OCaml Scientific Computing

1st Edition (in progress)
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Constants and Metric System

In many scientific computing problems, numbers are not abstract but reflect the realistic meanings. In other words, these numbers only make sense on top of a well-defined metric system.

What Is A Metric System

For example, when we talk about the distance between two objects, I write down a number 30. But what does 30 mean in reality? Is it meters, kilometers, miles, or lightyears? Another example, what is the speed of light? Well, this is really depends on what metrics you are using, e.g., km/s, m/s, mile/h

Things can get really messy in computation if we do not unify the metric system in a numerical library. The translation between different metrics are often important in real-world application. I do not intend to dig deep into the metric system here, so please read online articles to find out more, e.g., Wiki: Outline of the metric system.

Four Systems in Owl

There are four metrics adopted in Owl, and all of them are wrapped in Owl.Const module.

All the metrics defined in these four systems can be found in the interface file owl_const.mli.

In general, SI is much newer and recommended to use. International System of Units (French: Système international d’unités, SI), historically also called the MKSA system of units for metre–kilogram–second–ampere. The SI system of units extends the MKS system and has 7 base units, by expressing any measurement of physical quantities using fundamental units of Length, Mass, Time, Electric Current, Thermodynamic Temperature, Amount of substance and Luminous Intensity, which are Metre, Kilogram, Second, Ampere, Kelvin, Mole and Candela respectively.

Here is a nice one-page poster from NPL to summarise what have talked about SI.

Figure 1: Owl constants

SI Prefix

As a computer scientist, you must be familiar with kilo, mega, giga these prefixes. SI system includes the definition of these prefixes as well. But be careful (especially for computer science guys), the base is 10 instead of 2.

These prefixes are defined in Const.Prefix module.


  Const.Prefix.peta;;
  Const.Prefix.tera;;
  Const.Prefix.giga;;
  Const.Prefix.mega;;
  Const.Prefix.kilo;;
  Const.Prefix.hecto;;

Examples: Physics and Math constants

Now we can safely talk about the distance between two objects, light of speed, and many other real-world stuff with atop of a well-defined metric system in Owl. See the following examples.


  Const.SI.light_year;;     (* light year in SI system *)
  Const.MKS.light_year;;    (* light year in MKS system *)
  Const.CGS.light_year;;    (* light year in CGS system *)
  Const.CGSM.light_year;;   (* light year in CGSM system *)

How about Planck’s constant?


  Const.SI.plancks_constant_h;;     (* in SI system *)
  Const.MKS.plancks_constant_h;;    (* in MKS system *)
  Const.CGS.plancks_constant_h;;    (* in CGS system *)
  Const.CGSM.plancks_constant_h;;   (* in CGSM system *)

The table below shows some physical constants that the SI module includes:

Table 1: Physical constants
Constant name Explanation
speed_of_light speed of light in vacuum
gravitational_constant Newtonian constant of gravitation
plancks_constant_h Planck constant
plancks_constant_hbar reduced Planck constant
astronomical_unit one astronomical unit in meters
light_year one light year in meters
parsec one light year in meters
grav_accel standard acceleration of gravity
electron_volt electron volt
mass_electron electron mass
mass_muon muon mass
mass_proton proton mass
mass_neutron neutron mass
rydberg Rydberg constant
boltzmann Boltzmann constant
molar_gas molar gas constant
standard_gas_volume molar volume of ideal gas (273.15 K, 100 kPa)
bohr_radius Bohr radius
stefan_boltzmann_constant Stefan-Boltzmann constant
thomson_cross_section Thomson cross section in square metre
bohr_magneton Bohr magneton in Joules per Tesla
nuclear_magneton Nuclear magneton in Joules per Tesla
electron_magnetic_moment electron magnetic moment in Joules per Tesla
proton_magnetic_moment proton magnetic moment in Joules per Tesla
faraday Faraday constant
electron_charge electron volt in Joules
vacuum_permittivity vacuum electric permittivity
vacuum_permeability vacuum magnetic permeability
debye one debye in coulomb metre
gauss one gauss in maxwell per square metre

Some basic mathematical constants are also provided in Owl, though some constants in advanced mathematics are not yet included such as the golden ratio or Euler–Mascheroni constant.

