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\begin{document}
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Subject: \hspace*{0.25in}\=Physical or engineering units\\
From: \>Van Snyder\\
Reference: \>03-258r1, section 1.2\\
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\section{Number}
TBD
\section{Title}
Physical or engineering units
\section{Submitted By}
J3
\section{Status}
For consideration.
\section{Basic Functionality}
Provide for numeric variables and named constants to have physical or
engineering units such as length, mass, area, temperature\dots.
``Compute'' and check units for expressions and assignment at compile
time. Verify that units for corresponding actual and dummy arguments are
identical at compile time if an explicit interface is available. (A
processor could provide for checking units for corresponding actual and
dummy arguments at run time, which would be useful when procedures
without explicit interface are used, but this should not be required by
the standard.) Input and output units. Convert numeric values during
formatted, namelist or list-directed input and output if the item has
units, a conversion expression is defined, and unit input/output is
requested.
It may be possible to apply units to derived types, which would be useful
for such user-defined types as intervals or extended-precision numbers.
This proposal does not include a provision for doing so. It does not
appear to be sensible to apply units to objects of character or logical
type.
\section{Rationale}
Incorrect use of physical units is the second-most-common error in
scientific or engineering software, coming immediately after mismatching
the types of actual and dummy arguments. Explicit interfaces largely
solve the second problem, but do nothing directly for the first. (One
can use derived types to provide a physical units system, at the expense
of redefining assignment and all of the necessary operations and
intrinsic functions.) A particularly expensive ($\approx$
\$$3\times10^8$) and embarrassing example of an units mistake was the the
cause of the loss of the Mars Climate Orbiter. The loss resulted because
the NASA contract required small forces, e.g. from attitude-control
maneuvers, to be reported in Newton-Seconds, but Lockheed nonetheless
reported them in Pound-Seconds. (This was quite inscrutable, as Lockheed
had had contracts with JPL for over thirty years, and they've
\emph{always} specified SI units.)
\section{Estimated Impact}
In terms of changes to the standard, this is a substantial project,
requiring changes in Sections 4, 5, 6, 8, 10, and 13. Determining the
units of an expression, checking units of subexpressions, and doing
conversions automatically, is not tremendously difficult --- I've already
done it in an expression evaluator for an input routine.
\section{Detailed Specification}
Define a new UNIT attribute or type parameter (call it what you will)
that can be specified for any numeric variable or named constant.
Literal constants are unitless except in one case explained below.
Four varieties of units are defined: \emph{Atomic} units are not defined
in terms of any other units. \emph{Composite} units are defined in terms
of atomic units or other composite units. \emph{Value-converting} units
convert values according to specified linear conversions. \emph{Abstract
units} are provided to specify the relation between units for dummy
arguments and function results. They can be either atomic or composite,
but not value-converting. Non-abstract units participate in generic
resolution; abstract units do not.
Multiplication and division operations on units are defined.
Exponentiation by an unitless integer constant is defined to be
equivalent to repeated multiplication or division.
Each unit declaration creates a \emph{unit coercion function} having the
same name as the unit, that takes an argument with any unit, and coerces
it to have the unit specified by the name of the function. If the unit
of the argument and the result are related by a linear conversion
expression, value conversion is also implied. There is an intrinsic
UNITLESS coercion function.
Quantities can be added, subtracted, assigned, or compared by relational
operations only if they have equivalent units. Atomic and
value-converting units are equivalent by name. Composite units are
equivalent by structure. If a dummy argument has a unit that is not
abstract, the associated actual argument shall have an equivalent unit.
Abstract units allow enforcing a particular relation between the units,
without requiring particular units. For example, the SQRT intrinsic
function result has abstract units A, and its argument has abstract units
A*A. If a dummy argument has an abstract unit, the associated actual
argument can have any unit, but if several dummy arguments have abstract
units, the units of their corresponding actual arguments shall be related
in the same way as the units of the dummy arguments. If a function
result has an abstract unit, and that unit is related to units of dummy
arguments, the unit of the result of invoking the function is related to
the actual arguments in the same way.
There is an intrinsic RADIAN unit, and a parallel set of generic
intrinsic trigonometric functions that take RADIAN arguments and produce
unitless results. All of the remaining intrinsic procedures have
arguments with abstract units and results that are unitless (e.g.
SELECTED\_INT\_KIND) or have the same units as their argument (e.g.
