HyperDualNumbers.jl: Julia Implementation Of HyperDualNumbers

JuliaDiff/HyperDualNumbers

Hyper-dual numbers can be used to compute first and second derivatives numerically without the cancellation errors of finite-differencing schemes.

The initial Julia implementation (up to v3.0.1) is directly based on the C++ implementation by Jeffrey Fike and Juan J Alonso, both of Stanford University, department of Aeronautics and Astronautics as described in the paper:

The Development of Hyper-Dual Numbers for Exact Second Derivative Calculations

Up to v3.0.1 the Julia versions have been derived/written by Rob J Goedman (goedman@icloud.com).

HyperDualNumbers.jl v4.0.0 has been completely redone by Benoit Pasquier and follows the structure of the JuliaDiff/DualNumbers package.

For a quick intro, see STEPBYSTEP.md

Latest tagged versions:

• v1.1.0 (Julia 0.5 & 0.6, Oct 2017)
• v2.0.0 (Julia v0.7-, Oct 2017)
• v3.0.1 (Julia v0.7 & Julia v1.0), Aug 2018, Pkg(3))
• v4.0.0 (Julia v1.0, Nov 2018)
• v4.0.9 (Julia 1.9-DEV, May 2022)

For details see VERSION.md

The following functions are specific to hyperdual numbers:

• Hyper,
• Hyper256,
• Hyper128,
• ishyper,y2
• hyper_show
• realpart,
• ε₁part(), replaces eps1,
• ε₂part(), replaces eps2,
• ε₁ε₂part(), replaces eps1eps2

In the future it is my intention to deprecate:

• hyper,
• hyper256,
• hyper128,
• eps1,
• eps2,
• eps1eps2

Documentation

• STABLE — documentation of the most recently tagged version.
• DEVEL — documentation of the in-development version.

Questions and issues

Question and contributions are very welcome, as are feature requests and suggestions. Please open an issue if you encounter any problems or have a question.

Download Details:

Author: JuliaDiff
Source Code: https://github.com/JuliaDiff/HyperDualNumbers.jl 
License: View license

#julia #number 

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HyperDualNumbers.jl: Julia Implementation Of HyperDualNumbers

JuliaDiff/HyperDualNumbers

Hyper-dual numbers can be used to compute first and second derivatives numerically without the cancellation errors of finite-differencing schemes.

The initial Julia implementation (up to v3.0.1) is directly based on the C++ implementation by Jeffrey Fike and Juan J Alonso, both of Stanford University, department of Aeronautics and Astronautics as described in the paper:

The Development of Hyper-Dual Numbers for Exact Second Derivative Calculations

Up to v3.0.1 the Julia versions have been derived/written by Rob J Goedman (goedman@icloud.com).

HyperDualNumbers.jl v4.0.0 has been completely redone by Benoit Pasquier and follows the structure of the JuliaDiff/DualNumbers package.

For a quick intro, see STEPBYSTEP.md

Latest tagged versions:

• v1.1.0 (Julia 0.5 & 0.6, Oct 2017)
• v2.0.0 (Julia v0.7-, Oct 2017)
• v3.0.1 (Julia v0.7 & Julia v1.0), Aug 2018, Pkg(3))
• v4.0.0 (Julia v1.0, Nov 2018)
• v4.0.9 (Julia 1.9-DEV, May 2022)

For details see VERSION.md

The following functions are specific to hyperdual numbers:

• Hyper,
• Hyper256,
• Hyper128,
• ishyper,y2
• hyper_show
• realpart,
• ε₁part(), replaces eps1,
• ε₂part(), replaces eps2,
• ε₁ε₂part(), replaces eps1eps2

In the future it is my intention to deprecate:

• hyper,
• hyper256,
• hyper128,
• eps1,
• eps2,
• eps1eps2

Documentation

• STABLE — documentation of the most recently tagged version.
• DEVEL — documentation of the in-development version.

Questions and issues

Question and contributions are very welcome, as are feature requests and suggestions. Please open an issue if you encounter any problems or have a question.

