CP-Algorithms Library
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Characteristic Polynomial (verify/linalg/characteristic.test.cpp)
- View this file on GitHub
- Last update: 2025-05-31 15:35:36+02:00
- Problem: https://judge.yosupo.jp/problem/characteristic_polynomial
Depends on
- cp-algo/linalg/frobenius.hpp
- cp-algo/linalg/matrix.hpp
- cp-algo/linalg/vector.hpp
- cp-algo/math/affine.hpp
- cp-algo/math/combinatorics.hpp
- cp-algo/math/common.hpp
- cp-algo/math/cvector.hpp
- cp-algo/math/fft.hpp
- cp-algo/math/poly.hpp
- cp-algo/math/poly/impl/div.hpp
- cp-algo/math/poly/impl/euclid.hpp
- cp-algo/number_theory/discrete_sqrt.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/random/rng.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/checkpoint.hpp
- cp-algo/util/complex.hpp
- cp-algo/util/simd.hpp
Code
// @brief Characteristic Polynomial
#define PROBLEM "https://judge.yosupo.jp/problem/characteristic_polynomial"
#pragma GCC optimize("Ofast,unroll-loops")
#define CP_ALGO_MAXN 1 << 10
#include <bits/stdc++.h>
#include "blazingio/blazingio.min.hpp"
#include "cp-algo/linalg/frobenius.hpp"
using namespace std;
using namespace cp_algo::math;
using namespace cp_algo::linalg;
const int64_t mod = 998244353;
using base = modint<mod>;
using polyn = poly_t<base>;
void solve() {
size_t n;
cin >> n;
matrix<base> A(n);
A.read();
auto blocks = frobenius_form(A);
reduce(begin(blocks), end(blocks), polyn(1), multiplies{}).print();
}
signed main() {
//freopen("input.txt", "r", stdin);
ios::sync_with_stdio(0);
cin.tie(0);
int t = 1;
//cin >> t;
while(t--) {
solve();
}
}
#line 1 "verify/linalg/characteristic.test.cpp"
// @brief Characteristic Polynomial
#define PROBLEM "https://judge.yosupo.jp/problem/characteristic_polynomial"
#pragma GCC optimize("Ofast,unroll-loops")
#define CP_ALGO_MAXN 1 << 10
#include <bits/stdc++.h>
#line 1 "blazingio/blazingio.min.hpp"
// NOLINTBEGIN
// clang-format off
// DO NOT REMOVE THIS MESSAGE. The mess that follows is a minified build of
// https://github.com/purplesyringa/blazingio. Refer to the repository for
// a human-readable version and documentation.
// Options: cbfoiedrhWLMXaIaAn
#define M$(x,...)_mm256_##x##_epi8(__VA_ARGS__)
#define $u(...)__VA_ARGS__
#if __APPLE__
#define $m(A,B)A
#else
#define $m(A,B)B
#endif
#if _WIN32
#define $w(A,B)A
#else
#define $w(A,B)B
#endif
#if __i386__|_M_IX86
#define $H(A,B)A
#else
#define $H(A,B)B
#endif
#if __aarch64__
#define $a(A,B)A
#else
#define $a(A,B)B
#endif
#define $P(x)void F(x K){
#define $T template<$c T
#define $c class
#define $C constexpr
#define $R return
#define $O operator
#define u$ uint64_t
#define $r $R*this;
#line 41 "blazingio/blazingio.min.hpp"
