fricas
(1) -> <spad>
++++++++++++++++++++++++
++ $Id: 01-AUG-2015$ +++
++++++++++++++++++++++++
fricas
)abbrev domain RIA Q_INTERVAL
++ Author: Kurt Pagani
++ Date Created: 2012
++ License: BSD (same as Axiom)
++ ===================================
++ Rational Interval Arithmetics (RIA)
++ ===================================
++ This is an implementation of RIA as a FriCAS domain.
++ Ref. "Guaranteed Proofs Using Interval Arithmetic"
++ by Marc Daumas, Guillaume Melquiond, and Cesar Munoz
++ See docs for more information.
++
++
++
Q_INTERVAL : Exports == Implementation where
Q ==> Fraction(Integer)
R ==> DoubleFloat
I ==> Integer
OF ==> OutputForm
DF ==> DoubleFloat
Exports == Join(CoercibleTo OutputForm, ConvertibleTo String) with
----------------------------
-- Basic interval operations
----------------------------
"+" : (%,%) -> %
"-" : (%,%) -> %
"*" : (%,%) -> %
"/" : (%,%) -> %
"^" : (%,I) -> %
"-" : % -> %
abs : % -> %
elt : (Q, %) -> %
coerce: % -> OutputForm
coerce: List Q -> %
coerce: Float -> %
coerce: DF -> %
concise : % -> OutputForm
----------------------------------
-- Constructors and query functions
-----------------------------------
mki : (Q,Q) -> %
lb : % -> Q
ub : % -> Q
mid : % -> Q
len : % -> Q
frep : % -> OF
tm_frep : % -> OF
Implementation == add
--------------------
-- Represent as Rec
--------------------
Rep := Record(lb:Q, ub:Q)
-----------------------
-- Implement functions
-----------------------
mki(a,b) == b>=a => [a,b]$Rep
lb(r:%) == r.lb
ub(r:%) == r.ub
x+y == mki(x.lb + y.lb, x.ub + y.ub)
x-y == mki(x.lb - y.ub, x.ub - y.lb)
x*y ==
a:Q:=min(min(x.lb*y.lb,x.lb*y.ub),min(x.ub*y.lb,x.ub*y.ub))
b:Q:=max(max(x.lb*y.lb,x.lb*y.ub),max(x.ub*y.lb,x.ub*y.ub))
mki(a,b)
x/y == if y.lb*y.ub > 0 then x*mki(1/y.ub,1/y.lb)
-x == mki(-x.ub,-x.lb)
abs(x) ==
if x.lb*x.ub >= 0 then
mki(min(abs(x.lb), abs(x.ub)), max(abs(x.lb), abs(x.ub)))
else
mki(0::Q, max(abs(x.lb), abs(x.ub)))
x^n ==
zero? n => mki(1,1)
one? n => x
x.lb>=0 or odd? n => mki(x.lb^n,x.ub^n)
x.ub<=0 or even? n => mki(x.ub^n,x.lb^n)
mki(0,max(x.lb^n,x.ub^n))
coerce(r : %) : OutputForm ==
hconcat ["[",r.lb::OF,",",r.ub::OF,"]"]
coerce(l : List Q) : % ==
#l ~= 2 => error "Format: [lb,ub]"
l.1 > l.2 => error "Must have lb <= ub"
mki(l.1,l.2)
coerce(r:DF):% ==
rc:Q:=retract r
mki(rc,rc)
coerce(r:Float):% ==
rc:Q:=retract r
mki(rc,rc)
coerceSTR(s:String):% ==
s1 := split(s,char ".")
