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eigen
Haskel binding for Eigen library. Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms.
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README
units
The units package provides a mechanism for compiletime dimensional analysis
in Haskell programs. It defines an embedded type system based on
unitsofmeasure. The units and dimensions defined are fully extensible, and
need not relate to physical properties. This package exports definitions
only for Dimensionless
and Number
. The set of units and dimensions from
the International System (SI) are exported from the companion package unitsdefs
.
This package supports independent notions of dimension and unit. Examples of dimensions include length and mass. Examples of unit include meter and gram. Every unit measures a particular dimension, but a given dimension may be measured by many different units. For example, both meters and feet measure length.
The package supports defining multiple interconvertible units of the same
dimension, such as Meter
and Foot
. When extracting a numerical value from
a quantity, the desired unit must be specified, and the value is converted
into that unit.
The laws of nature have dimensions, and they hold true regardless of the units
used. For example, the gravitational force between two bodies is
(gravitational constant) * (mass 1) * (mass 2) / (distance between body 1 and
2)^2
, regardless of whether the distance is given in meters or feet
or centimeters. In other words, every law of nature is unitpolymorphic.
The units package supports unitpolymorphic programs through the coherent
system of units (CSU) mechanism. A CSU is essentially a mapping from
dimensions to the units. All dimensioned quantities (generally just called
quantities) are expressed using the Qu
type. The Qu
type constructor takes
a (perhaps compound) dimension, a CSU and a numerical value type as arguments.
Internally, the quantity is stored as a number in the units as specified in
the CSU  this may matter if you are worried about rounding errors. In the
sequence of computations that works within one CSU, there is no unit
conversion. Unit conversions are needed only when putting values in and out of
quantities, or converting between two different CSUs.
Checking out units
Units has a git submodule, so you'll want to use git clone recursive
. Example:
git clone recursive https://github.com/goldfirere/units
User contributions
It is easy to imagine any number of builtin facilities that would go well with this package (sets of definitions of units for various systems, vector operations, a suite of polymorphic functions that are commonly needed but hard to define, etc.). Yet, I (Richard) don't have the time to imagine or write all of these. If you write code that is sufficiently general and might want to be included with this package (but you don't necessarily want to create your own new package), please write me!
Modules
The units package exports several key modules. Note that you will generally
import only one of Data.Metrology
, Data.Metrology.Poly
, or
Data.Metrology.Vector
.
Data.Metrology.Poly
This is the main exported module. It exports all the necessary functionality for you to build your own set of units and operate with them.
Data.Metrology
This reexports most of the definitions from
Data.Metrology.Poly
, but restricts a few operators to work only with the default LCSU, as this is simpler for new users tounits
.Data.Metrology.Vector
This also reexports a similar set of definitions as
Data.Metrology.Poly
, but provides numerical operations based onvectorspace
instead of the standard numerical classes.Data.Metrology.Linear
This exports a set of definitions for interoperability with the
linear
package. This is /not/ a toplevel import; generally, import this withData.Metrology.Poly
as well.Data.Metrology.Internal
This module contains mostlyinternal definitions that may appear in GHC's error messages. Users will generally not need to use these definitions in their code. However, by exporting this module from within
Data.Metrology.Poly
, we can reduce the moduleprefix clutter in error messages.Data.Metrology.Unsafe
This module exports the constructor for the central datatype that stores quantities. With this constructor, you can arbitrarily change units! Use at your peril.
Data.Metrology.Show
This module defines a
Show
instance for quantities, printing out the number stored along with its canonical dimension. This behavior may not be the best for every setting, so it is exported separately. Importing this module reduces the guaranteed unitsafety of your code, because it allows you to inspect (in a roundabout way) how your quantities are stored.Data.Metrology.Parser
This module allows users to create custom unit parsers. The user specifies a set of prefixes and a set of units to parse, and then a quasiquoting parser is generated. See the module documentation for details.
Data.Metrology.TH
This module exports several functions, written with Template Haskell, that make programming with
units
somewhat easier. In particular, seedeclareMonoUnit
, which gets rid of a lot of the boilerplate if you don't want unit polymorphism.Data.Metrology.Quantity
This module defines a
Quantity
class to enable easy, safe conversions with nonunits
types. See the module for more documentation.
Examples
We will build up a full working example in several sections. It is awkward to explain the details of the pieces until the whole example is built, so please read on to see how it all works. For more complete(ish) examples, see this test case (for examples of how to use units) and unitsdefs (for examples of how to define units).
