ParaMonte Fortran 2.0.0 Parallel Monte Carlo and Machine Learning LibrarySee the latest version documentation.
pm_mathConst Module Reference

This module contains relevant mathematical constants. More...

Data Types

type  ninf_type
This is the indicator type for generating instances of objects that indicate the use of the negative infinity $$-\infty$$ as an input argument to the generic interfaces of the ParaMonte library.
More...

type  origin_type
This is the derived type origin_type representing the geometric origin of the coordinates. More...

type  pinf_type
This is the indicator type for generating instances of objects that indicate the use of the positive infinity $$+\infty$$ as an input argument to the generic interfaces of the ParaMonte library.
More...

Variables

character(*, SK), parameter MODULE_NAME = "@pm_mathConst"

real(RKB), parameter PI = acos(-1._RKB)
The scalar real constant of kind with highest available precision RKB representing the irrational number $$\pi$$. More...

real(RKB), parameter TWO_PI = 2 * PI
The scalar real constant of kind with highest available precision RKB representing twice the irrational number $$\pi$$. More...

real(RKB), parameter HALF_PI = .5_RKB * PI
The scalar real constant of kind with highest available precision RKB representing half the irrational number $$\pi$$. More...

real(RKB), parameter INVERSE_PI = 1._RKB / PI
The scalar real constant of kind with highest available precision RKB representing the inverse of the irrational number $$\pi$$. More...

real(RKB), parameter QUARTER_PI = .25_RKB * PI
The scalar real constant of kind with highest available precision RKB representing a quarter of the irrational number $$\pi$$. More...

real(RKB), parameter LOG_PI = log(PI)
The scalar real constant of kind with highest available precision RKB representing $$\log(\pi)$$. More...

real(RKB), parameter SQRT_PI = sqrt(PI)
The scalar real constant of kind with highest available precision RKB representing $$\sqrt{\pi}$$. More...

real(RKB), parameter LOG_TWO_PI = log(TWO_PI)
The scalar real constant of kind with highest available precision RKB representing $$\log(2\pi)$$. More...

real(RKB), parameter SQRT_TWO_PI = sqrt(TWO_PI)
The scalar real constant of kind with highest available precision RKB representing $$\sqrt{2\pi}$$. More...

real(RKB), parameter SQRT_HALF_PI = sqrt(HALF_PI)
The scalar real constant of kind with highest available precision RKB representing $$\sqrt{\frac{\pi}{2}}$$. More...

real(RKB), parameter INVERSE_SQRT_PI = sqrt(INVERSE_PI)
The scalar real constant of kind with highest available precision RKB representing $$\frac{1}{\sqrt{\pi}}$$. More...

real(RKB), parameter INVERSE_SQRT_TWO_PI = 1._RKB / SQRT_TWO_PI
The scalar real constant of kind with highest available precision RKB representing $$\frac{1}{\sqrt{2\pi}}$$. More...

real(RKB), parameter LOG_INVERSE_SQRT_TWO_PI = log(INVERSE_SQRT_TWO_PI)
The scalar real constant of kind with highest available precision RKB representing $$\log\left(\frac{1}{\sqrt{2\pi}}\right)$$, frequently appearing in distributions (e.g., Normal). More...

real(RKB), parameter NAPIER = exp(1._RKB)
The scalar real constant of kind with highest available precision RKB representing the Napier constant (a.k.a. Euler number) $$e = \exp(1)$$. More...

real(RKB), parameter LOG_TWO = log(2._RKB)
The scalar real constant of kind with highest available precision RKB representing $$\log(2)$$. More...

real(RKB), parameter LOG_TEN = log(1.e1_RKB)
The scalar real constant of kind with highest available precision RKB representing $$\log(10)$$. More...

real(RKB), parameter LOG_HALF = log(0.5_RKB)
The scalar real constant of kind with highest available precision RKB representing $$\log\left(\frac{1}{2}\right)$$. More...

real(RKB), parameter SQRT_TWO = sqrt(2._RKB)
The scalar real constant of kind with highest available precision RKB representing $$\sqrt{2}$$. More...

