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?

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);
}
$$```$$
``````

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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acegen – Newton iteration method without multiplier in AceFem

Is there a way in AceFem to use a Newton iteration scheme without using the multiplier λ or the time step t? For example, when trying to calculate a solution for a nonlinear equation, no multiplier is needed.
For the problem I'm facing, I try to use AceFem to compute the geometry of a surface composed of surface patches. Each surface field has a finite element that has three degrees of freedom in each node. These degrees of freedom are twist vectors, and with their variation, the shapes of patches can be changed. To determine the values ​​of twist vectors, I defined the energy potential in the finite element, which must be minimized. This potential is composed of the mean curvature in combination with the Gaussian curvature (2H ^ 2-2K). Node positions and normal vectors in nodes are always fixed, so no constraints are defined in the analysis (even twist vectors can not be defined as constraints because they are all unknown in each node).

Any help is welcome. Thanks. Tomo