d e t ( A ) = ∑ i 1 , i 2 , ⋯ , i n ( − 1 ) σ ( i 1 , ⋯ , i n ) a 1 , i 1 a 2 , i 2 , ⋯ , a n , i n det(A) =\sum_{i_1,i_2,\cdots,i_n } (-1)^{\sigma(i_1,\cdots,i_n)} a_{1,i_1}a_{2,i_2},\cdots, a_{n,i_n} det(A)=i1,i2,⋯,in∑(−1)σ(i1,⋯,in)a1,i1a2,i2,⋯,an,in
i 1 , ⋯ , i n i_1,\cdots,i_n i1,⋯,in 是 1 , ⋯ , n 1,\cdots,n 1,⋯,n 的排列.
计算复杂度
O ( n ⋅ n ! ) O(n\cdot n!) O(n⋅n!)
{ 1 , 2 } \{1,2\} {1,2} 的全排列如下
| 排列 | 逆序数 | 奇偶性 |
|---|---|---|
| 1 , 2 1,2 1,2 | 0 | 偶 |
| 2 , 1 2,1 2,1 | 1 | 奇 |
[ a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 ] \left[\begin{matrix}a_{1,1} &a_{1,2}\\ a_{2,1} & a_{2,2}\end{matrix}\right] [a1,1a2,1a1,2a2,2]
{ 1 , 2 , 3 } \{1,2,3\} {1,2,3} 的全排列如下
| 排列 | 逆序数 | 奇偶性 |
|---|---|---|
| 1 , 2 , 3 1,2,3 1,2,3 | 0 | 偶 |
| 1 , 3 , 2 1,3,2 1,3,2 | 1 | 奇 |
| 2 , 1 , 3 2,1,3 2,1,3 | 1 | 奇 |
| 2 , 3 , 1 2,3,1 2,3,1 | 2 | 偶 |
| 3 , 1 , 2 3,1,2 3,1,2 | 2 | 偶 |
| 3 , 2 , 1 3,2,1 3,2,1 | 3 | 奇 |
[ a 1 , 1 a 1 , 2 a 1 , 3 a 2 , 1 a 2 , 2 a 2 , 3 a 3 , 1 a 3 , 2 a 3 , 3 ] \left[\begin{matrix}a_{1,1} &a_{1,2} & a_{1,3}\\ a_{2,1} & a_{2,2}& a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3}\end{matrix}\right] a1,1a2,1a3,1a1,2a2,2a3,2a1,3a2,3a3,3
递归法生成全排列
julia
using CSV, DataFrames, Tables
function OE(sigma) # 判断奇偶置换
n=length(sigma); out=1
for i=1:n
for j=1:n-1
if sigma[j] > sigma[j+1]
out=-out; sigma[j],sigma[j+1]=sigma[j+1], sigma[j]
end
end
end
return(out)
end
function remove_last(seq) # 删除序列最后一个元素
lth=length(seq)
z=copy(seq[1:lth-1])
return(z)
end
function rankfull(seq,n,cpl) # 递归全排列生成器
lth=length(seq)
if lth==1
m=[cpl; seq]
open("rankfull_sol.txt","a") do io
for i=1:n
s=m[i]; write(io, "$s,")
end
write(io,"\n");
end
else
for i=1:lth
s=seq[i]; idx=0;
list=zeros(Int,lth-1)
for j=1:lth
if !(seq[j]==s)
idx=idx+1; list[idx]=seq[j]
end
end
cpl=[cpl; s];
rankfull(list,n,cpl)
cpl=remove_last(cpl)
end
end
end
function det_defn(A) # 定义法求行列式
lth=size(A,1)
seq=1:lth
global ranksol=[seq];
lth=length(seq); cpl=[]; iters=factorial(lth)
try
rankfull(seq,lth,cpl)
catch nothing
end
io=open("rankfull_sol.txt","r")
data=read(io,String)
file = CSV.File(IOBuffer(data), header=false)
CSV.write("fullrank.csv", file)
close(io)
sigma=zeros(Int64,iters,lth);
for i=1:factorial(lth)
for j=1:lth
sigma[i,j]=file[i][j]
end
end
DETs=0*A[1,1]
for i=1:iters
Prs=1
for j=1:lth
Prs=Prs*M[j,sigma[i,j]]
end
DETs=DETs+ OE(sigma[i,:])*Prs
end
return(DETs)
end
M=[4//1 3 2 1;
3 2 1 4;
2 1 4 3;
1 4 3 2]
a= det_defn(M)
println("\n")
println("defn=",a)-160//1
邻近对换算法生成全排列
julia
function ranking(lth) # 邻近对换全排列生成器
prd=1; arr=zeros(Int64,lth);
for i=1:lth arr[i]=i end
for i=1:lth prd=prd*i end
jx=lth ; dr=-1; s=0;
for j=1:prd
if dr<0
if jx>1
arr[jx],arr[jx+dr] = arr[jx+dr],arr[jx]
jx=jx+dr
else
arr[lth],arr[lth-1] = arr[lth-1],arr[lth]
dr=-dr
end
else
if jx<lth
arr[jx],arr[jx+dr] = arr[jx+dr],arr[jx];
jx=jx+dr
else
arr[1],arr[2] = arr[2],arr[1]; dr=-dr
end
end
open("ranking_sol.txt","a") do io
for i=1:lth
s=arr[i]; write(io, "$s,")
end
write(io,"\n");
end
end
end
function det_defn2(A) # 定义法求行列式
lth=size(A,1)
seq=1:lth
global ranksol=[seq];
lth=length(seq); cpl=[]; iters=factorial(lth)
try rm("ranking_sol.txt")
catch
nothing
end
ranking(lth)
io=open("ranking_sol.txt","r")
data=read(io,String)
file = CSV.File(IOBuffer(data), header=false)
CSV.write("fullrank.csv", file)
close(io)
sigma=zeros(Int64,iters,lth);
for i=1:factorial(lth)
for j=1:lth
sigma[i,j]=file[i][j]
end
end
DETs=0*A[1,1]
for i=1:iters
Prs=1
for j=1:lth
Prs=Prs*M[j,sigma[i,j]]
end
DETs=DETs+ OE(sigma[i,:])*Prs
end
return(DETs)
end
M=[4//1 3 2 ;
3 2 1 ;
2 1 4]
a= det_defn2(M)-160//1