real(RKB), parameter LOG10_NAPIER = log10(NAPIER)
The scalar real constant of kind with highest available precision RKB representing $$\log_{10}(e)$$. More...

real(RKB), parameter INVERSE_LOG_TWO = 1._RKB / LOG_TWO
The scalar real constant of kind with highest available precision RKB representing $$\frac{1}{\log(2)}$$. More...

real(RKB), parameter INVERSE_SQRT_TWO = 1._RKB / SQRT_TWO
The scalar real constant of kind with highest available precision RKB representing $$\frac{1}{\sqrt{2}}$$. More...

type(origin_type), parameter ORIGIN = origin_type()
The scalar constant object of type origin_type representing the geometric origin of the coordinates. More...

real(RKB), parameter EULER_CONST = 0.577215664901532860606512090082402431042159335939923598805767234884867726777664670936947063291746749_RKB
The scalar real constant of kind with highest available precision RKB representing the Euler-Mascheroni constant. More...

real(RKB), parameter APERY_CONST = 1.20205690315959428539973816151144999076498629234049888179227155534183820578631309018645587360933525814619915_RKB
The scalar real constant of kind with highest available precision RKB representing the Apery constant. More...

real(RKB), parameter PRIME_CONST = .414682509851111660248109622154307708365774238137916977868245414488640960619357334196290048428475777939616_RKB
The scalar real constant of kind with highest available precision RKB representing the irrational Prime constant. More...

real(RKB), parameter GOLDEN_RATIO = .5_RKB * (1._RKB + sqrt(5._RKB))
The scalar real constant of kind with highest available precision RKB representing the Golden Ratio constant. More...

real(RKB), parameter SILVER_RATIO = 1._RKB + sqrt(2._RKB)
The scalar real constant of kind with highest available precision RKB representing the Silver Ratio constant. More...

real(RKB), parameter SUPER_GOLDEN_RATIO = (2._RKB * cosh(acosh(29._RKB / 2._RKB) / 3._RKB) + 1._RKB) / 3._RKB
The scalar real constant of kind with highest available precision RKB representing the Supergolden Ratio constant.
More...

type(ninf_type), parameter ninf = ninf_type()
The scalar constant object of type ninf_type that indicates the use of the negative infinity $$-\infty$$ as an input argument to the generic interfaces of the ParaMonte library.
More...

type(pinf_type), parameter pinf = pinf_type()
The scalar constant object of type pinf_type that indicates the use of the positive infinity $$+\infty$$ as an input argument to the generic interfaces of the ParaMonte library.
More...

Detailed Description

This module contains relevant mathematical constants.

Note
The constants of this module are saved with the highest available real precision kind.
To use the constants at expressions involving lower-precision real kinds, simply convert the numbers to the desired kind via the Fortran intrinsic real(x, kind = RKG) where RKG refers to the target kind parameter used in the expression.

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Todo:
Low Priority: Th contents of this module can be expanded to include more mathematical constants.
Author:
Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

◆ APERY_CONST

 real(RKB), parameter pm_mathConst::APERY_CONST = 1.20205690315959428539973816151144999076498629234049888179227155534183820578631309018645587360933525814619915_RKB

The scalar real constant of kind with highest available precision RKB representing the Apery constant.

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number

\begin{aligned} \zeta(3) &= \sum _{n=1}^{\infty} \frac{1}{n^{3}} \\ &= \lim _{n\to\infty} \left( {\frac{1}{1^{3}}} + {\frac {1}{2^{3}}} + \cdots + {\frac{1}{n^{3}}} \right) ~, \end{aligned}

where $$\zeta$$ is the Riemann zeta function.
It has an approximate value of,

$$\zeta(3) = 1.202056903159594285399738161511449990764986292\ldots ~.$$

The constant is named after Roger Apéry.
It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics.
It also arises in the analysis of random minimum spanning trees and in conjunction with the Gamma function when solving certain integrals involving exponential functions in a quotient.
These appear occasionally in physics, for instance, when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law.