#include $a(<arm_neon.h>,<immintrin.h>)
#line 43 "blazingio/blazingio.min.hpp"
#include $w(<windows.h>,<sys/mman.h>)
#include<sys/stat.h>
#include $w(<io.h>,<unistd.h>)
#include $w(<ios>,<sys/resource.h>)
#if _MSC_VER
#define __builtin_add_overflow(a,b,c)_addcarry_u64(0,a,b,c)
#define $s
#else
$H(,u$ _umul128(u$ a,u$ b,u$*D){auto x=(__uint128_t)a*b;*D=u$(x>>64);$R(u$)x;})
#define $s $a(,__attribute__((target("avx2"))))
#endif
#define $z $a(16,32)
#define $t $a(uint8x16_t,__m256i)
#define $I $w(__forceinline,__attribute__((always_inline)))
#define $F M(),
#define E$(x)if(!(x))abort();
$w(LONG WINAPI $x(_EXCEPTION_POINTERS*);,)namespace $f{using namespace std;struct B{enum $c A:char{}c;B&$O=(char x){c=A{x};$r}$O char(){$R(char)c;}};$C u$ C=~0ULL/255;struct D{string&K;};static B E[65568];template<int F>struct G{B*H,*S;void K(off_t C){$w(char*D=(char*)VirtualAlloc(0,(C+8191)&-4096,8192,1);E$(D)E$(VirtualFree(D,0,32768))DWORD A=C&-65536;E$(!A||MapViewOfFileEx(CreateFileMapping(GetStdHandle(-10),0,2,0,A,0),4,0,0,0,D)==D)E$(VirtualAlloc(D+A,65536,12288,4)==D+A)E$(~_lseek(0,A,0))DWORD E=0;ReadFile(GetStdHandle(-10),D+A,65536,&E,0);,int A=getpagesize();char*D=(char*)mmap(0,C+A,3,2,0,0);E$(D!=(void*)-1)E$(mmap(D+((C+A-1)&-A),A,3,$m(4114,50),-1,0)!=(void*)-1))H=(B*)D+C;*H=10;H[1]=48;H[2]=0;S=(B*)D;}void L(){H=S=E;}$I void M(){if(F&&S==H){$w(DWORD A=0;ReadFile(GetStdHandle(-10),S=E,65536,&A,0);,$a($u(register long A asm("x0")=0,D asm("x1")=(long)E,G asm("x2")=65536,C asm($m("x16","x8"))=$m(3,63);asm volatile("svc 0" $m("x80",):"+r"(A),"+r"(D):"r"(C),"r"(G));S=launder(E);),off_t A=$H(3,$m(33554435,0));B*D=E;asm volatile($H("int $128","syscall"):"+a"(A),$H("+c"(D):"b","+S"(D):"D")(0),"d"(65536)$H(,$u(:"rcx","r11")));S=D;))H=S+A;*H=10;if(!A)E[1]=48,E[2]=0;}}$T>$I void N(T&x){while($F(*S&240)==48)x=T(x*10+(*S++-48));}$T>$I decltype((void)~T{1})O(T&x){M();int A=is_signed_v<T>&&*S==45;S+=A;N(x=0);x=A?1+~x:x;}$T>$I decltype((void)T{1.})O(T&x){M();int A=*S==45;S+=A;$F S+=*S==43;u$ n=0;int i=0;for(;i<18&&($F*S&240)==48;i++)n=n*10+*S++-48;int B=20;int C=*S==46;S+=C;for(;i<18&&($F*S&240)==48;i++)n=n*10+*S++-48,B-=C;x=(T)n;while(($F*S&240)==48)x=x*10+*S++-48,B-=C;if(*S==46)S++,C=1;while(($F*S&240)==48)x=x*10+*S++-48,B-=C;int D;if((*S|32)==101)S++,$F S+=*S==43,O(D),B+=D;static $C auto E=[](){array<T,41>E{};T x=1;for(int i=21;i--;)E[40-i]=x,E[i]=1/x,x*=10;$R E;}();while(B>40)x*=(T)1e10,B-=10;while(B<0)x*=(T)1e-10,B+=10;x*=E[B];x=A?-x:x;}$I void O(bool&x){$F x=*S++==49;}$I void O(char&x){$F x=*S++;}$I void O(uint8_t&x){$F x=*S++;}$I void O(int8_t&x){$F x=*S++;}$T>$s void P(string&K,T C){M();B*G=S;C();K.assign((char*)G,S-G);while(F&&S==H&&($F H!