Question

An \(n \times n\) linear system of equations \(A \mathbf{x}=\mathbf{b}\) is modified in the following manner: for each \(i, i=1, \ldots, n\), the value \(b_{i}\) on the right-hand side of the \(i\) th equation is replaced by \(b_{i}-x_{i}^{3}\). Obviously, the modified system of equations (for the unknowns \(x_{i}\) ) is now nonlinear. (a) Find the corresponding Jacobian matrix. (b) Given that \(A\) is strictly diagonally dominant with positive elements on its diagonal, state whether or not it is guaranteed that the Jacobian matrix at each iterate is nonsingular. (c) Suppose that \(A\) is symmetric positive definite (not necessarily diagonally dominant) and that Newton's method is applied to solve the nonlinear system. Is it guaranteed to converge?

Short answer

Question: Determine the Jacobian matrix of the modified linear system \(Ax = b - x^3\), where \(A\) is an \(n \times n\) matrix. Analyze whether this matrix is nonsingular when matrix \(A\) is strictly diagonally dominant with positive diagonal elements. Finally, check if Newton's method converges when matrix \(A\) is symmetric positive definite and the modified system is nonlinear. Answer: The Jacobian matrix of the modified system is given by: $J = \begin{pmatrix} a_{11} + 3x_1^2 & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22}+ 3x_2^2 & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn}+ 3x_n^2 \end{pmatrix}$ The Jacobian matrix is guaranteed to be nonsingular when matrix \(A\) is strictly diagonally dominant with positive diagonal elements. However, it is hard to conclude whether Newton's method converges for the nonlinear system for the case when matrix \(A\) is symmetric positive definite (not necessarily diagonally dominant) since the convergence depends on specific properties and the structure of the matrix, as well as the initial guess.

Step-by-step solution

Modified System

\(A \mathbf{x} = \mathbf{b} - \mathbf{x}^3\) Now, we need to find the Jacobian matrix of this modified system. The Jacobian matrix of a system of functions \(F(x)\) with respect to the variables \(x_i\) is given by: \(J=\frac{dF_{i}}{dx_{j}}\) Here, \(F_i(\textbf{x}) = \sum_{j=1}^n a_{ij}x_j - b_i + x_i^3\). So the Jacobian matrix will be: $ J = \begin{pmatrix}\frac{dF_{1}}{dx_{1}} & \frac{dF_{1}}{dx_{2}} & \cdots & \frac{dF_{1}}{dx_{n}}\\ \frac{dF_{2}}{dx_{1}} & \frac{dF_{2}}{dx_{2}} & \cdots & \frac{dF_{2}}{dx_{n}}\\ \vdots & \vdots & \ddots & \vdots \\ \frac{dF_{n}}{dx_{1}} & \frac{dF_{n}}{dx_{2}} & \cdots & \frac{dF_{n}}{dx_{n}} \end{pmatrix}$ Since \(\frac{dF_i}{dx_j} = a_{ij}\) for \(i \neq j\) and \( \frac{dF_i}{dx_i} = a_{ii} + 3x_i^2\), the Jacobian matrix is:

Jacobian Matrix

$J = \begin{pmatrix} a_{11} + 3x_1^2 & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22}+ 3x_2^2 & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn}+ 3x_n^2 \end{pmatrix}$ #b) Analyzing nonsingularity of Jacobian matrix# We need to check whether it is guaranteed that the Jacobian matrix is nonsingular, given that the original matrix \(A\) is strictly diagonally dominant with positive elements on its diagonal. A matrix is nonsingular if its determinant is non-zero. We have to find the determinant of the Jacobian matrix:

