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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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Discuss the problem of determining a polynomial of degree at most $2$ for which $p(0)$, $p(1)$, and $p'(\zeta)$ are prescribed

Discuss the problem of determining a polynomial of degree at most $2$ for which $p(0)$, $p(1)$, and $p'(\zeta)$ are prescribed, $\zeta$ being any preassigned point. So, I started by using the interpolation polynomials that I know: Newton's…
Username Unknown
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Finite volume method

Using finite volume method, Consider the problem: $-U_{xx}=f(x)$ for all $x$ in $(0,1)$ and $u(0)=u(1)=0$. And consider the admissible mesh. Let $U^*i$ denotes the mean value of $U$ on $K_i($control volume$)$, How can I show that…
Math1995
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Choosing bounds to construct cumulative frequency table for Newton's forward interoplation.

If we have to generate a cumulative frequency table in the context of applying Newton's forward interpolation, how do we know when to construct the cumulative frequency table with lower interval and when to construct with upper interval. For…
yathish
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On changing the base of $x<1$ from $10$ to everything else

Assume that we have a number $x$ which is less than $1$ and is written in base 10. For example, if we want to write $x$ in base $4$, The algorithm says that each time we multiply $x$ by $4$. Then $4x$ has an integer part and the rest is the…
Arman Malekzadeh
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Apply Bairstow's method with the initial point $(u,v)=(3,1)$. Compute the corrections $\delta u$ and $\delta v$.

For the polynomial $$p(z)=3z^5-7z^4-5z^3+z^2-8z+2$$ apply Bairstow's method with the initial point $(u,v)=(3,1)$. Compute the corrections $\delta u$ and $\delta v$. Since $u=3, v=1$, then we have $z^2-3z-1$. If we divide $p(z)$ by $z^2-3z-1$, then…
Username Unknown
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Is this how to obtain discretization for a second order derivative (backward difference, second order accurate)?

The backward difference formula with second order accuracy (BDF2) for a first order derivative is $$d_tx = \frac{3x_{n+1}-4x_n + x_{n-1}}{2\Delta{t}}$$ I am attempting to generate the BDF2 formula for a second order derivative. I can't find anything…
Cogicero
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Midpoint method, error and tolerance

I am asked to find the root of a function, using the midpoint technique. Now I have some definition questions. In one question I am asked to find the root with error $\delta \le 10^{-3}$. In another question I am asked to find a root wit tolerance…
Sylvester Stallone
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Truncation error in approximation of $\frac{e^x - e^{-x}}{2x}$

I have a function $\frac{e^x - e^{-x}}{2x}$. In order to avoid loss of significance when calculating values of this function near $x=0$, I represent $e^x$ as Taylor series. The truncation error of $e^x$ is $\frac{x^5}{5!}*e^x$ (If i use 4 members of…
John
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numerical integration and integrable singularities

Consider the following integral $\int_{0}^{1} f(x) dx$ If $f(x)= \text{exp}(x)/ \sqrt{x}$, we have an integrable singularity at the origin. If $f(x)= \text{exp}(x)/ x$, we have a non-integrable singularity at the origin. There is no point in…
Smirk
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Find $y = P(x) $ for $x=4,18$

Let $P(x)= ax^2+bx+c$ whose coefficients are approximate numbers: $a= 2,51$, $b=-0,89$, $c=2,84$ Find $y= p(x) for x=4,1800000000$ and find $\delta_y$ the absolute error of the answer. My attempt: To find $y$ all I have to do is plug in the value.…
Sylvester Stallone
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Calculate population after time

I want to know how to calculate following We have one female that gives birth to one child every year for 10 years. Out of 10 children 5 are female and after 20 years they can reproduce as well. How could I calculate total population after specified…
user3256672
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spacing between two floating point values in a normalized system?

Let x be a given nonzero floating-point number in a normalized system, and let y be an adjacent floating-point number, also non-zero. a) What's the minimum possible spacing between x and y? b) What's the maximum possible spacing between x and y? I…
solobyy
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Numeric mathematics, solve an equation with two unknowns.

Given the relation $k=450nl+55n+205l+25$ where $k,n,l$ are all integers with $n,l
bijan karimi
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Numerical Analysis resources

What are some of the best sources for a course of Introduction to Numerical Analysis? I am looking for something which will start from basic and goes up to the advanced level. Books/ Lecture notes/ Video lectures any suggestions are welcome. Thanks…
Germain
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Taylor series and approximations of functions

I am suppose to approximate it in two different ways. one by just using one taylor series at one point and the other way by breaking it down into segments and integrating. Which is suppose to be more accurate. The second way is supposed to be better…
Tiger Blood
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