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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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Curve adaptation with a finite number of measured values

I consider a function $f(t)=\alpha\cdot 2^{-t}+\beta\cdot 2^{-2t}$ and a finite number of measured values $f_0,...,f_n$. The coefficients $ \alpha, \beta $ are calcualted by the least squares method which means $$ \sum\limits_{i=0}^n…
hallo007
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$g \colon [0,1] \to [0,1]$ be a continuous map and consider the iteration $x_{n+1}=g(x_n)$.

I came across the following problem: Let $g \colon [0,1] \to [0,1]$ be a continuous map and consider the iteration $x_{n+1}=g(x_n)$.Then Which of the following maps will yield a fixed point for $g$? The options are as follows: a.…
learner
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Howuse this $R_{l}=\frac{1}{n}\left(\frac{(-1)^l}{2n}+\sum\limits_{m=1}^{n-1}\frac{1}{m}\cos{\frac{ml\pi}{n}}\right)$ and MATLAB get this four fig?

we consider Tikhonov's regularization method for $\delta =0.1, 0.01,0.001,$ and $\delta =0$ The Tikhonov's regularization method you can see:http://en.wikipedia.org/wiki/Tikhonov_regularization and apply this investigated regulaization strategies…
math110
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Definite Integral by Bernoulli numbers

How we can show the following definite integral on $[0,1]$? $\begin{eqnarray}‎ ‎ \int _0^1\left(\ln \frac{x}{1-x}\right)^kdx=(2^k-2)\pi ^k|B_k|, ‎\end{eqnarray}‎$ where $B_k$ are the $k$-th Bernoulli numbers, $k=1,2,\ldots$, respectively. Note that…
M. Raha
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Calculating $F(x)=\int_{-\infty}^{x} \frac{1}{\sqrt{2 \pi}} e^{-y^{2} / 2} d y$ with a two decimal precision

Consider the distribution function of the normal distribution, $$ F(x)=\int_{-\infty}^{x} \frac{1}{\sqrt{2 \pi}} e^{-y^{2} / 2} d y $$ We want to calculate the $F(1.96)$ with a two decimal precision. Devise a method for doing so, and find the number…
Sorry
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Finding where f is well conditioned

I am looking for help with part b. I'm not sure where f is well conditioned in a relative sense given that $\kappa(x)$ is a constant. Is f merely well condition at this point ($\frac13$)? If so, how do I know this? Is $\kappa(x)$ of "moderate size"?…
user551155
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How many digits of precision are necessary to get the integer part of exponentiation?

For a positive real number $x$, and an integer $n$, i'd like to compute $\lfloor x^n \rfloor$. The $n$ here will be quite large, so I want to know how precise my approximation of $x$ needs to be to guarantee $\lfloor x^n \rfloor$ takes on the…
eeegnu
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LU factorisation of singular matrix

Is it possible to LU factorise a singular matrix. If this can be done in multiple ways, then how would you go about finding the different decompositions?
user76025
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3-digit chopping arithmetic

I have to solve the following system of equations: $$3.3330x_1 + 15920x_2 + 10.333x_3 = 7953$$ $$2.2220x_1 + 16.710x_2 + 9.6120x_3 = 0.965$$ $$−1.5611x_1 + 5.1792x_2 − 1.6855x_3 = 2.714$$ should I chop off the numbers at the beginning like $$3.3330…
Hasan
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FIxed Point Iteration (numerical analysis)

The equation $ e^{x} - 4x^{2}=0 $ has a root between 4 and 5. Fixed point iteration with iteration function $ \frac{1}{2}e^{\frac{x}{2}} $ (A) diverges (B) converges (C) oscillates (D) converges monotonically I know that if we use the fixed point…
Mohit Sharma
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Finding a polynomial

Find a polynomial $q(a)$ of degree less than equal to $2$ that saitsifies the condition $q(a_0)=b_0, q'(a_0)=b'_0, \ \text{and} \ q'(a_1)=b'_1,$ where $a_0,a_1,b_0,b'_0,b'_1\in \mathbb{R}$, where $a_0\ne a_1$. And give a formula of the form…
Tom
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Lagrange basis function finding orthogonality

How can I find three points such that $t_0,t_1,t_2\in [0,1]$ are so that the interpolation of Lagrange basis functions $k_0(t),k_1(t),k_2(t)$ associate with them are orthogonal to each other, that is $\int_0^1k_i(t)k_j(t)dt=0$ for any…
user60514
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Secant Method; how many iterations needed for a certain accuracy?

Currently, I am following a numerical methods course. I came across the following question on an old exam, and don't know how to approach it: We have the function $f(x)=e^{-x} -5x+10$. First, I had to calculate $x_2$, while given that $x_0$ =2.0 and…
Math420
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Computing a numerical Hessian matrix and 2 questions on SAS v8 finite difference approximations for the entries

I am investigating the maximum value for $h$ for a multivariate function: $h = f(x_i)$ for $x=1 \ldots n$ and using the gradient method: $x_i = x_0 + \lambda \nabla x_i$ with $\lambda$ a small scalar to update $x_i$. However the function is…
Randall
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Interpolation polynomial derivative error

The chapter I'm looking at is 'numerical differentiation'. There's a formula that just got thrown in there out of nowhere and I've got no idea where it comes from. Anyway, let's get the notation out of the way. $f[x_i, ..., x_{i+k}]$ - denotes the…
Koy
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