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Chapter 13: Problem 41

The matrix in Newton's method is the Jacobian: $$J=\left[\begin{array}{ll}\partial g / \partial x & \partial g / \partial y \\\ \partial h / \partial x & \partial h / \partial y\end{array}\right] \quad \text { and } \quad J\left[\begin{array}{l}\Delta x \\ \Delta y\end{array}\right]=\left[\begin{array}{l}-g_{n} \\\\-h_{n}\end{array}\right]$$ Find \(J\) and \(\Delta x\) and \(\Delta y\) for \(g=e^{x}-1, h=e^{y}+x\).

Short Answer

Expert verified
\(J = \begin{bmatrix} e^x & 0 \\ 1 & e^y \end{bmatrix}\), \(\Delta x = \frac{-g_n}{e^x}, \Delta y = \frac{-h_n + \frac{g_n}{e^x}}{e^y}\)."

Step by step solution

01

Calculate the Partial Derivatives for g

The function given is \(g(x, y) = e^x - 1\). The partial derivatives are:- \(\frac{\partial g}{\partial x} = e^x\)- \(\frac{\partial g}{\partial y} = 0\). This is because \(g\) does not depend on \(y\).
02

Calculate the Partial Derivatives for h

The function given is \(h(x, y) = e^y + x\). The partial derivatives are:- \(\frac{\partial h}{\partial x} = 1\)- \(\frac{\partial h}{\partial y} = e^y\).
03

Form the Jacobian Matrix J

Using the partial derivatives calculated:\[J = \begin{bmatrix}\frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \\frac{\partial h}{\partial x} & \frac{\partial h}{\partial y}\end{bmatrix} = \begin{bmatrix}e^x & 0 \1 & e^y \end{bmatrix}\]
04

Set Up the System of Linear Equations

The equation given is:\[\begin{bmatrix}e^x & 0 \1 & e^y \end{bmatrix}\begin{bmatrix}\Delta x \\Delta y\end{bmatrix} = \begin{bmatrix}-g_n \-h_n\end{bmatrix}\].
05

Solve for \(\Delta x\)

From the equation \(e^x \Delta x = -g_n\), solve for \(\Delta x\):\[\Delta x = \frac{-g_n}{e^x}\].
06

Solve for \(\Delta y\)

From the equation \(e^y \Delta y + \Delta x = -h_n\), substitute \(\Delta x = \frac{-g_n}{e^x}\):\[e^y \Delta y = -h_n - \frac{-g_n}{e^x}\].Solve for \(\Delta y\):\[\Delta y = \frac{-h_n + \frac{g_n}{e^x}}{e^y}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Method
Newton's Method is an iterative numerical technique used to find approximate solutions to complex equations. It is particularly useful for solving nonlinear equations where analytical solutions might be difficult to obtain.
Here is how it generally works:
  • Initially, an approximation is made to the root or solution of a function.
  • Then, the method uses derivatives to refine this approximation.
  • Gradually, the solution converges to a more accurate estimate.
In a multidimensional context, like systems of equations, it utilizes the Jacobian matrix to adjust the variables incrementally closer to the solution with each iteration.
Partial Derivatives
Partial Derivatives are derivatives of functions of multiple variables with respect to one variable at a time, holding the other variables constant. This concept helps understand how a function changes as each input variable is altered individually.
For example, if a function is given by:
  • \( g(x, y) = e^x - 1 \), then:
    • \( \frac{\partial g}{\partial x} = e^x \) - indicating the rate of change of \( g \) with respect to \( x \).
    • \( \frac{\partial g}{\partial y} = 0 \) - showing no change with respect to \( y \) since \( y \) is not present in the expression.
  • \( h(x, y) = e^y + x \), then:
    • \( \frac{\partial h}{\partial x} = 1 \) - the rate of change of \( h \) with respect to \( x \).
    • \( \frac{\partial h}{\partial y} = e^y \) - the change concerning \( y \).
These derivatives are foundational for constructing the Jacobian matrix.
Linear Equations
Linear Equations form a part of mathematical models that describe a straight line in algebraic terms. In the context of systems of equations, they appear in generalized forms involving matrices, which can be solved to find variable values.
  • Each equation in the system can be represented with a matrix equation: \( A\mathbf{x} = \mathbf{b} \), where \( A \) is a matrix, \( \mathbf{x} \) is a vector of variables, and \( \mathbf{b} \) is a vector of constants.
  • For instance, the system given by our Jacobian analysis can be structured as:
  • \[ \begin{bmatrix} e^x & 0 \ 1 & e^y \end{bmatrix} \begin{bmatrix} \Delta x \ \Delta y \end{bmatrix} = \begin{bmatrix} -g_n \ -h_n \end{bmatrix} \]
  • Solving this set of linear equations requires using techniques such as substitution or matrix operations.
Functions
Functions are the backbone of mathematical analysis. They express relationships between one set of variables (inputs) and another set (outputs). Understanding functions is crucial as they are used to model a wide variety of real-world phenomena.
  • A function like \( g(x, y) = e^x - 1 \) is defined for each \( x \) and \( y \) within a domain and outputs results based on these inputs.
  • Similarly, \( h(x, y) = e^y + x \) combines the effect of both variables simultaneously, showing the collaborative nature of functions in multivariable problems.
  • Functions help in defining systems of equations that can later be analyzed or solved using tools such as derivatives and matrices.
Matrix Algebra
Matrix Algebra is a key element in modern mathematical computation, essential for solving systems of equations, particularly when they involve multiple variables. A matrix is a rectangular array of numbers, and understanding how to manipulate these arrays is crucial.
  • In our exercise, the Jacobian matrix \( J \) is formed using partial derivatives to show small changes in functions relative to their input variables.
  • By setting up matrices such as \[ \begin{bmatrix} e^x & 0 \ 1 & e^y \end{bmatrix} \], we can model complex systems in a manageable form.
  • The Matrix Algebra involves operations like addition, subtraction, and multiplication, as well as the inversion of matrices, which are crucial for finding the solution to sets of linear equations.
Mastering matrices allows for efficient computation and analysis, facilitating solutions to otherwise cumbersome problems.

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Most popular questions from this chapter

At the absolute minimum of \(f(x, y)\), the derivatives __________ are zero. If this point happens to fall on the curve \(g(x, y)=k\) then the equations \(f_{x}=\lambda g_{x}\) and \(f_{y}=\lambda g_{y}\) hold with \(\lambda=\) _____________.

Find the velocity \(v\) and the tangent vector \(T\). Then compute the rate of change \(d f / d t=\operatorname{grad} f \cdot \mathbf{v}\) and the slope \(d f / d s=\operatorname{grad} f \cdot \mathbf{T}\). \(f=x y \quad x=t^{2}+1 \quad y=3\)

If \(x=x(t, u, v)\) and \(y=y(t, u, v)\) and \(z=z(t, u, v),\) find the \(t\) derivative of \(f(x, y, z)\)

Allow inequality constraints, optional but good. Find the minimum of \(f=3 x+5 y\) with the constraints \(g=\) \(x+2 y=4\) and \(h=x \geqslant 0\) and \(H=y \geqslant 0,\) using equations like (7). Which multiplier is zero?

If a train approaches a crossing at \(60 \mathrm{mph}\) and a car approaches (at right angles) at \(45 \mathrm{mph}\), how fast are they coming together? (a) Assume they are both 90 miles from the crossing. (b) Assume they are going to hit.
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