Download Details:

Author: JuliaDiff
Source Code: https://github.com/JuliaDiff/HyperDualNumbers.jl 
License: View license

#julia #number 

A Julia Implementation Of Barycentric interpolation

BarycentricInterpolation   

This Julia package implements the Barycentric formula for polynomial interpolation on equispaced points and Chebyshev points of the first and second kind. The formulae used are taken from the paper of Berrut and Trefethen, SIAM Review, 2004.

This is not a general purpose interpolation package but is intended to be used as a base for other numerical methods, such as numerical collocation. For a general use interpolation package see Interpolations.jl

Usage

There are various types of polynomials defined based on the locations of their nodes (zeros).

• Equispaced (Equispaced{N}()) — a common choice when data is equispaced but suffers from Runge phenomenon for high degree polynomials. When used as part of a collocation scheme with Gauss-Legendre collocation points, they provide the benefit of super-convergence. By default the nodes are equispaced over [-1, +1].
• Chebyshev type 1 (Chebyshev1{N}()) — nodes distributed according to cos(π(2j + 1)/(2N + 2)) where N is the degree of the polynomial, for j in [0, N].
• Chebyshev type 2 (Chebyshev2{N}()) — nodes distributed according to cos(πj/N) where N is the degree of the polynomial, for j in [0, N].
• Legendre (Legendre{N}()) — nodes distributed according to the zeros of the corresponding Legendre polynomial where N is the degree of the polynomial.
• Arbitrary nodes (ArbitraryPolynomial(nodes)) — nodes distributed as specified.

By default, each of the polynomials are defined over the range [-1, +1]. This can be modified by specifying a start and stop for the range, e.g., Equispaced{10}(0, 1) will generate a 10th order polynomial with equispaced nodes over the range [0, 1].

Polynomials with nodes asymptotically clustered towards the end points (such as Chebyshev) are optimal for avoiding the Runge phenomenon (see Trefethen, Spectral Methods in MATLAB, SIAM 2000).

Once a polynomial has been defined it can be used with the nodes(poly) and weights(poly) functions to return the locations of the nodes and the values of the Barycentric weights respectively. To interpolate a set of y values (defined at the nodes) use interpolate(poly, y, x_new); x_new can be either a scalar or a vector. If x_new is omitted, the interpolate function returns a function y(x) which can be used to evaluate the interpolant at any point.

To obtain the interpolant as a linear combination of the y values, use interpolation_matrix(poly, x); this returns a matrix which can be multiplied by a vector of y values to calculate the interpolated value.

Finally, the derivative of the polynomial at the nodes can be obtained using differentiation_matrix(poly). Similar to interpolation_matrix, this returns a matrix which can be multiplied by a vector of y values to calculate the derivative of y.

Simple example

using BarycentricInterpolation

p = Chebyshev2{20}()           # create a Chebyshev type 2 polynomial of order 20
x = nodes(p)                   # get the nodes
y = sinpi.(x)                  # generate y values at the nodes
x_new = rand()*2 -1            # a random number in [-1, +1]
println(interpolate(p, y, x_new) ≈ sinpi(x_new))       # hopefully true!
D = differentiation_matrix(p)  # get the differentiation matrix
println(interpolate(p, D*y, x_new) ≈ pi*cospi(x_new))  # hopefully true!

More complicated example

For an example with Barycentric.jl applied to the simulation of a PDE (in combination with DifferentialEquations.jl) see https://cityinthesky.co.uk/posts/2018/barycentricinterpolation.jl/.

Download Details:

Author: Dawbarton
Source Code: https://github.com/dawbarton/BarycentricInterpolation.jl 
License: View license

#julia #implemented 

ProtoBuf.jl: Julia Protobuf Implementation

ProtoBuf.jl

Protocol buffers are a language-neutral, platform-neutral, extensible way of serializing structured data for use in communications protocols, data storage, and more.

ProtoBuf.jl is a Julia implementation for protocol buffers.

