reduce.jl: Symbolic parser generator for Julia language expressions using REDUCE algebra term re-writer, automatically mirrored from https://github.com/chakravala/Reduce.jl https://github.com/chakravala/Reduce.jl
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Symbolic parser generator for Julia language expressions using REDUCE algebra term rewriter
The premise behind Reduce.jl is based on the idea that Symbol
and Expr
types can be translated into computer algebra rewrite commands and then automatically parsed back into Julia ASTs, essentially extending the Julia language into a fully programable symbolic AST rewrite environment.
REDUCE is a system for general algebraic computations of interest to mathematicians, scientists and engineers:
Interface for applying symbolic manipulation on Julia expressions using REDUCE's term rewrite system:
RExpr
objects that can parse
into julia Expr
objects and vice versa;rcall
method;reduce>
REPL within the Julia terminal window activated by }
key;+
,-
,*
,^
,/
,//
compute on Symbol
and Expr
types;Additional packages that depend on Reduce.jl are maintained at JuliaReducePkg.
The upstream REDUCE software created by Anthony C. Hearn is maintained by collaborators on SourceForge.
This package is a heavily modifed version of Nathan Smith's Maxima.jl with many additional features.
The Reduce
package provides the base functionality to work with Julia and Reduce expressions, provided that you have redcsl
in your path. On GNU/Linux/OSX/Windows, Pkg.build("Reduce")
will automatically download a precompiled binary for you. If you are running a different Unix operating system, the build script will download the source and attempt to compile redcsl
for you, success depends on the build tools installed. Automated testing for Travis CI and appveyor using Linux, OSX, and Windows are fully operational using Reduce
.
julia> Pkg.add("Reduce"); Pkg.build("Reduce")
julia> using Reduce
Reduce (Free CSL version, revision 4521), 11-March-2018 ...
For users who wish to experimentally apply additional precompilation, it is possible to enable extra precompilation scripts by setting the environment variable ENV["REDPRE"] = "1"
in julia (only effective when Reduce
is being compiled).
View the documentation stable / latest for more features and examples.
This Reduce
package for the Julia language was created by github.com/chakravala for mathematics and computer algebra research with the upstream developed REDUCE software.
Please consider donating to show your thanks and appreciation to this Julia project for interfacing the upstream REDUCE language at liberapay, GitHub Sponsors, Patreon, or contribute (documentation, tests, examples) in the repository.
The extended algebraic symbolic expression mode of Reduce.jl is activated with ForceImport.jl by
@force using Reduce.Algebra
This locally extends native Julia functions to Symbol
and Expr
types in the current module without extending global methods. Alternatively, the methods it provides can be accesed by prefixing Algebra.
in front of the method.
Reduce expressions encapsulated into RExpr
objects can be manipulated within julia using the standard syntax. Create an expression object either using the RExpr("expression")
string constructor or R"expression"
. Additionally, arbitrary julia expressions can also be parsed directly using the RExpr(expr)
constructor. Internally RExpr
objects are represented as an array that can be accessed by calling *.str[n]
on the object.
When Reduce
is used in Julia, standard arithmetic operations are now extended to also work on Symbol
and Expr
types.
julia> 1-1/:n
:((n - 1) // n)
julia> ans^-:n
:(1 // ((n - 1) // n) ^ n)
julia> limit(ans,:n,Inf)
ℯ = 2.7182818284590...
Julia abstract syntax trees are automatically converted into sequences of reduce statements (using RExpr
constructor) that are in return parsed into julia quote
blocks usig parse
.
The rcall
method is used to evaluate any type of expression.
julia> :(int(sin(im*x+pi)^2-1,x)) |> rcall
:((1 - (ℯ ^ (4x) + 4 * ℯ ^ (2x) * x)) // (8 * ℯ ^ (2x)))
However, there are often multiple equivalent ways of achieving the same result:
julia> int(sin(im*:x+π)^2-1,:x)
:((1 - (ℯ ^ (4x) + 4 * ℯ ^ (2x) * x)) // (8 * ℯ ^ (2x)))
The output of rcall
will be the same as its input type.