=E)){C();K.append(E,S);}}$s void O(string&K){P(K,[&]()$s{B*p=S;$w(ULONG R;,)$t x;$a(uint64x2_t A;while(memcpy(&x,p,16),A=uint64x2_t(x<33),!(A[0]|A[1]))p+=16;S=p+(A[0]?0:8)+$w((_BitScanForward64(&R,A[0]?A[0]:A[1]),R),__builtin_ctzll(A[0]?A[0]:A[1]))/8;,int J;$t C=M$(set1,32);while(memcpy(&x,p,32),!(J=M$(movemask,M$(cmpeq,C,_mm256_max_epu8(C,x)))))p+=32;S=p+$w((_BitScanForward(&R,J),R),__builtin_ctz(J));)});}$s void O(D&A){P(A.K,[&](){S=(B*)memchr(S,10,H-S+1);});if(A.K.size()&&A.K.back()==13)A.K.pop_back();if(A.K.empty()||S<H)S+=*S==10;}$T>$I void O(complex<T>&K){T A,B{};if($F*S==40){S++;O(A);if($F*S++==44)Q(B),S++;}else O(A);K={A,B};}template<size_t N>$s void O(bitset<N>&K){if(N>4095&&!*this)$R;ptrdiff_t i=N;while(i)if($F i%$z||H-S<$z)K[--i]=*S++==49;else{B*p=S;for(int64_t j=0;j<min(i,H-S)/$z;j++){i-=$z;$t x;memcpy(&x,p,$z);$a(auto B=(uint8x16_t)vdupq_n_u64(~2ULL/254)&(48-x);auto C=vzip_u8(vget_high_u8(B),vget_low_u8(B));auto y=vaddvq_u16((uint16x8_t)vcombine_u8(C.val[0],C.val[1]));,u$ a=~0ULL/65025;auto y=$w(_byteswap_ulong,__builtin_bswap32)(M$(movemask,M$(shuffle,_mm256_slli_epi32(x,7),_mm256_set_epi64x(a+C*24,a+C*16,a+C*8,a))));)p+=$z;memcpy((char*)&K+i/8,&y,$z/8);}S=p;}}$T>$I void Q(T&K){if(!is_same_v<T,D>)while($F(uint8_t)*S<33)S++;O(K);}$O bool(){$R!!*this;}bool $O!(){$R S>H;}};struct U{G<0>A;G<1>B;U(){struct stat D;E$(~fstat(0,&D))(D.st_mode>>12)==8?A.K(D.st_size):B.L();}U*tie(nullptr_t){$R this;}void sync_with_stdio(bool){}$T>$I U&$O>>(T&K){A.S?A.Q(K):B.Q(K);$r}$O bool(){$R!!*this;}bool $O!(){$R A.S?!A:!B;}};short A[100];char L[64]{1};struct
V{char*D;B*S;int J;V(){$w(E$(D=(char*)VirtualAlloc(0,536870912,8192,4))E$(VirtualAlloc(D,4096,4096,260))AddVectoredExceptionHandler(1,$x);,size_t C=536870912;$m(,rlimit E;getrlimit(RLIMIT_AS,&E);if(~E.rlim_cur)C=25165824;)D=(char*)mmap(0,C,3,$m(4162,16418),-1,0);E$(D!=(void*)-1))S=(B*)D;for(int i=0;i<100;i++)A[i]=short((48+i/10)|((48+i%10)<<8));for(int i=1;i<64;i++)L[i]=L[i-1]+(0x8922489224892249>>i&1);}~V(){flush($w(!J,));}void flush($w(int F=0,)){$w(J=1;auto E=GetStdHandle(-11);auto C=F?ReOpenFile(E,1073741824,7,2684354560):(void*)-1;DWORD A;E$(C==(void*)-1?WriteFile(E,D,DWORD((char*)S-D),&A,0):(WriteFile(C,D,DWORD(((char*)S-D+4095)&-4096),&A,0)&&~_chsize(1,int((char*)S-D)))),auto G=D;ssize_t A;while((A=write(1,G,(char*)S-G))>0)G+=A;E$(~A))S=(B*)D;}$P(char)*S++=K;}$P(uint8_t)*S++=K;}$P(int8_t)*S++=K;}$P(bool)*S++=48+K;}$T>decltype((void)~T{1})F(T K){using D=make_unsigned_t<T>;D C=K;if(K<0)F('-'),C=1+~C;static $C auto N=[](){array<D,5*sizeof(T)/2>N{};D n=1;for(size_t i=1;i<N.size();i++)n*=10,N[i]=n;$R N;}();$w(ULONG M;,)int