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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

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How do you find a coefficient experimentally and graphically?

If I did not know that the coefficient $K$ of the frequency of the pendulum; $\frac{1}{t} = K*(g/L)^{\frac{1}{2}}$ is $\frac{\pi}{2}$, how would I go about finding it? Is there any textbook that I could take a look at? I found some equations through…
Humberto
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$h=0.15, y(0.6)=?, y'=x(y+x)-2,y(0)=2$ , anwer correct upto 5 decimal places using Euler method

use euler's method with step size $h=0.15$ to compute the approximate value of $y(0.6)$,correct upto five decimal places from initial value problem $$y'=x(y+x)-2,\, \, y(0)= 2$$ Actually, I have found the answer for $y(0.6)$ with $h=0.15$ but i…
abhinav singh
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Determine the constants $b,c,d$ so that the succession $x_{n+1}=g(x_n)$ has a cubic convergence towards $\sqrt{z}$

Consider the function $$g(x)=\frac{x^3+bx}{cx^2+d}$$How can I determine the constants $b,c,d$ so that the succession $x_{n+1}=g(x_n)$ has a cubic convergence towards $\sqrt{z}$?. My professor solved it calculating…
codingnight
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Numerical optimization

Suppose that I need to find the global maximum of $f(x)$ but $f'(x)=0$ and $f''(x)=0$ can only be solved numerically. How do I am 100% sure that I have found the correct optimums. My doubt is because the algorithms do not find the roots with…
Quema
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Why i have got two different answer with two logical method?

I have to find the root of following function $$f(x)=81.45472 \cdot x^2+ 96.5008\cdot x-283.6563.$$ There are two methods: finding by mathematical formula, delta method, which gives this answer: delta= 96.5008^2+4*81.45472*283.6563=101733, so …
sepideh
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Newton's method and fixed point

Can anyone explain this problem? I am wondering what it means for evaluate $g'(x^∗)$, isn't that zero? Newton's method can be viewed as a way of transforming a root-finding problem $f(x)=0$ into a fixed-point problem $x=g(x)$, where…
Cong Yang
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If p(z) has a double root $s_1$, show that $s_1^n$ and $ns_1^n$ are both solutions of $\sum_{j=0}^r \alpha_j U^{(n+j)}=0$

If p(z) has a double root $s_1$, show that $s_1^n$ and $ns_1^n$ are both solutions of $\sum_{j=0}^r \alpha_j U^{(n+j)}=0$ I was able to prove the no repeated root case. I have a hint to use that $\rho'(S_1)=0$ but I can't figure out the significance…
MathIsHard
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Prove all solutions of $\sum_{j=0}^r \alpha_j U^{(n+j)}=0$ can be written in the form $U^{(n)}=c_1s^n_1+...+c_rs_r^n$

Consider a linear constant coefficient rth order difference equation $\sum_{j=0}^r \alpha_j U^{(n+j)}=0$ with $a_r \neq 0$ and $a_0 \neq 0$. I know that if s is a root of the polynomial $p(s)= \sum_{j=0}^r \alpha_j s^j=0$ then $U^{(u)}=Cs^n$…
MathIsHard
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Secant and Bisection Method

What would be the example of a function for which a Secant Method fails but Bisection Method converges (to the root). In particular, if we are checking the interval $[a,b]$, then starting points for the Secant Method are $a$ and $b$.
Ekber
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Central difference quotient in two variables.

I was asked to compute the difference quotient approximation for a function $u(x,y)$ of two variables namely $$ \frac{\partial^2 u}{\partial x \partial y}(x,y) $$ Here is what I did: use the approximation for $\partial y$ $$ \frac{\partial…
Riemann-bitcoin.
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The expression cannot be written?

I am solving exercises, and I found this one. The problem is that my result was: $2 * 10^{-9}$ , but this answer not appear in the alternatives, what is wrong ?
Eduardot
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Finding N formula (Backward Difference)

I want to find the N formula of $▽^n f(x)=x^2+x$ Where $h=1$ So far, I reached to this point, $▽ f(x)=f(x)-f(x-h)$ $ =x^2+x-[(x-h)^2+(x-h)]$ $= x^2+x-[x^2-2xh+h^2+x-h]$ $= x^2+x-x^2+2xh-h^2-x+h$ $= 2xh-h^2+h$ What to do next?
Eman Yusuf
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Hermite Interpolation - Algorithm explanation

I am struggling to understand the Hermite interpolation algorithm using divided differences. I am reading from Pg.23/24 here In their notation, $m_j$ denotes the "number of derivatives that we have to interpolate for each point $x_j$" So, for…
Taln
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Showing $\sum_{j=0}^{n} x_j \ell_j(x) = x$ from the Lagrange Interpolating polynomial

I am asked to show that $\sum_{k=0}^{n} x_k \ell_k(x) = x$ where $\ell_k(x)$ is the $k$th cardinal function $\ell_k(x) = \prod_{\substack{j = 0 \\ j \neq k}}\frac{x - x_j}{x_k - x_j}$. I have tried doing some elementary expanding, but I don't see…
Taln
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Derivatives of Divided Differences

I want to show that $$ \frac{d}{dx} f[x_0,\ldots, x_k, u_1,\ldots, u_n] = \sum_{v=1}^n f[x_0,\ldots, x_k, u_1,\ldots, u_n, u_v] \frac{d u_v}{d x} $$ where $x_0,\ldots x_k$ are constants and $u_1,\ldots, u_n$ are differentiable functions of…
Alain
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