julia> "int(sin(y)^2, y)" |> rcall
"( - cos(y)*sin(y) + y)/2"
Use rcall(expr,switches...)
to evaluate expr
using REDUCE mode switches
like :expand
, :factor
, and :latex
.
julia> :((x+im+π)^2; int(1/(1+x^3),x)) |> RExpr
^(+(x,i,pi),2);
int(/(1,+(1,^(x,3))),x);
julia> rcall(ans,:horner) |> parse
quote
((π + 2x) * π + 2 * (π + x) * im + x ^ 2) - 1
((2 * sqrt(3) * atan((2x - 1) // sqrt(3)) - log((x ^ 2 - x) + 1)) + 2 * log(x + 1)) // 6
end
Mathematical operators and REDUCE modes can be applied directly to Expr
and RExpr
objects.
julia> Expr(:function,:(fun(a,b)),:(return 4x^4-44x^3+61x^2+270x-525)) |> horner
:(function fun(a, b)
return ((4 * (x - 11) * x + 61) * x + 270) * x - 525
end)
Additionally, REDUCE switch statements can be used as macros to control evaluation of expressions.
julia> @rounded @factor x^3-2x+1
:((x + 1.61803398875) * (x - 1) * (x - 0.61803398875))
Most core features have a corresponding Julia method, but language features that have not been implemented yet can also be directly evaluated with rcall
using a synergy of julia syntax.
julia> Expr(:for,:(i=2:34),:(product(i))) |> rcall
:(@big_str "295232799039604140847618609643520000000")
The squash
function provides a way to reduce full program blocks into simplified functions, e.g.
julia> Expr(:function,:(example(a,b)),quote
z = 3
target = z * :a * :b
z -= 1
target += z*(1-:a)*(1-:b)
end) |> squash |> factor
:(function example(a, b)
(5b - 2) * a - 2 * (b - 1)
end)
where z
is a program variable and :a
and :b
are symbolic variables.
Packages which come shipped with REDUCE can be loaded with the load_package
method. For example, the optimize
method is available with
julia> load_package(:scope)
julia> Algebra.optimize(:(z = a^2*b^2+10*a^2*m^6+a^2*m^2+2*a*b*m^4+2*b^2*m^6+b^2*m^2))
quote
g40 = b * a
g44 = m * m
g41 = g44 * b * b
g42 = g44 * a * a
g43 = g44 * g44
z = g41 + g42 + g40 * (2g43 + g40) + g43 * (2g41 + 10g42)
end
Other packages can be loaded, but not all of them come with pre-defined Julia dispatch methods.
Some special support for symbolic matrices has also been added to Reduce.Algebra
methods,
julia> [:x 1; :y 2]^-1
2×2 Array{Any,2}:
:(2 / (2x - y)) :(-1 / (2x - y))
:(-y / (2x - y)) :(x / (2x - y))
The jacobian
method has been added to the ReduceLinAlg package, which is dedicated to the LINALG extra package included with Reduce binaries.
julia> using ReduceLinAlg
julia> eqns = [:x1-:x2, :x1+:x2-:x3+:x6t, :x1+:x3t-:x4, 2*:x1tt+:x2tt+:x3tt+:x4t+:x6ttt, 3*:x1tt+2*:x2tt+:x5+0.1*:x8, 2*:x6+:x7, 3*:x6+4*:x7, :x8-sin(:x8)]
8-element Array{Expr,1}:
:(x1 - x2)
:(x1 - ((x3 - x6t) - x2))
:((x3t - x4) + x1)
:(x4t + x6ttt + x3tt + x2tt + 2x1tt)
:((10x5 + x8 + 20x2tt + 30x1tt) // 10)
:(2x6 + x7)
:(3x6 + 4x7)
:(x8 - sin(x8))
julia> vars = [:x1, :x2, :x3, :x4, :x6, :x7, :x1t, :x2t, :x3t, :x6t, :x7t, :x6tt, :x7tt];
julia> jacobian(eqns, vars) |> Reduce.mat
8×13 Array{Any,2}:
1 -1 0 0 0 0 0 0 0 0 0 0 0
1 1 -1 0 0 0 0 0 0 1 0 0 0
1 0 0 -1 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 2 1 0 0 0 0 0 0 0
0 0 0 0 3 4 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
The package also provides a demonstration of how additional Reduce
methods can be imported into Julia.
