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# This file is a part of Julia. License is MIT: https://julialang.org/license
# Factorials
const _fact_table64 = Vector{Int64}(20)
_fact_table64[1] = 1
for n in 2:20
_fact_table64[n] = _fact_table64[n-1] * n
end
const _fact_table128 = Vector{UInt128}(34)
_fact_table128[1] = 1
for n in 2:34
_fact_table128[n] = _fact_table128[n-1] * n
end
function factorial_lookup(n::Integer, table, lim)
n < 0 && throw(DomainError(n, "`n` must not be negative."))
n > lim && throw(OverflowError(string(n, " is too large to look up in the table")))
n == 0 && return one(n)
@inbounds f = table[n]
return oftype(n, f)
end
factorial(n::Int128) = factorial_lookup(n, _fact_table128, 33)
factorial(n::UInt128) = factorial_lookup(n, _fact_table128, 34)
factorial(n::Union{Int64,UInt64}) = factorial_lookup(n, _fact_table64, 20)
if Int === Int32
factorial(n::Union{Int8,UInt8,Int16,UInt16}) = factorial(Int32(n))
factorial(n::Union{Int32,UInt32}) = factorial_lookup(n, _fact_table64, 12)
else
factorial(n::Union{Int8,UInt8,Int16,UInt16,Int32,UInt32}) = factorial(Int64(n))
end
function gamma(n::Union{Int8,UInt8,Int16,UInt16,Int32,UInt32,Int64,UInt64})
n < 0 && throw(DomainError(n, "`n` must not be negative."))
n == 0 && return Inf
n <= 2 && return 1.0
n > 20 && return gamma(Float64(n))
@inbounds return Float64(_fact_table64[n-1])
end
# Basic functions for working with permutations
"""
isperm(v) -> Bool
Returns `true` if `v` is a valid permutation.
# Examples
```jldoctest
julia> isperm([1; 2])
true
julia> isperm([1; 3])
false
```
"""
function isperm(A)
n = length(A)
used = falses(n)
for a in A
(0 < a <= n) && (used[a] ⊻= true) || return false
end
true
end
isperm(p::Tuple{}) = true
isperm(p::Tuple{Int}) = p[1] == 1
isperm(p::Tuple{Int,Int}) = ((p[1] == 1) & (p[2] == 2)) | ((p[1] == 2) & (p[2] == 1))
function permute!!(a, p::AbstractVector{<:Integer})
count = 0
start = 0
while count < length(a)
ptr = start = findnext(p, start+1)
temp = a[start]
next = p[start]
count += 1
while next != start
a[ptr] = a[next]
p[ptr] = 0
ptr = next
next = p[next]
count += 1
end
a[ptr] = temp
p[ptr] = 0
end
a
end
"""
permute!(v, p)
Permute vector `v` in-place, according to permutation `p`. No checking is done
to verify that `p` is a permutation.
To return a new permutation, use `v[p]`. Note that this is generally faster than
`permute!(v,p)` for large vectors.
See also [`ipermute!`](@ref).
# Examples
```jldoctest
julia> A = [1, 1, 3, 4];
julia> perm = [2, 4, 3, 1];
julia> permute!(A, perm);
julia> A
4-element Array{Int64,1}:
1
4
3
1
```
"""
permute!(a, p::AbstractVector) = permute!!(a, copymutable(p))
function ipermute!!(a, p::AbstractVector{<:Integer})
count = 0
start = 0
while count < length(a)
start = findnext(p, start+1)
temp = a[start]
next = p[start]
count += 1
while next != start
temp_next = a[next]
a[next] = temp
temp = temp_next
ptr = p[next]
p[next] = 0
next = ptr
count += 1
end
a[next] = temp
p[next] = 0
end
a
end
"""
ipermute!(v, p)
Like [`permute!`](@ref), but the inverse of the given permutation is applied.
# Examples
```jldoctest
julia> A = [1, 1, 3, 4];
julia> perm = [2, 4, 3, 1];
julia> ipermute!(A, perm);
julia> A
4-element Array{Int64,1}:
4
1
3
1
```
"""
ipermute!(a, p::AbstractVector) = ipermute!!(a, copymutable(p))
"""
invperm(v)
Return the inverse permutation of `v`.
If `B = A[v]`, then `A == B[invperm(v)]`.
# Examples
```jldoctest
julia> v = [2; 4; 3; 1];
julia> invperm(v)
4-element Array{Int64,1}:
4
1
3
2
julia> A = ['a','b','c','d'];
julia> B = A[v]
4-element Array{Char,1}:
'b'
'd'
'c'
'a'
julia> B[invperm(v)]
4-element Array{Char,1}:
'a'
'b'
'c'
'd'
```
"""
function invperm(a::AbstractVector)
b = zero(a) # similar vector of zeros
n = length(a)
@inbounds for (i, j) in enumerate(a)
((1 <= j <= n) && b[j] == 0) ||
throw(ArgumentError("argument is not a permutation"))
b[j] = i
end
b
end
function invperm(p::Union{Tuple{},Tuple{Int},Tuple{Int,Int}})
isperm(p) || throw(ArgumentError("argument is not a permutation"))
p # in dimensions 0-2, every permutation is its own inverse
end
invperm(a::Tuple) = (invperm([a...])...,)
#XXX This function should be moved to Combinatorics.jl but is currently used by Base.DSP.
"""
nextprod([k_1, k_2,...], n)
Next integer greater than or equal to `n` that can be written as ``\\prod k_i^{p_i}`` for integers
``p_1``, ``p_2``, etc.
# Examples
```jldoctest
julia> nextprod([2, 3], 105)
108
julia> 2^2 * 3^3
108
```
"""
function nextprod(a::Vector{Int}, x)
if x > typemax(Int)
throw(ArgumentError("unsafe for x > typemax(Int), got $x"))
end
k = length(a)
v = ones(Int, k) # current value of each counter
mx = [nextpow(ai,x) for ai in a] # maximum value of each counter
v[1] = mx[1] # start at first case that is >= x
p::widen(Int) = mx[1] # initial value of product in this case
best = p
icarry = 1
while v[end] < mx[end]
if p >= x
best = p < best ? p : best # keep the best found yet
carrytest = true
while carrytest
p = div(p, v[icarry])
v[icarry] = 1
icarry += 1
p *= a[icarry]
v[icarry] *= a[icarry]
carrytest = v[icarry] > mx[icarry] && icarry < k
end
if p < x
icarry = 1
end
else
while p < x
p *= a[1]
v[1] *= a[1]
end
end
end
# might overflow, but want predictable return type
return mx[end] < best ? Int(mx[end]) : Int(best)
end