Table 2: Math constants
Constant name Explanation
pi Pi
e Natural constant
euler Euler constant

Besides these constants, we also provide some frequently used computations based on them, including:

  • log2e (\(\log_2 e\))
  • log10e (\(\log_10 e\))
  • loge2 (\(\log_e 2\))
  • loge10 (\(\log_e 10\))
  • logepi (\(\log_e \pi\))
  • pi2 (\(2\pi\))
  • pi4 (\(4\pi\))
  • pi_2 (\(\pi / 2\))
  • pi_4 (\(\pi / 4\))
  • sqrt1_2 (\(\sqrt{\frac{1}{2}}\))
  • sqrt2 (\(\sqrt{2}\))
  • sqrt3 (\(\sqrt{3}\))
  • sqrtpi (\(\sqrt{\pi}\))

International System of Units

Now that you know how to use constants, we will use the International System of Units (SI) module as an example to show the constants we include in Owl. These units are all derived from the seven basic units we have mentioned, and can be categorised according to different application fields.

Time

The base SI unit for time measurement is second.

Table 3: Time units
Constant name Explanation
minute one minute in seconds
hour one hour in seconds
day one day in seconds
week one week in seconds

Length

The base SI unit for length measurement is metre.

Table 4: Length units
Constant name Explanation
inch one inch in metres
foot one foot in metres
yard one yard in metres
mile one mile in metres
mil one mil in metres
fathom one fathom in metres
point one point in metres
micron one micron in metres
angstrom one angstrom in metres
nautical_mile one nautical mile in metres

Area

Measuring area and volume still relies SI base unit metre.

Table 5: Area units
Constant name Explanation
hectare one hectare in square meters
acre one acre in square meters
barn one barn in square meters

Volume

Table 6: Volume units
Constant name Explanation
liter one liter in cubic meters
us_gallon one gallon (US) in cubic meters
uk_gallon one gallon (UK) in cubic meters
canadian_gallon one Canadian gallon in cubic meters
quart one quart (US) in cubic meters
cup one cup (US) in cubic meters
pint one pint in cubic meters
fluid_ounce one fluid ounce (US) in cubic meters
tablespoon one tablespoon in cubic meters

Speed

The base units for speed are that of time and length.

Table 7: Speed units
Constant name Explanation
miles_per_hour miles per hour in metres per second
kilometers_per_hour kilometres per hour in metres per second
knot one knot in metres per second

Mass

The base unit for presenting mass is kilogram (kg).

Table 8: Mass units
Constant name Explanation
pound_mass one pound (avoirdupous) in kg
ounce_mass one ounce in kg
metric_ton 1000 kg
ton one short ton in kg
uk_ton one long ton in kg
troy_ounce one Troy ounce in kg
carat one carat in kg
unified_atomic_mass atomic mass constant
solar_mass one solar mass in kg

Force

Measuring force relies on the SI derived unit: newton, and one newtong equals to 1 kilogram metre per squared second.

Table 9: Force units
Constant name Explanation
newton SI derived unit (\(kg \cdot m \cdot s^{-2}\))
gram_force one gram force in newtons
kilogram_force one kilogram force in newtons
pound_force one pound force in newtons
poundal one poundal in newtons
dyne one dyne in newtons

Energy

The unit of measuring energy level is joule, which equals to one kilogram square metre per square second.

Table 10: Energy units
Constant name Explanation
joule SI base unit
calorie one calorie (thermochemical) in Joules
btu one British thermal unit (International Steam Table) in Joules
therm one therm (US) in Joules
erg one erg in Joules

Power

The unit of power is watts, a SI derived unit. One watts equals to one kilogram square metre per cubic second, or one Joule per second.

Table 11: Power units
Constant name Explanation
horsepower one horsepower in watts

Pressure

To measure we often use pascal as a standard unit. One pascal equals to a kilogram per metre per square second, or a newton per square metre.

Table 12: Pressure units
Constant name Explanation
bar one bar in pascals
std_atmosphere standard atmosphere in pascals
torr one torr (mmHg) in pascals
meter_of_mercury one metre of mercury in pascals
inch_of_mercury one inch of mercury in pascals
inch_of_water one inch of water in pascals
psi one psi in pascals

Viscosity

The poise is a unit in dynamic viscosity and the stokes is for kinematic viscosity. They are actually included in the CGS-based system for electrostatic units.

Table 13: Viscosity units
Constant name Explanation
poise base unit
stokes base unit

Luminance

Candela is the base unit for luminance, and both lumen and lux are derived units.

Table 14: Luminance units
Constant name Explanation
stilb Candela per square metre
lumen luminous flux, Candela square radian, SI derived unit
phot base unit
lux one lux in phots, SI derived unit
footcandle one footcandle in phots
lambert base unit
footlambert one footlambert in lambert

Radioactivity

The SI unit of radioactivity is becquerel, named in honour of the scientist Henri Becquerel, defined as one transformation (or decay or disintegration) per second. The other base unit ampere, second, and kilogram are also used.

Table 15: Radioactivity units
Constant name Explanation
curie one curie in becquerel
roentgen one ampere second per kilogram
rad erg per gram

Next: Chapter 27Internal Utility Modules