Various output modes are supported. While in the REPL, the default nat
output mode will be displayed for RExpr
objects.
julia> :(sin(x*im) + cos(y*MathConstants.φ)) |> RExpr
(sqrt(5) + 1)*y
cos(-----------------) + sinh(x)*i
2
This same output can also be printed to the screen by calling print(nat(r))
method.
It is possible to direclty convert a julia expression object to LaTeX code using the latex
method.
julia> print(@latex sin(x) + cos(y*MathConstants.φ))
\begin{displaymath}
\cos \left(\left(\left(\sqrt {5}+1\right) y\right)/2\right)+\sin \,x
\end{displaymath}
Internally, this command essentially expands to rcall(:(sin(x) + cos(y*MathConstants.φ)),:latex) |> print
, which is equivalent.
In IJulia
the display output of RExpr
objects will be rendered LaTeX with the rlfi
REDUCE package in latex
mode.
Similar to ? help and ; shell modes in Julia, Reduce
provides a reduce>
REPL mode by pressing shift+] as the first character in the julia terminal prompt. The output is in nat
mode.
reduce> df(atan(golden_ratio*x),x);
2 2
sqrt(5)*x + sqrt(5) - x + 1
-------------------------------
4 2
2*(x + 3*x + 1)
If the reduce>
REPL is not appearing when }
is pressed or the Reduce pipe is broken, the session can be restored by simply calling Reduce.Reset()
, without requiring a restart of julia
or reloading the package. This kills the currently running Reduce session and then re-initializes it for new use.
Otherwise, questions can be asked on gitter/discourse or submit your issue or pull-request if you require additional features or noticed some unusual edge-case behavior.
By importing the AbstractTensors.jl module, the Reduce
is able to correctly bypass operations on TensorAlgebra
elements to the correct methods within the scope of the Reduce.Algebra
module.
This requires no additional overhead for the Grassmann.jl or Reduce
packages, because the AbstractTensors
interoperability interface enables separate precompilation of both.
The Reduce
package currently provides a robust interface to directly use the CSL version of REDUCE within the Julia language and the REPL. This is achieved by interfacing the abstract syntax tree of Expr
objects with the parser generator for RExpr
objects and then using an IOBuffer
to communicate with redpsl
.
REDUCE is a system for doing scalar, vector and matrix algebra by computer, which also supports arbitrary precision numerical approximation and interfaces to gnuplot to provide graphics. It can be used interactively for simple calculations but also provides a full programming language, with a syntax similar to other modern programming languages. REDUCE has a long and distinguished place in the history of computer algebra systems. Other systems that address some of the same issues but sometimes with rather different emphasis are Axiom, Macsyma (Maxima), Maple and Mathematica. REDUCE is implemented in Lisp (as are Axiom and Macsyma), but this is completely hidden from the casual user. REDUCE primarily runs on either Portable Standard Lisp (PSL) or Codemist Standard Lisp (CSL), both of which are included in the SourceForge distribution. PSL is long-established and compiles to machine code, whereas CSL is newer and compiles to byte code. Hence, PSL may be faster but CSL may be available on a wider range of platforms.
Releases of Reduce.jl
enable the general application of various REDUCE functionality and packages to manipulate the Julia language to simplify and compute new program expressions at run-time. Intended for uses where a symbolic pre-computation is required for numerical algorithm code generation.
Julia is a high-level, high-performance dynamic programming language for numerical computing. It provides a sophisticated compiler, distributed parallel execution, numerical accuracy, and an extensive mathematical function library. Julia’s Base library, largely written in Julia itself, also integrates mature, best-of-breed open source C and Fortran libraries for linear algebra, random number generation, signal processing, and string processing. The strongest legacy of Lisp in the Julia language is its metaprogramming support. Like Lisp, Julia represents its own code as a data structure of the language itself. Since code is represented by objects that can be created and manipulated from within the language, it is possible for a program to transform and generate its own code. This allows sophisticated code generation without extra build steps, and also allows true Lisp-style macros operating at the level of abstract syntax trees.