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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

14158 questions
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Numerical analysis condition number

I have the following problem. Let $p = p(a)$ be the positive unique root of the equation $x^n − a*e^{−x} = 0$ where $n$ is a natural number and $a > 0$. Show that the condition number $κ_{p}(a) <1/n$. Attempted to start the Proof and I realized I…
Mihai.Mehe
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numerically solve differential equation with square root term

I have a problem of the following form to be solved numerically. Can someone suggest a numerical scheme that can be used for this problem? $$\frac{∂y}{∂t}= -C \frac{∂}{∂x} \left[\sqrt\frac{∂y}{∂x}\right]$$ where C is a constant [$L^2/T$] Thank you!
Nadi
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Implicit Trapezoidal Rule

Integrate $y' = \sin(y)$, $y_0 = 2$; using $h = 0.1$ with the implicit Trapezoidal rule, to compute $y_1$ within approximate relative error $e<10^{-5}$. I first tried to apply the Trapezoidal rule which then resulted in an implicit equation. To…
Damini
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Use graphic representation to determine the zeroes of the function

I want to solve the following question: Use graphic representation to determine the zeroes of the function to one correct decimal $f(x)=(x+1)e^{x-1}-1$. This is a problem from the course book Numerical Methods, but since the book only has a answer…
kabin
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Order Notation question $\sin(x) = O(1-x)$ as $x\to 1$

Is the problem $\sin(x) = O(1-x)$ as $x\to 1$ true or false? I think that it is false, but I don't know how to prove it. I know that for some ε there exists a constant C so that $|x-a|\leq \epsilon$ implies that $|f(x)-a|\leq C|g(x)|.$ I don't know…
eilema
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Represent the following numbers in normalised floating point

Using an 8-decimal digit floating point representation (4 digits for mantissa, 2 for exponent and 1 each for sign for exponent and sign for mantissa) represent the following numbers in normalised floating point from (using chopping if required) (i)…
chikku
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Trying to solve a set of non-linear equations

I am looking for a solution to a set of non-linear equations. Let $K$ and $S$ be $N\times N$ real matrices. Given $K$ I would like to solve the following for $S$: $$K_{jk} = \sum_a S_{ka} |S_{ja}|^\delta \mathrm{sign}(S_{ja}).$$ The exponent is $0 <…
poldpold
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Ruled surface by skinning

I have the following problem. I have one curve and the article I am analyzing says: we can generate a ruled surface by skinning the curve in a direction D. I don't know how I can obtain a ruled surface by a process of skinning one single curve.…
hoping
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Bounding error of derivative when using Lagrange interpolation

I have a function $f:[a,b] \to \mathbb{R}$, where $f \in C^{2}([a,b])$. Admit that $p_{1}$ resulting from Lagrange interpolation of f in $x_{0},x_{1} \in [a,b]$. Then I must show that for any $x \in [a,b]$ we have $\lvert f'(x) - p'_{1}(x) \rvert…
Pedro Silva
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Weighting Soccer ability

I'd assume there's a few methods on how to weight soccer ability between different leagues but when I asked for the maths behind it in a different websites forum, I was basically just told "It's to do with the elo rankings". Which didn't really help…
JayRSP
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Trying different methods

I was working on the following problem: Two trains depart towards each other from $650\,\mathrm{km}$ apart. If they leave at the same time, they will meet after $10$ hours, but if one of them leaves $4$ hours and $20$ minutes after the other, they…
Daniel
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prove statement about fixed point iteration

I have the following fixed point iteration: $$ p_{n+1} = \frac{p_n^3 + 3ap_n}{3p_n^2 + a} $$ By defining $$g(x) = \frac{x^3 + 3ax}{3x^2 + a}$$ en some algebra I found that the fixed point is $x = \sqrt{a}$. So $g(\sqrt{a}) = \sqrt(a)$. Now I need…
user54297
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How to use the Runge-Kutta 4th order method to integrate the acceleration?

Suppose we have the following dynamic equation for acceleration We can then apply any of several known numerical integration techniques to integrate the acceleration to compute future positions and velocities. Given initial conditions on the motion,…
Eng Eng
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Numerical analysis - Gaussian quadrature

I'm trying to solve a problem related to the Gaussian quadrature. The first step of the problem is to prove that the following claim holds: I was able to prove the $\theta = \pi k$ part very easily by using some simple trigonometric identities, but…
RandyMarsh
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A numerical analysis problem

I was looking at old exam papers and was stuck on the following problem: I have hardly any idea how to progress with the problem. Can some give some explanation about how to progress with the problem?
learner
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