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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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Checking to see if is correct

Suppose Newton’s method is applied to the function f(x) = 1/x. If the initial guess is x0 = 1, find x50. Solution F(x) =1/x x0=1 where n=1,2,3,... F'(x) =-1/(x^2) Xn+1=xn-(f(xn))/(f'(xn)) And I had x50=1125899907*10^6
Andrew Odonkor
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A Scheme for Finding Several Zeros of a Function

Let $f(x) = x-tan(x)$. I am trying to develope a scheme to find its zeros using a particular numerical technique. Let: $$ g(x) = x -mf(x) $$ then $g(r)=r-f(r)=r-0=r$, where $r$ is any of the zeroes of $f$. So $r$ is a fixed point of $g$. For a given…
A Slow Learner
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showing a function has exactly one root using Numerical Analysis Methods

Im trying out the Banach Caccioppoli Contradiction Principle but having a few problems.. f(x) = exp(x/2) - 25x^2 How would i show that this function f has exactly one root x^ in (-Infinity,0) ? Some explanation would be much appeciated! Many thanks
edna1974
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Numerical method for fitting 2D data in regular rectangular mesh

I'm implementing a system, that is receiving on the input imprecise (due to various external influences) coordinates in 2D. My goal is to assign to each coordinate position in regular rectangular mesh. I don't have knowledge on how many rows and…
Biggles
  • 101
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Approximate a curve shape from given parameters

I'm trying to approximate a velocity curve from the following parameters: initial velocity final velocity the total displacement (the area under the velocity curve) the duration of the acceleration We assune that the curve is monotically…
A. Khalef
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Show $\tan(\pi x) − x − 6 = 0$ has a root between 0 and 1.

Show $tan(\pi x) − x − 6 = 0$ has a root between $0$ and $1$ for this question when I plug in $0$ or $1$ as values for $x$, $f(x)$ gives the values of $-6$ and $-7$, respectively so because it does not appear to cross the x-axis, I cannot say there…
number8
  • 515
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eigenvalues of Jacobi matrix and convergence of Jacobi method.

Let $J$ be the Jacobian iteration matrix of the linear system $\begin{bmatrix} 1&2&1\\ 2&1&2\\ -4&2&1 \end{bmatrix}\begin{bmatrix} x\\y\\z \end{bmatrix} = \begin{bmatrix} 1\\2\\3 \end{bmatrix}$. Consider the following statements: (P): one of…
PAMG
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Understanding convergence rate for a iterative scheme

For a given we denote the error $e_k = x_k - x^*$, and the convergence rate of the iterative scheme is $r$ so that $$ \lim \frac{ ||e_{k+1} || }{ || e_k ||^r } = C $$ where $C$ is finite and positive. Im trying to understand how to effectively use…
Mikey Spivak
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What's the difference between feasible and pareto optimal? Do I need to prove feasibility?

What's the difference between feasible and pareto optimal? Do I need to prove feasibility? If I search for pareto optimal through Lagrange multipliers, then would this mean that it's also feasible, since it's the optimal in analytic sense?
mavavilj
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global error in Euler's method

I learned how to find local error in Euler's method and it is proportional to $\frac {h^2} {2}$ . I can understand this. But when we calculate the global error, why do we just multiply by the number of steps and say global error is proportional to…
whwjddnjs
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Lagrange Interpolation Matlab programming

Consider the function $f(x)=|x|$ on $[-1,1]$. Given $n\neq0$, define $x_n^k=-1+\frac{2k}{n}$ for $0\leq k \leq n$. Let $g_n(x)$ be the unique polynomial of degree n which results by interpolating the data with $g_n(x_n^k)=f(x_n^k)$ for all $0\leq k…
rrz.math
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Numerical approximations of derivatives --- source?

Could someone point me to a source (preferably a numerical methods textbook) where they derive / show the formulas to approximate derivatives of a function at a given point $x_0$ based on the values of the function at $x_0$, and the values at…
Cal-linux
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Relative error in numerical solutions

Suppose that $x_0,x_1,\dots$ are the successive approximations of a solution of some iterative scheme (can think of successive approximations in Newton-Rapson method in finding root of a non-linear equation, for example). In general the following…
Kumara
  • 706
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how to improve numerical stability

I'm trying to improve the numerical stability of $(x+1)(x-1)(x-2)/(x^2+4)$. I already have turned it into $(x+1)(x-1)/(x+2)$ and NaN if $x=2$. I realize that I can't make it better around $x=-2$ due to bad condition, but I'm still trying to improve…
IrDe
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How to numerically solve this ODE: $y''' \left(x\right)=-0.5\cdot y \left(x\right) \cdot y''\left( x \right) -0.05 $

I am trying to numerically solve the following ordinary differential equation that I encountered in one article: $$ y''' \left( x \right) =-0.5\cdot y \left( x \right) \cdot y'' \left( x \right) -0.05 $$ This equation has the following boundary…
t387
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