Slicing and Indexing¶

Accessing elements and submatrices in sparse matrices.

Element Access¶

1. Single Element¶

```python from scipy import sparse

def main(): A = sparse.csr_matrix([[1, 0, 2], [0, 3, 0], [4, 0, 5]])

print(f"A[0, 0] = {A[0, 0]}")
print(f"A[0, 2] = {A[0, 2]}")
print(f"A[1, 0] = {A[1, 0]}")  # Zero element

if name == "main": main() ```

Row and Column Slicing¶

1. Row Slicing (CSR efficient)¶

```python from scipy import sparse

def main(): A = sparse.csr_matrix([[1, 0, 2], [0, 3, 0], [4, 0, 5]])

row_0 = A[0, :]
rows_0_1 = A[0:2, :]

print("Row 0:")
print(row_0.toarray())
print()
print("Rows 0-1:")
print(rows_0_1.toarray())

if name == "main": main() ```

2. Column Slicing (CSC efficient)¶

```python from scipy import sparse

def main(): A = sparse.csc_matrix([[1, 0, 2], [0, 3, 0], [4, 0, 5]])

col_0 = A[:, 0]
cols_0_1 = A[:, 0:2]

print("Column 0:")
print(col_0.toarray())

if name == "main": main() ```

Fancy Indexing¶

1. Row Selection¶

```python from scipy import sparse import numpy as np

def main(): A = sparse.csr_matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12]])

rows = [0, 2, 3]
B = A[rows, :]

print("Selected rows [0, 2, 3]:")
print(B.toarray())

if name == "main": main() ```

2. Boolean Indexing¶

```python from scipy import sparse import numpy as np

def main(): A = sparse.csr_matrix([[1, 2], [3, 4], [5, 6]])

mask = np.array([True, False, True])
B = A[mask, :]

print("Boolean selected rows:")
print(B.toarray())

if name == "main": main() ```

Modifying Elements¶

1. Set Single Element¶

```python from scipy import sparse

def main(): A = sparse.lil_matrix((3, 3))

A[0, 0] = 1
A[1, 1] = 2
A[2, 2] = 3

print(A.toarray())

if name == "main": main() ```

2. LIL for Modification¶

```python from scipy import sparse

def main(): # CSR is inefficient for element-by-element modification # Use LIL, then convert

lil = sparse.lil_matrix((5, 5))
for i in range(5):
    lil[i, i] = i + 1

csr = lil.tocsr()
print(csr.toarray())

if name == "main": main() ```

Performance¶

1. Format Matters¶

Summary¶

• Use CSR for row operations
• Use CSC for column operations
• Use LIL for building/modifying
• Avoid element-by-element access on CSR/CSC

Exercises¶

Exercise 1. Create a \(10 \times 10\) sparse CSR matrix from the dense matrix where entry \((i, j) = i \cdot 10 + j\) if \(|i - j| \leq 1\), and 0 otherwise. Extract rows 3 through 6 using slicing and print the result as a dense array. Also extract the single element at position \((5, 4)\).

import numpy as np
from scipy import sparse

n = 10
dense = np.zeros((n, n))
for i in range(n):
    for j in range(max(0, i - 1), min(n, i + 2)):
        dense[i, j] = i * 10 + j

A = sparse.csr_matrix(dense)

rows_3_6 = A[3:7, :]
print("Rows 3-6:")
print(rows_3_6.toarray())

element = A[5, 4]
print(f"\nA[5, 4] = {element}")

Exercise 2. Build a \(100 \times 100\) sparse random matrix in CSR format (density 0.1, np.random.seed(7)). Use fancy indexing to select rows [0, 10, 20, 50, 99] and print the resulting submatrix shape and number of nonzeros. Then use boolean indexing to select all rows whose row sum exceeds 5.

import numpy as np
from scipy import sparse

np.random.seed(7)
A = sparse.random(100, 100, density=0.1, format='csr')

# Fancy indexing
selected_rows = A[[0, 10, 20, 50, 99], :]
print(f"Selected rows shape: {selected_rows.shape}")
print(f"Selected rows nnz: {selected_rows.nnz}")

# Boolean indexing
row_sums = np.array(A.sum(axis=1)).flatten()
mask = row_sums > 5
filtered = A[mask, :]
print(f"\nRows with sum > 5: {mask.sum()}")
print(f"Filtered shape: {filtered.shape}")

Exercise 3. Create a \(50 \times 50\) sparse matrix using LIL format. Set the diagonal entries to their row index (0 through 49), then set entries \((i, i+5)\) to -1 for valid indices. Convert to CSR, extract column 10 using CSC format (convert first), and print the nonzero entries of that column.

import numpy as np
from scipy import sparse

n = 50
lil = sparse.lil_matrix((n, n))
for i in range(n):
    lil[i, i] = i
    if i + 5 < n:
        lil[i, i + 5] = -1

csr = lil.tocsr()
csc = csr.tocsc()

col_10 = csc[:, 10]
nz_rows = col_10.nonzero()[0]
print("Column 10 nonzero entries:")
for r in nz_rows:
    print(f"  Row {r}: {col_10[r, 0]}")