Paradoxes in Numerical Calculations

Abstract

Precise wind energy potential assessment is vital for wind energy generation and planning and development of new wind power plants. This work proposes and evaluates a novel two-stage method for location-specific wind energy potential assessment. It combines accurate statistical modelling of annual wind direction distribution in a given location with supervised machine learning of efficient estimators that can approximate energy efficiency coefficients from the parameters of optimized statistical wind direction models. The statistical models are optimized using differential evolution and energy efficiency is approximated by evolutionary fuzzy rules.

Relativistic perihelion shift of Mercury revisited

“…Finally, from Equations (7) and (6), and Kepler's third law $$$ {a}^3/{T}^2={GM}_{\odot }/\left(4{\pi}^2\right) $$$, it follows that after one period (more precisely, between two successive perihelion passages) the perihelion shifts about the angle $$$$ {\displaystyle \begin{array}{cc}\varepsilon & =2\left(\phi -\pi \right)=3\pi \frac{\alpha }{a\left(1-{e}^2\right)}=3\pi \frac{2{GM}_{\odot }}{ac^2\left(1-{e}^2\right)}\\ {}& =24{\pi}^3\frac{a^2}{T^2{c}^2\left(1-{e}^2\right)},\end{array}} $$$$ that is, the relationship (1), which according to (2) yields the idealized value of the relativistic perihelion shift of Mercury 43″ per century. Here, one subtracts two numbers of almost the same size, namely ϕ – π , which is a delicate numerical operation, see Brandts et al (2016).…”

“…that is, the relationship (1), which according to (2) yields the idealized value of the relativistic perihelion shift of Mercury 43 ′′ per century. Here, one subtracts two numbers of almost the same size, namely 𝜙 -𝜋, which is a delicate numerical operation, see Brandts et al (2016).…”

“…In the current astrophysical community, it is generally accepted that $$$$ O-C=E, $$$$ where E is the value (2) predicted by Einstein. However, this is again an ill‐conditioned problem due to the subtraction of two quite inexact numbers of almost equal magnitude, see Brandts et al (2016). An appropriate mixed polynomial‐trigonometric interpolation of observed values leads to O , see Bretagnon & Francou (1988) (also Folkner et al 2014), whereas C is obtained by numerical solution of the classical N ‐body problem (see, e.g., Narlikar & Rana 1985; Rana 1987).…”

Do Einstein's Equations Describe Reality Well?

“…However, this is an ill-conditioned problem due to the subtraction of two quite inexact numbers of almost equal magnitude [7]. Moreover, the quantities O and C are not uniquely defined.…”

“…Further problems arise from the nonlinearity and instability of Einstein's equations, division by zero, subtraction of two inexact numbers of almost the same size. This usually leads to a catastrophic loss of accuracy [7]. Finally note that many conclusions in cosmology are not in the form of mathematical implications, since various simplifications and approximations are done.…”