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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 method should I use to numerically compute this integral?

$$\int_{r_{min}}^{\infty} \frac{dr}{r^2 \sqrt{1-\frac{V(r)}{E_c}-\frac{p^2}{r^2}}} $$ where $r_{min}$ is the root of the denominator. $$V(r)=\frac{Z_1 Z_2 q^2}{4 \pi \epsilon_0 r} \Phi(r)$$ where $a_U$, $q$, $\epsilon_0$, $a_0$, $Z_1$, $Z_2$, $p$…
ThreeStarProgrammer57
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Find $L$ from a function $\phi$ such that $\lim\limits_{h→0} \phi(h) = L$ and $L − \phi(h) \approx ce^{−1/h}$

Consider a function $ϕ$ such that $$\lim_{h→0} ϕ(h) = L$$ and $$L − ϕ(h) ≈ ce^{−1/h}$$ for some constant $c$. By combining $ϕ(h)$, $ϕ(h/2)$, and $ϕ(h/3)$, find an accurate estimate of $L$. Isn't $ϕ(h)=-ce^{−1/h}+L$? I think I am over-simplfying…
user530646
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Is it possible to use Eulers method for lower this initial value problem

I have the following initial value problem $$k(x):=0.00025+10^{(6.3+0.027 \cdot x-10)}$$ $$s(x):= 0.0006+10^{(4.716009+0.06\cdot x-10)}$$ $$ \frac{dy}{dx} = (\log(1+0.05)+s(x)+k(x))y(x)+1, y(68)=0$$ I solved this differential equation and found…
Viktor Jeppesen
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Loss Of Significance $f(x)=\sqrt{x+1}-\sqrt{x}$

Let $f(x)=\sqrt{x+1}-\sqrt{x}$ and $x=0.12345\cdot 10^5$ if the numbers are stored in decimal mantissa with $6$ significant digits we will get $$f(0.12345\cdot 10^5)=\sqrt{0.12346\cdot 10^5}-\sqrt{0.12345\cdot 10^5}=\\=0.111113\cdot…
gbox
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Relative Propagation Error

I understand that the absolute error of a function $f(x)$ where $f(x^*)$ is an approximation and $f$ is continuously differentiable can be written as $$f(x)-f(x^*)\approx f'(x^*)(x-x^*)$$ To find the relative error we…
gbox
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Precision degree numerical integration

How should I understand precision degree? I know that if the number of nodes $n$ is even the precision degree is $n+1$ and if it's odd its $n$. Does this mean that if for a example we are calculating for a polynomial of the $8$th degree, the error…
Ayoub Rossi
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find the Errors for $ \ n=2 \ $ and $ \ n=20 \ $

Let $ \ I=\large \int_a^b f(x) dx=7.621372 \ $ and let the trapezoidal approximation for $ \ n=2 \ $ is given by $ I_T=\int_a^b f(x) dx \approx 7.362011 \ $ . Then find the Errors for $ \ n=2 \ $ and $ \ n=20 \ $ Answer: $n $ = number of…
MAS
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What are the possible ways to write an equation in $x=\phi(x)$ form for Iteration method?

Iterative method For $x=\phi(x)$ , $x_{n+1}=\phi(x_n)$ $x\sin x+\cos x=0$ $[x=2.7984]$ I tried the following forms but they are not provide…
Smart Manoj
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Stability region of forward Euler method

I have problem with determining the stability region for forward Euler method with equation: $$ y' = -100y + 100 \cos t - \sin t \quad t \in[0, \pi], \quad y(0)= 1$$ Forward Euler method is defined as: $$ y_{n+1} = y_{n} + hf(t_n, y_n)$$ How to find…
joel_embiid
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Iteration problem

Suppose that we have the iteration method $x_{k+1}=f(x_{k})$ and $f(x)=Ax+Bx^{2}+Cx^{3}$.For given $a>0$ , determine $A$, $B$, $C$ in order to have local convergence to $\dfrac{1}{a}$ with convergence rate $p=3$. So I suppose that we have to…
Jonathan1234
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Halley's Method for square roots

I'm a computer engineering student and I have a design problem. I'd like to be able to use a numerical method to find the square root of a number with only logic gates on a physical hardware level and I'm restricted to binary numbers. If I set…
jake mckenzie
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How many multiples of 6 are there from **1 to 200** which are not multiple of 4?

Options: 1)16 2)17 3)19 4)15 How did we get that answer by solving??
Priyanka
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How we can find $n$ positive real numbers, $x_1,\ldots,x_n$ such that $x_i>\sum\limits_{j\neq i}x_j$?

We let $n\geq 3$. How we can find $n$ positive real numbers $x_1, x_2, \ldots x_n$, such that \begin{align}x_1&>\displaystyle\sum_{i\neq 1} x_i, \\ x_2&>\displaystyle\sum_{i\neq 2} x_i,\\ &\vdots \\ x_n&>\displaystyle\sum_{i\neq n} x_i.…
M. Raha
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Successive substitution

Successive substitution: Why is the successive substitution proces $x=2(1-e^{-x})$ monotone convergent for start value $x_0=0,5$ while in the tabel the value for $f'(x_0)=1,2121>1$? So we would think it is monotone divergent. We took …
WinstonCherf
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Use Bisection Method for a function that is always positive

I am working on a project for my 'Numerical Methods' course, and I came across a function to which I need to find roots using Bisection Method, also I don't have the initial values $X_u$ and $X_l$. Usually i can pick two values for which…
Samer Hajjar
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