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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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Requirement(s) for consistency of Runga Kutta methods?

I know that for the RK2 method to be consistent we must have $a + b = 1$ in the following equation $$\begin{aligned} y_{i+1} &= y_i + h(ak_1 + bk_2)\\ k_1 &= f(x_i, y_i)\\ k_2 &= f(x_i + \alpha h, y_i + \beta hk_1) \end{aligned} $$ However, for…
sonicboom
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How many iterations are necessary to reduce error in the Jacobi and Gauss-Seidel method?

I reedited the question: We want to use Jacobi method to solve the following system: $$ \begin{bmatrix} \pi^2 & -1 & 0 & 0 \\ -1 & \pi^2 & -1 & 0 \\ 0 & -1 & \pi^2 & -1 \\ 0 & 0 & -1 & \pi^2 \end{bmatrix} \times \begin{bmatrix} x_1 \\ x_2 \\ x_3…
Silas
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Finding the upper bound on an expression.

I am trying to find the smallest $N_0$ such that $$\max_{n\leq N}\frac{1}{4NL}\bigg(\bigg(1+\frac{h}{N}\bigg)^n-1\bigg) \leq 10^{-4}$$ for $N \geq N_0$ I guess that this is equivalent to…
user197848
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Solving linear systems with LU factorization, defining numerical stability?

Lets say you have a linear system $Ax = b$ you want to solve and you use LU factorization for it. Can this system ever become numerically unstable? We know that if the condition number of $A$ is $1$, the system is numerically stable but if the…
osk
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Solving $y' = \ln\ln(4+y^2)$ using Euler's method

Consider the following problem $$y' = \ln\ln(4+y^2), \quad x\in[0,1], y(0)=1$$ We can formulate the problem the approximate the solution $$y_{n+1} = y_n + h \ln \ln(4+y_n^2), \qquad n = 0,1,...,N-1, \quad y_0 = 0$$ with mesh points $x_n = nh$. I am…
user197848
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Problem related to finding roots using Newton-Raphson method

I am really stuck on problems $9$ and $10$. Here is what I tried to do: I can use Lagrange's interpolation formula to get the approximating polynomial, however, in order to apply Newton Raphson's method to find roots, I must get exactly the…
Abhishek Shrivastava
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Newton Raphson method initial guess

I am facing problems in making the initial guess while applying Newton-Raphson method to equations of more than one variable.Is ther some thumb rule for making the initial guess in the general case?To be specific let the system of equations be…
AgnostMystic
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Error in 4th order RK method , given two different solution with different step sizes

Using 4th order RK-method the computed solution to an ODE at $ t=10 \ s $ computed from $ \ t=9.9 \ s \ $ with time step $ \ \Delta t=0.1 \ $ is $ \ 24.1 \ $. But with half -step size $ \Delta t=\frac{0.1}{2}=0.05 \ $ the same value becomes $ \ \…
MAS
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roots of shifted Legendre polynomial

How can we find roots of shifted Legendre polynomials? I found table of roots of Legendre polynomials but nothing about shifted ones.
Maths Ruba
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Least Squares Problem with function

My professor gave us this problem and I'm still trying to figure it out. Given $f(x)=c_{1}sin(\pi x)+c_{2}cos(\pi x)$ Minimize $(f(0)-5)^{2}+(f(\frac{1}{4}))^2+(f(\frac{1}{2})+1)^{2}$ He gave us some partial work, but I don't understand it. $\left…
Erock Brox
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How do I graph the region of absolute stability for the block mode in a continuous linear multistep method?

solving the block mode I am trying to find the region of stability for the block mode given in the image
OdunayoO
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The fourth order accuracy of $f''(x_0)$

Could you please give me formulas for the fourth order of accuracy $O(h^4)$ of the $``$forward and backward$"$ difference methods to find the approximation of $f''(x_0)$?
Ahmed
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2 point quadrature calculation

I was curious about how to calculate an approximate value of a definite integral with the 2 point quadrature formula. So I did a test. We all know that: $$\int_{0}^{\pi} \sin(x)dx = 2$$ Now I tested the quadrature approximation with.…
Erock Brox
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Lagrange Cardinal Function Proof

How can I use the Lagrange interpolation polynomial $$p(x) = \sum_{i=0}^n ℓ_i(x)f(x_i)$$ that interpolates $f(x)$ at distinct points: $x_0 , x_1, ..., x_n$ where $ℓ_i(x)$’s are cardinal functions to show that $$\sum^n_{i=0}ℓ_i(x) = 1$$, for all x?
StealzHelium
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Iterative method for finding a tangent to circle, through a given point

As in the title. Alternatively, the problem can be stated as, given $a, b \in \mathbb{R}^2$ and $r \in \mathbb{R}$, find $x \in \mathbb{R}^2$, such that $$ \begin{align*} (x - a) \cdot (x - b) &= 0 \\ |x - a|^2 &= r^2 \end{align*} $$ Here $a$ would…
Lieven
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