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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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Necessary conditions for convergence in the fixed-point iteration algorithm?

What are the necessary conditions for convergence of the fixed point iteration algorithm? One condition I have come across is that if $|g'(x)|<1$ for all $x$ in some interval $[a,b]$ where g is continuously differentiable in [a.b] then the iteration…
Jamminermit
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Numerical Differentiation Using Lagrange Polynomials

Here is a snapshot of Richard Burden's "Numerical Analysis" where he is discussing numerical differentiation using Lagrange polynomials with node points $x_0,x_1$ and $x_2$. I cannot fathom how he derived the formulas using the variable…
mali1234
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About initial value $x_0$ in Newton's method

For the root $a$ of the function $\displaystyle{f(a)=\frac{a}{\sqrt{1+a^2}}=0}$ I want to use the sequence $\{x_n\}_{n\geq 0}$ such that $x_{k+1}=x_k-f(x_k)/f'(x_k)$ then for what initial value $x_0$ that the sequence converge to $a$ and how to…
user76608
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How to calculate the error of $x-(x^3-1)^{1/3}$ substituting $x$=35

The equalation $f=\ln({x-\sqrt[3]{x^3-1}})$ is calculated for $x=35$. The root is calculated by taking 8 significiant digits (after the decimal point). What is the error in calculating $f$? find at least one solution which gives better…
user65985
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Solving a fifth power equation without using Newton's method

Is there a way to solve the following equation without using Newton's method: $$x^5-x-1=0$$
Andrea
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Order of convergence for Crank-Nicolson method for classical Black-Scholes problem

I was trying to solve numerically the classic Black-Scholes (B-S) problem $$\frac{\partial{u}}{\partial{\tau}}=\frac{1}{2}\sigma^2x^2\frac{\partial^2{u}}{\partial{x}^2}+rx \frac{\partial{u}}{\partial{x}}-ru,~x\in (0,\infty),~\tau\in(0,T),$$ with …
Riaz
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Iterative method for solving a linear system - convergence

I have to solve the following linear system using an iterative method : $\begin{cases} &4x_1+x_2 = b_1 \\ &x_{i-1}+4x_i + x_{i+1}=b_i,\ i=2 \cdots n-1\\ &x_{n-1} + 4x_n=b_n \end{cases}$ The iterative method is $\begin{cases} x_1^{k+1} =…
Artemus
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A numerical inverse problem

I have a function of several variables defined by k different couples. I want to invert it. I guess this is an inverse problem. But I don't know what to look for to solve it. Here is a more formal explanation of my problem: Let f be a bijective…
ths1104
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Is there a formula for this interest over a period

Say I have £100 overdraft and my balance is currently £0. when I go into my overdraft I am charged 1.25 % at the end of each day, but I have to divide the rest of the money equally for the next 10 days to see what I can realistically afford. Is…
Dave
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How to estimate maximum possible error if plug in $\sqrt2$ instead of $1.41$ in the expression?

Suppose we want to calculate, $$0.2-0.8\times \frac{4-1.41^2}{20}$$If I use the approximation $1.41\approx\sqrt2$ the above expression will be equal to $0.12$. I'm wondering how to estimate maximum possible error we have made by using this…
Etemon
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Proof that relative error $\leq \frac{5 \times 10^{-n}}{a_m}$

Assume that $a = a_m \times 10^m + \dots > 0$ is an approximation of $A$ with $n$ correct significant digits. Prove that: $$ \delta(a) \leq \frac{5 \times 10^{-n}}{a_m} $$ I know that if $a$ has $n$ correct significant digits then $$ \delta(a) < 5…
Mathemagician
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estimate solution for a 'almost' monotone increasing function

Here is an interview question: $f:[0,1]\rightarrow [0,1],$ a 'almost' monotone increasing function. Given 10 inputs and their outputs, how can you find the best estimation of $f(x) = 0.5?$ I think using interpolation is best here. However almost…
user6703592
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Linear regression least square method

Right now, I'm studying linear regression using the least squares method. So, if $f(x)=ax+b$, I have to find $$\min\sum(y-ax-b)^2.$$ But why does this minimum exist, and is there only one minimum or multiple of them?
Nick_Dowson
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If $x_0$ is sufficiently close to $z$, the sequence ${x_n}$ converges to $z$

I have some troubles with the following problem: Suppose for any $x_0$ in reals, the sequence$\{x_n\}_{n=0}^\infty$ satisfies the inequality $|x_{n+1}−z|≤4|x_n−z|^2$ for n= 0,1,2,...Thus if $x_0$ is sufficiently close to $z$, the sequence ${x_n}$…
Mihai.Mehe
  • 369
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A clarification on "Runge–Kutta–Fehlberg" method

To apply the RK45 method to a system of ODEs represented as: $$\dot{\overline{z}}=\overline{F}$$ At each time step you should first compute \begin{align*} R_1 &= h\cdot \overline{F}(t_n,\overline{z}_n)\\ R_2&= h\cdot…
Charith
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