SciPy - solve_toeplitz() Function

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The scipy.linalg.solve_toeplitz() method efficiently solves linear systems Ax=b, where A is a Toeplitz matrix, using the Levinson-Durbin recursion. With respect to generic solvers, this method uses recursion to find solutions much faster and with less complexity.

Systems of equations having structured matrices, which appear very commonly in signal processing, time series, and control systems, are typically solved using this method. It is quite versatile in use since it accommodates the possibility of more than one right-hand side.

The technique utilizes Levinson-Durbin recursion in order to optimize efficiency for Toeplitz matrices. It is very compatible with other numerical methods and allows for both single and many right-hand sides.

solve_toeplitz() is often used in combination with eigenvalue decomposition (eigvals()) for stability analysis or matrix reconstruction (toeplitz()) for solution verification. Strong solutions and system insights are ensured with this combination.

Syntax

Following is the syntax of the SciPy solve_toeplitz() method −

.solve_toeplitz(c_or_cr, b, check_finite=True)

Parameters

Following are the parameters of solve_toeplitz() method

• c_or_cr (array_like) − The Toeplitz matrix A is represented by its first column c and optionally its first row r. If a tuple (c,r) is provided, the first element of c and r must be the same to ensure consistency.

• b array_like, shape (n,) or (n, k) − Input equation of Right-hand side.

• lower (array_like) − The matrix to multiply with Toeplitz matrix.

• check_finite (bool, optional)− If True, checks whether the input contains only finite numbers. Disabling this can improve performance but risks issues with invalid inputs.

Return Value

x (ndarray) − The solution to the linear system T. x=b, where x matches the dimensions of b.

Example 1

In the below example we have created a Toeplitz matrix using its first column.The function solves the system T.

import numpy as np
import scipy.linalg
from scipy.linalg import solve_toeplitz

# First column of the Toeplitz matrix
c = [2, 1, 0]

# Right-hand side vector
b = [4, 5, 6]

# Solve the system
x = scipy.linalg.solve_toeplitz(c, b)
print(x)

When we run above program, it produces following result

[2.0000000e+00 4.4408921e-16 3.0000000e+00]

Example 2

In this code, we defined the Toeplitz matrix using both its first column [3, 2, 1] and first row [3, 4, 5] and solved the system Ax=b using solve_toeplitz() method.

import numpy as np
from scipy.linalg import solve_toeplitz
# First column and first row of the Toeplitz matrix
c_or_cr = ([3, 2, 1], [3, 4, 5])

# Right-hand side vector
b = [9, 8, 7]

# Solve the system
x = solve_toeplitz(c_or_cr, b)
print(x)

Following is an output of the above code

[-1. -2.  4.]