s2 := split(s1.2, char "(")
vs := concat [s1.1,".",s2.1]
v:=float(convert vs)
v::Rep
elt(q:Q,x:%):% == mki(q*x.lb, q*x.ub)
mid(x:%):Q == (x.lb + x.ub)/2::Q
len(x:%):Q == (x.ub - x.lb)
frep(x:%):OF ==
pm:Character:=char(177)
sep:=concat [" ",pm::String," "]::Symbol
hconcat [mid(x)::DF::OF, sep::OF, (len(x)/2::Q)::DF::OF]
tm_frep(x:%):OF ==
sep:=" <pm> "::Symbol
hconcat [mid(x)::DF::OF, sep::OF, (len(x)/2::Q)::DF::OF]
concise(x:%):OF == frep(x) -- to do
fricas
)abbrev category PUNIT PhysicalUnit
++
PhysicalUnit() : Category == with
"*" : (%,%) -> %
"/" : (%,%) -> %
"^" : (%,Integer) -> %
"=" : (%,%) -> Boolean
%one : -> %
coerce : % -> OutputForm
fricas
)abbrev domain SI SI_UNIT
++ Author: Kurt Pagani
++ Date Created: 2012
++ License: BSD (same as Axiom)
++ ================================
++ Physical Quantities and SI Units
++ ================================
++ This is an implementation of physical quantities and SI units as FriCAS
++ domains.
++
++ See docs for more information.
++
++ Revision:
++ - removing outputForm, Ring
++ - added negation, elt
++
++
SI_UNIT : Exports == Implementation where
EI ==> Expression Integer
VI ==> Vector Integer
LS ==> List Symbol
LEI ==> List EI
LVI ==> List VI
Exports == Join(PhysicalUnit,CoercibleTo OutputForm) with
----------------------------
-- (de)-construct an SI unit
----------------------------
mksi: List(Integer) -> %
si2l: % -> List(Integer)
--------------------------------
-- basic operation with SI units
--------------------------------
--"*" : (%,%) -> %
--"/" : (%,%) -> %
--"^" : (%,Integer) -> %
--"=" : (%,%) -> Boolean
-----------------------------------------------------------------
-- constructor for each base unit: e.g. %m(2)=m^2 -> square meter
-----------------------------------------------------------------
%m : Integer -> %
%kg : Integer -> %
%s : Integer -> %
%A : Integer -> %
%K : Integer -> %
%mol : Integer -> %
%cd : Integer -> %
--%one : -> %
-------------------
-- SI derived units
-------------------
SI_derived : String -> %
----------------
-- Buckingham Pi
----------------
buck : Table(Symbol,%) -> List(Expression(Integer))
Implementation == add
---------------------------------------
-- Represent as 1-Array [0,0,0,0,0,0,0]
---------------------------------------
Rep := OneDimensionalArray(Integer)
---------------------------------
-- Implementation of constructors
---------------------------------
mksi(l) ==
r:Rep:=new(7,0$Integer)$Rep
for i in 1..7 repeat r.i := l(i)
r
si2l(r) ==
l:List(Integer):=[0,0,0,0,0,0,0]
for i in 1..7 repeat l(i) := r.i
l
x * y ==
z:Rep:=new(7,0$Integer)$Rep
for i in 1..7 repeat z.i := x.i + y.i
z
x / y ==
z:Rep:=new(7,0$Integer)$Rep
for i in 1..7 repeat z.i := x.i - y.i
z
x ^ n ==
z:Rep:=new(7,0$Integer)$Rep
for i in 1..7 repeat z.i := x.i * n
z
x = y ==
res:Boolean:=true
for i in 1..7 repeat res:=(res and test(x.i=y.i)$Boolean)
res
------------------------
-- Coerce to output form
------------------------
coerce(r : %) : OutputForm ==
SYM ==> Symbol
OF ==> OutputForm
sym:List(String):=["m","kg","s","A","K","mol","cd"]
f:OutputForm:=empty()
for i in 1..7 repeat
if not zero? r.i then
f:=hconcat(f,super(sym(i)::SYM::OF, r.i::OF))
f
------------------------------------------
-- Base unit constructors (implementation)
------------------------------------------
%m(n:Integer) ==
mksi([n,0,0,0,0,0,0])
%kg(n:Integer) ==
mksi([0,n,0,0,0,0,0])
%s(n:Integer) ==
mksi([0,0,n,0,0,0,0])
%A(n:Integer) ==
mksi([0,0,0,n,0,0,0])
%K(n:Integer) ==
mksi([0,0,0,0,n,0,0])
%mol(n:Integer) ==
mksi([0,0,0,0,0,n,0])
%cd(n:Integer) ==
mksi([0,0,0,0,0,0,n])
%one() ==
mksi([0,0,0,0,0,0,0])
SI_derived(s:String):% ==
