There are n number of intermediate train stations between point X and Y. Count the number of different ways trains can be arranged to stop at s stations such that no two stations are next to each other. So in this article, we will explain every possible approach to find out the number of stopping stations. Looking at the problem, we can find that we need to find combinations by which trains can be stopped at s number of stations.Approaches to Solve the ProblemLet's take an example that there are eight intermediate stations and we need to find the ways ... Read More

The linear function named scipy.linalg.solve_toeplitz is used to solve the Toeplitz matrix equation. The form of this function is as follows −scipy.linalg.solve_toeplitz(c_or_cr, b, check_finite=True)This linear function will solve the equation Tx = b for x where T is the Toeplitz matrix.ParametersBelow are given the parameters of the function scipy.linalg.solve_toeplitz()c_or_cr− array_like or tuple of (array_like, array_like)This parameter is the vector c or tuple of arrays (c, r). Despite the actual shape of c, it will always be converted to a one-dimensional array. If r is not given, the assumption made is r = conjugate(c). Below are given two cases −    ... Read More

In this article we are going to find number of solution of equation n = x + n ⊕ x, i.e we need to find number of values of x possible with given value n such that n = x + n ⊕ x where ⊕ represents XOR operation.Now we will discuss the complete information regarding number of solutions of n = x + n ⊕ x with an appropriate examples.Brute Force MethodWe can simple use brute force approach in order to find number of solution, i.e for given value of n we apply every integer value of x starting ... Read More

In this article, we will explain the approaches to find the number of solutions for the equation x+y+z

In this article, we will describe the important information on solving the number of sinks nodes in a graph. We have a Directed Acyclic Graph with N nodes (1 to N) and M edges in this problem. The goal is to find how many sink nodes are there in the given graph. A sink node is a node that does not produce any outgoing edges. So here is a simple example −Input : n = 4, m = 2 Edges[] = {{2, 3}, {4, 3}} Output : 2Simple Approach to Find the SolutionIn this approach, we will go through ... Read More

In this article, we will provide complete information to determine the number of siblings of a given node in the n-ary tree. We need to find the node's siblings with the value of key as given by the user; if it is non, then output as -1. There is only one approach that we can use −Simple ApproachIn this approach, we will go through all the nodes and check if a child has the same value as the user. If it exists, we answer a number of children - 1(the given value).Example #include using namespace std; class Node { ... Read More

The linear function named scipy.linalg.solveh_triangular is used to solve the banded matrix equation. In the below given example we will be solving the triangular system ax = b where −$$\mathrm{a} = \begin{bmatrix} 3 & 0 & 0 & 0\ 2 & 1 & 0 & 0\ 1 &0 &1 &0 \ 1& 1& 1& 1 \end{bmatrix};\; \mathrm{b} =\begin{bmatrix} 1\ 2\ 1\ 2 \end{bmatrix}$$Examplefrom scipy.linalg import solve_triangular import numpy as np a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 1]]) b = np.array([1, 2, 1, 2]) x = solve_triangular(a, b, lower=True) print (x)Outputarray([ 0.33333333, 1.33333333, 0.66666667, -0.33333333])

The linear function named scipy.linalg.solve_triangular is used to solve the triangular matrix e8quation. The form of this function is as follows −scipy.linalg.solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False, overwrite_b=False, debug=None, check_finite=True)This linear function will solve the equation ax = b for x where a is a triangular matrix.P ParametersBelow are given the parameters of the function scipy.linalg.solve_triangular() −a− (M, M) array_likeThis parameter represents the triangular matrix.b− (M, ) or (M, N)array_likeThis parameter represents the right-hand side matrix in the equation ax = b.lower− bool, optionalBy using this parameter, we will be able to use only the data that is contained in the ... Read More

The linear function named scipy.linalg.solveh_banded is used to solve the banded matrix equation. In the below given example we will be solving the circulant system Cx = b −Examplefrom scipy.linalg import solve_circulant, solve, circulant, lstsq import numpy as np c = np.array([2, 2, 4]) b = np.array([1, 2, 3]) solve_circulant(c, b)Output array([ 0.75, -0.25, 0.25])ExampleLet’s see a singular example, it will raise an LinAlgError −from scipy.linalg import solve_circulant, solve, circulant, lstsq import numpy as np c = np.array([1, 1, 0, 0]) b = np.array([1, 2, 3, 4]) solve_circulant(c, b)Output -------------------------------------------------------------------------- LinAlgError Traceback (most recent call last) in ... Read More

The linear function named scipy.linalg.solve_circulant is used to solve the circulant matrix equation. The form of this function is as follows −scipy.linalg.solve_circulant(c, b, singular=’raise’, tol=None, caxis=-1, baxis=0, outaxis=0)This linear function will solve the equation Cx = b for x where C is a Circulant matrix associated with the vector c.The circulant matrix equation is solved by doing division in Fourier space as follows −x = ifft(fft(b) / fft(c))Here fft is the fast Fourier transform and ifft is the inverse fast Fourier transform.ParametersBelow are given the parameters of the function scipy.linalg.solve_circulant() −c− array_likeThis parameter represents the coefficient of the circulant matrix.b− ... Read More