## 110 Reputation

4 years, 155 days

## Nontrivial solution for large matrix...

Maple

Hi

I got an square matrix (70×70) from MATLAB (please download attached text file 1.txt). Following codes are used in MAPLE, but an unknown error is occurred. It seems that matrix is divided in two submatrices. Please hint me or run the codes to obtain fiirst three minimum real positive roots of the determinant.

Thank you for taking your time

Maple codes

Digits:=150:

LinearAlgebra:-Determinant(M):

fsolve(%,P=0..0.5)*100;

## Laplacian and Euler Lagrange equation...

Maple

Hi

I try to find a relation between EL and Lap(EL) in polar coordinate for one variable function w(r), where Lap is laplacian and EL is Euler Lagrange equation. Please check the Maple code and help me to do some manipulations to find a general relation (if any relation exists!).

In fact I need the inverse of Euler Lagrange equation to obtain f(r) for an arbitrary function g(r) in equation below

EL(f) = Lap(EL(g))

Or f=inverseEL(Lap(EL(g)))

Thank you for taking your time

 (1)

## Divergence or Gauss Theorem...

Maple

Hi

I want to write a code to show that

```restart;
with(VectorCalculus):
SetCoordinates(cartesian[x, y, z]):
g1 := proc (u1, u2, u3, s)
local N, u, n, intr1, intr2, intr3, R1, R2:
u := VectorField([u1, u2, u3]):
n := N/sqrt(add(N[k]^2, k = 1 .. 3)):
intr1 := solve(subs(z = 0, s), y):
intr2 := solve(subs(z = 0, y = 0, s)):
R1 := int(int(subs(z = solve(s, z)[1], u . n), y = intr1[1] .. intr1[2]), x = intr2[1] .. intr2[2], numeric):
intr1 := solve(s, z):
intr2 := solve(subs(z = 0, s), y):
intr3 := solve(subs(z = 0, y = 0, s)):
R2 := evalf(int(int(int(Del . u, z = intr1[1] .. intr1[2]), y = intr2[1] .. intr2[2]), x = intr3[1] .. intr3[2])):
print(R1, R2)
end proc```

It seems that different answers are obtained.

g1(x, 1, z, x^2+y^2+z^2-2);
6.664324407, -23.69537567

## Inverse of Euler-Lagrange equation...

Maple

Hi

In mathematics, the inverse problem for Lagrangian mechanics (Helmholtz inverse problem) is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function.

http://www.unilim.fr/pages_perso/loic.bourdin/Documents/bourdin-thesis2013.pdf

________________________________________________________________________

I need some hints or procedures (if it is possible) for similar (but a little more complex) problem:

1- Assume that you have one ordinary differential equation, ode1(r) in polar coordinate system (i.e. (r, theta)). The ODE is taken to be independent from theta (It is not a PDE).

2- Assume that "Euler" is an operator that gives the Euler-Lagrange equation, I need a procedure to calculate ode2(r) such that

1/(2r)*Euler (ode2(r)) -Laplacian (1/(2r)*Euler(ode1(r)))=0

It is obvious that we need inverse of Euler operator (say IE) to calculate ode2(r).

ode2(r) =IE( 2r*Laplacian (1/(2r)*Euler(ode1(r))))

I calculate ode2(r) for some simpler cases via trial and error method.

```s := proc (S)
subs(w = w(r), w1 = diff(w(r), r), w2 = diff(w(r), r\$2), S)
end proc:
Euler := proc (f)
s(diff(f, w))-(diff(s(diff(f, w1)), r))+diff(s(diff(f, w2)), r\$2)
end proc:```

Example:

ode1(r) = -r*(diff(w(r),r))^2:

ode2(r) = (diff(w(r),r))^2/r+r*(diff(w(r),r\$2))^2:

-1/(2*r)*Euler(w1^2*r):

simplify(1/(2*r)*Euler(w1^2/r+r*w2^2)-VectorCalculus:-Laplacian(%,('polar')[r,theta]))

I will be grateful if you can hint me to write an appropriate procedure.

Thanks

## Minimum positive real root(s) of large m...

Maple

Please explain a method (including corresponding commands) to find first positive real root(s) of a large matrix determinant, briefly. Since each entry or element of the matrix involves with the large formula, probably Newton Iterative method is suitable. The main question is, how it is possible to define some matrices like the Hessian or gradient matrices, when calculating parametric solution of determinant is a very time consuming procedure.

Let assume that only one independent variable exists. After evaluating matrix value at a certain amount of independent variable, the determinant can be calculated swiftly. The problem is that the derivative of determinant in each step will be calculated slowly. How one can rectify this deficiency in Newton Iterative method? For example, is there another iterative method to find derivative of determinant at a certain amount of independent variable to use it in the Newton iterative method?

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