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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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Error term with trapezoidal rule

I have the integral $\int_{0}^{h} e^{x}cos(x) dx$ and then I am asked to find the highest error term for this integral with trapezoidal. My textbook denotes the highest error term |E| to be $- \frac{h^3}{12}f''(c)$ Since $f''(x)= -2sin(x) e^{x}$ I…
user420309
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Using trapezoidal rule to calculate an improper integral.

I have numerically calculated the integral $$\int_{-1}^{1}\frac{e^{-x^2}}{\sqrt{1-x^2}}dx$$ using Gauss-Legendre and Gauss-Chebyshev quadrature. Now, I am asked to calculate the integral using the trapezoidal rule and compare the different methods.…
John Keeper
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How to implement Steffenson method using Goertzel method for evaluating function values?

The task is to implement function in Matlab that will solve numerically following function: $\sum _{k=0}^na_k\cos \left(kx\right)$ using Steffenson method and Goertzel for evaluating the series. I have working Steffenson method part, but no idea…
Piotr Pilis
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Convert Linear Multistep Method into one-step method

When given an nth-order ordinary differential equation, it is possible to make a simple algebraic manipulation to convert the differential equation into a first-order "autonomous" differential equation. Is there a similar transformation from a…
TylerMasthay
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Convergence Rate

So I have the polynomial $f(x)=8x^{4}-12x^{3}+6x^{2}-x$ And I was trying to find the convergence rate of the newton iteration when started close to the root r=0 My attempt: Since $f'(r) \neq 0$ we have quadratic convergence with the convergence…
user420309
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calculating rational approximations of arbitrary radical expressions

I need a way to find a rational approximation of any radical expression (say, $\sqrt{1+\sqrt[3]{2}}$) in such a way that I know a bound on the error. Given a rational number, I think I can always use Newton's method to approximate its nth root…
Alex Kindel
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Can Von Neumann (fourier) stability analysis be used for $v_t=Bv_{xx}$ with homogeneous dirichlet boundary conditions?

My book says that Von Neumann analysis can be used when boundary conditions are periodic, in other words it says when $v(0,t)=v(1,t)$. I am confused if pdes with homogeneous Dirichlet boundary conditions can be analyzed with the fourier analysis or…
MathIsHard
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Degree of precision of a quadrature formula (numerical integration)

I'm working on the Newton cote formula, and I would like to be sure if the only method to find the degree of precision is to compute each integral for each monomial. Example : $\displaystyle \int_{-1}^1f(x)\cdot dx =…
Stu
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Interpolate the values of $ \ f \ $ at $ \ x_0-h , \ \ x_0, \ \ x_0+2h \ $.

Interpolate the values of $ \ f \ $ at $ \ x_0-h , \ \ x_0, \ \ x_0+2h \ $. Use the Interpolant to find an approximation of $ \ f'(x_0+\frac{h}{2}) \ $. Answer: Then the Lagrange interpolant $ \ L(x) \ $ is given…
MAS
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2 Points, 3 points, 4 points Numerical Differentiation (forward)

http://www.shodor.org/cserd/Resources/Algorithms/NumericalDifferentiation/ How did they get the three points $$y'(x)=\frac{y(x+h)-y(x-h)}{2h}$$ to $$y'(x)=\frac{-3y(x)+4y(x+h)-y(x+2h)}{2h}$$ Where does the $-3y + 4y$ come from? It does not make…
Mikeez
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Explain '' Lagrangian interpolating method has complexity $ \ O(N^2) \ $ ''

Building an interpolation polynomial by solving a linear system has a complexity $ \ O(N^3) \ $ and that of Lagrangian interpolating method has complexity $ \ O(N^2) \ $. What does mean it ? Explain in short way. Answer: I think when solving a…
MAS
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Whats so special about the Landweber method that it justifies an individual name for ordinary gradient descent

the Landweber Method is imho nothing more than gradient descent on a specific quadratic form. https://en.wikipedia.org/wiki/Landweber_iteration so why does it warrant an own name?
CD86
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Truncation error of Backward Euler applied to IVP

I am looking at the IVP $u'(t)=f(u(t),t)$, $u(0)=n$. I am using this method $$\frac{U^{n+1}-U^n}{k}=f(U^{n+1},t_{n+1})$$ where $U^n$ is meant to approximate the solution $u(t)$ at $t=t_n=nk$. So far, I have that the location truncation error…
MathIsHard
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Find $α_0, α_1, β_0$ and $β_1$: Quadrature Rule

If $\int_0^1 f(x)dx ≈ \alpha_0f(0) + \alpha_1f(1) + \beta_0f'(0) + \beta_1f'(1)$, find $\alpha_0 , \alpha_1 , \beta_0$ and $\beta_1$. I have attempted this and when I work it out I don't find a value for $\beta_0$? Can someone tell me from my…
IanWalter
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Find a quadrature rule that is exact for all polynomials of degree 2 or less

Where am I going wrong? I want to prove that it's accurate for deg 2 polynomials or less and my answer doesn't suggest that? Any tips on where I'm wrong ? Ive attached an image of my work below!
IanWalter
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