Mathematics Stack Exchange
Mathematics Stack Exchange

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
1
vote
0 answers

Booles Rule derivation ( closed Newton cotes formulae)

Can anyone show me the derivation of Booles Rule. I've been studying the Closed Newton cotes formula and derived the rules up to degree n=3 Simpsonson Rule 3/8 but would love to see a derivation of Booles Rule. I'm aware it stems from the 3/8…
IanWalter
  • 41
1
vote
2 answers

How to find the growth rate for a certain year?

The following is the question. The world population growth rate at a certain reference year was 3.5%. Assuming exponential growth of population, after how many years, will the population of the world would have increased by a factor 16 ? I…
Venkat
  • 209
1
vote
0 answers

Proving cubic convergence of Halley and Euler-Chebyshov methods

It's my homework assigment and I don't expect written ready proof, but would like some guidance and hints about how to go on doing this. It is asked to prove that Euler-Chebyshov and Halley's iteration methods are cubic convergent, and for solution…
flowian
  • 541
1
vote
1 answer

Using Steffensen's method trying to prove $ \phi ' ( x ^ * ) = 0 $

So I have been reading numerical analysis and I encountered this: Assume that $ g $ is continuously differentiable, $ x ^ * = g ( x ^ * ) $, $ g ' ( x ^ * ) \ne 0 $, $ g ' ( x ^ * ) \ne 1 $. Previously given formula does not define $ \phi ( x ^ * )…
John.Doh
  • 101
  • 5
1
vote
1 answer

Name of the numerical method

I came across one numerical method, however I have lost the link or is not able to find it on my history. It used the following variables: $$G=\frac{p'(x_k)}{p(x_k)} \ \& \ \ H=G^2-\frac{p''(x_k)}{p(x_k)}$$ Or something to that effect, not entirely…
John Glenn
  • 2,323
  • 11
  • 25
1
vote
1 answer

Determine $x$-value given 3 points on a curve (Quadratic approximation)

I have 3 points ($x_0$, $y_0$), ($x_1$, $y_1$) and ($x_2$, $y_2$) that lie on a monotonically-increasing (asymptotic) curve (whose function is not known a priori). The only unknown value is $x_2$, which corresponds to reaching an asymptote (in the…
compbiostats
  • 115
  • 6
1
vote
1 answer

Fixed Point: Finding rate of convergence

I have $g(x) = \sqrt{1+\log(x)}$, I want to find the rate of convergence using fixed point iteration. I have confirmed that this is linearly convergent, because the absolute value of its derivative is less than $1$, but I want to know how fast it…
Corp. and Ltd.
  • 423
1
vote
1 answer

Calculating the total derivative for a Taylor method

I am attempting to set up a second-order Taylor method for a second-order ODE $$y''(x) = f(y'(x),y(x))$$ I have substituted a vector $\textbf{u}=(y,y')$ and then a new function $\textbf{g}(x,\textbf{u})=(u^{(2)},f(u^{(2)},u^{(1)}))$ To set up the…
James Hunt
  • 21
1
vote
2 answers

Alternating series numerical evaluation

Im trying to solve a question that is similar to this one, so any help with this one will be greatly appreciated. I know that the final answer is 84 terms. Consider the Taylor…
MathsRookie
  • 225
1
vote
0 answers

Different way of expressing the interpolation error

So I have got the problem about another way of expressing the interpolating error. And unfortunately I am clueless. The question is that I should show that it is possible to Express the representation error as…
user420309
1
vote
3 answers

How to find x in this case $\frac{x}{2}-1=e^x$?

$$\frac{x}{2}-1=e^x$$ Let say we can write above function as: $$\frac{x}{2(e^x-1)}=1$$ In Wolframalpha I get a solution $x\approx -1.59362$. How can I find x?
Melina
  • 937
1
vote
2 answers

$6.23 \times 5.4 = 33.642$ write down two more multiplications with the answer of $33.642$

For some reason I’m really stuck on how to work this out. I don’t know whether I’m just over complicating it for myself but I genuinely cannot for the life of me think of how to get the answers
Daisy west
  • 11
1
vote
0 answers

Stability of multiplication

Let $\alpha_i$ be a floating point number (say, single precision). Say I want to compute $\alpha = \alpha_1\alpha_2\alpha_3...\alpha_{n-1}\alpha_n$. Suppose I do the multiplication from left to right, and $x_c$ is the result of multiplication. Then…
Ted
  • 365
1
vote
1 answer

Discrete measures of orthogonal polynomials

Disclaimer this is a hw problem I DO NOT want a full answer I just want to know if I am on the right track. If d$\lambda(t)=$d$\lambda_N(t)$ is a discrete measure with exactly $N$ support points $t_1, t_2,...,t_N$, and…
Riemann-bitcoin.
  • 441
  • 2
  • 15
1
vote
0 answers

Absolute And Relative Error

I am given the following formulas for absolute and relative error for floating point $$\Delta=|x-fl(x)|\leq\begin{cases} \frac{1}{2}B^{e-p+1} & \text{For Rounding } \\ B^{e-p+1} & \text{For Chopping} \end{cases}$$ The relative error: $$\delta=\mid…
gbox
  • 12,867
1 2 3
…
99 100