G=L[$w(($H(_BitScanReverse(&M,ULONG((int64_t)C>>32))?M+=32:_BitScanReverse(&M,(ULONG)C|1),_BitScanReverse64(&M,C|1)),M),63^__builtin_clzll(C|1))];G-=C<N[G-1];short H[20];if $C(sizeof(T)==2){auto n=33555U*C-C/2;u$ H=A[n>>25];n=(n&33554431)*25;H|=A[n>>23]<<16;H|=u$(48+((n&8388607)*5>>22))<<32;H>>=40-G*8;memcpy(S,&H,8);}else if $C(sizeof(T)==4){auto n=1441151881ULL*C;$H(n>>=25;n++;for(int i=0;i<5;i++){H[i]=A[n>>32];n=(n&~0U)*100;},int K=57;auto J=~0ULL>>7;for(int i=0;i<5;i++){H[i]=A[n>>K];n=(n&J)*25;K-=2;J/=4;})memcpy(S,(B*)H+10-G,16);}else{$H($u(if(C<(1ULL<<32)){$R F((uint32_t)C);}auto J=(u$)1e10;auto x=C/J,y=C%J;int K=100000,b[]{int(x/K),int(x%K),int(y/K),int(y%K)};B H[40];for(int i=0;i<4;i++){int n=int((429497ULL*b[i]>>7)+1);B*p=H+i*5;*p=48+char(n>>25);n=(n&~0U>>7)*25;memcpy(p+1,A+(n>>23),2);memcpy(p+3,A+((n&~0U>>9)*25>>21),2);}),$u(u$ D,E=_umul128(18,C,&D),F;_umul128(0x725dd1d243aba0e8,C,&F);D+=__builtin_add_overflow(E,F+1,&E);for(int i=0;i<10;i++)H[i]=A[D],E=_umul128(100,E,&D);))memcpy(S,(B*)H+20-G,20);}S+=G;}$T>decltype((void)T{1.})F(T K){if(K<0)F('-'),K=-K;auto G=[&](){auto x=u$(K*1e12);$H($u(x-=x>999999999999;uint32_t n[]{uint32_t(x/1000000*429497>>7)+1,uint32_t(x%1000000*429497>>7)+1};int K=25,J=~0U>>7;for(int i=0;i<3;i++){for(int j=0;j<2;j++)memcpy(S+i*2+j*6,A+(n[j]>>K),2),n[j]=(n[j]&J)*25;K-=2;J/=4;}S+=12;),$u(u$ D,E=_umul128(472236648287,x,&D)>>8;E|=D<<56;D>>=8;E++;for(int i=0;i<6;i++)memcpy(S,A+D,2),S+=2,E=_umul128(100,E,&D);))};if(K==0)$R F('0');if(K>=1e16){K*=(T)1e-16;int B=16;while(K>=1)K*=(T).1,B++;F("0.");G();F('e');F(B);}else if(K>=1){auto B=(u$)K;F(B);if((K-=(T)B)>0)F('.'),G();}else F("0."),G();}$P(const char*)$w(size_t A=strlen(K);memcpy((char*)S,K,A);S+=A;,S=(B*)stpcpy((char*)S,K);)}$P(const uint8_t*)F((char*)K);}$P(const int8_t*)F((char*)K);}$P(string_view)memcpy(S,K.data(),K.size());S+=K.size();}$T>$P(complex<T>)*this<<'('<<K.real()<<','<<K.imag()<<')';}template<size_t N>$s $P(const bitset<N>&)auto i=N;while(i%$z)*S++=48+K[--i];B*p=S;while(i){i-=$z;$a(short,int)x;memcpy(&x,(char*)&K+i/8,$z/8);$a(auto A=(uint8x8_t)vdup_n_u16(x);vst1q_u8((uint8_t*)p,48-vtstq_u8(vcombine_u8(vuzp2_u8(A,A),vuzp1_u8(A,A)),(uint8x16_t)vdupq_n_u64(~2ULL/254)));,auto b=_mm256_set1_epi64x(~2ULL/254);_mm256_storeu_si256(($t*)p,M$(sub,M$(set1,48),M$(cmpeq,_mm256_and_si256(M$(shuffle,_mm256_set1_epi32(x),_mm256_set_epi64x(0,C,C*2,C*3)),b),b)));)p+=$z;}S=p;}$T>V&$O<<(const T&K){F(K);$r}V&$O<<(V&(*A)(V&)){$R A(*this);}};struct W{$T>W&$O<<(const T&K){$r}W&$O<<(W&(*A)(W&)){$R A(*this);}};}namespace std{$f::U i$;$f::V o$;$f::W e$;$f::U&getline($f::U&B,string&K){$f::D A{K};$R B>>A;}$f::V&flush($f::V&B){if(!i$.A.S)B.flush();$R B;}$f::V&endl($f::V&B){$R B<<'\n'<<flush;}$f::W&endl($f::W&B){$R B;}$f::W&flush($f::W&B){$R B;}}$w(LONG WINAPI $x(_EXCEPTION_POINTERS*A){auto C=A->ExceptionRecord;auto B=C->ExceptionInformation[1];if(C->ExceptionCode==2147483649&&B-(ULONG_PTR)std::o$.D<0x40000000){E$(VirtualAlloc((char*)B,16777216,4096,4)&&VirtualAlloc((char*)(B+16777216),4096,4096,260))$R-1;}$R 0;},)
