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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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electrostatic potential and shooting method

Letting u be the electrostatic potential between 2 concentric metal spheres, with $R_1 < R_2$ and if we have a ODE: $$ \frac{d^2 u}{dr^2} + \frac{2}{r} \frac{du}{dr} = 0, $$ for $R_1 \le r \le R_2$, with $u(R_1) = V_1$, and $u(R_2) = 0$ and where…
mary
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Stability Region of ODE Numerical Method (Runge-Kutta)

I am self-studying numerical methods solving ODEs, and I came across the following question: Consider the 4th order Runge Kutta method with $\Delta t = 1$ applied to $\frac{dx}{dt}=i\omega x$. Find the range of $\omega$ for which the method is…
xf16
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How to find the Lipschitz constant that verifies that g is a contraction?

Suppose you have $f(x)=e^{-2x}-1=0$ then an iterative form of this equation would be $x_{n+1} = g( x_n )=\ln(2 x_n +1)$, by the IVT I know $f(x)$ has a root t in the interval $[1,2]$ and the fixed point of $g(x)$ between $1$ & $2$ is $t$. Hence, how…
LaurenceMeister
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Problem in understand Lagrange integration coefficients

The picture is a extract of Handbook of Mathematical Functions by Milton Abramowitz and Irene A. Stegun. What is the meaning of $m$ in this table and how $DA_k^n(m)$ is related to $A_k(m)$ ? Take for example $n=5$ . I know that …
amilton moreira
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Why can a variable's dependance be neglected in a function?

In this derivation of the 2nd order Runge-Kutta method it's said that from a certain point $y$'s dependence on $t$ in $f(t,y)$ will be neglected. Why is this possible? Source: http://www.math.iit.edu/~fass/478578_Chapter_3.pdf
Noah
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Find the quadratic polynomial

Find the quadratic polynomial $p(x)$ for given data points $$p(x_0)=y_0, p'(x_1)=y_1', p(x_2)=y_2 \text{ with } x_0 \neq x_2.$$ My approach I tried the problem taking $p(x)=a+bx+c x^2$ but I am not sure about. Any help is appreciated.
Atul Anurag Sharma
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Proof for interval within contraction mapping theorem

So, when proving the contraction mapping theorem this book that I am using seemed to imply that if g is a contraction on [a,b], then if we take any two points inside this interval say [c,d], then g is also a contraction on c and d. Now if g is…
user681443
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Algorithms for numerical integration of functions which are zero in a "large part" of its domain

I need to evaluate a definite integral of a function which is zero almost everywhere, for instance something like a Gaussian with an artificial cutoff, such that any function value below some threshold is returned as 0. I don't know beforehand where…
OD IUM
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Is this exercise on "Numerical Analysis" well written or it's just me?

Let $v(t)$ the velocity of an object in the direction of a forze defined by: $$v(t) = \left\{ \begin{array}{ll} 4t & 0 \leq t\leq 4\\ 16+(4-t)^2 & 4 \leq t\leq 14 \\ \end{array} \right.$$ i) Find the Work if apply $200N$ for any $t$ (Use…
sango
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Estimate the absolute error

Assume that we know the approximate value a* of the real number a, and the approximate value b* of the real number b, i.e. a≈a* and b≈b*. Assuming that a, b ∈ [−2, 4], estimate the absolute error abserr(a · b, a* · b*). Can someone help me with this…
user747758
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Construct an expression that bounds the truncation error for a given value of x.

The function that has been approximated with a Taylor Series is: $$f(x) = \frac{log(x) - x + 1}{(x-1)^2} $$ It resulted in: $$f(x) = \sum_{n=2}^{\infty} \frac{(-1)^{k-1}}{k} (x-1)^{k-2}$$ I know truncation error is: $E = f(x) - T_n (x)$ From this,…
datanodes
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Error bound in Simpson's formula

I have and extra information that error has to be smaller than $10^{-2}$ , is there any formula for error bounds that I could use for different types of functions?
NotEuler
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Solving a system of three quadratic equations using numerical methods - (Netwon's method / Jacobian Matrix?)

This is an extension of a previous question involving geometric positioning. I've been able to reduce this problem to a system of three quadratic equations but I need to implement a generalized solution for this that I can solve via computer program…
Dave Franchino
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Inaccuracy when solving problems using FEA

I hope this is the correct forum to post this question because I did not find one that really suits this question. Basically I was reading this paper: 2D Triangular Elements I like to implement things myself in Java and I did implement the formulas…
Finn Eggers
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How to find the weight coefficients in the Gaussian type quadrature formula?

I am trying to determine the node values in the Gaussian type quadrature formula given by: I need to find the weight coefficients for the Gaussian quadrature when the weight function is $ w(x) =\frac {1}{\sqrt{1-x^2}} $ In order to do so, I use…
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