Todo:
Critical Priority: The current implementation of this constant hardcodes the first 109 significant digits.
This is more than sufficient for all existing use pm_sampleCov, only: getCovMerged kinds supported by compilers as of 2022.
However, the situation may change in the distant future, where the precision of some use pm_sampleCov, only: getCovMerged kinds surpasses the number of decimal digits hardcoded here.
As such the implementation of this constant should be improved when better mechanisms of arbitrary precision implementation become available.

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 222 of file pm_mathConst.F90.

◆ EULER_CONST

 real(RKB), parameter pm_mathConst::EULER_CONST = 0.577215664901532860606512090082402431042159335939923598805767234884867726777664670936947063291746749_RKB

The scalar real constant of kind with highest available precision RKB representing the Euler-Mascheroni constant.

The Euler constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter $$\gamma$$. It is defined as the limiting difference between the harmonic series and the natural logarithm,

\begin{aligned} \gamma &= \lim _{n \to \infty} \left( -\log(n) + \sum _{k=1}^{n} \frac{1}{k} \right) \\ &= \int _{1}^{\infty} \left( -\frac{1}{x} + \frac{1}{\lfloor x \rfloor} \right) ~ dx ~. \end{aligned}

Here, $$\lfloor x \rfloor$$ represents the Fortran intrinsic floor() function.

The area of the blue region converges to the Euler constant

Applications
The Euler constant appears in many areas of science, including,

1. Expressions involving the exponential integral.
2. The Laplace transform of the natural logarithm.
3. The first term of the Laurent series expansion for the Riemann zeta function, where it is the first of the Stieltjes constants.
4. Calculations of the Digamma function.
5. A product formula for the Gamma function.
6. The asymptotic expansion of the Gamma function for small arguments.
7. An inequality for the Euler totient function.
8. The growth rate of the divisor function.
9. In dimensional regularization of Feynman diagrams in quantum field theory.
10. The calculation of the Meissel–Mertens constant.
11. The third of the Mertens theorems.
12. Solution of the second kind to the Bessel equation.
13. In the regularization/renormalization of the harmonic series as a finite value.
14. The mean of the Gumbel distribution.
15. The information entropy of the Weibull and Lévy distributions, and, implicitly, of the Chi-squared distribution for one or two degrees of freedom.
16. The answer to the Coupon Collector problem.
17. In some formulations of the Zipf law.
18. A definition of the cosine integral.
19. Lower bounds to a prime gap.
20. An upper bound on Shannon entropy in quantum information theory.
21. Fisher–Orr model for genetics of adaptation in evolutionary biology.
The Euler Number
The Euler Constant
Todo:
Critical Priority: The current implementation of this constant hardcodes the first 99 significant digits.
This is more than sufficient for all existing use pm_sampleCov, only: getCovMerged kinds supported by compilers as of 2022.
However, the situation may change in the distant future, where the precision of some use pm_sampleCov, only: getCovMerged kinds surpasses the number of decimal digits hardcoded here.
As such the implementation of this constant should be improved when better mechanisms of arbitrary precision implementation become available.

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 189 of file pm_mathConst.F90.

◆ GOLDEN_RATIO

 real(RKB), parameter pm_mathConst::GOLDEN_RATIO = .5_RKB * (1._RKB + sqrt(5._RKB))

The scalar real constant of kind with highest available precision RKB representing the Golden Ratio constant.

In mathematics, two quantities are in the Golden Ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.
Expressed algebraically, for quantities $$a$$ and $$b$$ with $$0 < a < b$$,

$$\frac{a + b}{a} = \frac{a}{b} = \phi ~,$$

where the Greek letter $$\phi$$ denotes the Golden Ratio.
The constant $$\phi$$ satisfies the quadratic equation $$\phi^{2} = \phi + 1$$ and is an irrational number with a value of,

$$\phi = \frac{1 + \sqrt{5}}{2} = 1.618033988749\ldots ~.$$

The Golden Ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, among other names.
Mathematicians have studied the properties of the Golden Ratio since antiquity.
It is the ratio of a diagonal of a regular pentagon to its side and thus appears in the construction of the dodecahedron and icosahedron.
A golden rectangle — that is, a rectangle with an aspect ratio of $$\phi$$ — may be cut into a square and a smaller rectangle with the same aspect ratio.
The Golden Ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data.
The Golden Ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.
Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the Golden Ratio, believing it to be aesthetically pleasing.
These usages frequently appear in the form of a golden rectangle, as illustrated below.

