Solve the tensor equation in Python

To solve tensor equations in Python, use the numpy.linalg.tensorsolve() method. This function solves the tensor equation by finding the solution where all indices of the unknown tensor are summed over in the product with the coefficient tensor.

Syntax

numpy.linalg.tensorsolve(a, b, axes=None)

Parameters

The function accepts the following parameters:

• a − Coefficient tensor of shape b.shape + Q, where Q is a tuple representing the shape of the rightmost indices
• b − Right-hand tensor that can be of any shape
• axes − Axes in tensor 'a' to reorder before inversion (optional, default is None)

Basic Example

Let's create a simple tensor equation and solve it ?

import numpy as np

# Create coefficient tensor using identity matrix
coeff_tensor = np.eye(2*3*4)
coeff_tensor.shape = (2*3, 4, 2, 3, 4)

# Create right-hand tensor with random values
rhs_tensor = np.random.randn(2*3, 4)

print("Coefficient tensor shape:", coeff_tensor.shape)
print("Right-hand tensor shape:", rhs_tensor.shape)

# Solve the tensor equation
solution = np.linalg.tensorsolve(coeff_tensor, rhs_tensor)
print("Solution shape:", solution.shape)
print("Solution:\n", solution)

Coefficient tensor shape: (6, 4, 2, 3, 4)
Right-hand tensor shape: (6, 4)
Solution shape: (2, 3, 4)
Solution:
[[[-0.09878127 -1.13343506 -0.18818199 -0.06808653]
  [ 0.93755673  1.55346988 -1.12108994 -1.52624341]
  [ 0.56097842 -1.21092616 -0.96901414 -0.91459123]]

 [[ 0.64827935  1.04503893  0.67154924  0.33616993]
  [-0.88831754  0.99542901  1.58906859 -1.04906518]
  [-1.18002076 -1.46936715 -0.41749688 -0.29938698]]]

Understanding How It Works

The tensorsolve function works by reshaping the coefficient tensor and solving the resulting linear system ?

import numpy as np

# Simple 2x2 example for clarity
a = np.array([[3, 1], [1, 2]])
b = np.array([9, 8])

# Using tensorsolve
solution_tensor = np.linalg.tensorsolve(a, b)
print("Tensorsolve solution:", solution_tensor)

# Verification using regular linear solve
solution_linear = np.linalg.solve(a, b)
print("Linear solve solution:", solution_linear)

# Verify the solution
verification = np.dot(a, solution_tensor)
print("Verification (should equal b):", verification)
print("Original b:", b)

Tensorsolve solution: [2. 3.]
Linear solve solution: [2. 3.]
Verification (should equal b): [9. 8.]
Original b: [9. 8.]

Complex Tensor Example

Here's an example with higher-dimensional tensors ?

import numpy as np

# Create a 3D coefficient tensor
a = np.random.rand(4, 4, 4, 4)
# Make it well-conditioned by adding identity structure
a = a + 0.1 * np.eye(4).reshape(4, 4, 1, 1)

# Create right-hand side tensor
b = np.random.rand(4, 4)

try:
    # Solve the tensor equation
    x = np.linalg.tensorsolve(a, b)
    print("Solution found successfully!")
    print("Solution shape:", x.shape)
    print("First few elements:", x.flat[:6])
    
    # Verify solution using tensordot
    verification = np.tensordot(a, x, axes=x.ndim)
    print("Verification error:", np.allclose(verification, b))
    
except np.linalg.LinAlgError as e:
    print("Linear algebra error:", e)

Solution found successfully!
Solution shape: (4, 4)
First few elements: [ 0.79394609  0.51718634 -1.67309843  0.87626007  0.54738235 -0.68524653]
Verification error: True

Key Points

• The coefficient tensor must be invertible for a unique solution to exist
• The shape relationship prod(Q) == prod(b.shape) must be satisfied
• Use np.tensordot() to verify your solution
• For ill-conditioned problems, consider using np.linalg.lstsq() instead

Conclusion

The numpy.linalg.tensorsolve() function provides an efficient way to solve tensor equations in Python. It's particularly useful for multi-dimensional linear algebra problems where traditional matrix solving methods are insufficient.