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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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help me to prove this condition number!

I want to prove this, but I have some questions. If we define $f_n$ like this. $$ f_n = (-1)^n \frac{d^n}{dx^n}(\frac{e^{-x}}{x}) \quad n =0,1,2,... \quad f_0 = \frac{e^{-x}}{x} $$ Prove this. $$ (cond f_n)(f_0) =\frac{1}{|e_n(x)|} , \quad e_n(x) =1…
haleh
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Find the polynomial p of least degree that takes these values

Find the polynomial p of least degree that takes these values: $p(0) = 2$, $p(2) = 4$, $p(3) = −4$, $p(5) = 82.$ Use divide differences to get the correct polynomial. It is not necessary to write the polynomial in the standard form…
Shammy
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Estimate the value $ \int_{1}^{2}\int_{3}^{4}{f(x,y)dydx} $

Using the Gauss-Legendre form. Estimate the value of$$\displaystyle\int_{1}^{2}\displaystyle\int_{3}^{4}{f(x,y)dydx}$$ where, $f(x,y)=x^3y$. My approach: We can approximate the integral $\int_{-1}^{1}{f(x)dx}=\sum_{i=1}^{n}{A_{i}f(x_{i})}$ with the…
Ahna Akbar
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Numerical Instability with this cumulate

I am trying to compute this cumulate. However, with very high or very low values of x, the computation is affected by numerical errors. $$p=1-\frac{2 u}{\sqrt{2 \pi } l \sqrt{t} e^{\frac{(l t+u x)^2}{2 t u^2}} \left(\text{erf}\left(\frac{l t+u…
Vaaal88
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Showing the leapfrog scheme is consistent for $u_t+au_x=0$.

To show that a scheme is consistent we define a smooth function $\phi$, and consider the following: $$\frac{\phi_m^{n+1}-\phi_m^{n-1}}{2k}+a\left(\frac{\phi_{m+1}^{n}-\phi_{m-1}^{n}}{2h}\right)=0$$ I'm stuck on showing how to demonstrate this. I…
emka
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Really confused on how to implement the Secant Method

I'm really confused on how to implement the Secant Method. If the secant method is used on $f(x) = x^5 + x^3 + 3$ and if $x_{n-2} = 0$ and $x_{n-1} = 1$ what is $x_n$? I know I need to use $x_{n+1} = x_n - \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})} *…
Shammy
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Newton method for solving this problem

Problems involving the amount of money required to pay off a mortgage over a fixed period of time involve the formula $$ A = \frac Pi\cdot [1 − (1 + i)^{−n}], $$ known as an ordinary annuity equation. In this equation, $A$ is the amount of the…
Isma Haro
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What does the notation 5(4) , 3(2), etc. mean exactly in Runge-Kutta pairs?

The Runge Kutta methods usually are listed as pairs like 5(4), 3(2), and the like. Do these mean the maximum value of $i$ in $k_i$ in the RK methods. But yet that would be only 1 number. What is the other? And do these pairs occur only as adjacent…
newtonian
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Horner's Method

This is a question from a computational mathematics course. An error analysis of Horner's rule shows that the computed value of the polynomial satisfies $$\hat{c}_{0} = (1 + \theta_1)\alpha_0 + (1 + \theta_3)\alpha_1 x + \ldots + (1 +…
Wolfy
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Numerical Mathematics, Ch.2 problem 5

Let $\hat{x}\neq 0$ be an approximation of a non null quantity $x$. Find the relation between the relative error $\epsilon = |x - \hat{x}|/|x|$ and $\tilde{E} = |x - \hat{x}|/|\hat{x}|$ I believe we need to study the conditioning number of both…
Wolfy
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Numerical Methods: Float Aritmetic Sum

I am in a course of numerical methods and I have a question: If I have the harmonic sum: $\sum_{i=1}^{n}\dfrac{1}{i}$ and I can approximate it by "rounding" it. This rounding can be defined as: $S_n = fl(S_{n-1} +a_n)$ where the $a_n$ are the…
Adolf
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show a derivation of this recurrence relation

One of the approximations for $\sqrt3$ is $x_{n+1} = \frac{x_{n} + 3}{x_{n} + 1}$. I can see clearly why this is true, since if the sequence converges, $x_{n}$ approaches $x_{n+1}$ (then multiply out). But its really bugging me because I can't find…
jg mr chapb
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Fixed Point for finding a root

I need to solve this equation (find $\lambda$) using numerical methods: $\displaystyle N_0e^{\lambda}+v\frac{e^{\lambda}-1}{\lambda}-N_1 = 0$ All other terms are constant and known. N0 = 1000000; v = 435000; N1 = 1564000; I need to solve it by using…
dpalma
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Find an algorithm to calculate the following function

I'm solving questions from an exam I failed, and I would love some help with the following question: Question We want to calculate the following function in Matlab: $$ f(x) = \frac{e^{x^2} - (1 + x^2)}{x^4} $$ We know that as $x$ approaches $0$,…
Hila
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What is the sufficient global/local convergence condition of inverse quadratic interpolation?

Given a root-finding function f(x)=0, what is the sufficient global/local convergence condition of inverse quadratic interpolation?
Bill Locke
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