Dimension definitions
When setting up your welltyped unitsofmeasure program, the first step is
to define the dimensions you will be working in. (If your application involves
physical quantities, you may want to check Data.Dimensions.SI
in the
unitsdefs
package first.)
data LengthDim = LengthDim  each dimension is a datatype that acts as its own proxy
instance Dimension LengthDim
data TimeDim = TimeDim
instance Dimension TimeDim
We can now build up dimensions from these base dimensions:
type VelocityDim = LengthDim :/ TimeDim
Unit definitions
We then define units to work with these dimensions. Here, we define two different
interconvertible units for length. (Note that just about all of this boilerplate
can be generated by functions in the Data.Metrology.TH
module.)
data Meter = Meter
instance Unit Meter where  declare Meter as a Unit
type BaseUnit Meter = Canonical  Meters are "canonical"
type DimOfUnit Meter = LengthDim  Meters measure Lengths
instance Show Meter where  Show instances are optional but useful
show _ = "m"  do *not* examine the argument!
data Foot = Foot
instance Unit Foot where
type BaseUnit Foot = Meter  Foot is defined in terms of Meter
conversionRatio _ = 0.3048  do *not* examine the argument!
 We don't need to specify the `DimOfUnit`;
 it's implied by the `BaseUnit`.
instance Show Foot where
show _ = "ft"
data Second = Second
instance Unit Second where
type BaseUnit Second = Canonical
type DimOfUnit Second = TimeDim
instance Show Second where
show _ = "s"
A unit assignment
To perform computations with units, we must define a socalled local coherent set of units, or LCSU. This is a mapping from dimensions to units, and it informs exactly how the quantities are stored. For example:
type LCSU = MkLCSU '[(LengthDim, Meter), (TimeDim, Second)]
This definition says that we wish to store lengths in meters and times in seconds.
Note that, even though Meter
is defined as the Canonical
length, we could have
used Foot
in our LCSU. Canonical units are used only in conversion between
units, not the choice of how to store a quantity.
Value types
To use all these pieces to build the actual type that will store quantities, we
use one of the MkQu_xxx
type synonyms, as follows:
type Length = MkQu_DLN LengthDim LCSU Double
 Length stores lengths in our defined LCSU, using `Double` as the numerical type
type Length' = MkQu_ULN Foot LCSU Double
 same as Length. Note the `U` in `MkQu_ULN`, allowing it to take a unit
type Time = MkQu_DLN TimeDim LCSU Double
Some computations
We now show some example computations on the defined types:
extend :: Length > Length  a function over lengths
extend x = redim $ x + (1 % Meter)
inMeters :: Length > Double  extract the # of meters
inMeters = (# Meter)  more on this later
conversion :: Length  mixing units
conversion = (4 % Meter) + (10 % Foot)
vel :: Length %/ Time  The `%*` and `%/` operators allow
 you to combine types
vel = (3 % Meter) / (2 % Second)
Explanation
Let's pick this apart. The data LengthDim = LengthDim
declaration creates both the
type LengthDim
and a termlevel proxy for it. It would be possible to get away
without the proxies and use lots of type annotations, but who would want to?
We must define an instance of Dimension
to declare that LengthDim
is a dimension.
Why suffix with Dim
? To distinguish the length dimension from the length type.
Generally, the type is mentioned more often and should be the shorter name.
We then create a TimeDim
to operate alongside the LengthDim
. Using the
:/
combinator, we can create a VelocityDim
out of the two dimensions defined
so far. See below for more information on unit combinators.
Then, we make some units, using similar data
definitions. We define an
instance of Unit
to make Meter
into a proper unit. The Unit
class is
primarily responsible for handling unit conversions. In the case of Meter
,
we define that as the canonical unit of length, meaning that all lengths
will internally be stored in meters. It also means that we don't need to
define a conversion ratio for meters. You will also see that we say that
Meter
s measure the dimension LengthDim
, through the DimOfUnit
declaration.
We also include a Show
instance for Meter
so that lengths can be printed
easily. If you don't need to show
your lengths, there is no need for this
instance.
When defining Foot
, we say that its BaseUnit
is Meter
, meaning that
Foot
is interconvertible with Meter
. This declaration also says that
the dimension measured by a Foot
must be the same as the dimension for
a Meter
. We must then define the conversion
ratio, which is the number of meters in a foot. Note that the
conversionRatio
method must take a parameter to fix its type parameter, but
it must not inspect that parameter. Internally, it will be passed
undefined
quite often.
The definition for Second
is quite similar to that for Meter
.
The next section of code constructs an "LCSU"  a local coherent set of units. The idea is that we wish to be able to choose a set of units which are to be used in the internal, concrete representation. An LCSU is just an association list giving a concrete unit for each dimension in your domain. The particular LCSU here says that length is stored in meters and time is stored in seconds. It would be invalid to specify an LCSU with repeats for either dimension or unit.
With all this laid out, we can make the types that store values. units
exports several MkQu_xxx
type synonyms that vary in the arguments they
expect. MkQu_DLN
, for example, takes a dimension, an LCSU, and a
numerical type. With the definition above, Length
is now a type suitable
for storing lengths.
Note
that Length
and Length'
are the same type. The MkQu
machinery notices
that these two are interconvertible and will produce the same dimensioned
quantity.
