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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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Is there a way of approximating a polynomial's inverse?

Suppose I have a fifth degree polynomial: $f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5$ and that it does not factor nicely or have any nice roots. Is there a way to approximate the inverse function $f^{-1}$? Whether by hand or by computer…
Joseph DiNatale
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Why is $(1+x)^2$ less accurate than $ (x+2)x+1$ for small $x$?

I've known that accuracy is based on the amount of roundings (or multiplications) that occur, but from what I can tell, both equations will require the same amount. My first thought was to related $(1+x)^2$ to $1+2x+x^2$ and say that there are two…
em_
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Determining $a$ such that adjacent loops in the parameterized curve $(x,y) = (t+a\sin2t, \sin t)$ are tangent

Consider the parametric equations of the independent parameter $t$ as follows: $$\begin{cases} x = t + a \sin(2t) \\ y = \sin(t)\end{cases}$$ When $a = 0$, we get the usual sine curve. When $a = 0.4$ we get a near saw-tooth curve. When $a =…
Dottard
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Why is this numerical method more accurate and stable?

Suppose I have the current position, velocity and acceleration of a particle. As I see it, the best estimate of its velocity and position at a time $\Delta t$ from now should be $$\vec{v}(t+\Delta t) = \vec{v}(t) + \vec{a}\Delta…
WordP
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Best way of computing the decimal representation of a fraction with an arbitrary precision?

Say you are given a fraction, e.g. $\frac{1}{37}$. What is the best way to compute its decimal notation given an arbitrary precision? Is there a better way than to use a numerical algorithm, e.g. Newton's method? Is this what your calculator does in…
user12205
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Approximating a function with a piecewise constant function

I have some distribution X of values (which I don't know exactly but I can sample many times). I also have a function $f : X \to Y$ which may be complicated. I want to approximate $f$ with a piecewise constant function $g$, where the number of…
Andrey
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Butcher's Tableau and Runge-Kutta 4

I have been reading about RK4 and one thing that I am not able to understand is how to determine the coefficients to use? I found extensive information about Butcher's table (see here and here for instance). Though some properties must be satisfied…
Sos
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How to solve simultaneous equations using Newton-Raphson's method?

I understand how to find roots of a polynomial equation programmatically using Newton-Raphson method as explained here. How to find the values of $x$ and $y$ from the simultaneous equation given below, using Newton-Raphson method. Any body…
Martin
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Rate of convergence of fixed-point iteration in higher dimensions

Consider the fixed-point iteration process in $\mathbb{R}^n$. Given a sufficiently smooth function $f:\mathbb{R}^n\to\mathbb{R}^n$ and an initial value $x_0\in\mathbb{R}^n$, define the iteration sequence $x_{k+1}=f(x_k)$. Suppose…
Niwatori
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Numerical Analysis: Why this function is giving Sage / WolframAlpha so much trouble to integrate?

I am trying to find the numerical value of $$\int_{-\pi}^\pi \sin(2\cos\theta)\cos(4\theta)\,d\theta$$. The integrand looks like a nice, continuous function that is finite, bounded, differentiable, etc. Furthermore, I am just looking for a numerical…
yoyostein
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Interpolating polynomials.

Let $$f(x) = a_{n+1}x^{n+1} + a_nx^n + ... + a_1x + a_0$$ be a polynomial of degree $n+1$ defined for all $x\in\mathbb{R}$. Show that: (a)The derivative of order $n+1$ of $f$ is equal to: $$f^{(n+1)}(x) = (n+1)!a_{n+1}$$ First we deal with the…
user197848
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Does averaging always provide faster converging numerical methods?

So I am studying SICP (Structure and Interpretation of Computer Programs) and doing one of the excercises which is based on the fixed-point method for finding the fixed-point of $f(x)$. In a particular excercise question it asks to find the root of…
abkds
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Iterative approaches to the solution of nonlinear equations $f(x) = 0$

I am currently studying for a midterm, and I am review over the following methods: Fixed point method Bisection method Regula Falsi method Newton-Raphson Accelerated Newton-Raphson Secant I know how to use the methods, however I am more…
ChickSentMeHighE
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How to find the interval $[a,b]$ on which fixed-point iteration will converge of a given function $f(x)$?

Theorem 1 If $g \in [a,b]$ and $g(x) \in [a,b] \forall x \in [a,b]$, then $g$ has a fixed point in $[a,b].$ If in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k < 1$ exists with $$|g'(x)| \leq k, \text{ for all } \in (a, b)$$…
roxrook
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Derive error term by using Taylor series expansions.

Using Taylor series expansions, derive the error term for the formula \begin{equation} f''(x)\approx \frac{1}{h^{2}}\left [ f(x)-2f(x+h)+f(x+2h) \right ]. \end{equation} I've tried it on my own way. We see…
UnknownW
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