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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 to find the convergence rate of the Newton method for the function $f(x)=x^a$?

The Newton method is $x_n=x_{n-1}-f(x_{n-1})/f'(x_{n-1})$? But in the function $f(x)$ there is no $n$ component for substitution, so I am unsure how to go about this?
chloe loughan
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Asymptotic Order versus Order of Convergence.

I'm taking my first numerical analysis course. I'm trying to understand the difference between order of convergence and Big O notation (asymptotic order). If I have two algorithms and algorithm A is Big O of $h$ while algorithm B is Big O of $h^2$…
Idle Fool
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Intuition on $L$ in Euler's error

An upper bound for the global error in Euler's method to solve a first order ODE numerically is given by the equation \begin{equation} \frac{Mh}{2L} ( e^{L(t_{i+1}-a)} - 1), \end{equation} where $t_{i+1}$ is the $i+1$ cell in the abcsisa and $a$…
Herman Jaramillo
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Example of hyperquadratic convergence through Newton method.

I took a class about numerical analysis. And when f(x)=sin(x) and what to do is find approximation of Pi. Then in Newton method, it has hyperquadratic convegergence, and actually its order is 3. I don't know why. (I tried to solve this problem…
최선웅
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Theorem on Horner's Method: Prove that $p(x)=\alpha_0$, $p'(x)=\beta_0$, and $p''(x)=2\gamma_0$.

Concerning the polynomial $p(x)=a_0+a_1x+\dots+a_nx^n$, prove the following result. For a given $x$, we set $(\alpha_n,\beta_n,\gamma_n)=(a_n,0,0)$ and define inductively…
Username Unknown
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Rewriting a function to avoid loss of significance

Let $f(x) = \sqrt{1+x} - \sqrt{1-x}$ where $\vert x \vert \le 1$ Would this mean that if $\vert x \vert \lt \epsilon$ where $\epsilon$ is the machine epsilon we would get $f(x) = 0?$ which would mean a loss of significant digits? I'm thinking of…
user137481
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Approximate $e$ with $4$-digit chopping and $5$-degree Maclaurin-Taylor polynomial

Number $e$ is defined as $$ e=\sum^\infty_{n=0} \frac{1}{n!}.$$ Use four-digit chopping arithmatic to compute the following Approximations to $e$ $$ e\approx \sum^5_{n=0} \frac{1}{n!}$$ Attempt: \begin{align} \sum^5_{n=0} \frac{1}{n!} &=…
Tiger Blood
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Linear and cubic interpolation

There are four $(x,y)$ coordinates $(-5,-1), (0,0), (5,1), (8,4)$ and using linear and cubic interpolation I intended to find the value of y at x=3. My solution For linear interpolation I picked $(0,0)$ and $(5,1)$ as they are close to each other.…
etet112
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How many significant decimal digits are lost in performing the subtraction?

In the Floating point number system $ \ F(10,10,-98,100) \ $ , subtract $ \ \sqrt{1001 } \ $ from $ \ \sqrt{1002} \ $ . How many significant decimal digits are lost in performing the subtraction ? Answer: In $ F(10,10,-98,100) \ $ , $…
MAS
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Numerical computation of a formula with erf() and exp()

I need to compute the following formula numerically using double precision: $\frac {\frac{1}{2 \pi^{2} \sigma} \left(\pi^{2} \sigma \operatorname{erf}{\left (\frac{1}{\sqrt 2} \left(\frac{r}{\sigma} + \sigma\right) \right )} - \pi^{2} \sigma…
Pastafarianist
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For the series $e^{\tan x}$, estimate the series finding the best integer value possible using o- and O-notation.

Consider the series: $$e^{\tan x} = 1+x+\frac{x^2}{2!}+\frac{3x^3}{3!}+\frac{9x^4}{4!}+\quad...\qquad (|x|\le\frac{\pi}{2})$$ Retaining three terms, estimate the remaining series with o-notation for the best integer value possible, as $ x \to 0…
liveFreeOrπHard
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Calculate each of the following rounding to **the number of digits shown**

Calculate each of the following rounding to the number of digits shown . Further find the smallest interval using true values instead of rounded values, must lie : (a) $ \ 23.46-12.753 \ $ and (b) $ \ 2.747*6.83 \ $ Answer: (a) $23.46-12.753…
MAS
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Showing stability of a multi step method (numerical analysis)

How would I show the stability of the following three step method $y_{n+3}-y_{n}=h[\frac{3}{8}f(t_{n+3},y_{n+3})+\frac{9}{8}f(t_{n+2},y_{n+2})+\frac{9}{8}f(t_{n+1},y_{n+1})+\frac{3}{8}f(t_n,y_n$)] using the root method I am having trouble doing this
Fernando Martinez
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Find the points $x$ at which $y-\ln(x+y)=0$ defines $y$ implicitly as a function of $x$

Find the points $x$ at which the equation $$y-\ln(x+y)=0$$ defines $y$ implicitly as a function of $x$. I'm not really sure what is meant by this question. If I define the function $g(x,y) = y - \ln(x+y)$ then it is implicit. Does it mean for…
Dragonite
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