TINY). Because function results do not participate in generic
resolution, it is not possible to have a parallel set of generic
intrinsic inverse trigonometric functions that return RADIAN results. It
may be useful to provide an intrinsic module that has some public units
and procedures, e.g. units TICK and SECOND and a SYSTEM\_CLOCK module
procedure that has arguments with units TICK, TICK/SECOND and SECOND.
When quantities are added or subtracted, the units of the result are the
same as the units of the operands. When quantities are multiplied or
divided, the units of the result are the units that result from applying
the operation to the operands' units. Multiplication or division by an
unitless operand produces a result having the same units as the other
operand. Exponentiation by an unitless integer constant is defined to be
equivalent to multiplication or division. In an exponentiation
operation, the exponent shall be unitless. For intrinsic assignment, the
units shall be the same.
There are two forms of the UNIT statement. One without
(~\si{unit-name}~) declares and defines units. Units are declared to be
atomic by appearing without a definition. Units are declared to be
composite or value-converting by having a defining expression that uses
literal constants, unitless named constants, multiplication, division,
exponentiation by an unitless integer, previously-declared unit names,
and, in the case of value-converting units, addition or subtraction.
Units are declared to be abstract by having the ABSTRACT attribute in
their declarations. A nonatomic abstract unit shall be defined in terms
of abstract units.
Value-converting units are defined using linear expressions equivalent in
form to $U = [\,a\,]\, U^\prime\, [\, \pm \, b\, ]$, where $U$ is thereby
declared to be an unit name, $U^\prime$ is required to be a previously
declared nonabstract unit name, and $a \neq 0$ and $b$ are unitless
numeric initialization expressions. A value-converting unit is defined
even if $a$ is absent or one and $b$ is absent or zero. In these
expressions, $a$ is considered to have units $U/U^\prime$ and $b$ is
considered to have units $U$, but we can't say that since $U$ isn't
defined yet. This is the only case where literal constants are not
unitless. This explicitly defines a conversion function from $U$ to
$U^\prime$ and implicitly defines its inverse (always possible because
$a$ is required to be nonzero). Although $U^\prime$ need not be atomic,
if it is atomic it remains atomic even though an inverse conversion is
defined. Value-converting functions are generic, since there may be
several implicitly-defined inverse conversions. Neither $U$ nor
$U^\prime$ shall be abstract.
Value conversions are transitive. Since each unit can only be declared
once, and cannot be referenced before being declared, an explicitly
circular dependence of conversions is impossible. There will usually be
an atomic unit involved in one of the value conversions, but a composite
unit is possible. There are examples of transitive conversions, multiple
implicit conversions, and conversions that depend on nonatomic units
below.
Value conversion is implied during input or explicit units conversion
using the units conversion function implied by the unit declaration or
its inverse, provided the argument or input value is related to the
result or input list item by a sequence of explicit or implicit
conversions. Value conversion does not occur during intrinsic assignment
(this could be an extension), argument association or output. When
conversion is applied, all constants within $a$ and $b$ are converted to
the kind of the argument expression or input list item. This
specification could interact with the proposal to include function result
types in the criteria for generic resolution.
\begin{boxit}
The Mars Climate Orbiter crashed because quantities with the wrong units
were written by one program, and assumed to have the correct units by
another program. Automatic units checking and value conversion would
have let Lockheed use whatever units they wanted to use, so long as the
JPL software had the unit name, and the appropriate conversion,
available. If JPL software insisted on units, and the Lockheed data were
unitless, or used units that JPL software did not specify, the error
would have been detected.
\end{boxit}
In all other unit-defining expressions, the only constants allowed are 1
and 1.0, or named constants having those values.
Variables or named constants are declared to have units by specifying
UNIT(\emph{unit-name}) in their declarations, *\emph{unit-name} after
their names in declaration statements, or by an UNIT(~\si{unit-name}~)
statement.
UNIT(~\si{unit-name}~) is allowed in the \si{prefix} of a
\si{function-stmt}.
There is an optional U[w] suffix to numeric format descriptors, that
causes units to be output by write statements, or input and checked by
read statements. The text of the unit that is output, or checked during
input, is the same as the unit name, except that case of letters is not
significant. There is a specification in OPEN, READ and WRITE statements
that controls whether units for numeric quantities that have units are to
be output (checked) by namelist or list-directed output (input). There
is a specification for the INQUIRE statement to inquire whether this mode
is set by an OPEN statement for a connection. A single specification,
rather than separate ones for namelist and list-directed transfers, is
adequate.
