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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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Newton-Horner Method Example

I'm struggling to understand Newton-Horner's method and to find information about it so pardon me if I seem a bit lost in expressing my question, my goal is to implement it in racket as a learning exercise. Given a polynomial $P(x)$ of $n$ degree…
Tristian
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Deducing that the rate of convergence of the Secant method is 1.6.18 (Golden Ratio?

I have a function $$f(x) = e^x - (2-x)^3$$ And I am using the Secant method to find a root between $0$ and $5$. I know that the value of this root is $t = 0.7261444\ldots$ Here's my output from matlab for 14 iterations where I show $k$,…
Jim_CS
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What does the symbol $\ll$ mean?

I came across, If $|y| \ll 1$, then $x/y$ may have large relative and absolute errors. I'm not sure what the symbol $\ll$ means. I looked it up on Google and nothing came up.
user12312
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Newton's method - error bounds

I just have a very brief question regarding the formula for error bounds in Newton's method. Depending on where you look, this will either be written as: $$e_{n+1} \approx \frac{f^{\prime \prime}(r)}{2 f^{\prime}(r)}e_{n}^2$$ or: $$e_{n+1} \approx…
Kristian
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Numerical method with convergence greater than 2

It is a well-known fact that, for solving algebraic equations, the bisection method has a linear rate of convergence, the secant method has a rate of convergence equal to 1.62 (approx.) and the Newton-Raphson method has a rate of convergence equal…
IgotiT
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When is Newton's Method guarenteed to converge

From what I understand, for Newton's Method to converge in the equation $$g(x_n) = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}$$ (1) $g(x)$ and $g'(x)$ must be continuous around the root. (2) The initial guess must fall within this interval. (3) The…
Richard Slabbert
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Numerical approximation of $\sin(11/8)$

How one can prove by pen and paper that $0.98089<\sin\left (\frac{11}{8}\right )<0.9809$? I was thinking some series proof but I'm not sure how to prove that the error is small enough. I also thought to use the formulas $\sin 3x=3 \sin x - 4 \sin^3…
studying
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Newton's method arctan

Why does it oscillate? I looked up the graph of it and I think it is convergent? And when the function is $0$, the solution is also 0. What is the difference of choosing diverse starting values? Thanks!
J.doe
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Precision check for the solution of equations in Numerical Analysis

In order to have precision of 5 decimal points in a Numerical Analysis method for the solution of an equation then: $$|x_{n+1}-\rho|\leq \frac{1}{2}\cdot 10^{5}$$ where $x_{n+1}$ the current approximation of the solution and $\rho$ the actual…
Adam
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Maximum accuracy in IEEE standard floating point arithmetic using bisection method

Just wanna make sure that I didn't make any mistakes. I use the bisection method to calculate $P_n$ and find out a pattern, which is $P_n= \left(-1\right)^{n+1}2^{-n}$. So the largest number that n can be is 1022 to avoid underflow. Is this…
J.doe
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Summed Newton-Cotes formula for 4 points applied to interval with 5 points

Given the Newton-Cotes formula for 4 points, can I do an integration over an interval where there are arbitrary many points (at least 4 points) given (so e.g. 4,5,6,7,...)? To be concrete, I have the Newton-Cotes formula $\int_a^b f(x)dx =…
user3185454
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Solving coupled 2nd order ODEs with Runge-Kutta 4

I'm having a hard time figuring out how coupled 2nd order ODEs should be solved with the RK4 method. This is the system I'm given: $x'' = f(t, x, y, x', y')$ $y'' = g(t, x, y, x', y')$ I'll use the notation $u = x',\ w = y'$, thus $u' = x'',\ w' =…
Paul Manta
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Convergence rate of the secant method

I'm having a little trouble doing the part the needs to be proven. The first part is the definition of approximate error and I just need some help showing that it is equal to the second part of this problem. If the Secant Method converges to $r$,…
J_Smith88
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Find roots of equation $(x^2+1)\cdot \arccos\left(\frac{2x}{1+x^2}\right)+2x\cdot \mathrm{sgn}(x^2-1)=0$

Find roots of equation $(x^2+1)\cdot \arccos\left(\frac{2x}{1+x^2}\right)+2x\cdot \mathrm{sgn}(x^2-1)=0$ One root is $x=1$ (checking functions $\arccos$ and $\mathrm{sgn}$). Second root is $x=0.442$. How to find the second root? How to check if…
user300045
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Calculating the x-intercept of the line that passes through the points $ (x_0,y_0)$ and $(x_1,y_1)$

I got this problem from the book Numerical Analysis 8-th Edition (Burden): Suppose two points $(x_0,y_0)$ and $(x_1,y_1)$ are on a straight line with $y_1\neq y_0$, Two formulas are available to find the x-intercept of the…
MathNerd
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