Generating Code (from .proto files)

ProtoBuf.jl includes the protoc compiler version 3 binary appropriate for your operating system. The Julia code generator plugs in to the protoc compiler. It is implemented as ProtoBuf.Gen, a sub-module of ProtoBuf. The callable program (as required by protoc) is provided as the script plugin/protoc-gen-julia for unix like systems and plugin/protoc-gen-julia_win.bat for Windows.

For convenience, ProtoBuf.jl exports a protoc(args) command that will setup the PATH and environment correctly for the included protoc. E.g. to generate Julia code from proto/plugin.proto, run the command below which will create a corresponding file jlout/plugin.jl, simply run (from a Julia REPL):

julia> using ProtoBuf

julia> ProtoBuf.protoc(`-I=proto --julia_out=jlout proto/plugin.proto`)

Each .proto file results in a corresponding .jl file, including one each for other included .proto files. Separate .jl files are generated with modules corresponding to each top level package.

If a field name in a message or enum matches a Julia keyword, it is prepended with an _ character during code generation.

If a package contains a message which has the same name as the package itself, optionally set the JULIA_PROTOBUF_MODULE_POSTFIX=1 environment variable when running protoc, this will append _pb to the module names.

ProtoBuf map types are generated as Julia Dict types by default. They can also be generated as Array of key-values by setting the JULIA_PROTOBUF_MAP_AS_ARRAY=1 environment variable when running protoc.

Julia Type Mapping

.proto Type	Julia Type	Notes
double	Float64
float	Float32
int32	Int32	Uses variable-length encoding. Inefficient for encoding negative numbers – if your field is likely to have negative values, use sint32 instead.
int64	Int64	Uses variable-length encoding. Inefficient for encoding negative numbers – if your field is likely to have negative values, use sint64 instead.
uint32	UInt32	Uses variable-length encoding.
uint64	UInt64	Uses variable-length encoding.
sint32	Int32	Uses variable-length encoding. Signed int value. These more efficiently encode negative numbers than regular int32s.
sint64	Int64	Uses variable-length encoding. Signed int value. These more efficiently encode negative numbers than regular int64s.
fixed32	UInt32	Always four bytes. More efficient than uint32 if values are often greater than 2^28.
fixed64	UInt64	Always eight bytes. More efficient than uint64 if values are often greater than 2^56.
sfixed32	Int32	Always four bytes.
sfixed64	Int64	Always eight bytes.
bool	Bool
string	ByteString	A string must always contain UTF-8 encoded or 7-bit ASCII text.
bytes	Array{UInt8,1}	May contain any arbitrary sequence of bytes.
map	Dict	Can be generated as Array of key-value by setting environment variable JULIA_PROTOBUF_MAP_AS_ARRAY=1

Well-Known Types

The protocol buffers well known types are pre-generated and included in the package as a sub-module ProtoBuf.google.protobuf. The version of the code included with this package have additional changes to make them compatible with Julia.

You can refer to them in your code after including the following statements:

using ProtoBuf
using ProtoBuf.google.protobuf

While generating code for your .proto files that use well-known types, add ProtoBuf/gen to the list of includes, e.g.:

julia> using ProtoBuf

julia> ProtoBuf.protoc(`-I=proto -I=ProtoBuf/gen --julia_out=jlout proto/msg.proto`)

Though this would generate code for the well-known types along with your messages, you just need to use the files generated for your messages.

Generic Services

The Julia code generator generates code for generic services if they are switched on for either C++ (cc_generic_services), Python (py_generic_services) or Java (java_generic_services).

To use generic services, users provide implementations of the RPC controller, RPC channel, and service methods.

The RPC Controller must be an implementation of ProtoRpcController. It is not currently used by the generated code except for passing it on to the RPC channel.

The RPC channel must implement call_method(channel, method_descriptor, controller, request) and return the response.

RPC method inputs or outputs that are defined as stream type, are generated as Channel of the corresponding type.

