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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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What does the function domain with letter C stand for?

I am reading a mathematics textbook on the subject of numerical analysis. In one theory the author says let us assume $f$ to be a function in $C^{n+1}[a,b]$. I understand that $[a, b]$ is the interval limits. What I don't understand is "what the…
Dr. Ehsan Ali
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Error and residue in linear system

Consider the linear system $A x = b$ with a computed solution, what will be the relation between error and residue?
Neha
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How many Gauss points are required to provide exact value for the Gauss quadrature rule

How many Gauss points are required if the Gauss quadrature rule should provide the exact value of the integral $I=\int_{-1}^1f(x)dx$ for $f(x)=(x^2-1)^2$? I am really not sure what theorem to use to solve this problem. What I can think of is a…
user71346
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Some trivial but confusing terms about numerical integration

Some terminological questions about numerical integration: When a question states trapezoidal rule with 2 points, does that mean 2 subintervals or 3 subintervals? Since 3 subintervals have 2 points other than the end points. How about a Gaussian…
user71346
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Avoiding loss of significance without series.

How could the function $$f(x)=\frac{\sin x}{(x^2+1)^{1/2}-1}$$ be computed to avoid loss of significance? I know that $$f(x)=\frac{\sin x((x^2+1)^{1/2}+1)}{x^2}$$ But $x^2$ has a problem.... How to solve this problem??
asfajaf
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What is the typical $\epsilon$?

In doing some self-learning in numerical methods I having come across the following a number of times. $\epsilon$ is the smallest computational unit such that, $x + \epsilon = x$. What value does $\epsilon$ typically take? (Am not sure if this is…
Sharky
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Find condition number for vector of roots

Consider $f(z)=z^2+az+b$ we put it's root in vector $[z_1, z_2]^T \in \mathbb{C}^2$. Find condition number in maximum norm of finding the root vector when changing variable a where $a=-2, b=3$ I know in general that for function $g$ finding it's…
Jessy
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Proving an equivalent to a summation formula

Let us recall the summation formula $$\sum_{k=1}^nk=\frac{n(n+1)}{2}$$ How do we show that $$\sum_{k=1}^nk=\frac{1}{2}n^2+\mathcal{O}(n) ?$$ I started by stating the definition of "big-o" notation where we let $f$ and $g$ be two functions defined on…
cryptomath
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Polynomial interpolation with evenly spaced data.

Given the following data table, x 1.20 1.25 1.3 1.35 1.40 1.45 1.50 f(x) 0.1823 0.2231 0.2624 0.3001 0.3365 0.3716 0.4055 What degree of polynomial required to exactly fit all 7 points to within 3 significant digits? I…
k7dy
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Discovering unknown function input by evaluation

Suppose that we have a function: $\tilde U(\tau ,\omega ) = U(\tau ,\omega )f(\tau ,\omega ,Q)$ By evaluating this function, I would like to be able to find $Q$. Suppose that I don't know the explicit form of $U(\tau ,\omega )$ but I do know the…
Nicholas Kinar
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Solve the equation by Jacobi's method : $5x + 2y +4z = 12$ ;$x+4y=2z=15$ ;$x+2y+5z=20$

Solve the equation by Jacobi's method : $$5x + 2y +7z = 30$$ $$x+4y+2z=15$$ $$x+2y+5z=20$$ I am getting the values as $$x=4.08,y=3.375,z=4.12.$$ My equations : $$x=\frac{1}{5} \{30-2y-7z\}$$ $$y=\frac{1}{4} \{15-x-2z\}$$ $$z=\frac{1}{5}…
PurpleShark
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Backward Euler's Method

This question was asked in CSIR. please help me to find out correct choice Let $y(t)$ satisfy the differential equation $$y'=\lambda y;y(0)=1$$. Then the backward Euler method for $n\geq 1$ and $h>0$ $$\frac{y_n-y_{n-1}}{h}=\lambda y_n;\quad…
Learner
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Proof of error propagation formula?

In my course we have stated and used the error propagation formula: $$|y-y_0|\approx|f^\prime(x)|\cdot|x-x_0|$$ But it was presented with no proof and I wonder if you can help me understand the formula holds?
Niklas Rosencrantz
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If $y=(3x+1)(3x+4)\cdots(3x+22)$ prove that $\Delta^4 y=136080(3x+13)(3x+16)(3x+19)(3x+22)$

This is an assignment question from the topic 'Interpolation' I tried to begin this sum by calculating $\Delta^4=y_4-4y_3+6y_2-4y_1+y_0$
Jignesh Shah
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How to find numerically stable version of a function?

What's the general procedure for finding a numerically stable form for a function? Such as: $$\frac{1}{1+2x}-\frac{1-x}{1+x}$$ when $x≈0$ or $$ln(x)-1$$when $x≈e$ I've only seen examples, but cannot grasp the "procedure" for what to look for in…
mavavilj
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