Some thought and debate will be necessary to decide what to do about
input and output of arrays that have units. Should the value in the
input for every element be required to specify its units, or is it enough
that at least one does? If one is enough should it be specified to be
first (or last)? Should units be provided for every element of output, or
is one enough? There is not a problem for array constructors, since they
can be wrapped with an units coercion or conversion function.
\subsection{Examples}
\subsubsection{Atomic and composite units}
\begin{alltt}
UNIT :: INCH, SECOND
UNIT :: CM, INCH_TO_CM = CM / INCH
REAL, PARAMETER, UNIT(INCH_TO_CM) :: CONVERT = INCH_TO_CM(2.54)
UNIT :: SQINCH = INCH * INCH ! or INCH ** 2
UNIT :: IPS = INCH / SECOND, Hz = 1 / SECOND ! or SECOND ** (-1)
REAL, UNIT(SQINCH) :: A
REAL, UNIT(Hz) :: F
REAL, UNIT(INCH) :: L, L2, C*CM
REAL, UNIT(SECOND) :: T
REAL :: V
UNIT(IPS) :: V
V = A + L ! INVALID -- SQINCH cannot be added to INCH,
! and neither one can be assigned to IPS
V = IPS(A + SQINCH(L)) ! VALID -- I'm screwing this up intentionally
V = (A / L + L2) / T ! VALID -- IPS is compatible with INCH / SECOND
A = L * L2 ! VALID -- SQINCH is compatible with INCH * INCH
F = V / L ! VALID -- units of LHS and RHS are both 1/SECOND
C = CONVERT * L ! VALID -- CM / INCH * INCH = CM
L = SQRT(A) * 5.0e-3 ! VALID -- exercise for reader
\end{alltt}
\subsubsection{Abstract units}
\begin{alltt}
INTERFACE
REAL UNIT(UR) FUNCTION CBRT ( A )
UNIT, ABSTRACT :: UR, UA = UR**3
REAL, UNIT(UA) :: A
END FUNCTION CBRT
END INTERFACE
\end{alltt}
\subsubsection{Value-converting units}
\begin{alltt}
UNIT :: MHz = 1.0e6 * Hz, GHz = 1000 * MHz, KHz = 0.001 * MHz
UNIT :: F, C = ( F - 32 ) * 5.0 / 9.0 ! Also defines (atomic) F = 1.8 * C + 32.0
UNIT :: DEGREES = 45.0 * RADIAN / ATAN(1.0)
REAL, UNIT(Radian) :: Angle = Degrees(45) ! = Radian(0.785398163)
REAL, UNIT(MHz) :: Frq = MHz(3310.0)
REAL, UNIT(C) :: Temp = F(212.0) ! = C(100.0)
...
CHARACTER(32) :: LINE
WRITE(LINE,*,UNITS="yes") GHz(Frq), F(Temp), Degrees(Angle)
READ(LINE,*,UNITS="yes") Frq, Temp, Angle
\end{alltt}
Execution of the {\tt WRITE} statement causes LINE to have the value {\tt
"3.31 GHz 212.0 F 45.0 Degrees"} (approximately). Execution of the {\tt
READ} statement causes the variables {\tt Frq}, {\tt Temp} and {\tt
Angle} to get their original values (approximately).
Notice that {\tt MHz(3.31)}, {\tt C(212)} and {\tt Radian(45)} have
values 3.31, 212 and 45 respectively, with units {\tt MHz}, {\tt C} and
{\tt Radian} respectively, not 3310.0, 100.0 and 0.785398163
respectively, since generic resolution selects the simple unit-coercion
functions, not the conversion ones, for unitless arguments.
Input of the form {\tt "3.3021148036e-10 Second 212 F 45 Degrees"} is not
permitted, since the declaration of {\tt Hz} does not define a
value-converting unit.
If we have
\begin{alltt}
REAL(kind(0.0d0)), UNIT(Radian) :: AngleD
\end{alltt}
then in {\tt Degrees(AngleD)} the {\tt 1.0} that is the argument of {\tt
ATAN} in the definition of the {\tt Degrees} conversion function is
converted to {\tt 1.0d0}.
\section{History}
\label{lastpage}
\end{document}