Determinant of Jacobian Matrix

\(det(J) \neq 0\) for all \(x_i\) By observing the Jacobian, we can see that the diagonal elements of the Jacobian are strictly larger than the corresponding diagonal elements of Matrix \(A\) since \(3x_i^2 \geq 0\) for all \(x_i\). So, the Jacobian matrix is diagonally dominant with positive diagonal elements. A diagonally dominant matrix is nonsingular, so the determinant of the Jacobian matrix is nonzero, and we can conclude that it is guaranteed that the Jacobian matrix at each iterate is nonsingular. #c) Analyzing convergence of Newton's method# Finally, we need to check the convergence of the Newton's method for solving the nonlinear system if the given matrix \(A\) is symmetric positive definite. Newton's method converges for a linear system if the matrix is nonsingular. From part (b), we have already seen that the Jacobian matrix at each iterate is nonsingular given that matrix \(A\) is diagonally dominant with positive elements on its diagonal. However, we do not have a direct guarantee for symmetric positive definite (i.e., not necessarily diagonally dominant). In that case, it is hard to conclude whether Newton's method is guaranteed to converge for the nonlinear system where A is symmetric positive definite. The convergence can depend on specific properties and the structure of the matrix, as well as the initial guess.

Key concepts

Nonlinear System of Equations

A nonlinear system of equations occurs when at least one equation in the system is non-linear. This means that the equations do not represent straight lines when graphed. Instead, they could represent curves or complex shapes depending on the nature of the nonlinearity. In this scenario, the modification of the system involves replacing the term on the right side of the equations with non-linear expressions.
For example, in the given exercise, the modification from a linear to nonlinear system is seen when each element in the vector \( \mathbf{b} \) is replaced by \( b_i - x_i^3 \). The addition of the \( x_i^3 \) term creates nonlinearity in the equations.

• Linear equations often allow for straightforward calculations and solutions.
• Nonlinear equations can be more complex and require iterative approaches for solutions.

Nonlinear systems can describe many real-world phenomena more accurately than linear systems because they can accommodate a wider variety of interactions and dependencies between variables.

Newton's Method

Newton's Method is a popular technique used to solve systems of nonlinear equations. It is an iterative approach, which means it builds successive approximations to the solution by updating guesses based on specific calculations. The main idea is to start from an initial guess and refine it to get closer to the actual solution.
Newton's Method relies heavily on the Jacobian matrix, which provides information about the gradients of the functions involved. This matrix must be calculated at each step to update the guesses effectively. When this matrix is nonsingular, or has a nonzero determinant, it indicates that the matrix can invert, which is necessary for finding new approximations.

• Newton's method converges quickly for solutions when starting guesses are relatively close to the actual solution.
• If the starting point is far from the solution, Newton's method may not converge, or it could converge to an incorrect result.

Choosing a good initial guess and ensuring the Jacobian matrix remains nonsingular throughout the iterations are crucial for the success of Newton's Method.

Diagonal Dominance

A matrix is diagonally dominant if the absolute value of each diagonal element in a matrix is larger than the sum of the absolute values of all the other (non-diagonal) elements in that same row. Diagonal dominance is a useful property in solving linear and nonlinear systems of equations because it suggests numerical stability and well-conditioned problems.
In the context of the given exercise, the concept of diagonal dominance helps ensure the Jacobian matrix remains nonsingular. This condition is met when the modified matrix has diagonal elements that are strictly larger than the sum of the off-diagonal elements.

• Strict diagonal dominance provides strong guarantees for the nonsingularity of matrices, which aids in converging solutions.
• It is closely related to the stability of solving systems analytically or numerically.

Being aware of diagonal dominance allows one to predict the behavior of a system better and to diagnose potential issues with convergence or solution accuracy.

Symmetric Positive Definite Matrices

A symmetric positive definite matrix is an important concept in both linear algebra and machine learning. A matrix is symmetric positive definite if it is symmetric (its transpose is equal to itself) and positive definite (all its eigenvalues are positive).
In the realm of numerical solutions, symmetric positive definite matrices occur frequently because they possess nice properties that simplify calculations. However, when using Newton's method for a nonlinear system, as in the exercise, symmetric positive definiteness does not guarantee convergence like diagonal dominance might.

• The symmetric positive definite property ensures that solutions to linear equations involving such matrices are unique and stable.
• These matrices don't necessarily ensure convergence for nonlinear systems without additional properties like diagonal dominance.

Understanding when and why symmetric positive definite matrices are useful can lead to better implementations and insights when working with complex systems.