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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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Comparing stability of two methods.

In my textbook, there's an example in which we try to determine if $x^2-y^2$ or $(x+y)(x-y)$ is a more stable method. We do this by computing $\Delta_sr = \bar{F}(g) - F(g)$ and $\delta_sr = \frac{\bar{F}(g) - F(g)}{F(g)}$ in which $F(g)$ is the…
Auberon
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Order of accuracy of a numerical method

From this article, if some numerical method is second order accurate then we will obtain four times smaller error $E$ given that the step size $h$ is halved. Let say we have a numerical method for some partial differential equation. This method is…
Sukan
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Numerical derivative of sensor data in the presence of noise

I have a sensor producing bandlimited data at a predictable periodic rate, corrupted by IID white noise (at least over relatively short periods of time). There is also a slowly time-varying bias, which can safely be ignored as it is several orders…
Damien
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Determine Value of Constant for Iterative Convergence

The question I am trying to answer is stated as follows: The iteration $x_{n+1} = 2 - (1+c)x_n + cx_n^3$ will converge to $\alpha = 1$ for some values of $c$ (provided that $x_0$ is sufficiently close to $\alpha$). Find the values of $c$ for which…
cnolte
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numerical-methods, Fixed point theorem.

I am just looking to apply a result, so can someone confirm the following for me. Let say I have a equation of the form below: $V_1(x) = a + bV_0(x)$, where in theory $V_1(x) = V_0(x)$. I have an algorithm as follow: Start with initial guess…
chuck
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Condition number interpretation

I have a (nonlinear) problem with two variables, for which I computed a relative condition number as $$K_{rel}(x_1,x_2) = \max\{1, c\},$$ where I had $$\Bigg| \frac{f(x_1, x_2) - f(\tilde{x_1},\tilde{x_2})}{f(x_1, x_2)} \Bigg| = \Bigg| 1 \cdot…
jenna
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Relation between iterative methods and preconditioning , your thoughts

Let's suppose we have an invertible matrix P ( P from preconditioning ) : $Ax=b \Leftrightarrow Px = Px -Ax + b$ or $ Px = (P-A)x + b$ The iterative method which produced by the above is : $Px^{k+1}= (P-A)x^{k} + b$ . So the preconditioning is to…
chaviaras michalis
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Padé approximation

How do I use Padé approximation to transform a given function? Pade approximation is the best approximation in mathematics, compared to Taylors approximation & others. I have learnt this approximation transform d given function to another equivalent…
Oderinde Seun
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Numerical Solutions of ordinary differential equations

Numerically solving $y'=f(x,y)$ with $k_1=f(x_n,y_n), k_2=f(x_n+c_2*h, y_n+c_2*h*k_1), y_{n+1}=y_n+h(b_1*k_1+b_2*k_2)$ what would the local truncation error be? Also, how would we perform a Taylor expansion on this to show that it is second order,…
R.M
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Of significant figures and truthworthy computation

I have a question I picked on the internet, but I am not sure about the term truth-worthiness part of the question. Find the product of 346.1 and 865.2. State how many figures of the result are trustworthy, given that the numbers are correct to four…
Sylvester
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How to find the numerical error when we don't know the exact solution?

When some quantity $x$ (e.g., the values of a solution of a PDE, using a finite difference method) is calculated numerically, we get its approximate value $x^*$. The error is $|x-x^*|$. But since we don't know $x$ itself, how is it possible to find…
C.HICHAM
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Numerical method to solve $\sum\limits_{k=0}^{n}\frac{1}{2k+1} \lim\limits_{\sigma \to 0} \frac{d^{2k}}{d\sigma^{2k}} \left(f(\sigma,\alpha)\right)$

As part of a larger physical model I am currently searching for a solution to the following expression, a numerical solution is fine as I am ultimately really after the numerical result. $\alpha$ is a physical value and I need solutions for upto $n…
MarcF
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What degree should my hermite polynomial be

I constructed with a degree 6 polynomial but apparently even a degree 5 would suffice. $$\array{f(x)&=&ax^6&+&bx^5&+&cx^4&+&dx^3+ex^2+fx+g \\ f'(x)&=& &&6ax^5&+&&&\cdots \\ f''(x)&=&&&&&30ax^4&+&\cdots}$$ So it is now just a matter of substitution…
stackdsewew
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Where am I going wrong in my cubic spline, work out a,b and c

So I know $f^{'}_1(x)=6x+3ax^2$ and $f'_2$(x)=6x+3bx^2$ and $f^{"}_1(x)=6+6ax$ and $f^{"}_2(x)=6+6bx$ Now, $f_1(0)=0$ and $f_2(0)=c$, therefore $c=0$ And $f^{"}_1(0)=0$ and $f^{"}_2(0)=0$ But they don't seem to help me solve for a and b.
stackdsewew
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Fixed point iteration question

This image was taken from this youtube video https://www.youtube.com/watch?v=OLqdJMjzib8 Since only one of them converges, how do we know in advance which formula to work with?
stackdsewew
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