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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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Applying Richardson Extrapolation to Euler Method

When applying Richardson Extrapolation to Euler Method, when isn't the local truncation error of h^2 used to get (4*Euler(h/2) - Euler(h)) / 3 instead of using the global error of h to get 2*Euler(h/2) - Euler(h)?
languageoftheuniverse
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Find smallest natural n>0 such that $2^n > (100 \times n)^{(100^{100})}$

I am studying complexity theory and have reduced one of the exercises to the inequality. $$2^n > (100 \times n)^{(100^{100})}$$ Are there any numeric methods (possibly, programming libraries) that will be able to find smallest natural $n >0$…
Evgenii.Balai
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Higher order Taylor method error

In numerical differential equation method Taylor high order 2 formula in initial value problem $ y'=f(t,y), y(a)=\alpha, a\leq t \leq b,$ $h$ is step size of $N=2$ $$w_0=\alpha, w_{i+1}=w_i + hf(t_i,w_i)+ \frac{h^2}{2}f'(t_i,w_i) $$ is it…
seyunkim
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tridiagonal form for householder's

How can we use Householder's method to place the following matrix in tridiagonal form? $$A =\pmatrix{4 & -1 & -1 & 0 \\ -1 & 4 & 0 & -1 \\ -1 & 0 & 4 & -1 \\ 0 & -1 & -1 & 4} $$ Basically, I'm…
Buddy Holly
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Local truncation error and convergence

I am trying to find the local truncation error and the order of convergence of the finite difference scheme $$ \frac{3U^m_n -4U^{m-1}_n + U^{m-2}_n}{2 \Delta t} - \frac{a}{ h^2} \lbrace U^m_{n+1} -2 U^m_n + U^m_{n-1} \rbrace = f(x_n,…
rioneye
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How the roots are calculated for this function?

It says that the roots of $$f(x) = x^{-1} \sin(x^{-1}\log(x))$$ are defined as $$1 > a_{1} > a_{2} > \cdots > 0$$ where $a_{i} = \exp(-b_{i})$ and $b_{i}$ is the unique solution to the equation $b \exp(b) - i\pi = 0$, $1 < b < \infty$. I am…
Joshua Leung
  • 555
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Deriving Newton's divided differences from sub-interval interpolation

I am reading about Newton's divided differences and I am confused by the following derivation of the coefficients of the Newton's polynomial. The Newton's polynomial is given as $P_n(x) = a_0 + a_1(x - x_0) + a_2(x-x_0)(x-x_1) + \dots +…
tmaric
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fixed point iteration of a single variable with unknown constants

Okay so given this simple looking fixed point iteration, i.e., $x_{n+1}=g(x_n)$ $$ x_{n+1} = g(x) = -b - \frac{c}{x_n}$$ The idea is to find a region in the space of $(b,c) \in \mathbb{R^2}$ that will converge for all good starting points $x_0$…
guitarphish
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Does polynomial interpolation using Chebyshev nodes always convege?

On my lecture nodes, it says "Polynomial interpolation on Chebyshev nodes using the barycenteric form is extremely robust". Does that mean it always converges? Here, "robustness" refers to "the effect of round-off error, in particular as the number…
Joshua Leung
  • 555
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How do I construct a second order convergent fixed point iteration?

Say I am dealing with $f(x) = x^3+2x+1$ and I need to come up a fixed point iteration to find its root, that has a second order convergence. How should one approach this question? Should I just compute a couple formulae and see which one converge…
UnsinkableSam
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How to use trial and error

I am trying to solve the same problem as asked here. In short, I am trying to find the values for which $$8n^2 \lt 64n\,\log_2(n)$$ To find a solution in the real numbers, it involves using the Lambert W function, but as the domain of the functions…
jsj
  • 535
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What does this expression $C^2$ mean?

I saw this expression on my numerical analysis class. What does $C^2[a,b]$ mean?
Joshua Leung
  • 555
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Operation counts in gaussian elimination

According to my textbook, step 6 takes $(n-i)(n-i+1)$ multiplications and $(n-i)(n-i+1)$ subtractions. I don't understand where these numbers are coming from. From my understanding, step 6, the multiplication bit should take, $$\sum_{j=i+1}^n…
Joshua Leung
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Show that $x^3-\lambda x+1=0$ has an approximate root $x=\frac{1}{\lambda}$ when $\lambda$ is positive and large

I'm currently trying to figure out a way to show that $\frac{1}{\lambda}$ is an approximate root for $$x^3-\lambda x +1=0$$ when $\lambda$ is positive and large. My idea is to rearrange the equation as $x=(\lambda x-1)^\frac{1}{3}$ and use a…
Loo Soo Yong
  • 515
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Is Von Neumann stability analysis for homogenous PDE different from inhomogenous one?

I'm working on nonhomogenous heat equation and the stability using Von Neumann for the implicit Euler method on $u_{t}-cu_{xx}=f(t,x)$, from the title is there any difference between the homogenous and nonhomogenous case? when applying…
S.N.A
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