## optimization – Newton method using automatic differentiation

I wrote a large Matlab code which solves partial differential equations. Now I would like to test the code on nonlinear problems, for which I need the Newton-Raphson iteration for systems of nonlinear algebraic equations.

How do I employ automatic differentiation in Matlab to use the Newton-Raphson iteration?

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## Newton Raphson question

Find an intersection of the curves y = x3-4x-5 and y = ex-4x-5 by selecting the starting point X0 = 3 with the Newton-Raphson method with an error of 10 ^ -3

I cannot solve the question. Can you help me?

## Beginners – Newton Raphson and polynomials in C.

I have the following code that defines:

1. A polynomial structure with some useful functions.
2. The Newton-Raphson algorithm for polynomials.

and calculated `sqrt(2)`.
What can i improve?

There are a few things I'm not sure about:

1. Is `size_t` A good data type for my purposes? I had to be very careful when I put the loop condition in the `eval` Function. Is this a good place to switch to signed arithmetic?

2. Is my definition of `my_nan` a good way to ensure portability when switching to another floating point type?

Things I am aware of:
I know that the Newtonian Raphson for polynomials does not require the explicit construction of a derived polynomial, and I am a little wasteful of memory.

``````#include
#include
#include
#include

typedef double real;
const real my_nan = ((real) 0.0) / ((real) 0.0);

typedef struct Polynomial {
real* coeffs;
size_t capacity;
} Polynomial;

Polynomial* create_poly(size_t capacity)
{
real* coeffs;

coeffs = malloc(capacity * sizeof(real));
if (coeffs == NULL) return NULL;

Polynomial* poly = malloc(sizeof(Polynomial));

if (poly == NULL) {
free(coeffs);
return NULL;
}
poly->coeffs = coeffs;
poly->capacity = capacity;

return poly;
}

void delete_poly(Polynomial* p)
{
free(p->coeffs);
free(p);
p = NULL;
}

size_t deg(const Polynomial*const p)
{
for (size_t i = p->capacity - 1; i > 0; i--) {
/* Here we actually want to compare reals exactly instead of |a - b| < eps */
if (p->coeffs(i) != 0.0) return i;
}
return 0;
}

void print(const Polynomial*const p)
{
size_t i;
for (i = 0; i < deg(p); ++i) {
printf("%f * x^%zu + ", p->coeffs(i), i);
}
printf("%f * x^%zun", p->coeffs(i), i);
}

real eval(const Polynomial*const p, real x)
{
/* Use Horner Scheme for evaluation */
size_t i = deg(p);
real res = p->coeffs(i);

for (; i-- > 0;) {
res = res * x + p->coeffs(i);
}
return res;
}

Polynomial* derive(const Polynomial*const p)
{
Polynomial* Dp = create_poly(p->capacity);
if (Dp == NULL) return NULL;

for (size_t i = 1; i < p->capacity; ++i) {
Dp->coeffs(i - 1) = ((real) i) * p->coeffs(i);
}
return Dp;
}

real newton_raphson_poly(const Polynomial*const p, real x0, real eps)
{
Polynomial* Dp = derive(p);
real x, prev = x0;
const int max_iter = 100;

for (int i = 0; i < max_iter; ++i) {
x = prev - eval(p, prev) / eval(Dp, prev);
if (fabs(x - prev) < eps) {
return x;
} else {
prev = x;
}
}

return my_nan;
}

int main()
{
Polynomial* p;
const real EPS = pow(10, -7);
p = create_poly(3);

p->coeffs(0) = -2;
p->coeffs(1) = 0;
p->coeffs(2) = 1;

printf("The result of sqrt(2) is given by the root ofn");
print(p);
printf("Its value is: %f n", newton_raphson_poly(p, 1.0, EPS));

delete_poly(p);
}
$$```$$
``````

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## Newton method and machine learning

There is some debate about why the Newtonian method is not widely used in machine learning. Instead, people tend to use a gradient descent.

• Some people claim that the Newton method is not used because it contains the second derivative. As? Indirectly? Why? Doesn't the Newton method neglect the second derivative?

• Is there a name for Newton's cubic convergence method?

• Can we say that Newton's method is a form of gradient descent?

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## Statistics – translation of the Newton Raphson code into the evaluation method

``````
require(pracma)     # to access Norm

x0 <- c(0.7, 0.3) # starting value for the iteration
data <- c(-0.1,0.3,48.7,-1.5,-0.1,-8.3,-0.3,7.3)
g <- c(1, 1)    # starting value for the 'while'
h <- matrix(0, 2, 2)    # command to create 2x2 matrix of 0s

while(Norm(g) > 1e-05) {
a <- x0(1)
b <- x0(2)
b2 <- b^2
x1 <- data - a
den <- b2 + x1^2
den2 <- den^2
g(1) <- 2 * sum(x1/den) # the gradient vector components
g(2) <- sum((x1^2 - b2)/(b * den))
h(1, 1) <- 2 * sum((x1^2 - b2)/den2)
h(1, 2) <- -4 * b * sum(x1/den2)
h(2, 1) <- h(1, 2)
h(2, 2) <- -5/b^2 - h(1, 1) # completes specification of the
xn <- x0 - solve(h) %*% cbind(g)    # Hessian matrix
x0 <- xn
}

cat("maximum likelihood estimate:", x0, "n") # the maximum-likelihood estimate
cat("gradient at the maximum:", g, "n")  # the gradient at the maximum
cat("eigenvalues/vectors of the hessian:n"); print(eigen(h))
``````

Hello, I have this code, which was taken from the MATLAB code from Applied Stochastic Modeling and Data Analysis by Byron JT Morgan (2008), and I want to be able to easily modify it so that instead of Newton-Raphson, an evaluation method is used instead. I understand that instead you have to take the expectation of the Hessian matrix and the result:

$$E (A ( theta)) = - frac {n} {2 beta ^ 2} begin {bmatrix} 1 & 0 \ 0 & 1 end {bmatrix}$$

Any help on how this would be translated into this R code would be appreciated.

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## [ Politics ] Open question: Why do liberals dispute the fact that Trump's intelligence is on a par with that of Isaac Newton, Nikola Tesla and Albert Einstein?

[Politics] Open question: Why do liberals dispute the fact that Trump's intelligence is on a par with that of Isaac Newton, Nikola Tesla and Albert Einstein?

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