# How to Translate Univariate Newton's Approximation to Multidimensional Systems?

• MHB
• Purplepixie
In summary, the conversation discusses the difficulty of translating the univariate Newton's approximation to the multidimensional case. The person has attempted to translate it but is struggling with the concept of inverse matrices in the context of a nx1 column vector. They also mention the use of Newton's approach to solve equations and the need to consider a matrix of derivatives in the n-variate case. The conversation ends with a comment on the potential for mathematics to be simplified with improved representational systems.
Purplepixie
Hello,
I am having difficulty in translating the univariate Newton's approximation {Xn = Xn-1 - [ f(Xn-1) / f'(Xn-1)]} into the multidimensional case. My multidimensional equation system is y = F.x where y and x are (nx1) column vectors and the coefficients matrix F is (nxn), so that (nx1) = (nxn).(nx1) = (nx1).

I have translated the univariate Newton's approximation to the n-variate case as:
x2 = x1 - (F.x1 / F'.x1) <=> x1 - (F.x1).Inv(F'.x1)

But F'.x1 is a nx1 column vector and has no inverse. I then thought that perhaps the x1's cancel out . But if so then we would have x2 = x1 - F.Inv(F') with the last term a nxn matrix, so (nx1) = (nx1) - (nxn), which is not possible.

Your assistance would be greatly appreciated!

(This is my first post incidentally, so pls excuse any breaches of protocol)

Purplepixie said:
Hello,
I am having difficulty in translating the univariate Newton's approximation {Xn = Xn-1 - [ f(Xn-1) / f'(Xn-1)]} into the multidimensional case. My multidimensional equation system is y = F.x where y and x are (nx1) column vectors and the coefficients matrix F is (nxn), so that (nx1) = (nxn).(nx1) = (nx1).

I have translated the univariate Newton's approximation to the n-variate case as:
x2 = x1 - (F.x1 / F'.x1) <=> x1 - (F.x1).Inv(F'.x1)

But F'.x1 is a nx1 column vector and has no inverse. I then thought that perhaps the x1's cancel out . But if so then we would have x2 = x1 - F.Inv(F') with the last term a nxn matrix, so (nx1) = (nx1) - (nxn), which is not possible.

Your assistance would be greatly appreciated!

(This is my first post incidentally, so pls excuse any breaches of protocol)

Hi Purplepixie, welcome to MHB!

First we have to establish what $F'$ is.
If it is a matrix that does not depend on any variable, then $F'$ is the matrix that contains only zeroes.
Consequently we won't really get any useful result.

Newton's approach aims to solve $f(x)=0$, and makes use of the fact that $f(x) \approx f(x_0) + f'(x)(x-x_0) \implies x_0\approx x - \frac{1}{f'(x)}\cdot f(x)$.
We might try to solve $F(x_1,\ldots,x_n)=(0,\ldots, 0)$ in a similar fashion. In this case $F$ is not a matrix, but a set of $n$ functions. And each function has $n$ parameters.
If we take the derivative, we don't have just 1 derivative, but instead we have the derivatives of $n$ functions with respect to each of the $n$ parameters.
The result is a matrix of $n\times n$ derivatives. Let's call it $DF(x)$ to denote that it's a matrix of functions that depend on $x$.
Now we can write Newton's approximation as $x^{(k+1)} = x^{(k)} - \Big(DF(x^{(k)})\Big)^{-1}\cdot F(x^{(k)})$, where $x^{(k)}$ denote the successive approximations of $x$.

Dear Klaas,
Thank you for your clear and succinct explanation. I can now see where I was wrong. So much of mathematics would be simplified if representational systems could be improved!
All the best,
PP

## 1. What is Multidimensional Newton Raphson?

Multidimensional Newton Raphson is a numerical method used for finding the roots of a system of nonlinear equations. It is an extension of the one-dimensional Newton Raphson method, which is used for finding the root of a single equation.

## 2. How does Multidimensional Newton Raphson work?

This method uses an iterative process to approximate the roots of a system of equations. It starts with an initial guess and then uses the Jacobian matrix, which contains the partial derivatives of the equations, to improve the guess in each iteration until a satisfactory solution is found.

## 3. What are the advantages of using Multidimensional Newton Raphson?

One of the main advantages of this method is its fast convergence rate, meaning it can find the roots of a system of equations in fewer iterations compared to other numerical methods. It is also relatively easy to implement and has a wide range of applications in various fields of science and engineering.

## 4. What are the limitations of Multidimensional Newton Raphson?

This method may fail to converge if the initial guess is too far from the actual root or if the Jacobian matrix is not well-behaved. It also requires the equations to be differentiable, which may not always be the case in real-world problems.

## 5. How is Multidimensional Newton Raphson different from other root-finding methods?

Unlike other methods, such as the bisection method or the secant method, Multidimensional Newton Raphson can find multiple roots of a system of equations simultaneously. It also has a higher convergence rate compared to these methods, making it a more efficient choice for solving complex systems of equations.

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