s = "hertz" => %s(-1)
s = "newton" => %kg(1)*%m(1)*%s(-2)
s = "pascal" => %kg(1)*%m(-1)*%s(-2)
s = "joule" => %kg(1)*%m(2)*%s(-2)
s = "watt" => %kg(1)*%m(2)*%s(-3)
s = "coulomb" => %s(1)*%A(1)
s = "volt" => %kg(1)*%m(2)*%s(-3)*%A(-1)
s = "farad" => %kg(-1)*%m(-2)*%s(4)*%A(2)
s = "ohm" => %kg(1)*%m(2)*%s(-3)*%A(2)
s = "siemens" => %kg(-1)*%m(-2)*%s(3)*%A(2)
s = "weber" => %kg(1)*%m(2)*%s(-2)*%A(-1)
s = "tesla" => %kg(1)*%s(-2)*%A(-1)
s = "henry" => %kg(1)*%m(2)*%s(-2)*%A(-2)
s = "lumen" => %cd(1)
s = "lux" => %m(-2)*%cd(1)
s = "becquerel" => %s(-1)
s = "gray" => %m(2)*%s(-2)
s = "sievert" => %m(2)*%s(-2)
s = "katal" => %s(-1)*%mol(1)
error "Expected string name of a derived SI unit."
-- Helper function
mkxpr(s:LS,v:VI):EI ==
r:EI:=1
for j in 1..#s repeat r:=r*(s.j)::EI^(v.j)
r
-- Buckingham Pi
buck(t:Table(Symbol,%)):LEI ==
M:Matrix(Integer):=matrix [si2l(t.x) for x in keys(t)]
ns:LVI:=nullSpace transpose(M)
r:LEI:=[]
for j in 1..#ns repeat r:=append(r,[mkxpr(keys(t),ns.j)])
r
-- End of SI_UNIT --
-------------------------------------------------------------------------------
fricas
)abbrev domain PQTY PhysQty
++ Author: Kurt Pagani
++ Date Created: 2012
++ License: BSD (same as Axiom)
++
PhysQty(U): Exports == Implementation where
Q ==> Fraction(Integer)
F ==> Float
R ==> DoubleFloat
I ==> Integer
QI ==> Q_INTERVAL
U:PhysicalUnit
Exports == Join(CoercibleTo OutputForm) with
--------------------------------------
-- Physical quantities base operations
--------------------------------------
"*": (%,%) -> %
"/": (%,%) -> %
"+": (%,%) -> %
"-": (%,%) -> %
"^": (%,Integer) -> %
"-": % -> %
elt: (Q, %) -> %
elt: (QI,U) -> %
elt: (Union(Q,F,R,I),U) -> %
coerce : % -> OutputForm
---------------------------------------------------
-- Main constructor: pqty(value, uncertainty, unit)
---------------------------------------------------
pqty: (Q, Q, U) -> %
pqty_value: % -> Q -- get the value
pqty_error: % -> Q -- get the uncertainty
pqty_unit: % -> U -- get the unit
pqty_interval: % -> QI -- get the interval [v-delta,v+delta]
pqty_scale: (%,Q) -> % -- scale by rational number
pqty_tm_toggle: U -> Boolean
Implementation == add
----------------------
-- Represent as Record
----------------------
Rep := Record(ival:QI, unit:U)
----------------------
-- Implement functions
----------------------
pqty(v,e,u) == [[v-e,v+e]::List(Q)::QI,u::U]$Rep
pqty_value(r) == mid(r.ival)
pqty_error(r) == len(r.ival)/2::Q
pqty_unit(r) == r.unit
pqty_interval(r) == r.ival
----------------------------------------------------
-- Implement basic operations (interval arithmetic))
----------------------------------------------------
x*y == [x.ival*y.ival, x.unit*y.unit]$Rep
x/y == [x.ival/y.ival, x.unit/y.unit]$Rep
x+y == x.unit=y.unit => [x.ival+y.ival, x.unit]$Rep
x-y == x.unit=y.unit => [x.ival-y.ival, x.unit]$Rep
x^n == [x.ival^n, x.unit^n]$Rep
- x == [-x.ival, x.unit]$Rep
-----------------------------------------------------------
-- scale
-----------------------------------------------------------
pqty_scale(x,s) == [s x.ival,x.unit]$Rep
--elts
elt(q:Q,x:%):% == pqty_scale(x,q)
elt(qi:QI,u:U):% == [qi,u]$Rep
elt(s:Union(Q,F,R,I),u:U):% == [[s::Q,s::Q]::List(Q)::QI,u]$Rep
pqty_tm := false
pqty_tm_toggle(u:U) : Boolean == -- bad hack
pqty_tm := not pqty_tm
pqty_tm
coerce(r : %) : OutputForm ==
OF ==> OutputForm
pqty_tm = true =>7
hconcat [tm_frep r.ival, " "::OF, r.unit::OF ]
hconcat [frep r.ival, " "::OF, r.unit::OF ]
-- End of PhysQty</spad>
fricas
Compiling FriCAS source code from file
/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/8367240618814005156-25px001.spad
using old system compiler.
