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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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Studying numerical methods

Is there a book which I can self-study numerical methods needed in engineering and which proves results rigorously? I would like to learn engineering mathematics needed in every day life as well as prove the methods.
selfstudent
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Numerical problem

The value of 1001 to the power 3?. Any trick for quick answer?
Vikash
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Bound on numerical integral

I'm running a numerical integral over a function which I can only know on specific points as the problem is defined on a lattice. I've been using a trapezoidal method which has for known error: $$\Delta = \frac{1}{12} \text{mesh}^3 \;*\text{length}…
Learning is a mess
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Taylor Approximation

For $f(x)=e^x$, find a Taylor approximation that is in error by at most $10^-7$ on [-1,1]. Using this approximation, write a function program to evaluate $e^x$. Compare it to the standard value of $e^x$ obtained from the MATLAB function exp(x);…
user141745
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How to find convergence point for a given iterative scheme

The equation $x^2+ax+b=0$ has two real roots $\alpha$ and $\beta$. Show that the iterative method given by $\displaystyle x_{k+1}=-\frac{(ax_k+b)}{x_k}$ is convergent near $x=\alpha$, if $|\alpha|>|\beta|$
prat
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Elementary Numerical Analysis2

Just wanted to confirm my answer here again: Q: Evaluate $$p(x)=1-\frac{x^3}{3!}+\frac{x^6}{6!}-\frac{x^9}{9!}+\frac{x^{12}}{12!}-\frac{x^{15}}{15!}$$ as efficiently as possible. How many multiplications are necessarY? Assume all coefficients have…
user141745
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Numerical analysis Taylor $1/(1-x)$

How does accuracy depend on the degree of the Taylor Polynomial and the distance from the point its being expanded about (say $x=0$). So I'm considering the function $f(x) =\frac{1}{1-x}$ centered at $0$. I have found up to the 4th degree Taylor…
Daniel
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Show that $\phi'(x)=0$

Let $ f \in C^2([a,b], R)$ and $$\phi(x)=x-\frac{f(x)}{f'(x)}$$ Such that 1.$ f'(x)\neq 0, \forall x \in [a,b]$ $\exists \alpha \in [a,b]: f(\alpha)=0$ Show that $\phi'(x) = 0$ My attempt: $$\phi'(x) = 1 -\frac{f'(x)f'(x) - f(x)f''(x)}{(f'(x))^2}…
Giiovanna
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relation between euler function and divisors $n$

Please hint me. Let $S$ be the number non trivial divisors $n$. prove that $ S<\phi(n)+1$. $\phi$ is euler function. hint: we know $ \sum_{d|n}\phi(d)=n$ so $ \phi(1)+\phi(n)+\sum_{1,n\not=d|n}\phi(d)=n$, thus…
M.Mehranian
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Finding the fixed point of $\exp(-x^2)$ numerically

Consider the function $g(x)=e^{-x^2}$. With a starting approximation of $p_0=0$, use the iteration scheme $p_n=e^{-p_{n-1}^2}$ to approximate the fixed point on $[0,1]$ to within $5 \times 10^{-7}$. So, our teacher did not go over this section,…
Brian
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difference equation-numerical analysis

Assume $E$ is the difference equation. We have that $E^3(E-2)=0$ The solution should be $x=(a_1\ a_2\ a_3\ 2a_4\ 4a_4 … 2^n a_4…)$ But what is the basis of the solution space? I think it should be $(0\ 0\ 0\ 2\ 4\ 8 …2^n…)$. Because $a_1,a_2,a_3$…
claire
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interpolate a function in a little interval

I want to interpolate a function that I have amount of it in this points: $0:.001:1$ and I have interpolate it in this points: $0:.0005:.001$. When I use lagrange method, I have a very bad result. I think it happens because of my interval…
user169140
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Is there a formula that can be used to determine the number of iterations needed when using the Secant Method like there is for the bisection method?

The formula used to find the number of iterations needed to find a root of a function using the bisection method is this; $$|c_n-c|\le\frac{|b-a|}{2^n}.$$ Is there a formula that can be used to determine the number of iterations needed to find a…
Jed
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Jacobi Method for the linear systems (for first two iterations)

I have the following 3 linear systems: $$4x_1 + x_2 - x_3 = 5$$ $$-x_1 + 3x_2 + x_3 = -4$$ $$2x_1 + 2x_2 + 5x_3 = 1$$ for x^(0) = 0 Then, I write it as following: $$x_1 = -1/4x_2 + 1/4x_3 + 5/4$$ $$x2 = 1/3x_1 - 1/3x_3 - 4/3$$ $$x3…
user156407
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Solving $f(x) = e^{-\sin(4x)} - 3/4$ with 3 digits after the decimal point correction

I need to solve $f(x) = e^{-\sin (4x)} - 3/4$ with 3 digits after the decimal point correction, but cannot find out how. I'd really appreciate it if anyone could point me to the solution. I think I may need to use iterative methods , but it would be…
itamar
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