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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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Using bisection method

I have created the following code which does the bisection method. I am trying to use it to find the root for the function $f(x)=x^7-6x^6-28x^5+232x^4-336x^3-544x^2+1728x-1152$ on the interval $[1, 3.1]$ I have been given a hint that $x_0 = 2$ is a…
user123
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$f(x) = \pi + \frac{1}{2}\sin \left ( \frac{x}{2}\right) $ has a unique fixed point on $[0, 2\pi]$

I am looking for some guidance and help with the following question Prove that $$f(x) = \pi + \frac{1}{2}\sin \left ( \frac{x}{2}\right) $$ has a unique fixed point on $[0, 2\pi]$
mt12345
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Use the singular value decomposition of $A$ to prove that $||A||_2 = \sigma_1 = \sigma_{\max}(A)$

I have computed the singular value decomposition of the following matrix $$A= \begin{bmatrix}1&2\\0&1\\-1&0\\\end{bmatrix}$$ here are the important findings below. $$\Sigma=\left[\begin{matrix}1 & 0 \\ 0 & \sqrt 6 \\ 0 &…
jh123
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Verify that the left singular-vectors (the left columns of $U$) are the eigenvectors $AA^*,$

The singular value decomposition of a complex-values $n\times m$ matrix $A$ is defined as $$A = U \Sigma V^*$$ where $U$ and $V$ are unitary $n \times n$ matrices and $m \times m$ matrices respectively, and $Σ$ is an $n \times m$ rectangular…
user104
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compute the singular value decomposition of the matrix

Looking for some help with the question below, I have given it a try and looking to see if my workings make sense or where I have went wrong. Spent some time trying to figure this out and typing this up in the right format, all help is greatly…
jh123
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Compute two steps of the Jacobi and Gauss-Seidel methods starting with $(0,0)^T$

Compute two steps of the Jacobi and Gauss-Seidel methods starting with $(0,0)^T$ for the system $$\begin{bmatrix}4&1\\1&2\\\end{bmatrix} \begin{bmatrix}x\\y\\\end{bmatrix} = \begin{bmatrix}-1\\1\\\end{bmatrix}$$ Do you expect the iteration to…
jh123
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Iteration for solving $x=g(x)$.

$g(x) = \frac{x^2}{3}$ $P=3$ $p_0 = 3.5$ 1) Graph $g(x)$, the line $y=x$, and the fixed point $P$ (done) 2) Using the given starting value $p_0$, compute $p_1$ and $p_2$ (the answer might be $p_1 = 4.083333,p_2 = 5.537869$) Determine geometrically…
John Lennon
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Numerical ODE solving with very good initial guess?

I have a very good initial guess for the solution to my system of ordinary differential equations, which comes from a solution to a similar ODE system with slightly different coefficients (my program is simply covering parameter space). Are there…
Display Name
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How to speed up the convergence?

I should you Aitken's method and the following formula : $T_n = S_n - \dfrac{{A_{n+1}}^2}{A_{n+1}-A_{n+2}}$ in order to speed up $S_n=\sum_{k=1}^{n} (0.99)^k $ Please, help!
John Lennon
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Polynomial proof Newton

Let $f(x)=a_mx^{m}$ + (lower degree terms) be a polynomial. Show that $$f[x_0,...,x_n,x] = \begin{cases} {degree}[m-n-1], & n < m-1 & \\ a_m, &n =m-1 \\ 0 & n>m-1\end{cases} $$where $f[x_0,...,x_n,x]$ is Newton divided difference. I am having…
user60514
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Question regarding notation in numerical methods

I am sitting with a problem where I have to show that Heun's method is consistent w.r.t. properties which is not related to my question. Anyway. I am given two different definitions and the above statement is easy to prove with (1) $\phi (t,y;h) =…
Chengdu
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Loss of significance errors

Give exact ways of avoiding loss-of-significance errors in the following computations: a. $\tan x-\tan y$, with $x\approx y$ b. $\sin x - \sin y$, with $x\approx y$ I don't know how to do a but for b I derived…
Lays
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BFD2 Second Order Time Black Scholes Finite Difference Approximation

Given the Finite Difference approximation of the Black Scholes equation with time discretization $\Delta t*n, n = 1,2,3,...,N$ and in space $\Delta s*j, j = 1,2,3,...,M$ $$\frac{V^n_j-V^{n-1}_j}{\Delta t}…
Laplacinator
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$711.7$ base $10$ to base $2$

I am attempting to solve the following two questions. The first question I believe I have done the whole number calculation correctly but I am confused how to do the decimal portion. The second question I am not sure about. $1)$ Convert $711.7$ in…
user123
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Is there always a numerically stable algorithm for each problem?

Is there always a numerically stable algorithm for each problem? Maybe there is a problem(s) for which there is no numerically stable algorithm? Anyone can say something about it?
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