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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 difference actually between big O and small o in numerical methods?

I am studying numerical methods for ODE. I am confused about the "o"s there. What is the difference actually between big O and small o in numerical methods? I have read somewhere that the big O means the convergence order. Am I right? If so, what…
learner
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Solveing integral with numerical methods

I want to solve this integral with numerical method (for example Trapezoidal). $u$ in It's dominant is variable. My question is Can I use the numerical methods for solve these integrals?
user115927
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Which of the following iterations converges to the fixed point $x = 3$

Suppose that $x_0$ is sufficiently close to $3$. Which of the following iteration $x_{n+1}$ = $g(x_n)$ will converges to the fixed point $x = 3$ ? $x_{n+1} = -16 +6x_n +\dfrac{3}{x_n}$ $x_{n+1} = \sqrt{3+2x_n}$ $x_{n+1} = \dfrac{3}{x_n -…
user120386
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Choosing interpolation points

We want to approximate a function $f$ with a second-degree interpolating polynomial in the interval $[-1,1]$. I need to pick three interpolation points such that the interpolation polynomial $p$ has the best possible approximation. I need to argue…
user111199
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Inner product with machine numbers

Suppose we are working with a machine that does arithmetical calculations with a relative accuracy of $\xi, |\xi| \leq \xi '$, if we are working with machine numbers. We want to calculate the inner product $\langle x , y \rangle = \sum_{i=1}^{n} x_i…
user119470
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What is the maximum natural number n, so that all polynomials of degree n are integrated exactly with the formula

What is the maximum natural number n, so that all polynomials of degree n are integrated exactly with the formula $$ \int_a^b f(t) \approx \frac {b-a} {4} (f(a) +3f(\frac {a+2b} {3})) $$
user123763
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Why is $w$ a function of $x$?

I have a question. I am looking at the proof of the error of the Simpson's rule. Let $p_{3} \in P_{3}$ be the polynomial that interpolates $f$ at the points $x_{0},x_{1},x_{2}$. Then, from the interpolation error, we have…
evinda
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Upper bound for interpolation polynomial

I'm practising for my exam and here is a problem that I have difficulties to solve: For a given sequence of different nodes $\left\{x_k \right\}_{k=0}^{n} \subset [a,b]$, let $\left\{l_k\right\}_{k=0}^n$ denotes Lagrange basis polynomials and…
xan
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Minimum number of function evaluations to numerically estimate gradient

Let's say I have a function of $N$ input variables, $f(x_1,x_2,...,x_N)$. I want to numerically estimate the gradient of the function at a particular point in space, $\vec{x}_0$. Lets say my function takes a very long time to evaluate. At minimum,…
digbyterrell
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Composite Simpson's Rule vs. Adaptive Simpson's Method

I was playing in Matlab when the following occured: I had an integral $I$ which I computed with the (a) Composite Simpson's Rule and (b) Adaptive Simpson's Method (my teacher told me that the built-in function quad in Matlab uses this method). These…
bobbo
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Monotone convergence Newton-Raphson

Let $g(x) = x^2 \ln (x)$ and consider the equation $g(x) = a$ with $a = \frac{-3}{2e^3}$. The solution of this equation is given by $x = e^{-3/2}$. Note that $g''(e^{-3/2}) = 0$ and on a neighbourhood of $e^{-3/2}$ we have $f(x)f''(x) < 0$ with…
user119470
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navier stokes finite element method

I have discretized the navier stokes equation in 2D but when I used Newton method to deal with the nonlinear, I don't know the shape of jacobian matrix (you can look at this finite element for incompressible Navier Stokes equation. p28-p30)…
hung
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How would you solve this.

Find the constants $c_0, c_1,$ and $x_1$ so that the quadrature formula $$\int_0^1 f(x) dx = c_0 f(0) + c_1 f(x_1)$$ has the highest possible degree of precision.
eChung00
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If you want to know $\sin (x)$ within $0.5$ of its true value, then how accurately do you need to know $x$?

If you want to know $\sin (x)$ within $0.5$ of its true value, then how accurately do you need to know $x$? I don't really understand how to think about this question.
user87274
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Proving boundedness for a FTBS numerical scheme

Given an FTBS scheme $\phi_j^{(n+1)} = \phi_j^{(n)} - c \left(\phi_j^{(n)} - \phi_{j-1}^{(n)} \right)$ where $c$ is the courant number, $n$ is the timestep, and $x$ is the spatial index, how can I prove that no new extrema are created at timestep…
hertzsprung
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