np.einsum() for efficient computation in NumPy - Step-by-Step Execution

Concept Flow - np.einsum() for efficient computation

Input Arrays

↓

Define Subscripts

↓

np.einsum() Parses Subscripts

↓

Perform Summation/Product

↓

Return Result Array

np.einsum takes input arrays and a subscript string to specify element-wise operations and summations, then computes the result efficiently.

Execution Sample

NumPy

import numpy as np
A = np.array([[1,2],[3,4]])
B = np.array([[5,6],[7,8]])
result = np.einsum('ij,ij->ij', A, B)

Computes element-wise multiplication of two 2x2 arrays using np.einsum.

Execution Table

Step	Subscript Parsing	Operation	Intermediate Result	Output
1	'ij,ij->ij'	Parse indices: i,j for both arrays, output i,j	Arrays ready for element-wise multiply	None yet
2	'ij,ij->ij'	Element-wise multiplication	[[5,12],[21,32]]	None yet
3	No summation since output is 'ij'	Keep shape (2,2)	[[5,12],[21,32]]	[[5,12],[21,32]]

💡 Completed element-wise multiplication with output shape (2,2) as specified by 'ij' in subscript.

Variable Tracker

Variable	Start	After Step 2	Final
A	[[1,2],[3,4]]	[[1,2],[3,4]]	[[1,2],[3,4]]
B	[[5,6],[7,8]]	[[5,6],[7,8]]	[[5,6],[7,8]]
result	None	[Intermediate: element-wise multiply]	[[5,12],[21,32]]

Key Moments - 2 Insights

Why does the output keep the same shape as inputs?

Because the subscript 'ij,ij->ij' tells np.einsum to multiply elements with matching indices and keep the indices 'ij' in output, so no summation happens (see execution_table step 3).

What if we change the output subscript to 'i'?

np.einsum will sum over the 'j' index, reducing the output to 1D array with shape matching 'i'. This is a summation step different from element-wise multiply (not shown in current table).

Visual Quiz - 3 Questions

Look at the execution table, what is the intermediate result after step 2?

A [[1,2],[3,4]]

B [[6,8],[10,12]]

C [[5,12],[21,32]]

D None

💡 Hint

Check the 'Intermediate Result' column at step 2 in execution_table.

At which step does np.einsum parse the subscripts?

A Step 1

B Step 2

C Step 3

D No parsing step

💡 Hint

Look at the 'Subscript Parsing' column in execution_table.

If the output subscript changes to 'i', what happens to the output shape?

A Output shape becomes (2,2)

B Output shape becomes (2,)

C Output shape becomes (1,2)

D Output shape becomes scalar

💡 Hint

Refer to key_moments explanation about summation over 'j' index.

Concept Snapshot

np.einsum(subscripts, *arrays) computes sums/products using index notation.
Subscripts specify input and output indices.
Matching indices are multiplied element-wise.
Indices not in output are summed over.
Efficient for complex array operations without loops.

Full Transcript

np.einsum is a powerful numpy function that lets you specify how to multiply and sum elements of arrays using index labels. You write a string like 'ij,ij->ij' to tell it to multiply elements with matching i and j indices and keep the output shape the same. The function parses this string, multiplies elements accordingly, and sums over indices not in the output. This example showed element-wise multiplication of two 2x2 arrays, resulting in a 2x2 output. Changing the output indices can cause summation and change the output shape. This method is efficient and avoids explicit loops.