#define freopen(...)if(freopen(__VA_ARGS__)==stdin)std::i$=$f::U{}
#define cin i$
#define cout o$
#ifdef ONLINE_JUDGE
#define cerr e$
#define clog e$
#endif
// End of blazingio
// NOLINTEND
// clang-format on
#line 1 "cp-algo/linalg/frobenius.hpp"
#line 1 "cp-algo/math/poly.hpp"
#line 1 "cp-algo/math/poly/impl/euclid.hpp"
#line 1 "cp-algo/math/affine.hpp"
#include <optional>
#line 7 "cp-algo/math/affine.hpp"
namespace cp_algo::math {
// a * x + b
template<typename base>
struct lin {
base a = 1, b = 0;
std::optional<base> c;
lin() {}
lin(base b): a(0), b(b) {}
lin(base a, base b): a(a), b(b) {}
lin(base a, base b, base _c): a(a), b(b), c(_c) {}
// polynomial product modulo x^2 - c
lin operator * (const lin& t) {
assert(c && t.c && *c == *t.c);
return {a * t.b + b * t.a, b * t.b + a * t.a * (*c), *c};
}
// a * (t.a * x + t.b) + b
lin apply(lin const& t) const {
return {a * t.a, a * t.b + b};
}
void prepend(lin const& t) {
*this = t.apply(*this);
}
base eval(base x) const {
return a * x + b;
}
};
// (ax+b) / (cx+d)
template<typename base>
struct linfrac {
base a, b, c, d;
linfrac(): a(1), b(0), c(0), d(1) {} // x, identity for composition
linfrac(base a): a(a), b(1), c(1), d(0) {} // a + 1/x, for continued fractions
linfrac(base a, base b, base c, base d): a(a), b(b), c(c), d(d) {}
// composition of two linfracs
linfrac operator * (linfrac t) const {
return t.prepend(linfrac(*this));
}
linfrac operator-() const {
return {-a, -b, -c, -d};
}
linfrac adj() const {
return {d, -b, -c, a};
}
linfrac& prepend(linfrac const& t) {
t.apply(a, c);
t.apply(b, d);
return *this;
}
// apply linfrac to A/B
void apply(base &A, base &B) const {
std::tie(A, B) = std::pair{a * A + b * B, c * A + d * B};
}
};
}
#line 1 "cp-algo/math/fft.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#line 6 "cp-algo/math/common.hpp"
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
const int maxn = CP_ALGO_MAXN;
#else
const int maxn = 1 << 19;
#endif
const int magic = 64; // threshold for sizes to run the naive algo
auto bpow(auto const& x, auto n, auto const& one, auto op) {
if(n == 0) {
return one;
} else {
auto t = bpow(x, n / 2, one, op);
t = op(t, t);
if(n % 2) {
t = op(t, x);
}
return t;
}
}
auto bpow(auto x, auto n, auto ans) {
return bpow(x, n, ans, std::multiplies{});
}
template<typename T>
T bpow(T const& x, auto n) {
return bpow(x, n, T(1));
}
inline constexpr auto inv2(auto x) {
