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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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New algorithm for finding the inverse of a function?

Does anyone know the name of this algorithm for finding the inverse of a function? I came up with it for solving a particular problem, but the algorithm is so simple that I think it must be known already. I just can't find it anywhere. Given some…
questiondude
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rounding error bound and truncation error bound in forward Euler method

How should I derive the truncation error bound and rounding error bound in the forward Euler method? $f^{\prime}(x) \approx \frac{f(x+h)-f(x)}{h}$ I know that the bound for truncation error is 2M/h where h is the step size and rounding error bound…
Randy Chen
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Numerical Methods on Solving Systems of Differential Equations

I have the following set of differential equations $$\dot{x} = p_x$$ $$\dot{y} = p_y$$ $$\dot{p_x} = -\frac{\partial H}{\partial x}$$ $$\dot{p_y} = -\frac{\partial H}{\partial y}$$ For context, these are Hamilton's equations of motion so $H$ is the…
Dan
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Suppose $f(x)=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$. Then $\Delta^nf(x)=n!h^n$ and $\Delta^{n+r}f(x)=0,$ for $r=1,2,3,4,5... \infty$.

Suppose $f(x)=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$. Then $\Delta^nf(x)=n!h^n$ and $\Delta^{n+r}f(x)=0,$ for $r=1,2,3,4,5... \infty$.(Where we have equally spaced points $x,x+h,...,x+nh$ with corresponding values $f(x),f(x+h),...f(x+nh)$.…
Unknown x
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Order of convergence of the sequence $\{x_n\}_{n=0}^{\infty }$ given by $x_{n+1}=2x_n-\pi {x_n}^2$ with $x_0=1/3$

I would like to find the exact order of convergence of the sequence $\{x_n\}_{n=0}^{\infty }$ given by $x_{n+1}=2x_n-\pi {x_n}^2$ with $x_0=1/3$. I found that the limit of the sequence is $\frac{1}{\pi }$. But, I'm not sure if this method of finding…
john
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Hermite interpolation for 2 data points and one only one datapoint derivative available

Let $P(x)$ be a polynomial that satisfies: $P(0) = f(0)$, $P(2) = f(2)$, $P^{'} (2) = f^{'}(2)$ (a) Using Hermite interpolation, find $P(x)$. (b) Compute the expression $\int^∞_0 e^{−x}P(x)dx$ using the approximation from part (a). Proof: Using the…
Mihai.Mehe
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Is a problem well conditioned

I have a following task of determining whether this problem is well conditioned. I tried to solve it but I am stuck because I don't know how to understand the last step. Is a problem of calculating the root of the polynomial $p(x) = ax + b$ well…
Cahir7
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absolute error and relative error with at least 3 digits chopping for $\frac{{1}}{A}$ $A=\frac{{e}}{2.71828}-2.71828* e^{{-1}}$

absolute error and relative error with at least 3 digits chopping for $\frac{{1}}{A}$ $A=\frac{{e}}{2.71828}-2.71828* e^{{-1}}$ My solution: \begin{align} A=\frac{{e}}{2.71828}-2.71828* e^{{-1}} &= 1.34530544*10^{{-6}} \end{align} So \begin{align}…
tent123
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Given constants $c_i$ and $K$, find $b$ numerically such that $b^{c_1} + b^{c_2} + \dots + b^{c_n} = K$

Given constants $c_i > 0$ and $K > 0$, find $b > 0$ numerically such that $b^{c_1} + b^{c_2} + \dots + b^{c_n} = K$. I'd like to solve this with a non-iterative method if possible. My attempt at isolating b (applying the identity $\log(x + y) = \log…
Monty Thibault
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Prove that that the convergence is cubic when the sequence converges to $\xi$

I would appreciate help interpreting the suggested solution for the task below, I cannot see how they get that $\varphi(x)=x-\frac{1}{2}u(x)-\frac{1}{2}[u(x)/u^\prime(x)]$: Question: $f(x)$ is a four times continuously differentiable and have a…
kabin
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Computing functions with discontinuity

what is the best way to compute the value of the following function in a computer program, e.g., C/Fortran, in double precision? $$ g(a,b)={{e^a/a - e^b/b}\over{a-b}} + {1\over{ab}}. $$ The above function has removable discontinuities at a=0, b=0,…
oukore
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Generating digits of sqrt of 2

If I wanted to generate a billion digits of sqrt of 2, what is the quickest way of doing that? (I am talking about the mathematical methods and the programming issue, not hardware memory limit problem) Today I coded a program in php, using…
Jane N.
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Get a value using Richardson extrapolation

I would appreciate help with how to get a value using Richardson extrapolation. This is the question I have: Calculate $\int^{1/2}_{0}e^x$... (a) to a six decimals, by determing the primitive function (b) with the trapezoidal rule, step length…
kabin
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Good initial guesses for $25\times 25$ system of equations numerical solution

I need to solve a $25\times 25$ very non-linear system of equations, I have very superficial knowledge of numerical methods, and I've been struggling in finding a real solution to my system. For what I know, in order to find a solution first initial…
manifold
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How to solve the projectile problem with numerical method in matlab

i wanna ask how to solve the projectile problem using matlab? could you give me the source code in matlab? the equations is x"=-(1+0.1*x)^2 , with x(0)=0, x'(0)=1. thanks before.
landrex
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