. Which is the correct way to import the GMRES solver from SciPy?

easy

A. import scipy.linalg.gmres

B. from scipy.sparse.linalg import gmres

C. from scipy.linalg import gmres

D. import gmres from scipy.sparse

Solution

1. Step 1: Recall the module location of GMRES

   The GMRES solver is in scipy.sparse.linalg, so it must be imported from there.

2. Step 2: Check the import syntax

   Correct Python import syntax for a function is from module import function. from scipy.sparse.linalg import gmres matches this and the correct module.

3. Final Answer:

   from scipy.sparse.linalg import gmres -> Option B

4. Quick Check:

   Correct import syntax and module [OK]

Hint: Import gmres from scipy.sparse.linalg using 'from ... import' [OK]

Common Mistakes:

• Importing from scipy.linalg instead of sparse.linalg
• Using incorrect import syntax
• Trying to import gmres directly from scipy.sparse

3. What will be the output of the following code snippet?

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

A = diags([1, 2, 1], [-1, 0, 1]).toarray()
b = np.array([4, 6, 4])
x, info = cg(A, b)
print(np.round(x, 2))

medium

A. [1. 2. 1.]

B. [2. 1. 2.]

C. [0. 0. 0.]

D. Error due to matrix not positive definite

Solution

1. Step 1: Analyze the matrix and vector

   The matrix A is tridiagonal with diagonals [1,2,1], which is symmetric positive definite. Vector b is [4,6,4].

2. Step 2: Solve using conjugate gradient

   Using cg, the solution x satisfies Ax = b. The solution is approximately [1, 2, 1].

3. Final Answer:

   [1. 2. 1.] -> Option A

4. Quick Check:

   cg solves Ax=b with symmetric positive definite A [OK]

Hint: Check matrix symmetry and positive definiteness before cg [OK]

Common Mistakes:

• Assuming cg fails on this matrix
• Confusing the solution vector with b
• Not rounding output before comparing

4. Identify the error in this code using cg solver:

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

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

medium

A. Matrix A is not symmetric positive definite, so cg will fail.

B. Vector b has wrong shape.

C. cg function is not imported correctly.

D. No error; code runs fine.

Solution

1. Step 1: Check matrix properties

   Matrix A = [[0,1],[1,0]] is symmetric but not positive definite (its eigenvalues are 1 and -1).

2. Step 2: Understand cg requirements

   The conjugate gradient method requires A to be symmetric positive definite. Since A is not, cg will fail or not converge properly.

3. Final Answer:

   Matrix A is not symmetric positive definite, so cg will fail. -> Option A

4. Quick Check:

   cg needs symmetric positive definite matrix [OK]

Hint: Check matrix eigenvalues before using cg [OK]

Common Mistakes:

• Ignoring matrix definiteness
• Assuming cg works for any symmetric matrix
• Thinking vector shape causes error

5. You have a large sparse matrix that is not symmetric positive definite. Which solver should you use to solve Ax = b efficiently?

hard

A. Use cg because it is always faster.

B. Convert matrix to dense and use direct solver.

C. Use gmres because it works for general matrices.

D. Use cg after transposing the matrix.

Solution

1. Step 1: Identify matrix properties

   The matrix is large, sparse, and not symmetric positive definite, so cg is not suitable.

2. Step 2: Choose appropriate solver

   gmres can handle general matrices efficiently without requiring symmetry or positive definiteness.

3. Step 3: Avoid dense conversion

   Converting to dense wastes memory and time, so it is not efficient.

4. Final Answer:

   Use gmres because it works for general matrices. -> Option C

5. Quick Check:

   gmres handles general sparse matrices [OK]

Hint: Use gmres for non-symmetric or indefinite sparse matrices [OK]

Common Mistakes:

• Trying to use cg on non-symmetric matrices
• Converting sparse to dense unnecessarily
• Thinking transposing fixes definiteness