Service stubs are Julia types. Stubs can be constructed by passing an RPC channel to the constructor. For each service, two stubs are generated:

• Stub: The asynchronous stub that takes a callback to invoke with the result on completion
• BlockingStub: The blocking stub that returns the result on completion

Note:

• Extensions are not supported yet.
• Groups are not supported. They are deprecated anyway.
• Enums are declared as Int32 types in the generated code. For every enum, a separate named tuple is generated with fields matching the enum values. The lookup method can be used to verify valid values.
• In order to use the code generator, you must have installed ProtoBuf in the base Julia environment (]activate; add ProtoBuf).

Using ProtoBuf

Julia code for protobuf message types can be generated via protoc (see "Generating Julia Code from .proto Specifications"). Generated Julia code for a protobuf message look something like:

mutable struct Description <: ProtoType
    # a bunch of internal fields
    ...
    function Description(; kwargs...)
        # code to initialize the internal fields
    end
end # mutable struct Description
const __meta_Description = Ref{ProtoMeta}()
function meta(::Type{Description})
    # code to initialize the metadata
    __meta_Description[]
end
function Base.getproperty(obj::Description, name::Symbol)
    # code to get properties
end

Reading and writing data structures using ProtoBuf is similar to serialization and deserialization. Methods writeproto and readproto can write and read Julia types from IO streams.

julia> using ProtoBuf                       # include protoc generated package here

julia> mutable struct MyType <: ProtoType   # a Julia composite type generated from protoc that
         ...                                # has intval::Int and strval::String as properties
         function MyType(; kwargs...)
             ...
         end
       end
       ...

julia> iob = PipeBuffer();

julia> writeproto(iob, MyType(; intval=10, strval="hello world"));   # write an instance of it

julia> data = readproto(iob, MyType());  # read it back into another instance

julia> data.intval
10

julia> data.strval
"hello world"

Reading message from a file is very similar to reading from a stream. Here's an example that writes a message to file and then reads it back.

julia> include("test_type.jl")

julia> mktemp() do path, io
           tt1 = TestType(; a="abc", b=true) # construct a message
           writeproto(io, tt1)  # write message to file
           close(io) # close the file handle
           open(path) do io2 # open the file we just wrote in read mode
               tt2 = readproto(io2, TestType()) # read message from the file
               @info("read back from file", tt1.a, tt1.b, tt2.a, tt2.b) # print written and read messages
           end
       end
┌ Info: read back from file
│   tt1.a = "abc"
│   tt1.b = true
│   tt2.a = "abc"
└   tt2.b = true       

Contents of the generated code in test_type.jl:

using ProtoBuf
import ProtoBuf.meta

mutable struct TestType <: ProtoType
    __protobuf_jl_internal_meta::ProtoMeta
    __protobuf_jl_internal_values::Dict{Symbol,Any}
    __protobuf_jl_internal_defaultset::Set{Symbol}

    function TestType(; kwargs...)
        obj = new(meta(TestType), Dict{Symbol,Any}(), Set{Symbol}())
        values = obj.__protobuf_jl_internal_values
        symdict = obj.__protobuf_jl_internal_meta.symdict
        for nv in kwargs
            fldname, fldval = nv
            fldtype = symdict[fldname].jtyp
            (fldname in keys(symdict)) || error(string(typeof(obj), " has no field with name ", fldname))
            values[fldname] = isa(fldval, fldtype) ? fldval : convert(fldtype, fldval)
        end
        obj
    end
end #type TestType
const __meta_TestType = Ref{ProtoMeta}()
function meta(::Type{TestType})
    if !isassigned(__meta_TestType)
        __meta_TestType[] = target = ProtoMeta(TestType)
        allflds = Pair{Symbol,Union{Type,String}}[:a => AbstractString, :b => Bool]
        meta(target, TestType, allflds, [:a], ProtoBuf.DEF_FNUM, ProtoBuf.DEF_VAL, ProtoBuf.DEF_PACK, ProtoBuf.DEF_WTYPES, ProtoBuf.DEF_ONEOFS, ProtoBuf.DEF_ONEOF_NAMES)
    end
    __meta_TestType[]
end
function Base.getproperty(obj::TestType, name::Symbol)
    if name === :a
        return (obj.__protobuf_jl_internal_values[name])::AbstractString
    elseif name === :b
        return (obj.__protobuf_jl_internal_values[name])::Bool
    else
        getfield(obj, name)
    end
end

