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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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Convergence Newton's Method for this system

I'd like to prove that the following non-linear system $$ F(x) = \begin{pmatrix} x_1^3 + x_2^3 - 4 \\ x_1^3 - x_2^3 \\ \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ \end{pmatrix} $$ will converge when using the Newton's Method for the start…
Clash
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Deriving a New Iteration Method by Solving a Quadratic Equation

My Question: Derive a new iteration method for solving $f(x)=0$ by solving the quadratic equation $$f(x_k)+f'(x_k)(x-x_k)+\frac{1}{2}f''(x_k)(x-x_k)^2=0$$ Complete your algorithm by specifying which root to choose, and prove cubic convergence under…
user312492
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Fixed Point Iteration Proof

Given the fixed point iteration $$x_{n+1}=\frac{-x_n^2-c}{2b}$$ where $b$ and $c$ are fixed, $x_n\longrightarrow x$, what does $x$ solve? Additionally, what is the region for $(b,c)$ values where our iteration converges at a rate of $O(2^{-n})$ or…
user310867
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Symplectic integration of harmonic oscillator

I try to get numerical solution of ordinary harmonic oscillator with symplectic integrator. The problem is that what I obtain doesn't conserve energy (but symplectic integration should do). I consider 2 simplectic integrators - symplectic Euler…
newt
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Error analysis for Runge Kutta, how to take Big O of 2 variables?

For the standard 4th order Runge Kutta: where the system is assumed to be smooth (so that the RHS has no discontinuous points) $\mathbf{y'} = \mathbf{F}(t,\mathbf{y})$ $\mathbf{y(t_0)} = \mathbf{y_0}$ $\mathbf{y_{i+1} = y_n + 1/6(k_1 + 2k_2 + 2k_3…
Lemon
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Given $f(x) = \sqrt{x^2+1}-x$, determine the value of $f(42.545)$ in 4-digit rounding arithmetic

Suppose $f(x) = \sqrt{x^2+1}-x$ is to be computed in 4-digit rounding arithmetic. (i) Determine the value of $f(42.545)$ in 4-digit rounding arithmetic, and the relative error of this value. Notation: $fl(a)$ means 4-digit rounding of the value…
Danxe
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Newton's method of finding roots of an equation.

Consider Newton's method on finding the roots of $x^3-x=0$, how to show that $x_n$ converges to $1$ for any $x_0>1/\sqrt{3}$? My attempt: The Newton's method says $x_{k+1}=x_k-\frac{x_k^3-x_k}{3x_k^2-1}$. I tried to prove $|x_{k+1}-1|/|x_k-1|$ is…
Tony
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Showing that an iterative method solves a particular system

I have $A=\begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$, $b=\begin{pmatrix}3 \\ 5\end{pmatrix}$. The system to be solved is $Ax=b$. We're also given: $$B_\theta=\frac 1 4 \begin{pmatrix} 2\theta^2 + 2\theta + 1 & -2\theta^2 + 2\theta + 1…
Jack M
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Numerical Methods for Algebraic Equations - Non root finding

I'm researching a topic for solving general algebraic equations using numerical method. My numerical recipe knowledge is rather rusty with the Bisection to Newton's methods but I don't think those could be applied for equations such…
Display name
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Gauss Quadrature Proof.

Let $x_0
user214138
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Reduce a fraction by interpolation

I am trying to solve this problem: Use $ x^2+1$ (polynomial interpolation) to reduce $$ \frac{(x^2+1)}{x(x-1)(x-2)(x-3)}.$$ I don't know how I can reduce a fraction by interpolation method.
bad bishop
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Gauss quadrature rule that is exact up to cubic.

Obtain a Gauss quadrature formula for $$\int_{0}^{2}{f(x)}\sim A_1 f(x_1)+A_2 f(x_2)$$ that is exact up to cubics. I know how to find the solution of $$\int_{-1}^{1}{f(x)}\sim A_1 f(x_1)+A_2 f(x_2)$$, but how do I proceed when I change the…
Anna_85
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Is there a case where one may prefer a lower order method of numerical integration over a higher order one?

Suppose you have a limit for the accuracy you need to acquire. Using, say RK4, will allow you to achive this with less time steps than, say Euler. Is there a particular scenario where using the Euler method will be more (computationaly) efficient,…
turnip
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Deriving Simpson's rule error from Lagrange polynomial

Let $L$ be the quadratic Lagrange interpolating polynomial for a function $f$ such that $f(-h)=L(-h)$, $f(0)=L(0)$, and $f(h)=L(h)$. Then it is well-known that $$f(x)=L(x)+\frac{f'''(\xi(x))}{6}x(x-h)(x+h),$$ where $\xi(x)\in(-h,h)$. Integrating,…
Doug Torrance
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Globally Convergent Methods for Nonlinear Systems of Equations

We recently switched from the basic Newton-Raphson to a more advanced globally convergent Newton’s method with Line Searches and Backtracking (see Numerical Recipes, Chapter 9.7). For some special cases many we still need to try too many initial…
abenci
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