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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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System of nonlinear equations

I need to solve following system of nonlinear equations. Here are some characteristics of this system: It consists of n equation and n variables. Every equation is in similar form -> sum of products = constant. The lenght of every product is the…
Tomek Tarczynski
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Are there any techniques to narrow down intervals in interval-arithmetic

I need to calculate some fairly complex equations using interval arithmetic. However, the interval width of the final result is too large. I have a feeling that it can be narrowed down especially while performing operations on intervals which have…
v1p3r
  • 121
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Condition number matrix matrix multiplication

can anybody tell me the condition number of $X \cdot Y$ with $X \in \mathbb{R}^{M \times N}, Y \in \mathbb{R}^{N \times P}$. I tried to describe the matrix multiplication by a sum of many multiplications ($a=M \cdot N \cdot P$), which each has a…
Bamahama
  • 21
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implicit-explicit predictor-corrector scheme for non-linear parabolic PDE

I am working through a paper on implicit-explicit predictor-corrector scheme for non-linear parabolic PDE and having trouble understanding some concepts. I have this PDE: $\frac{\partial u}{\partial t}$ = $\frac{\partial^2 u}{\partial x^2}$ +…
David Mirville
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Nonlinear simultaneous equations

I am looking at a system of nonlinear simultaneous equations. The two variables are u>0 and b>0. How can I solve this problem with computer packages, such as Matlab, Python, or Fortran?…
Bill TP
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What type (explicit, Runge-Kutta, Taylor series, one-step, etc.) is the numerical method $y_{n+1}=y_n+(h/2)(y_n'+y_{n+1}')+(h^2/12)(y_n''-y_{n+1}'')$?

This exam question is asked every year, but I am struggling to understand the difference between numerical methods even though I can solve all the exercises. Thanks a lot in advance for your help! QUESTION: Consider the autonomous, scalar…
s1047857
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Numerically Calculating the solution of very complex equations

I wanted to confirm a question of my own, and I figured out if there is a solution of the following equations such that every variable is real and $x,y\ge 0$, my question could be partially verified. However, after two hours of running…
Gheehyun Nahm
  • 399
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Find $x_i$ and weights such that the following integration rule is exact for all polynomials of degree $\leq 5$

I'm going over an exam I failed. I was told that I can't use the method I used to solve the following question, and I don't know why. Can you please explain and suggest a correct solution? Question Find $x_i$ and $A_i$ such that the following…
Hila
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order of convergence PDE

If I have the generic PDE \begin{equation} u_t + u_x = f, \end{equation} approximated with a first order in time and a second order in space numerical scheme, how can I show that the solution effectively converges as $\Delta t$ and $\Delta x^2$,…
user36390
  • 311
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Smallest possible number of steps

We have the following Runge Kutta Butcher tableau: $$ \begin{array}{c|ccccc} \tau_1 =0 & a_{11}=0 & a_{12} = 0\\ \tau_2 =\frac{3}{2} & a_{21} = \frac{3}{2} & a_{22} = 0\\ \hline & b_1 = \frac{2}{3} & b_2 = \frac{1}{3} & \ \end{array} $$…
evinda
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Find error term of a quadrature

The three-point quadrature rule with error term is given by $$\int_{-1}^1f(x)dx=\frac59f\left(\frac{-\sqrt{15}}5\right)+\frac89f(0)+\frac59f\left(\frac{\sqrt{15}}5\right)+kf^{(6)}(c).$$ Find $k$. After using Lagrange's interpolation (interpolating…
asepe
  • 499
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What is the general idea of Nitsche's method in numerical analysis?

I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without the use of Lagrange multipliers. And its…
Anh-Thi DINH
  • 573
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Show that $x^{m+n}-x^{m}=...$ in Gauss Seidel Iteration Scheme

I am just working through some exam practice questions and I would like to show that for non-negative integers, $m,n$, $$x^{(m+n)}-x^{(m)} = S^m(I+S+...+S^{n-1})(x^{(1)}-x^{(0)})$$ in the Gauss-seidel iteration scheme. I have that the scheme can be…
JJJ
  • 874
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Searching for numerical algorithm

I have nonlinear system of 3 equations. Here it is: $$ \frac{s_2 - K_2}{ps_2^{\gamma_1} + (1 - p)s_2^{\gamma_2}} = \frac{K_1 - s_1}{ps_1^{\gamma_1} + (1 - p)s_1^{\gamma_2}} \\ \frac{s_2}{s_2 - K_2} = \frac{p\gamma_1s_2^{\gamma_1} + (1 -…
0dd_b1t
  • 155
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$17!=3556xy428096000,$then $(x+y)$ equals?(without using a calculator)

$17!=3556xy428096000,$then $(x+y)$ equals? a)$15$ b)$6$ c)$12$ d)$13$ With help of calculator $(x+y)$ can be easily calculated as $15$.But without a calculator,I can only conclude that the sum of digits $(3+5+5+6+x+y+4+2+8+9+6)$ is a multiple of…
Infinitely Undefined
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