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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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What are the biggest and smallest represent-able numbers with single precision floating points?

I am trying to understand the limits of the floating point representation. On a 32-bit computer with 7 bits for the exponent and 24 bits for the mantissa, I want to know the biggest and smallest numbers. My calculation: Base 2 Biggest positive…
Christophe B.
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use duplex method to find square root of given number

I was reading this Wikipedia page about finding square root of given number. But I did't understand the section on the "Vedic duplex method". Not because of lack of English vocabulary, but main problem is that it is very massive documentation and …
dato datuashvili
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Gauss Kronrod quadrature rule

Given the abscissae and weights for 7-point Gauss rule with a 15-point Kronrod rule (Wikipedia); Can anyone provide me a working example how to numerically integrate a function given below: $$\int_0^1 x^{-1/2}\log(x) \textrm{d}x = -4 $$ Provided…
fahad
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Is it possible to solve initial value problem with central difference scheme?

Lets say I have an ordinary differential equation: $$\frac{dy(x)}{dx}=a \cdot e^x$$ I would like to solve this equation numerically. I could use for example Euler method or other explicit or implicit scheme to solve initial value problem (having one…
Misery
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Gaussian quadrature three-point

Derive the one and two-point Gaussian quadrature formulas for $$I=\int^1_0xf(x)dx\approx \sum_{j=1}^nw_jf(x_j)$$ with weight function $w(x)=x$. Which I know how to do and which I attached below (I will re-edit this sometime in the future by deleting…
user60514
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Why does the weak form approximate a function in some interval?

In Finite element analysis, we turn a differential equation into its weak form by multiplying both sides of the DE by a test function and integrate over small sections of the interval of interest. I know that one reason we use the weak form is that…
S. Rotos
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Does an attractive fixed point always have a derivative between 1 and -1?

Suppose $x_{k+1}= g(x_k)$ is fixed point iteration for some continuously diffrentiable $g(x)$. The theorem im learning says that if $g(r) = r$ and $|g'(r)| < 1$ then the fixed point iteration converges to $r$ for initial guess $x_0$ sufficiently…
James
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What functions can computers evaluate directly?

From Wikipedia computers can directly evaluate polynomials What precisely does direct evaluation mean? As far as I know, function evaluation can be difficult in complexity theory. I was wondering if polynomials are the only functions that…
Tim
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Condition number in numerical analysis

I learned that condition number of an operator/function is a measure of how sensitive the output of the function is with respect to a small change in the input argument. But then, what is the significance of condition number in the context of…
yumiko
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Trapezoid Rule Using Taylor Series

I am well aware of five approximate integration techniques: 1) Left endpoint approximation 2) Right endpoint approximation 3) Midpoint rule 4) Trapezoidal Rule 5) Simpson's Rule It is very natural to dervive all of these rules using Taylor Series…
user463792
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How to round 0.45? Is it 0 or 1?

This question is inspired by How to round 0.4999... ? Is it 0 or 1? I didn't quite understand the logic of the answer. It seems you round every decimal place no matter how far back it goes? In the case of 0.49999, you're rounding up the 9 increases…
user9638
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How many decimal digits of accuracy gained on each iteration of the bisection method

On each iteration of the bisection method the error is halved. So we gain one binary digit of precision on each iteration. I want to find how many decimal digits of precision are gained. So does this look alright - $$E_{k+1} = \frac{1}{2}E_k =…
dukenukem
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Bisection Method for root finding

My university teacher gave us a project. He wants us to create a program which will execute a modified bisection method for a function but not by diving everytime the range by $2$, but by dividing it with a random number. The process stops when $6$…
Georgio3
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The existence of real SVD

I have known that if matrix A $\in C^{m*n}$,then there exists a SVD $A=U\Sigma V$.My question is if A is real,does there exist SVD which U and V real?
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Numerical Analysis

I am trying to determine some numerical difficulties that arise from a couple problems, and a good way to re-write them to avoid those errors. For instance, I have: 1) $\sqrt{x+\dfrac{1}{x}} - \sqrt{x-\dfrac{1}{x}}$ where $x\gg 1$ I think that…
Samuel Gregory
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