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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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Help setting up non-linear parabolic BVP for Newton's method for Non-linear Systems

I am trying to apply Newton's method for non-linear systems to this equation: $$\frac{\partial u}{\partial t}=\frac{\partial ^{2} u}{\partial x^{2}}+(1-u^{2})u+f(x,t) , x \in [-1,1],…
user75269
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Numerical integral with Richardson extrapolation

I was asked to do a numerical integration of a function using the Richardson extrapolation method, but when I tried to search for it in the internet I found out that this is a differentiation method. So is there a Richardson extrapolation method for…
Frogfire
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How to solve $x\log_2 (x) = 10^6$ using Newton's Method?

I know that there are a few questions on how to solve this particular equation, but I'm not very proficient in maths and would like to follow a method which I had a little bit more of a knowledge of and it did seem simpler and best way to put my…
Lukali
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need help in discretizing the fluid equations

I have the two fluid equations: $\frac{\partial \rho}{\partial t}$ = - $\frac{\partial (\rho v)}{\partial x}$ $\frac{\partial (\rho v)}{\partial t}$ = - $\frac{\partial (\rho v^2 + p)}{\partial x} + \rho g$ I need to find a discretized solution for…
Frogfire
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Hermite interpolation of even degree

Using Hermite interpolation determine the fourth degree polynomial $p(x)$ for which \begin{equation} p(0) = p'(0) = 0, \quad p(1) = p'(1) = 1, \quad p(2) = 1 \end{equation} I can only seem to find content on how to apply Hermite…
Math_Day
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diffusion equation approximation

I'm trying to code a matlab function that approximates a 1d diffusion equation with neumann boundary conditions, but I'm really struggling to understand the general approach for solving the problem. Could anyone tell me a general set of steps to…
Jeff Bezos
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Question on Sterbenz Lemma

I am self-learning basic applied numerical analysis. In the book, Numerical methods in Scentific computing Volume I by Dahlquist and Bjoerck on page 109, the Sterbenz lemma is stated as follows: (Sterbenz Lemma.) Let the floating point numbers $x$…
Quasar
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Finite Difference scheme for a random grid

I was asked to give a finite difference scheme to approximate the second derivative ${U''}$ on a random grid $(x_{i+1}-x_i = h_i$ and $x_i-x_{i-1} = h_{i-1})$ and give its order. As usual for this kind of exercice I try to write the Taylor series…
LeA
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Relation between zero stability and absolute stability of numerical methods for solving ODEs

I'm confused by the relationship between the notions of zero stability and absolute stability in numerical analysis. Consider the leapfrog method given by: $y_{k+2} =y_k + 2hf(t_{k+1},y_{k+1}) $ It is known that this method has stability region…
Othman El Hammouchi
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Matlab code for fixed point iteration

I want to write in Matlab a function that appreciates the fixed point iteration for a system of equations. The idea is: $\begin{bmatrix} x{_{1}}^{m+1}\\ x{_{2}}^{m+1} \end{bmatrix}= \begin{bmatrix}y{_{1}}^{n}+hf{_{1}}(x{_{1}},x{_{2}})\\…
Mary Star
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Bivariate Quasi-interpolants

Consider the bivariate spline quasi-interpolant $S2$ defined on a bounded rectangle $R$ with simple interior knots. It is known that, if $f \in C^k(R)$, then $||f-S_2f||_{\infty}$ behaves like $O(H^k)$ for $k=0, \dots 3$, where $H$ is the maximum of…
Simo
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Fixpoint iteration of $ x + \ln(x) = 0 $

I want to solve the equation $ x + \ln(x) = 0 $ which has a root $ x^* \approx 0.5$. I am given three iterations: $$ \quad x_{n+1} = - \ln(x_n)\\ \quad x_{n+1} = e^{-x_n} \\ x_{n+1} =\frac{x_n + e^{-x_n}}{2} $$ What method do I use to find…
Oskar
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Derive consistency of finite difference approximation in both, x- and y-direction?

I am used to deriving consistency for approximations of derivatives such as $\frac{\partial u(x,y)}{\partial x} = \frac{u(x+h,y) - u(x,y)}{h}$ and then going on with taylor series: $u(x+h,y) = u(x,y) + hu'(x,y) + \frac{h^2}{2}u''(x,y) ... $ but how…
binaryBigInt
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Finding the maximizer

Let be $ A\in \mathbb{R}^{2,2} $ a symmetrical and positive definite matrix with different eigenvalues $ \lambda_1>\lambda_2>0 $. Further we have the rayleigh quotient $$ \max_{x\neq 0}\frac{\overline{x}^T\cdot A\cdot x}{\overline{x}^T\cdot…
hallo007
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Avoid Loss of significance error

I want to compute $f(x)$to avoid loss of significance$$f(x)=\frac{1-\cos x}{x^2}$$ One way is writing Taylor series approximation for $\cos x$ about $x=0$. what about this approach: $$f(x)=\frac{2\sin^2(\frac x2)}{x^2}$$ Is this ok ?
Etemon
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