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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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Is it possible to solve the equation $\nabla_x F(x) = G(x)$ with respect to x, where F is a known scalar field and G is a known vector field?

Is there a numerical method to solve this equation, considering that we cannot write out the specific expressions for F or G, but can numerically compute the gradient or function value information at a certain point x? Can anyone provide some…
梁港林
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Issues with Lerch Phi and Abel-Plana formula branch cuts & integral divergence

Introduction I have taken the approach of using the Abel-Plana formula to evaluate the Lerch Phi as follows: $$ \Phi(z,s,a) = \frac{1}{2 a^s} + \frac{(-\log(z))^{s-1}}{z^a} \Gamma(1-s, -a \log(z)) - 2 \int_0^{\infty} \frac{\sin(t \log z - s…
Kamila Szewczyk
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How to rewrite $x^3 - 6 \sqrt{3}=0$ for faster numerical root finding process?

I want to find the root of $x^3 - 6 \sqrt{3}=0$ using the bisection method (implemented thru C++) and I found it to be approximately $2.18244...$ but the number of iterations is 36. Is it possible to rewrite the orginal equation so that there are…
user1158304
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Why does the fixed point algorithm only converge to one fixed point?

For example, the function $f(x)=x^2$ has fixed points $(0,0)$ and $(1,1)$, but the algorithm can only converge to $(0,0)$. Does anyone have a good explanation of why this is so?
Clay Lang
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Why is the only $k$ step method with stiff decay BDF?

I'm studying for a test and I'd like to know how justify why the only $k$-step method of order $k$ with stiff decay is BDF. By definition of stiff decay(Ascher & Petzold) a method has stiff decay if \begin{equation}|y_n-g(t_n)|\rightarrow \…
Dave
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Lagrange interpolating polynomial what should i do when question does not specify the degree

this question in my homework and this is not the first one that doesn't specify the the degree of Lagrange interpolating polynomial in this case, what should I do?
hn_gara
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Is this proof of the error of trapezoidal rule valid?

In my numerical analysis lecture notes, the following proof of the error of (simple) trapezoidal rule of $f\in C^2[a,b]$ is presented, based on the error of a linear polynomial interpolation $p$ (with nodes $a,b$): $$ \begin{align*} &\quad \int_a^b…
durianice
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Just a calculation

Every day I have a problem while having simple calculations. Can anyone give a Tip how to overcome? For Example : How one will calculate the remainder of $ 2^{546}\pmod{43} $ ? Without a calculator ?? And how to go about this?
joey rohan
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Left preconditioning: Why do we multiply with $P^{-1}$ and not with $P$?

In the numerical analysis class we learnt about preconditioning. What the professor explained is the following: If in the problem $Ax=b$ that we want to solve the matrix $A$ is ill-conditioned (so $\kappa (A)$ is "large"), we try to find the matrix…
mikasa
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How to discretize a equation using central differentiation method?

I am new to numerical methods, in my understanding, the central differentiation method is used to calculate derivatives such as $$\frac{dy}{dx}=\frac{y(x+h)-y(x-h)}{2h}$$ How could we use this to discretize equations?
jjk
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how to show that matrix B is invertible and specific to a particular summation

I need to show that if the matrix $A$ is invertible, and $\|B-A\| < \|A^{-1}\|^{-1}$, then $B$ is invertible. I was able to prove that B is invertible $$(A^{-1}B)^{-1} = B^{-1}A = (I-A^{-1}B)^k$$, thus $$B^{-1} = \sum_{k=0}^{\infty} {(I-A^{-1}B)^k…
user88616
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How do find rate of convergence in newton's method with multiple roots?

I'm trying to solve the following problem. I've proven the two equalities, but can't figure out how to prove that last bit about the rate of convergence being equal to $\log_{10}(2)$.
John Smith
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Question about stopping criteria for bisection method.

I was just told the following: The stopping criterion in the bisection method is $|b − a| < \epsilon$. However for most other algorithms it is is $|f(x)| < \epsilon$. These two stopping criteria are within a factor of approximately $|f'(x^*)|$…
John Smith
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Variational formulation and change of coordinate

I was wondering if a change of coordinate (e.g cylindrical change) could affect the variational formulation with respect to the metric. I mean, does the metric $\mathbf{dx}$ which appears in the following form, for any domain $\Omega \subset…
Amzocks
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Newton's method for $f(x)=\arctan (x)$

I am trying to apply Newton's method for $f(x)=\arctan (x)$. There is already a post showing how to solve this: As $g(x)$ is an odd function, if $x_0$ is a root, then $x_1=-x_0$ is a root as well. The periodic sequence…
Devita J
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