A golden rectangle with long side a and short side b (shaded red, right) and a square with sides of length a (shaded blue, left) combine to form a similar golden rectangle with long side a + b and short side a.
SILVER_RATIO
GOLDEN_RATIO
SUPER_GOLDEN_RATIO

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 283 of file pm_mathConst.F90.

◆ HALF_PI

 real(RKB), parameter pm_mathConst::HALF_PI = .5_RKB * PI

The scalar real constant of kind with highest available precision RKB representing half the irrational number $$\pi$$.

Definition at line 50 of file pm_mathConst.F90.

◆ INVERSE_LOG_TWO

 real(RKB), parameter pm_mathConst::INVERSE_LOG_TWO = 1._RKB / LOG_TWO

The scalar real constant of kind with highest available precision RKB representing $$\frac{1}{\log(2)}$$.

Definition at line 72 of file pm_mathConst.F90.

◆ INVERSE_PI

 real(RKB), parameter pm_mathConst::INVERSE_PI = 1._RKB / PI

The scalar real constant of kind with highest available precision RKB representing the inverse of the irrational number $$\pi$$.

Definition at line 51 of file pm_mathConst.F90.

◆ INVERSE_SQRT_PI

 real(RKB), parameter pm_mathConst::INVERSE_SQRT_PI = sqrt(INVERSE_PI)

The scalar real constant of kind with highest available precision RKB representing $$\frac{1}{\sqrt{\pi}}$$.

Definition at line 58 of file pm_mathConst.F90.

◆ INVERSE_SQRT_TWO

 real(RKB), parameter pm_mathConst::INVERSE_SQRT_TWO = 1._RKB / SQRT_TWO

The scalar real constant of kind with highest available precision RKB representing $$\frac{1}{\sqrt{2}}$$.

Definition at line 73 of file pm_mathConst.F90.

◆ INVERSE_SQRT_TWO_PI

 real(RKB), parameter pm_mathConst::INVERSE_SQRT_TWO_PI = 1._RKB / SQRT_TWO_PI

The scalar real constant of kind with highest available precision RKB representing $$\frac{1}{\sqrt{2\pi}}$$.

Definition at line 59 of file pm_mathConst.F90.

◆ LOG10_NAPIER

 real(RKB), parameter pm_mathConst::LOG10_NAPIER = log10(NAPIER)

The scalar real constant of kind with highest available precision RKB representing $$\log_{10}(e)$$.

Definition at line 71 of file pm_mathConst.F90.

◆ LOG_HALF

 real(RKB), parameter pm_mathConst::LOG_HALF = log(0.5_RKB)

The scalar real constant of kind with highest available precision RKB representing $$\log\left(\frac{1}{2}\right)$$.

Definition at line 69 of file pm_mathConst.F90.

◆ LOG_INVERSE_SQRT_TWO_PI

 real(RKB), parameter pm_mathConst::LOG_INVERSE_SQRT_TWO_PI = log(INVERSE_SQRT_TWO_PI)

The scalar real constant of kind with highest available precision RKB representing $$\log\left(\frac{1}{\sqrt{2\pi}}\right)$$, frequently appearing in distributions (e.g., Normal).

Definition at line 60 of file pm_mathConst.F90.

◆ LOG_PI

 real(RKB), parameter pm_mathConst::LOG_PI = log(PI)

The scalar real constant of kind with highest available precision RKB representing $$\log(\pi)$$.

Definition at line 53 of file pm_mathConst.F90.

◆ LOG_TEN

 real(RKB), parameter pm_mathConst::LOG_TEN = log(1.e1_RKB)

The scalar real constant of kind with highest available precision RKB representing $$\log(10)$$.

