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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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Euler Method Failed Approximation

I used the euler scheme to approximate the solution of the IVP $$\frac{du}{dt} = \frac{8e^{-t}}{2\pi u^2 \cos(\pi t/2)}, \qquad u(t_0) = 0.4$$ on the interval $I = [0,5]$. However, the method failed to approximate a solution and obtained…
Jas
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Thomas algorithm and pivoting

$L_E = \begin{bmatrix} 1 && 0 && 0\\ l_{21} && 1 && 0\\ 0 && l_{32} && 1\\ \end{bmatrix}$ and $U = \begin{bmatrix} u_{11} && u_{12} && 0\\ 0 && u_{22} && u_{23}\\ 0 && 0 && u_{33} \end{bmatrix}$ I am using Thomas algorithm but i don't know how can…
Luai Ghunim
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Why don't they use 4-point formula to approximate derivative of a function

I am currently taking a numerical analysis course, whose main reference is the book "Numerical Analysis" by Richard L. Burden and J. Douglas Faires I am preparing for a presentation about numerical differentiation. One of the main reference is the…
ElementX
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What should be the sign of the diagonal elements of the Hessian matrix?

I am doing an optimization using Newton-Raphson, when I am setting the initial values of the parameter vector that makes the Hessian matrix is negative semi-definite as necessary condition for maximal point, i got a Hessian matrix with negative…
N. I. ElZayat
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composite rule; trapezium vs. Simpson

Suppose the interval $[a,b]$ has been divided into $2m$ intervals by the points $x_i=a+ih$, and $$ h=\frac{b-a}{2m}. $$ The Composite Simpson rule says…
Sha Vuklia
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Understanding the graph of linear system solvers comparison

What do you think what is the vertical axis here? I would guess it's the error but it's increasing with the iterations... Thank you.
bencemeszaros
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Starting solution of a Newton Method

I was going through some computations and I found something strange about Newton's Method. If anyone can verify it, it would be of great help. Newton's method is given by $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. Now, say I have a non differentiable…
Abhinav Jha
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Order of Convergence- Iterative Methods

I have a review question that asks us to find the order of this iterative method: $U^{n+3}=U^n+\frac{3}{8}h[f^{n+3}+3f^{n+2}+3f^{n+1}+f^n]$. I really do not understand how to do this, since we've talked about the method of…
bananagurlz
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Check numerical stabilityand correct the equation

I am struggling with a function that supposedly has a numerical error. I checked for consistency which seems to be fine but I couldn't find the stability error nor attempt to correct it. where c = 1-r^(-d) and d =[1:100] The function is struggling…
Yrden
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Python function to find the order of a quadrature formula

I am required to code a function that finds the order of a given quadrature formula and then apply the function to the mid-point rule and the two point Gauss quadrature formula. I have the following theorem to work with- Let $b_i$ and $c_i$ be the…
Eiraus
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Find $ x_3 \ $ using iteration method

Answer: We have to use Newton Raphson method. $ x_1=0.75 \ \Rightarrow f(x_1) \approx 2.2 , \ \ f'(x_1)= ? $ I can not find out what is $ f'(x_1) \ $ ? Can anyone help me?
MAS
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Derive the approximation $\int_{a+h}^a f(x)\;dx =\frac{h}{2}[f(a) + f(a + h)] + \frac{h^2}{12} [f_0(a) − f_0(a + h)] + O(h^5)$

Let $f$ be a smooth function. Derive the approximation $\int_{a+h}^a f(x)\;dx =\frac{h}{2}[f(a) + f(a + h)] + \frac{h^2}{12} [f_0(a) − f_0(a + h)] + O(h^5)$
David
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estimation of error in numerical methods

I need to implement the Log function using Taylor series, and I need that the error will be smaller than epsilon, which is input by the user. How can I estimate the error in each iteration and validate when it is smaller than such…
user213158
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Derive Richardson's method to central divided difference

I was given this Numerical Analysis assignment but I can't understand it. It says derive Richard's Method to $$f'(X_0) = \frac {f(X_0+h)-f(X_0-h)}{2h}$$
Muhammad Ahsan Mukhtar
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Does determining the truncation error of a numerical scheme depend on the problem the scheme is applied to?

Does determining the truncation error of a numerical scheme depend on the problem the scheme is applied to? For example, If Crank-Nicolson is applied to heat equation, does that impact the truncation error?
MathIsHard
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