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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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Numerical methods for root curve finding

Suppose I have a continuous function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ and I want to numerically find the curve(s) of points $$ f(x, y) =0 $$ By finding I mean constructing an array of coordinates that represent the curve. Are there any well…
Eddy
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Numerically find minimum of 2 dimensional surface

I have a large equation that essentially takes two variables and returns a real number. I know that the two values are real numbers between 0 and 10. Is there a standard way to numerically find the minimum point on the surface? I'm multiplying two…
dan
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What algorithm is used to calculate $\exp(i 2 \pi/1024)$ in FFT?

For the fast Fourier transform algorithm, a prerequisite is calculating quantities like $\exp(i 2\pi/ 2^n)$. How is this done? Is there any specific algorithm tailored for this kind of special angles?
poisson
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Explain why the RK2 method should compute the exact solution

Suppose y solves the initial value problem y(1)=1, y′(t)=2t for t>1. Explain why the RK2 method should compute the exact solution. Use this fact to give an expression for α in terms of b. I'm going back through old exam papers in preparation for my…
EvanSwan92
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Best Mean Square approximation.

I am trying to teach myself about the best mean square approximation of a function. I can't seem to find much informtion on internet. Could someone help me go through the best mean square approximation of a function $f= \dfrac{1}{x+2}$. It would be…
Sylvester Stallone
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Is it common in numerical analysis to make a change of variable when the condition number is high?

I want to approximate a function $f(x)$ on [a,b]. The condition number for computing a function $f$ at a point $x$ is defined by $$\kappa=\frac{x f^{\prime}(x)}{f(x)}$$ For my function $f$ this grows without bound as $x \rightarrow b$. So I make…
AUK1939
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How to check convergence

Please help me how to proceed in this question is it correct if I use iteration formula and perform iteration....
taniya kapoor
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Precision of measurements and of a number

I know the definitions of both the precision of measurements and that of an approximate number (sort of). I want to know how they relate to each other. I understand the relationship between the accuracy of a measurement and that of an approximate…
Not Euler
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Converting data set (x,y) into a function f(x,y) .

Is there any way by which I can find an equation satisfying my data set , I know there are a number of curve fitting methods in numerical methods; but I want a new function ( equation ) for my data set . e.g. any function like log(a(tan(bx))) . I…
Anshuman Sinha
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Condition of zeros of a polynomial

To every parameter $ x \in ( - \infty , 1]$ let $f(x)$ be the biggest real zero of $ p_x (z) :=z^4 - 4z^3 + 6z^2 - 4z+x$. How can I determine $ f(x) $ by substitute $z=u+1$ und calculate the relative condition of $f$ at $x=1$? Would be very helpful…
wondering1123
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What are multiderivative numerical schemes?

I've been desperately looking for a clear definition, but I have not found one. Why the name multiderivative? What does it refer to? What are the unknowns that are updated from time step to time step?
BinaryBurst
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How (if possible) can I determine the intersection between a line, and a function (whose input arguments are free variables)?

I have a function f(x,z) that returns a height y in a terrain. I want to be able to determine where a line P intersects with the terrain. A point P=[px,py,pz] along the line is defined as such: P = O + D*t, where O is the origin of the line, D is…
birgersp
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Determine the order of the local truncation error (LTE) of a numerical integration method

Given $y(t)$ the initial-valu problem: $y'=f(t,y),$ $y(0)=y_0$ I need to find the order of the local truncation error for the following numerical integration method: $w_{n+1}=w_n+hf(t_n+\alpha h,w_n+\alpha hf(t_n,w_n)), \alpha ∈ [0,1]$ I know I…
Walt Janssens
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Develop integration method

I am asked to find the coefficent $C$ and the nodes $ x_i$ such that the following formula will be accurate for any cubic polynomial, $\int_{-1}^{1}f(x)dx=C[f(x_0)+f(x_1)+f(x_2)]+E$ applying the formula on $\{1,x,x^2,x^3\}$ yields 4 equations for…
Rubenz
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