Definition at line 68 of file pm_mathConst.F90.

◆ LOG_TWO

 real(RKB), parameter pm_mathConst::LOG_TWO = log(2._RKB)

The scalar real constant of kind with highest available precision RKB representing $$\log(2)$$.

Definition at line 67 of file pm_mathConst.F90.

◆ LOG_TWO_PI

 real(RKB), parameter pm_mathConst::LOG_TWO_PI = log(TWO_PI)

The scalar real constant of kind with highest available precision RKB representing $$\log(2\pi)$$.

Definition at line 55 of file pm_mathConst.F90.

◆ MODULE_NAME

 character(*,SK), parameter pm_mathConst::MODULE_NAME = "@pm_mathConst"

Definition at line 42 of file pm_mathConst.F90.

◆ NAPIER

 real(RKB), parameter pm_mathConst::NAPIER = exp(1._RKB)

The scalar real constant of kind with highest available precision RKB representing the Napier constant (a.k.a. Euler number) $$e = \exp(1)$$.

Definition at line 66 of file pm_mathConst.F90.

◆ ninf

 type(ninf_type), parameter pm_mathConst::ninf = ninf_type()

The scalar constant object of type ninf_type that indicates the use of the negative infinity $$-\infty$$ as an input argument to the generic interfaces of the ParaMonte library.

See the documentations of the generic interfaces that use this constant for example usage.

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Author:
Amir Shahmoradi, September 1, 2017, 12:00 AM, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin

Definition at line 423 of file pm_mathConst.F90.

◆ ORIGIN

 type(origin_type), parameter pm_mathConst::ORIGIN = origin_type()

The scalar constant object of type origin_type representing the geometric origin of the coordinates.

For example usage, see the corresponding PaaMonte generic interfaces that use this object.

The origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.
In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer.
This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry.

In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.
The origin divides each of these axes into two halves, a positive and a negative semiaxis.
Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction.
The coordinates of the origin are always all zero, for example $$(0,0)$$ in two dimensions and $$(0,0,0)$$ in three.

The origin of the complex plane can be referred as the point where real axis and imaginary axis intersect each other.
In other words, the origin in the complex plane is the complex number zero.

origin_type

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 131 of file pm_mathConst.F90.

◆ PI

 real(RKB), parameter pm_mathConst::PI = acos(-1._RKB)

The scalar real constant of kind with highest available precision RKB representing the irrational number $$\pi$$.

Definition at line 48 of file pm_mathConst.F90.

◆ pinf

 type(pinf_type), parameter pm_mathConst::pinf = pinf_type()

The scalar constant object of type pinf_type that indicates the use of the positive infinity $$+\infty$$ as an input argument to the generic interfaces of the ParaMonte library.

See the documentations of the generic interfaces that use this constant for example usage.

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Author:
Amir Shahmoradi, September 1, 2017, 12:00 AM, Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin

Definition at line 470 of file pm_mathConst.F90.

◆ PRIME_CONST

 real(RKB), parameter pm_mathConst::PRIME_CONST = .414682509851111660248109622154307708365774238137916977868245414488640960619357334196290048428475777939616_RKB

The scalar real constant of kind with highest available precision RKB representing the irrational Prime constant.

The prime constant is the real number $$\rho$$ whose $$n$$th binary digit is $$1$$ if $$n$$ is prime and 0 if $$n$$ is composite or $$1$$.
In other words, $$\rho$$ is the number whose binary expansion corresponds to the indicator function of the set of prime numbers.
That is,

$$\rho = \sum_{p} \frac{1}{2^{p}} = \sum_{n = 1}^{\infty} \frac{\chi_{\mathbb{P}}(n)}{2^{n}} \rho = \sum_{{p}} \frac{1}{2^{p}} = \sum_{{n = 1}}^{\infty} \frac{\chi_{{{\mathbb{P}}}}(n)}{2^{n}} ~,$$

where $$p$$ indicates a prime and $$\chi_{{{\mathbb{P}}}}$$ is the characteristic function of the set $$\mathbb{P}$$ of prime numbers.
The beginning of the decimal expansion of $$\rho$$ is: $$\rho = 0.414682509851111660248109622\ldots$$.