Setting and Getting Fields

Types used as protocol buffer structures are regular Julia types and the Julia syntax to set and get fields can be used on them. The generated type constructor makes it easier to set large types with many fields by passing name value pairs during construction: T(; name=val...).

Fields that are marked as optional may not be present in an instance of the struct that is read. Also, you may want to clear a set property from an instance. The following methods are exported to assist doing this:

• propertynames(obj) : Returns a list of property names possible
• setproperty!(obj, fld::Symbol, v) : Sets obj.fld.
• getproperty(obj, fld::Symbol) : Gets obj.fld if it has been set. Throws an error otherwise.
• hasproperty(obj, fld::Symbol) : Checks whether property fld has been set in obj.
• clear(obj, fld::Symbol) : clears property fld of obj.
• clear(obj) : Clears all properties of obj.

julia> using ProtoBuf

julia> mutable struct MyType <: ProtoType  # a Julia composite type
           ... # intval::Int
           ...
       end

julia> mutable struct OptType <: ProtoType # and another one to contain it
           ... #opt::MyType
           ...
       end

julia> iob = PipeBuffer();

julia> writeproto(iob, OptType(opt=MyType(intval=10)));

julia> readval = readproto(iob, OptType());

julia> hasproperty(readval, :opt)
true

julia> writeproto(iob, OptType());

julia> readval = readproto(iob, OptType());

julia> hasproperty(readval, :opt)
false

The isinitialized(obj::Any) method checks whether all mandatory fields are set. It is useful to check objects using this method before sending them. Method writeproto results in an exception if this condition is violated.

julia> using ProtoBuf

julia> import ProtoBuf.meta

julia> mutable struct TestType <: ProtoType
           ... # val::Any
           ...
       end

julia> mutable struct TestFilled <: ProtoType
           ... # fld1::TestType (mandatory)
           ... # fld2::TestType
           ...
       end

julia> tf = TestFilled();

julia> isinitialized(tf)      # false, since fld1 is not set
false

julia> tf.fld1 = TestType(fld1="");

julia> isinitialized(tf)      # true, even though fld2 is not set yet
true

Equality & Hash Value

It is possible for fields marked as optional to be in an "unset" state. Even bits type fields (isbitstype(T) == true) can be in this state though they may have valid contents. Such fields should then not be compared for equality or used for computing hash values. All ProtoBuf compatible types, by virtue of extending abstract ProtoType type, override hash, isequal and == methods to handle this.

Other Methods

• copy!{T}(to::T, from::T) : shallow copy of objects
• isfilled(obj) : same as isinitialized
• lookup(en, val::Integer) : lookup the name (symbol) corresponding to an enum value
• enumstr(enumname, enumvalue::Int32): returns a string with the enum field name matching the value
• which_oneof(obj, oneof::Symbol): returns a symbol indicating the name of the field in the oneof group that is filled

Thread safety

Most of the book-keeping data for a protobuf struct is kept inside the struct instance. So that does not hinder thread safe usage. However struct instances themselves need to be locked if they are being read and written to from different threads, as is expected of any regular Julia struct.

Protobuf metadata for a struct (the information about fields and their properties as mentioned in the protobuf IDL definition) however is best initialized once and reused. It was not possible to generate code in such a way that it could be initialized when code is loaded and pre-compiled. This was because of the need to support nested and recursive struct references that protobuf allows - metadata for a struct could be defined only after the struct and all of its dependencies were defined. Metadata initialization had to be deferred to the first constructor call. But in order to reuse the metadata definition, it gets stored into a Ref that is set once. A process wide lock is used to make access to it thread safe. There is a small cost to be borne for that, and it should be negligible for most usages.