Note that, as you can see in the function examples at the end, it is necessary
to specify the choice of unit when creating a quantity or
extracting from a quantity. Thus, other than thinking about the
vagaries of floating point wibbles and the Show
instance, it is completely
irrelevant which unit the concrete unit in the LCSU.
The type Length
defined here could be
used equally well in a program that deals exclusively in feet as it could in a
program with meters.
As a tangential note: I have experimented both with definitions like data
Meter = Meter
and data Meter = Meters
(note the s
at the end). The second
often flows more nicely in code, but the annoyance of having to remember
whether I was at the type level or the term level led me to use the former in
my work.
Other features
Prefixes
Here is how to define the "kilo" prefix:
data Kilo = Kilo
instance UnitPrefix Kilo where
multiplier _ = 1000
kilo :: unit > Kilo :@ unit
kilo = (Kilo :@)
We define a prefix in much the same way as an ordinary unit, with a datatype
and a constructor to serve as a proxy. Instead of the Unit
class, though,
we use the UnitPrefix
class, which contains a multiplier
method. As with
other methods, this may not inspect its argument.
Due to the way units are encoded, it is necessary to explicitly apply prefixes
with the :@
combinator (available at both the type and term level). It is often
convenient to then define a function like kilo
to make the code flow more
naturally:
longWayAway :: Length
longWayAway = 150 % kilo Meter
longWayAwayInMeters :: Double
longWayAwayInMeters = longWayAway # Meter  150000.0
Unit combinators
There are several ways of combining units to create other units.
Units can be multiplied and divided with the operators :*
and :/
, at either
the term or type level. For example:
type MetersPerSecond = Meter :/ Second
type Velocity1 = MkQu_ULN MetersPerSecond LCSU Double
speed :: Velocity1
speed = 20 % (Meter :/ Second)
The units package also provides combinators "%*" and "%/" to combine the types of quantities.
type Velocity2 = Length %/ Time  same type as Velocity1
There are also exponentiation combinators :^
(for units) and %^
(for
quantities) to raise to a power. To represent the power, the
units package exports Zero
, positive numbers One
through Five
, and
negative numbers MOne
through MFive
. At the term level, precede the number
with a p
(mnemonic: "power"). For example:
type MetersSquared = Meter :^ Two
type Area1 = MkQu_ULN MetersSquared LCSU Double
type Area2 = Length %^ Two  same type as Area1
roomSize :: Area1
roomSize = 100 % (Meter :^ pTwo)
roomSize' :: Area1
roomSize' = 100 % (Meter :* Meter)
Note that addition and subtraction on units does not make physical sense, so those operations are not provided.
Dimensionsafe cast
The haddock documentation shows the termlevel quantity
combinators. The only one deserving special mention is redim
, the
dimensionsafe cast operator. Expressions written with the units package can
have their types inferred. This works just fine in practice, but the types are
terrible, unfortunately. Much better is to use toplevel annotations (using
abbreviations like Length
and Time
) for your functions. However, it may
happen that the inferred type of your expression and the given type of your
function may not exactly match up. This is because quantities have
a looser notion of type equality than Haskell does. For example, "meter *
second" should be the same as "second * meter", even though these are in
different order. The redim
function checks (at compile time) to make sure its
input type and output type represent the same underlying dimension and then
performs a cast from one to the other. This cast is completely free at
runtime. When providing type annotations, it is good practice to start your
function with a redim $
to prevent the possibility of type errors. For
example, say we redefine velocity a different way:
type Velocity3 = (MkQu_ULN Number LCSU Double) %/ Time %* Length
addVels :: Velocity1 > Velocity1 > Velocity3
addVels v1 v2 = redim $ v1 + v2
This is a bit contrived, but it demonstrates the point. Without the redim
, the
addVels
function would not typecheck. Because redim
needs to know its
result type to typecheck, it should only be used at the top level, such as
here, where there is a type annotation to guide it.
Note that redim
is always dimensionsafe  it will not convert a time to a
length!
Monomorphic behavior
units provides a facility for ignoring LCSUs, if your application does not
need to worry about numerical precision. The facility is through the type
family DefaultUnitOfDim
. For example, with the definitions above, we could
say
type instance DefaultUnitOfDim LengthDim = Meter
type instance DefaultUnitOfDim TimeDim = Second
and then use the DefaultLCSU
for our LCSU. To make the use of the default
LCSU even easier, the MkQu_xxx
operators that don't mention an LCSU all
use the default one. So, we can say
type Length = MkQu_D LengthDim
and get to work. (This uses Double
as the underlying numerical representation.)
The module Data.Metrology.SI
from the unitsdefs package exports type
instances for DefaultUnitOfDim
for the SI types, meaning that you can use
definitions like this right away.
More examples
Check out some of the test examples we have written to get more of a feel for how this all works, here.