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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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interpolation numerical methods

I have translated this question into English so excuse me if it is not proper. I am struggling to figure out how to solve the following question any directions would be greatly appreciated Let $F$ be the function defined…
NeedsToKnow
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iterative solutions to linear system

In the book Numerical recipes in C, equation (2.5.10) reads as $$ x_{n+1}-x_{n}=B_0(b-A x_n) $$ could someone give a hint how this is derived? From my side, I could only understand the following. The problem is to find improved solutions…
sunxd
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Crank nicolson method for solving telegraph equation

I'm trying to solve telegraph equation (transmission line) with no losses. I got this equation (wave eq) $$LC\frac{\delta^2u}{\delta t^2}=\frac{\delta^2u}{\delta x^2}$$ that I wan't to solve using Crank-Nicolson in MATLAB, but I'm stuck with…
user1214542
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Could you please explain how to derive the following equation

since $-\Delta\mathbf{m}=\mathrm{curl}^2\mathbf{m}-\nabla\mathrm{div}\mathbf{m}$ integration by parts…
p yz
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Problem in Understanding an Illustrative Example concerning the "Effect of Error in a Tabular Value".

I am relatively new to the topic of "Numerical Methods and Analysis ". I was reading a book named, "Numerical Mathematical Analysis " by J Scarborough. I was studying about the effect of an error in the tabular value. The book had an example stating…
Thomas Finley
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Using Secant Method to Solve for Specific Function Value

I'm looking for some guidance on utilizing the secant method to calculate a particular value within a function. Here's the formula I'm using: $$x_n = x_{n-1} - f(x_{n-1}) \cdot \frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}$$ I was suggested…
Bishop_1
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Maximum value of sin(x/2)

I am trying to solve a fixed point problem and I have the $\sin(x/2)$ in my function. After checking that my function is defined in a closed interval which is convex and also maps elements onto itself. I am trying to define the interval in which I…
Georges Djimi
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Series acceleration with computable error of estimation

I am trying to numerically sum a series $a_n$ which converges slowly. Although a bound on the remainder $\sum_{k>N} a_k$ can be computed fast. To find an estimate of the sum $S = \sum_{k>=1} a_k$, I am using Aitken's method which transforms the…
Sounak
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A problem of solving nonhomogeneous equations：$\frac{1-e^{-a/x} }{1-e^{-b/x}} =c,$ where $a, b, c$ are all constants to solve for $x$

A problem of solving nonhomogeneous equations：$$\frac{1-e^{-a/x} }{1-e^{-b/x}} =c,$$ where $a, b, c$ are all constants to solve for $x$. I wonder if there's a good way to solve this equation, I try to approximate it using Newton's method and in…
yh l
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About GMRES and Polynomial approximation

I was reading a book Lloyd N. Trefethen: Numerical Linear Algebra about GMRES, and on page 269 of book the authors say: The GMRES solves an approximation problem, and space of polynomials is $P_n=\{ \text{polynomials $p$ of degree $\leq n$ with…
nezudem
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Derivation of Richardson Extrapolation update rule for arbitrary $k$

In the Wikipedia article on Richardson Extrapolation, a recursive formula is shown: $$ A_{i + 1}(h) = \frac{t^{k^i}A_i(h/t) + A_i(h)}{t^{k^i} - 1} $$ I'm finding this hard to generalize into an algorithm that accepts an estimator function because of…
rocksNwaves
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Using the BFGS method, how can I prove that if $B_k$ is symmetric and positive definite, then $B_{k+1}$ is also symmetric and positive definite?

The following problem came up in my class, and I'm having trouble solving it. Knowing that the BFGS method is defined by $$B_{k+1} = B_k - \frac{B_ks_ks_k^TB_k^T}{s_k^TB_ks_k } +\frac{ y_ky_k^T}{y_k^Ts_k }, $$ how can I prove that, if $B_k$ is…
tomasb21
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Solving a 4-coupled non-linear autonoumous ODE system numerically

I have the following generalized system of differential equations: $$ \left( \frac{du}{dt}, \frac{dh}{dt}, \frac{dv}{dt}, \frac{dy}{dt} \right) = \left( f_1(u, h, a), f_2(u, h), f_3(u, h, u'), v \right) $$ where $a = \frac{dv}{dt}$, $u' =…
Prajval K
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Order of Convergence of a Fixed Point Iteration

I'm trying to find the order of convergence of the fixed point iteration $p_{n+1} = g(p_n)$ for $n \geq 0 $ where $g(x) = (3+x-3x^2)^{1/4}$ and the point of convergence is at $x = p$ This is how I started solving the problem. $$ p - p_{n+1} = p -…
pa1
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Cubic polynomial and x,y points

Hope everyone is well. I am working on a problem that gives me a table of 6 values each of x and y. Now, I am supposed to justify if the data corresponds to the function values of a cubic polynomial. I am still conceptually working on this subject…
johnathan
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