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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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What does it mean when an integrand is singular at x = 0?

the integrand $f(x) = x^{−1}sin(x^{−1}log(x))$ has infinitely many oscillations in the interval [0, 1] and is also singular at x = 0. What does it mean "singular at x = 0"?
Joshua Leung
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How to show barycentric weight in this form?

Barycentric weight is defined as : $$w_j := \prod_{\substack{k=0\\k \ne j} }^{n} (x_j - x_k)^{-1}.$$ For any $n$, let $x_j = j$, $j = 0,\dots , n$. Show that $$w_j = \frac{(-1)^{n-j}}{j!(n-j)!}.$$ The only thing that I got is $$\biggl(…
Joshua Leung
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Find two variable function from its relative derivative data points numerically

Please help me solve this problem numerically: At discrete points of the unknown function $z = f (x, y)$, only its relative derivative information is available in the following table. Suggest a numerical method to calculate $f(x_i,y_i)$ ($0\leq…
SH SADEGH
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Computing the order of accuracy of a numerical method

I am solving an advection-diffusion equation \begin{equation} u_{t}=au_{x}+Du_{xx} \end{equation} by some numerical method in matlab, the numerical solution matrix is denoted by Ynum, while the exact solution matrix is denoted by Ytru, and both of…
Abood Ob
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Order criteria on explicit two-stage RK method

I have the following exercise: Consider an explicit two-stage Runge-Kutta method given by the Butcher table: $$ A = \begin{pmatrix} 0 & 0 \\ a & 0 \end{pmatrix}, \quad \vec{c} = \begin{pmatrix} 0 \\ c \end{pmatrix}, \quad \vec{b} = \begin{pmatrix}…
user43158
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Why does the computational error of this matrix exponential algorithm has this shape?

Matrix exponential is defined as: Why is that the computational error decreases until around k = 75, then it stays the same? Is it because the factorial is too big, and because Matlab only uses 16 digits of precision, the round-off error is…
Joshua Leung
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How many counting operations are for matrix exponential?

Let A be a m x m matrix and $$\sum_{k=0}^n \frac{A^k}{k!}$$ I have this following flops count: For $A^k$, $(n-1)(2m^3-m^2)$ flops $n$ multiplications for $k!$ $n$ divisions for $\frac{A^k}{k!}$ $n-1$ additions for the summation So the total is…
Joshua Leung
  • 555
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Find the roots of $f(x)$ that are found on the interval $[0{,}1 ; 3{,}3]$ in subintervals of amplitude $d = 0{,}4$

$$f(x)\ =\ \frac{\left(2x^2-\tan\left(x\right)\right)}{\ln\left(x-1.4\right)}$$ I'm not understanding what "subintervals of amplitude $d = 0{,}4$" means, if I solve this problem by Newton-Raphson method, how would I get started ?
Goun2
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interpolation tan

I'm trying to work out a tan interpolation passing through 3 points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, and which doesn't vanish on some interval including those three points. If anyone would have any tip on how to get to some similar result, I'd…
Eric Yatsus
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Don't know which iterative method to use for specific x = f(x)

I have this equation $$ f(x) = \sum_{k=0}^{n} {a_k x^k} + b*sin(nx) $$ for given $b,a_0,a_1,...,a_n$ ($n$ being dependant on haw many $a_k$ are there) and I have to solve $x=f(x)$ using an iterative method. The only problem I have is I don't know…
Dario
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Numerical smooth interpolation

I want to learn smooth interpolation but when I searched to find a website to read about this I could not find anything. Please suggest a source or put a link.
hussein
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I really don't understand "The Shooting Method"

I am attempting to learn from a textbook that has the following question: The boundary-value problem $$ y'' = 4(y-x), \qquad 0 \leq x \leq 1, \qquad y(0)=0, \, \, \, y(1)=2 $$ has the solution $y(x) = e^2(e^4 -1)^{-1} (e^{2x}-e^{-2x})+x$. Use…
Penguin47
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Theory of errors and confusion with roundings

With given: $k_1 = 37.1\pm0.3$ $k_2 = 9.87\pm0.11$ $k_3 = 6.052\pm0.016$ estimate the absolute and relative error for $w = k_1 \cdot k_2^2 \cdot k_3^3$ and round both $w$ and the error in such a way not to lose any precise figures. So my…
Straightfw
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I do not know where the formula comes from?

The following formula used in numerical integration but I do not understand where it comes from. Can somebody give me any hint? $$\int_{-\pi}^{\pi}f(x)\sin x…
user361960
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Are there implicit and explicit solution methods for elliptic equation?

I know implicit schemes means that the unknown values are given in terms of known-and-unknown values, which leads to a system of equations. Explicit schemes means that the unknown values are given in terms of the known values which can be solved for…
GRANZER
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