Todo:
Critical Priority: The current implementation of this constant hardcodes the first 105 significant digits.
This is more than sufficient for all existing use pm_sampleCov, only: getCovMerged kinds supported by compilers as of 2022.
However, the situation may change in the distant future, where the precision of some use pm_sampleCov, only: getCovMerged kinds surpasses the number of decimal digits hardcoded here.
As such the implementation of this constant should be improved when better mechanisms of arbitrary precision implementation become available.

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 246 of file pm_mathConst.F90.

◆ QUARTER_PI

 real(RKB), parameter pm_mathConst::QUARTER_PI = .25_RKB * PI

The scalar real constant of kind with highest available precision RKB representing a quarter of the irrational number $$\pi$$.

Definition at line 52 of file pm_mathConst.F90.

◆ SILVER_RATIO

 real(RKB), parameter pm_mathConst::SILVER_RATIO = 1._RKB + sqrt(2._RKB)

The scalar real constant of kind with highest available precision RKB representing the Silver Ratio constant.

In mathematics, two quantities are in the Silver Ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity.
This defines the Silver Ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623.
Its name is an allusion to the Golden Ratio.
Analogously to the way the Golden Ratio is the limiting ratio of consecutive Fibonacci numbers, the Silver Ratio is the limiting ratio of consecutive Pell numbers.
The Silver Ratio is denoted by $$\delta_S$$, which can be expressed algebraically as,

$$\frac {2a+b}{a} = \frac {a}{b} \equiv \delta _{S} ~,$$

or equivalently,

$$2 + \frac {b}{a} = {\frac {a}{b}} \equiv \delta _{S} ~.$$

The Silver Ratio can also be defined by the simple continued fraction $$[2; 2, 2, 2, ...]$$,

$$2 + {\cfrac{1}{2 + {\cfrac{1}{2 + {\cfrac{1}{2 + \ddots}}}}}} = \delta _{S} ~.$$

The convergents of this continued fraction are ratios of consecutive Pell numbers.
These fractions provide accurate rational approximations of the Silver Ratio, analogous to the approximation of the Golden Ratio by ratios of consecutive Fibonacci numbers.
The Silver Rectangle is connected to the regular octagon.
If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a Silver Rectangle with an aspect ratio of $$1:\delta_S$$.
The four sides of the trapezoids are in a ratio of $$1:1:1:\delta_S$$.
If the edge length of a regular octagon is $$t$$, then the span of the octagon (the distance between opposite sides) is $$\delta_{S}t$$, and the area of the octagon is $$2\delta_{S}t^2$$.

Silver Rectangle

Silver Ratio within the octagon.
SILVER_RATIO
GOLDEN_RATIO
SUPER_GOLDEN_RATIO

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 329 of file pm_mathConst.F90.

◆ SQRT_HALF_PI

 real(RKB), parameter pm_mathConst::SQRT_HALF_PI = sqrt(HALF_PI)

The scalar real constant of kind with highest available precision RKB representing $$\sqrt{\frac{\pi}{2}}$$.

Definition at line 57 of file pm_mathConst.F90.

◆ SQRT_PI

 real(RKB), parameter pm_mathConst::SQRT_PI = sqrt(PI)

The scalar real constant of kind with highest available precision RKB representing $$\sqrt{\pi}$$.

Definition at line 54 of file pm_mathConst.F90.

◆ SQRT_TWO

 real(RKB), parameter pm_mathConst::SQRT_TWO = sqrt(2._RKB)

The scalar real constant of kind with highest available precision RKB representing $$\sqrt{2}$$.

Definition at line 70 of file pm_mathConst.F90.

◆ SQRT_TWO_PI

 real(RKB), parameter pm_mathConst::SQRT_TWO_PI = sqrt(TWO_PI)

The scalar real constant of kind with highest available precision RKB representing $$\sqrt{2\pi}$$.

Definition at line 56 of file pm_mathConst.F90.

