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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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what is the meaning of asymptotic error constant

I got the definition of asymptotic error constant: $$A=\lim_{n\to\infty} \frac{\epsilon_{n+1}}{\epsilon_{n}^\phi}$$ The meaning of $\phi$ is well defined in the books, it says how fast the the result converge, but what is the meaning of the …
Mr.O
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Piecewise constant function explanation in finite-volume methods

Suppose we place a grid in xt-plane with points $(x_j,t_n)$ and mesh width $h= \Delta x$ and time step $k =\Delta t$ and $x_j=jh$ and $t_n = nk$ and call $U_j^n$ to be the approximation to some $u(x_j,t_n)$ at the discrete points. Also, notice that…
ILoveMath
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Gaussian Quadrature Rules

The integration formula is to have the general form $$\int_a^b f(x)\text{ }dx\approx\sum_{j=1}^nw_if(x_i).$$ Let $a=-1,b=1$. When $n=1$: The integration formula has the form $$\int_{-1}^1f(x)\text{ }dx \approx w_1f(x_1).$$ Using $f(x)\equiv 1$ and…
Schole
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Numerical Methods: calculate $b/a$ without division

Calculate $b/a$ in a calculator that only adds, subtracts and multiplies. This problem is in the textbook for my numerical methods class. Obviously you can calculate it by $$ \frac{1}{a} + \frac{1}{a} +\frac{1}{a} + ... \frac{1}{a} + = \frac{b}{a}…
Sigma
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find an interval of unit length which contains the negative root of smallest magnitude of the equation $3x^3+10x^2+10x+7=0$

I don't know how am I supposed to do it? Kindly help me with it. I have tried using bisection method but I don't have an idea regarding this.
user8454192
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Composite Trapezoidal Approximation

Prove that if $f \in C[a,b]$, and $T_n(f)$ is the composite trapezoidal approximation of $\int_{a}^{b}f(x)dx$, then $\lim_{n\to\infty}T_n=\int_{a}^{b}f(x)dx$. $T_n(f)=\frac{h}{2}\sum_{n=1}^{M}(f(x_{n-1})+f(x_n))$ at least I think this is what I am…
DHouse
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On the Lagrange Interpolation Error for the Trapezoidal Quadrature Rule

This is part of a document on numerical analysis. I have included its link below. In the derivation for the error in the linear interpolation approximation on page 84, the author "brings" the ξ(x) term of the integral outside using the mean value…
Panteleimon Tassopoulos
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Fixed point for a function. Numerical Analysis.

how might g defined such that the root of $f(x)=x^3-3x-1$ is a fixed point of $g$, for $x$ in the closed interval between $-\frac{1}{2}$ and $0$. Find two distinct functions $g_1$ and $g_2$ and investigate the rate of convergence for each case. I…
UnsinkableSam
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Condition for convergence of Newton-Raphson method.

Let $f :[a,b]\to\mathbb{R}$ be any function which is twice differentiable in $(a,b)$ with only one root $\alpha$ in $(a,b)$. Let $f'(x)$ and $f''(x)$ denote the first and second order derivatives of $f(x)$ with respect to $x$. If $\alpha$ is a…
PAMG
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Cubic interpolating spline

Consider calculating a cubic interpolating spline with the additional boundary conditions $s''(x_0)=0$ and $s''(x_n)=0$. Show that $$\int_{x_0}^{x_n}[s''(x)]^2dx \leq \int_{x_0}^{x_n}[g''(x)]^2dx$$ for any $g \in c^2[x_0,x_n]$ that satisfy…
user60514
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Applying Newton's method to $f(x) = x^n - a$

Attempt. Clearly, the root is $x=a^{1/n}$. Now, in the Newton method, the sequence given iteratively by $$ x_{n+1} = x_n - \frac{ f(x_n) }{f'(x_n)} $$ should converge to the root. We have $$ x_{n+1} = x_n - \frac{ x_n^n - a }{n x_n^{n-1} }…
James
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Numerical Analysis Taylor Method

using the taylor method perform two steps when $y' = -2t - y$ when $y(0) = -1$ and $h = 0.1$ what is $\frac{df}{dt},\quad \frac{d^2f}{dt^2}$ ... I found $\frac{df}{dt} = -2 - y'$ and I do not think my $\frac{d^2f}{dt^2}$ is correct please help. I…
user63574
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Least Squares Approx.

If I was using a least squares approximation of the form $y = A_1 + A_2\sin(wx) + A_3\cos(wx)$, would you be minimising the function $\sum_{i=0}^n (y_i - (A_1 + A_2\sin(wx) + A_3\cos(wx))^2$ ? I've never tried this for periodic data before!
Sanya
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Show that a sequence converges linearly

Given the sequence converges to $p^*$, show that it converges linearly: $p_{n+1}=\frac{1}{2}ln(p_n+1)$, $p_0=1$, and the limit is $p^*=0$. I want to use fixed point theorem and denote $p_{n+1}=g(p_n)$ to show that $g'(p^*)\neq0$. But I'm not sure if…
rrz.math
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Check if the problem is well condtioned

I'm trying to check if the problem of calculating the sum of two numbers a and b is well conditioned, provided that |a| > 2|b|. In my solution i split it into cases: 1: a>0 and b>0,then f(a,b)=a+b>3b C(a,b)=Ca(a,b)+Cb(a,b) where Ca(a,b)=|a(1)/(a+b)|…
Adam Stawarek
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