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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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Hundred-Digit Challenge - problem 2 - math's idea of solution

For you who don't know what 100 digits challenge is... The hundred-dollar, hundred-digits challenge problems are a set of ten problems in numerical analysis published in the January/February 2002 issue of SIAM News…
josf
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Order of convergence

How can find the values of $\alpha$ and $\beta$, such that $x_{n+1}=\alpha x_n\left(3-\dfrac{x_n^2}{a}\right)+\beta x_n\left(1+\dfrac{a}{x_n^2}\right)$ has $3^{\text{rd}}$ order convergence to $\sqrt a$ ? $\alpha=\dfrac{3}{8}$ and…
Surya Deep Mishra
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Anterpolation vs. Interpolation

What is anterpolation, and how does it differ from interpolation? I cannot find a definition for the former, but I see it sometimes used (e.g. in the context of Fast Multipole Methods, like here).
user310374
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Why does Aitken's method converge faster than this?

Improving $$x_{n+1}=g(x_n)=x_n-\frac{f(x_n)}{φ(x_n)} $$ we create Aitken's method $$ x_n*=\frac{x_{n+2}x_n-x_{n+1}}{x_{n+2}+x_n-2x_{n+1}}$$ But why does it converge faster than the initial method?
John Katsantas
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How can I prove this Euler method has a convergence order of two?

The method is the following and it's an altered version of Euler's method: $$y_{n+1}=y_n+hf(x_n+h/2,y_n+hy'_n/2) $$ How can I prove its order of convergence? All I can get from my textbook is how I can prove Euler's method order. But the way it's…
John Katsantas
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Proving the convergence of Jacobi method

Is there any elementary way of showing that the Jacobi method converges for the following system? $$\begin{pmatrix}10 & -2 & -11\\-3 & 10 & 2\\-1 & 1& 10\end{pmatrix}X= \begin{pmatrix} 13\\9\\11 \end{pmatrix}$$ The problem I face is that the…
user92596
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For the given function, find each fixed point and decide whether fixed point iteration is locally convergent to it

A theorem in my book states: Let $g$ be a function and $r$ a number fixed by the function (i.e. $g(r) = r$). Assume $g$ is continuously differentiable, $g(r) = r$ and $|g'(r)| < 1$, then the fixed point iteration is locally convergent to $r$. In…
Ninosław Brzostowiecki
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Numerical method for solving Irrational Equation

How to solve the following equation: $x \arcsin(x^2) + 2\sqrt{1-x^2}\arcsin(x) = 2x $ I think it is hard. Thanks.
newday
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Let $f(x)=x^4$ be approximated by a polynomial of degree less or equal to 2,

Let $f(x)=x^4$ be approximated by a polynomial of degree less or equal to 2, which interpolates $x^4$ at x = -1,0,1then the maximum absolute interpolation error over the interval[-1,1] is equal to? I use the result of error term in lagrange…
dipu
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Finding the maximum friendship quotient

Consider that there are N girls standing in a line..their positions range from 1 to N... and also given is their heights...we need to select two girls such that their friendship quotient is very high...friendship quotient is defined as the product…
Jayanth
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Divergence of approximation of roots by bisection method .

Is there any example of an equation(polynomial or transcendental) such that when we use bisection method for finding the root , the sequence of the approximation of the root diverges from the original root of the equation? Any kind of help or…
BAYMAX
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Using Horner's algorithm to evaluate AT x

I am using Numerical Mathematics and Computing by Cheney and Kincaid 7th edition. My problem is: use Horner's algorithm to evaluate 2x^4 + 9x^2 - 16x + 12 AT -6. Does 'at -6' mean I am to solve p(-6)? I've done it using synthetic division and I got…
Mint.K
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In a barter market (no currency), how to go about finding the "value" of each item based on list of trades of items?

First off I wasn't sure which exchange to ask this on. Eventually this will be turned into code using real data, but for now I just need some advice on how to start so I thought this might be the right place. Say you have a market where items are…
user3434233
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Writing $0.2$ with error less than $0.005$

Write $0.2$ in binary such that the relative error is less than $0.005$ I know that the requirement is that $$\frac{|0.2-x|}{0.2}\leq 0.005$$ But how does it help to find the biggest sum of power of 2 that will be less than $0.005$ ?
gbox
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$f(x)=ln(x-\sqrt[3]{x^3-1})$ Finding the error and achieve higher precision

$f(x)=ln(x-\sqrt[3]{x^3-1})$ is been calculated at $x=35$ and is store in $8$ significant numbers. 1.Find the size of the error 2.Suggest a better way to achieve higher precision and find the absolute error How should I approach it?
gbox
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