◆ SUPER_GOLDEN_RATIO

 real(RKB), parameter pm_mathConst::SUPER_GOLDEN_RATIO = (2._RKB * cosh(acosh(29._RKB / 2._RKB) / 3._RKB) + 1._RKB) / 3._RKB

The scalar real constant of kind with highest available precision RKB representing the Supergolden Ratio constant.

In mathematics, two quantities are in the superGolden Ratio if the quotient of the larger number divided by the smaller one is equal to

$$\psi = {\frac{1 + {\sqrt[{3}]{\frac{29 + 3{\sqrt{93}}}{2}}} + {\sqrt[{3}]{\frac{29 - 3{\sqrt{93}}}{2}}}}{3}} ~,$$

which is the only real solution to the equation $$x^{3} = x^{2} + 1$$.
It can also be represented using the hyperbolic cosine as,

$$\psi = \frac{2}{3} \cosh{\left( \tfrac{\cosh^{-1} \left(\frac{29}{2} \right)}{3} \right)} + \frac {1}{3} ~.$$

The decimal expansion of this number begins $$1.465571231876768026656731\ldots$$, and the ratio is commonly represented by the Greek letter $$\psi$$.

Many of the properties of the Supergolden Ratio are related to those of the Golden Ratio.
For example, the $$n$$th item of the Narayana sequence is the number of ways to tile a $$1\times n$$ rectangle with $$1\times 1$$ and $$1\times3$$ tiles.
Similarly, the $$n$$th term of the Fibonacci sequence is the number of ways to tile a $$1\times n$$ rectangle with $$1\times1$$ and $$1\times2$$ tiles.
The Supergolden Ratio satisfies $$\psi -1 = \psi^{-2}$$, while the Golden Ratio satisfies $$\phi -1 = \phi^{-1}$$.
In the Fibonacci Rabbit Problem, each pair breeds each cycle starting after two cycles, while in the Narayana Cow Problem, each pair breeds each cycle starting after three cycles.
There is a Supergolden Rectangle that has the property that if a square is removed from one side, the remaining rectangle can be divided into two Supergolden Rectangles of opposite orientations.

A Supergolden Rectangle is a rectangle whose side lengths are in the Supergolden Ratio, i.e. the length of the longer side divided by the length of the shorter side is equal to the Supergolden Ratio: $${\frac{1 + {\sqrt[{3}]{\frac{29 + 3{\sqrt{93}}}{2}}} + {\sqrt[{3}]{\frac{29-3{\sqrt{93}}}{2}}}}{3}}$$.
When a square with the same side length as the shorter side of the rectangle is removed from one side of the rectangle, the sides resulting rectangle will be in a $$\psi^2:1$$ ratio.
This rectangle can be divided into rectangles with side-length ratios of $$\psi:1$$ and $$1:\psi$$, two Supergolden Ratios of perpendicular orientations.
Their areas will be in a $$\psi^2:1$$ ratio.
In addition, if the line that separates the two Supergolden Rectangles from each other is extended across the rest of the original rectangle such that it – along with the side of the square that was removed from the original rectangle – divides the original rectangle into quadrants, then the larger Supergolden Rectangle has the same area as the opposite quadrant, its diagonal length is the length of the short side of the original rectangle divided by $$\sqrt{\psi}$$, the fourth quadrant is also a Supergolden Rectangle, and its diagonal length is $$\sqrt {\psi }$$ times the length of the short side of the original rectangle.

A triangle with sides lengths of the Supergolden Ratio, its inverse, and one has an angle of exactly 120 degrees opposite the ratio length.

This diagram shows the lengths of decreasing powers within a Supergolden Rectangle, and the pattern of intersecting right angles that appears as a result.
SILVER_RATIO
GOLDEN_RATIO
SUPER_GOLDEN_RATIO

Final Remarks

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 379 of file pm_mathConst.F90.

◆ TWO_PI

 real(RKB), parameter pm_mathConst::TWO_PI = 2 * PI

The scalar real constant of kind with highest available precision RKB representing twice the irrational number $$\pi$$.

Definition at line 49 of file pm_mathConst.F90.