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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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Intervals for Newton's method

I have a function $$ F(x)= \frac{x^3 - 14x^2 + 7x + 203}{(x-3)(8-x)} $$ I need to use Newton's Method to find the max interval such that a number of constraints are valid. • $3 < a < b < 8$, • $f\in C^2[ a, b ]$, • $f\left( \frac{2}{3}a +…
KevinCameron1337
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Newton Raphson Step Size

I am solving old exams and I came across the following question: Let $$ x_{n+1} = x_{n} - \alpha\frac{f(x_{n})}{f'(x_{n})} \;\;,\;\; f(x_{n}) \gt0 \;\;,\;\; f'(x_{n}) \neq0 $$ Is it true that there exists an $0 \lt \alpha \lt 1$ such that…
Hila
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What are two easiest numerical method to calculate $\sqrt[23]{123456789}$ by hand?

What are two easiest numerical methods to calculate $\sqrt[23]{123456789}$ by hand? I want to compare that two methods myself Just name the methods
Victor
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How to compute ratios when the numbers are extremely small and numerical issue might arise?

Consider that a set of numbers that can be written in the form $b^{x}$ (for some base): $$A = \{ a_1, ..., a_n\}$$ for example maybe $\{ 2^{-2000}, 2^{-2002}, 2^{-2500}\}$. Also assume we can compute the numbers indirectly because we know its…
Charlie Parker
  • 6,546
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How can I solve $x$ from $\int_0^x e^{t^3}dt=4$?

I'm interested in numerical analysis but I don't have experience on it. I was wondering how one can solve integral equations numerically like $\int_0^x e^{t^3}dt=4$? I was thinking whether there is some numerical differential equation solution…
curious
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Exponential curve fit

I want to fit a curve of the form $y = ab^x +c$ where a, b and c are constant whereby i have a data of points $(x_i, y_i)$ I can reduced my primary equation into a form $log(y - c) = log(a) + xlog(b)$ I need a $w_i$ such that $w_i = log(y_i - c)$…
Toye_Brainz
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construct $\mathcal{O}(h^2)$ finite difference scheme for $(a(x)\cdot u'(x))'$ operator

Obviously, Taylor expansion for $(a(x)\cdot u'(x))'$ is to be used somehow, but I'm not sure how to start at all... The scheme that I'm looking to derive is actually $\frac{(a\cdot u_{\bar{x}})_{x}+(a\cdot u_{x})_{\bar{x}}}{2}$, where…
Crni
  • 647
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Iterative equation

I have an equation that I want to try and solve iteratively. I don't have any background in numerical analysis so unsure as to how to go about it. Any help would be greatly appreciated. My equation is: $\hat{\tau}^2 = \left. \sum\limits_{i=1}^k…
denby47
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Numerical Analysis, divided differences

This is what I have to prove: $$f[x_0, x_1, \dots, x_n] = \frac{(-1)^n}{(x_0+a)(x_1+a) \dots (x_n+a)}$$ where $f(x) = \frac{1}{x+a}$ and $f[x_0, \dots, x_n]$ is the divided difference of $f$ in these points. I know that in the proof is used…
brick
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Numerical Stability

In my numerical analysis class we have been working on approximating functions with Maclaurin Series. I am sort of confused by the definition of what makes an algorithm numerically stable. I understand that if you introduce error in the initial…
freddrick
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Taylor Expansion on Newton's Method

I get that the Newton's Method is can be derive from the first Taylor Expansion, which is $$ f(r) = f(x_{0})+f'(x_{0})(r - x_{0}) + \frac{f''(\xi)}{2}(r - x_{0})^{2} $$ with $r$ is the root of $f(x)$, the question is, why only takes first term of…
Yagami
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Find the lowest value of a parameter for which two functions intersect

I am attempting to an equation to determine the lowest value of $\lambda$ for which $f(x) = \lambda \sin ( \pi x)$ and $y = x$ intersect outside of 0 on the interval $[0,1]$ for some numerical analysis I am doing. Would anyone be able to show me how…
sarah
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estimate for highly oscillatory superexponential integral

I would like to estimate $\int_{-\pi}^{\pi} e^{i n y} e^{-b e^{c y^{2}}} dy$ to within a RELATIVE error of better than 1%, if possible. Here, $n$ is an integer and $b$ and $c$ are positive. The imaginary component is zero since $\sin ny$ is odd…
user179959
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How does this equation hold (Secant method)?

Consider we are approxinating a root by the secant method. Then, the interation is given by $x_{n+1}=x_n - f(x_n)\frac{x_n - x_{n-1}}{f(x_n)-f(x_{n-1})}$. In my text (Atkinson), it's written that: After some manipulation, it's possible to show that…
Rubertos
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What is the easiest way to check whether the function is Globally Lipschitz continous or Locally?

What is the easiest way to check whether the function is Globally Lipschitz continuous or Locally? Say,for example the function, $sin(x^2)$? Many thanks in advance.
learner
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