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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 can we prove if $A = M - N$ (non-singular) and if the induced matrix norm $\|M^{-1}N\| > 1$ then the iteration error increases at some point?

For a system Ax = b, it seems like I am on the right track in proving the proposed statement, however, I believe I'm missing something and my proof has some issues in the second last, and last equation. For some reason, the markup is not working on…
sjain
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How to prove Preconditioned Richardson - Steepest Descend converges

Hi, I am trying to solve this problem but I am feeling like I am missing some prerequisites. Can help me solve this ? or at least a reference with a book where this is solved ? $\text{Let } A \in \mathbb{M}_n(\mathbb{R}) \text{ be a symmetric and…
johnyy
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numerical calculation on exponent of a matrix

I am using computer language (Matlab and mathematica) to compute the exponent of a matrix of the form $$B_n = \exp(n\hat{A}), \qquad n\in \{1, 2,3, \cdots, N\}$$ where $n$ is positive integer and N is some big positive number.$A$ is square matrix of…
user1285419
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Appropriate numerical scheme for eigen value problem

I am looking for a numerical scheme which can easily handle the following eigen value problem I already had the analytical results of this problem, now I want to know how to treat this problem numerically. Any idea, then please let me know. Thanks
zhk
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elliptic correction of magnetic field

I am playing around with implementing an elliptic divergence cleaning scheme for induction equation in MHD. My grid is staggered, i.e. the magnetic field components $B_x, B_y, B_z$ are at cell faces while the scalar $\psi$ (of the poisson equation…
NumberMan
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Wilkinson's Polynomial Derivative

I may be missing something simple here, but in the section for stability of Wikinson's polynomial in Wikipedia the derivative is defined as: $-\frac{\alpha^{19}}{\prod_{k \ne j}(\alpha_j - \alpha_k)}$ The denominator is defined as…
Sam Keays
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Stability of Linear FEM on $u_{x} \left( x \right) = f \left( x \right)$

I'm practicing with numerical schemes, and derived the linear spatial FEM scheme of the ODE $u_{x} \left( x \right) = f \left( x \right)$ on $a \le x \le b$ given $u \left( a \right) = U_{0}$. The scheme was linearized and outputs all interpolants…
Chloe
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Euler Forward step size numerical stability

I am tasked with the following exercise: Suppose we have 2 cars, and that car 2 has experienced engine failure and is stationary at position $x_2(t) = \bar{x}$. Let $x_1(0) = > \bar{x} - d$. Introduce the variable $\widetilde{x}(t) = \bar{x} - >…
rawestan
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Conditions of the error of Trapezoidal Rule

If $f\in C^2[a,b]$ and $f(a)=f(b)=0$,then exist $\xi \in (a,b)$,such that $\int_{a}^{b}f(x)dx=-\frac{1}{12}f''(\xi)(b-a)^3$ It is the error formula of trapezoidal rule and can be easily proved by partial integration or error formula of Lagrange's…
Mathscraft
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Proof that $a_{3}=f[x_{0}, x_{1}, x_{2}, x_{3}]$ (lagrange interpolation)

I want to show that $a_{3}=f[x_{0}, x_{1}, x_{2}, x_{3}]$, if $P_{n}(x)$ is the lagrange interpolation polynomial written in the form $$P_{n}(x)=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})(x-x_{1})+ \cdots+ a_{n}(x-x_{0})\cdots(x-x_{n-1})$$ from where…
Wrloord
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If $f(x) =\frac{1}{2} (x + a/x)$ such that $r_{n+1} = f(r_n)$ show that $0 < f^{'}(r_{n+1})< \frac{1}{2}$

If $f(x) =\frac{1}{2} (x + a/x)$ such that $r_{n+1} = f(r_n)$ Show that $0 < f^{'}(r_{n+1})< \frac{1}{2}$ I calculated that $f'(r_{n+1})=\frac{1}{2}(1-\frac{a}{r_{n+1}^2})$ but now I am stuck any hint?
Kevin Duran
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Runge-Kutta method for linear ODEs

I have a generic question: Can the Runge - Kutta method be used to solve every type of linear ordinary differential equation?
Steve
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Is it possible to find $|a|<\epsilon$ such that $(1+a)^n$ is equal to given $(1+\delta_1)(1+\delta_2)\dots(1+\delta_n)$ if $|\delta_i|<\epsilon$

Assume that $\epsilon$ is a very small positive number (for example $\epsilon$ is of order $10^{-200})$. Also $n$ numbers are given in form $\delta_i$ $(1\leq i\leq n)$ such that for every $i$, $|\delta_i|<\epsilon$. Now assume this expression is…
Mason Rashford
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Numerical stability for a non-homogeneous ODE

This is the definition of numerical stability I have learned: "A numerical method $ u_n $ is stable if $ |u_n|\to 0 $ as $ n \to \infty $ ". It is easy to show numerical stability if the IVP to show for takes the form of $ y' = \lambda y \\ y(0) =…
Alexander Jonsson
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A method to determine the order of convergence for a sequence

From Wikipedia: My question is: How I can get the formula? From the definition, the only thing I obtained was that $q \approx \frac{\log |\frac{x_{k+1}- L}{x_k-L}|}{\log |\frac{x_{k}-L}{x_{k-1}-L}|}$. How I can continue from here? Edit: A more…
MathLearner
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