How to solve recursion *T(n)=T(n/2)+T(n/3)+n*? I do not really know how to approach this kind of recurrence.

# Tag: Solve

## differential equations – Solve boundary value problem with NDSolve. How to print out approximations to a solution?

I solve particular boundary-value-problem for ODE with NDSolve “Shooting” method. The case is that solution is attained very slow, that seems to mean that boundary-value-problem which is supposed to be well-defined enough, in fact, is ill at some place. So i try figure out. First step is to see concrete values of a produced approximations to a solution. What language constructs should i use for that?

Simple example. Suppose we consider particle motion in vertical plane. We throw our particle from initial point with coordinates {x0,y0} := {0,0} and initial trajectory angle 45 degrees. And try to achieve point with coordinates {x1,y1} := {1,0} by varying particle initial velocity. We don’t know two things here: initial velocity and a duration of a motion. Here is how this toy problem can be presented and solved in *mathematica*:

```
gravity = 10;
bvpsol = NDSolve(
{
{
(* ODE system (5th order) *)
{x''(u) / c(u)^2 == 0,
y''(u) / c(u)^2 == -gravity,
c'(u) == 0},
(* boundary conditions (5 items) *)
{x(0) == y(0) == 0,
x(1) == 1,
y(1) == 0,
x'(0) == y'(0)}
}
}, {x(u), y(u), c(u)}, u,
Method -> {"Shooting",
"StartingInitialConditions" -> {x(0) == y(0) == 0,
x'(0) == y'(0) == 1, c(0) == 1}}) // Flatten;
{dxdu, dydu} = D({x(u), y(u)} /. bvpsol, u);
{vx0, vy0} = ({dxdu, dydu} / c(u) /. bvpsol) /. {u -> 0};
duration = c(u) /. bvpsol /. {u -> RandomReal()};
ivpsol = NDSolve({
(* ODE system *)
{x''(t) == 0, y''(t) == -gravity},
(* initial values *)
{x(0) == y(0) == 0, x'(0) == vx0, y'(0) == vy0}
}, {x(t), y(t)}, {t, 0, duration});
ParametricPlot({x(t), y(t)} /. ivpsol, {t, 0, duration},
GridLines -> Automatic, AspectRatio -> 1/2)
```

**Question:** Now what options or language construct should i use to see approximations which are produced NDSolve while solving boundary-value-problem?

## linear algebra – Need help to solve simple set of equations

I have below set of equations. All equations are inter-connected to each other and to find out the solution right now I am manually doing the tuning by trial-error method. To make this automated, I would like to know how to solve this for Xt and Xc for Yt=0.1 and Yc=-0.7.

Yt = -0.3759 * Xt + 0.2294

Yt = 0.2744 * Xc + 0.2294

Yc = 0.0325 * Xt – 0.1645

Yc = -0.1006 * Xc – 0.1645

Here,

Yt = 0.1

Yc = -0.7

And, I want to find the Xt and Xc which can satisfy above all equations.

Thanks in advance.

Darshan

## How to solve recurrence T(n) <= 2T(n/3) + (c)log3(n) using substitution method

The title wouldn’t let me format correctly, so here’s a better formatting of the question: Show by induction that any solution to a recurrence of the form

T(n) ≤ 2T(n/3) + c log_{3}n

is O(n log_{3} n).

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## How do I solve this integral equation with a DiracDelta?

I have to solve the following integral equation:

$$G_l(E^prime,E)=delta(E-E^prime)+int_E^{E_{max}}dE^{primeprime}G_l(E^{primeprime},E^prime) |E^{primeprime}-E^prime|$$

How can I do it? My problem is the `DiracDelta`

, I do not know how to treat it, if there were no delta I think I could simply derive, but because of the $delta$ this trick won’t work.

```
Gl(e1, e) == DiracDelta(e - e1) + Integrate(Gl(e2, e1) Abs(e2 - e1), {e2, e, emax})
```

**EDIT** This equation come out as an equation to determine the Green’s function for the solution of the Boltzmann equation (BE) for the transport of a low-energy electrons into a metal.

The solution of the BE can be written as:

$$psi_l(E)=int_E^{E_{max}}de^prime G_l(E,E^prime)tilde{S}(E_0,E^prime)$$

where $G_l$ is the Green function and $tilde{S}$ is a source term.

The full problem is treated in this book, in particular see page 45 for these equations.

The function $|E^{primeprime}-E^prime|$ above is not the real function in my equation, but that is too complicated to be added here (see it a MWE).

## optimization – Can LASSO algorithm solve for sparse $w$ in an under-determined system?

Consider the model $bf y=Xw+n$ where $bf w$ is a sparse complex number vector of length p, X is a N x p known complex number matrix, and $bf n$ is a complex Gaussian noise.

I’m confused because the literate answers yes to my question (in the title), but when implementing most of the functions provided in:

https://www.cs.ubc.ca/~schmidtm/Software/lasso.html

I get invalid solution.

Is LASSO supposed to solve an under-determined system?

## oc.optimization and control – How to solve this non-continuous optimization problem?

I hope you are well. I have a non-continuous optimization problem as follows;

```
*The Goal: max ∑i lotsize(i).(35f(i)-30)
Constraits: lotsize(i) >=0.1,
lotsize(i) is not continuous.*
```

where i represents i*th* trade , *f(i)* function is as follows;

```
*f(i)= 1 ; if a trade is successfull,
0 ; if a trade is not successfull.*
```

I try to create a lotsize function. So, I wonder if there are any ways to obtain lotsize function from the aboved optimization problem.

Thanks for your help!

Best,

Murat Y.

## How to solve this issue

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