RIA abbreviates domain Q_INTERVAL
------------------------------------------------------------------------
initializing NRLIB RIA for Q_INTERVAL
compiling into NRLIB RIA
compiling exported mki : (Fraction Integer,Fraction Integer) -> %
Time: 0.02 SEC.
compiling exported lb : % -> Fraction Integer
RIA;lb;%F;2 is replaced by QCAR
Time: 0 SEC.
compiling exported ub : % -> Fraction Integer
RIA;ub;%F;3 is replaced by QCDR
Time: 0 SEC.
compiling exported + : (%,%) -> %
Time: 0 SEC.
compiling exported - : (%,%) -> %
Time: 0 SEC.
compiling exported * : (%,%) -> %
Time: 0 SEC.
compiling exported / : (%,%) -> %
Time: 0 SEC.
compiling exported - : % -> %
Time: 0 SEC.
compiling exported abs : % -> %
Time: 0 SEC.
compiling exported ^ : (%,Integer) -> %
Time: 0 SEC.
compiling exported coerce : % -> OutputForm
****** comp fails at level 3 with expression: ******
error in function coerce
(|hconcat|
(|construct| | << [ >> | (|::| (|r| |lb|) (|OutputForm|)) ","
(|::| (|r| |ub|) (|OutputForm|)) "]"))
****** level 3 ******
$x:= [
$m:= (OutputForm)
$f:=
((((|r| # #) (|copy| #) (|setelt!| #) (|ub| # #) ...)))
>> Apparent user error:
Cannot coerce [
of mode [
to mode (OutputForm)Oscillation period pendulum
fricas
t:Table(Symbol,SI_UNIT):=table()
SI_UNIT is not a valid type.
Pressure loss in a pipe
fricas
r:Table(Symbol,SI_UNIT):=table()
SI_UNIT is not a valid type.
New: SI derived
fricas
SI_derived("newton")
There are no library operations named SI_derived
Use HyperDoc Browse or issue
)what op SI_derived
to learn if there is any operation containing " SI_derived " in
its name.
Cannot find a definition or applicable library operation named
SI_derived with argument type(s)
String
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
New: PQTY ELT
fricas
volume := 3.2 %m(3)
There are no library operations named %m
Use HyperDoc Browse or issue
)what op %m
to learn if there is any operation containing " %m " in its name.
Cannot find a definition or applicable library operation named %m
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Test internals:
fricas
pqty_value(rho)
There are no library operations named pqty_value
Use HyperDoc Browse or issue
)what op pqty_value
to learn if there is any operation containing " pqty_value " in
its name.
Cannot find a definition or applicable library operation named
pqty_value with argument type(s)
Variable(rho)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Test errors:
fricas
area:= pqty(2,2/10,%m(2))
There are no library operations named %m
Use HyperDoc Browse or issue
)what op %m
to learn if there is any operation containing " %m " in its name.
Cannot find a definition or applicable library operation named %m
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Unit to list (si2l)
fricas
si2l(SI_derived("siemens"))
There are no library operations named SI_derived
Use HyperDoc Browse or issue
)what op SI_derived
to learn if there is any operation containing " SI_derived " in
its name.
Cannot find a definition or applicable library operation named
SI_derived with argument type(s)
String
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
ok.