If an application wishes to eliminate that cost entirely, then the way to do it would be to call the constructors of all protobuf structs it wishes to use first and then switch the lock off by calling ProtoBuf.enable_async_safety(false). Once all metadata definitiions have been initialized, this would allow them to be used without any further locking overhead. This can also be set to false for a single threaded synchronous application where it is known that no parallelism is possible.

Both version 2 and 3 of the protobuf specification language are supported.

Author: JuliaIO
Source Code: https://github.com/JuliaIO/ProtoBuf.jl 
License: View license

#julia #protocol

AbstractFFTs.jl: A Julia framework for implementing FFTs

AbstractFFTs.jl

A general framework for fast Fourier transforms (FFTs) in Julia.

This package is mainly not intended to be used directly. Instead, developers of packages that implement FFTs (such as FFTW.jl or FastTransforms.jl) extend the types/functions defined in AbstractFFTs. This allows multiple FFT packages to co-exist with the same underlying fft(x) and plan_fft(x) interface.

Developer information

To define a new FFT implementation in your own module, you should

Define a new subtype (e.g. MyPlan) of AbstractFFTs.Plan{T} for FFTs and related transforms on arrays of T. This must have a pinv::Plan field, initially undefined when a MyPlan is created, that is used for caching the inverse plan.

Define a new method AbstractFFTs.plan_fft(x, region; kws...) that returns a MyPlan for at least some types of x and some set of dimensions region. The region (or a copy thereof) should be accessible via fftdims(p::MyPlan) (which defaults to p.region).

Define a method of LinearAlgebra.mul!(y, p::MyPlan, x) (or A_mul_B!(y, p::MyPlan, x) on Julia prior to 0.7.0-DEV.3204) that computes the transform p of x and stores the result in y.

Define a method of *(p::MyPlan, x), which can simply call your mul! (or A_mul_B!) method. This is not defined generically in this package due to subtleties that arise for in-place and real-input FFTs.

If the inverse transform is implemented, you should also define plan_inv(p::MyPlan), which should construct the inverse plan to p, and plan_bfft(x, region; kws...) for an unnormalized inverse ("backwards") transform of x.

You can also define similar methods of plan_rfft and plan_brfft for real-input FFTs.

The normalization convention for your FFT should be that it computes yₖ = ∑ⱼ xⱼ exp(-2πi jk/n) for a transform of length n, and the "backwards" (unnormalized inverse) transform computes the same thing but with exp(+2πi jk/n).

Documentation:

Download Details:

Author: JuliaMath
Source Code: https://github.com/JuliaMath/AbstractFFTs.jl 
License: MIT license

#julia #math #framework 

NGram.jl: Implement The NGram Model in Julia

NGram

Linear interpolation

This implementation uses the linear interpolation to build the model. For example, with a simple trigram model

p("book" | "the", "green") = count("the green book") / count("the green")

But there are some limitations

• We need a bigger corpus to efficiently train a trigram model compared to bigram or unigram
• Count(trigram) is often equal to zero
• With bigram or unigram we don't capture as much information

The idea is then to combine the results of trigram with bigram and unigram. We can generalize by saying that to compute ngram, we also use the results of (n-1)gram, ..., bigram, unigram. Here is an exemple in the case of a trigram model.

p("book" | "the", "green") = a * count("the green book") / count("the green")
                          +  b * count("the green") / count("the")
                          +  c * count("the") / count()
    where
        a + b + c = 1
        a >= 0
        b >= 0
        c >= 0

# For example: a = b = c = 1 / 3

Example

using NGram

texts = String["the green book", "my blue book", "his green house", "book"]

# Train a trigram model on the documents
model = NGramModel(texts, 3)

# Query on the model
# p(book | the, green)
model["the green book"]

Author: Remusao
Source Code: https://github.com/remusao/NGram.jl 
License: View license

#julia #model