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Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

Definitions: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).

Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss).

Applications: With the advent of the modern high speed electronic digital computers, the numerical methods are successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as biophysics, physics, atmospheric sciences and geo-sciences.

Possible topics include but are not limited to:

  1. Approximation theory, interpolations.
  2. Numerical ODE/PDE.
  3. Root finding algorithm.
  4. Numerical linear algebra, matrix computations.
  5. Discrete integral transform, FFT, etc.
  6. Linear/Non-linear programming, integer optimization.

For questions concerning matrices, please consider adding the tag.

For questions concerning optimization, please consider adding the tag.

For questions concerning Numerical ODE/PDE, please consider adding the // tag.

References:

https://en.wikipedia.org/wiki/Numerical_method

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain

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How to obtain $\gamma$ using a numerical solver?

$K = sign(\gamma)\cdot \{ \frac{A\cdot B}{R}[5-\frac{\gamma}{\pi}+8\theta \sinh(\frac{\gamma}{4\pi \theta})]-\frac{12A\cdot B\cdot \theta}{R}[4\sinh (\frac{\gamma}{8\pi \theta})] +\frac{A^2}{R}[2\theta \tanh \frac{1}{2\theta}-3] \}$ I want to obtain…
user203
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A simple query on numerical optimization.

Newton's method for $\frac1{\sqrt{a}}$ proceeds by iterating minimizing $\frac1{x^2}-a$. Why cant I do $\frac{x-ax^3}2$? Infact why cant I replace $2$ by any $k\in\Bbb R_{>0}$?
Turbo
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credit vs debit in a balance sheet dont add up to initial value

here is a very crazy question but i would like to know if the question is misleading me or am i being misled? credit of 100 in hand. i spent 40 so bal is 60 spent 30 so bal is 30 spent 18 so bal is 12 spent 12 so bal is 0 when i add up my spent…
coolstoner
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Ways of having in inverse proportionality without a divide by zero error?

Am formulating an equation like this: Dmax = 10000 meter (assumed max width of city) B = 2500 (number of people in a building) Distance1 = 10 (distance from a point in the city to B) Distance2 = 500 (distance from a point in the city to…
Nav
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Difficulty to prove some lost steps in a proof

prove that the order of convergence of the secant method is approximately 1.618 and asymptotic error constant is $$K= C^{\frac{1}{\alpha}}=(\frac{F"(P_n)}{2F'(P)})^{\alpha -1}$$. The proof is as follows:- *The secant method is the root finding…
Kavita
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Newton raphson convergence order

As you know convergence order of newton raphson method is: $ E_{t,i+1}=-\frac {f''(x_r)}{2f'(x_r)}E_{t,i}^2 $ What's newton raphson method convergence order if $f'(x_r)=0$.
H.H
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Need modified Euler's method equation to approximate exact solution of initial value problem.

I am asked to solve the following problem. Use Euler's modified method to solve the initial value problem $\frac{dx}{dt} =\frac{1+x^2}{t} ,1\le t \le4, x(1)=0$ The step size is not given here. Now, I can solve this using Euler's method but I am…
Kavita
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A basic question about iteration methods

I have two iteration methods say ite1 and ite2 . If ite1 is performing well in terms of number of iterations and accuracy while ite2 is slightly better than ite1 in terms of computing time only. Which method would be better one? Could anybody clear…
Srijan
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Chebyshev polynomial coefficients of $x^{n-2}$ and $x^{n-1}$ terms

I want to find a formula for the coefficients of the $x^{n-2}$ and $x^{n-1}$ terms of the chebyshev polynomial $T_0(x) = 1, T_1(x) = x; T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$. We've already shown the leading coefficient in $T_n$ is $2^{n-1}$, but i'm…
Taln
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Bisection Method - Example of strict inequality of left endpoints

I am asked to provide an example (or prove none exists) in which $a_0 < a_1 < \cdots$. It's basically only the left side of the interval changing. I cannot think of such an example, but I also don't know how to prove it. Can anyone offer advice? The…
Ozera
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Finding maximum error when interpolating polynomials

Using the points (0,2), (1,4), and (2,8) suppose $f(x)=2^{(x+1)}$ is an approximation. What is the maximum error on [0,3]? I know the error formula is $$\frac{f'''(\varepsilon(x))}{(n+1)!}(x-x_0)(x-x_1)(x-x_2)$$ so evaluating what I have I got…
ECollins
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Numerical Analysis Interpolation

I have a question that I would just like a little bit of clarification about. Find a and b, 0 < a 1, 0 < b 1 such that max x is element of [−1,1] |(x + b)(x + a)(x − a)(x − b)| = max x is element of [−1,1] |(x2 − a2)(x2 − b2)| be as small as…
Samuel Gregory
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Polygon and angles

A cyclic polygon is a polygon with vertices upon which a circle $C_0$ can be circumscribed. (All vertices lie on circle $C_0$). We are given the lengths of the cyclic polygon $\{L_1, L_2,..., L_n\}$. We need to find the coordinates of the vertices…
maverick
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Which numerical methods would you suggest for this nonlinear ODE

Consider the ODE $$ \begin{cases} x(0)=(x_1(0),x_2(0))\neq 0\\ x'(t)=\begin{pmatrix}x_1(t)+x_2(t)& x_1(t)+x_2(t)\\ -x_1(t)& x_2(t)\end{pmatrix}^{-1}v(t), \end{cases} $$ with some known function $v(t)$ (note that the matrix is invertible as long as…
Bananach
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Real World Need to Minimise Monthly SIM Data Costs

This is likely to be a bit basic for you guys but I'm struggling to come up with an automated way to calculate which price plan I should put my SIMs on. Here's the details: Plan A costs $1$ per SIM and is only for SIMs that use $<100$ kb of…
Stephen
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