assert(x % 2);
std::make_unsigned_t<decltype(x)> y = 1;
while(y * x != 1) {
y *= 2 - x * y;
}
return y;
}
}
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {
template<typename modint, typename _Int>
struct modint_base {
using Int = _Int;
using UInt = std::make_unsigned_t<Int>;
static constexpr size_t bits = sizeof(Int) * 8;
using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
constexpr static Int mod() {
return modint::mod();
}
constexpr static Int remod() {
return modint::remod();
}
constexpr static UInt2 modmod() {
return UInt2(mod()) * mod();
}
constexpr modint_base() = default;
constexpr modint_base(Int2 rr) {
to_modint().setr(UInt((rr + modmod()) % mod()));
}
modint inv() const {
return bpow(to_modint(), mod() - 2);
}
modint operator - () const {
modint neg;
neg.r = std::min(-r, remod() - r);
return neg;
}
modint& operator /= (const modint &t) {
return to_modint() *= t.inv();
}
modint& operator *= (const modint &t) {
r = UInt(UInt2(r) * t.r % mod());
return to_modint();
}
modint& operator += (const modint &t) {
r += t.r; r = std::min(r, r - remod());
return to_modint();
}
modint& operator -= (const modint &t) {
r -= t.r; r = std::min(r, r + remod());
return to_modint();
}
modint operator + (const modint &t) const {return modint(to_modint()) += t;}
modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
// Why <=> doesn't work?..
auto operator == (const modint &t) const {return to_modint().getr() == t.getr();}
auto operator != (const modint &t) const {return to_modint().getr() != t.getr();}
auto operator <= (const modint &t) const {return to_modint().getr() <= t.getr();}
auto operator >= (const modint &t) const {return to_modint().getr() >= t.getr();}
auto operator < (const modint &t) const {return to_modint().getr() < t.getr();}
auto operator > (const modint &t) const {return to_modint().getr() > t.getr();}
Int rem() const {
UInt R = to_modint().getr();
return R - (R > (UInt)mod() / 2) * mod();
}
constexpr void setr(UInt rr) {
r = rr;
}
constexpr UInt getr() const {
return r;
}
// Only use these if you really know what you're doing!
static UInt modmod8() {return UInt(8 * modmod());}
void add_unsafe(UInt t) {r += t;}
void pseudonormalize() {r = std::min(r, r - modmod8());}
modint const& normalize() {
if(r >= (UInt)mod()) {
r %= mod();
}
return to_modint();
}
void setr_direct(UInt rr) {r = rr;}
UInt getr_direct() const {return r;}
protected:
UInt r;
private:
constexpr modint& to_modint() {return static_cast<modint&>(*this);}
constexpr modint const& to_modint() const {return static_cast<modint const&>(*this);}
};
template<typename modint>
concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
template<modint_type modint>
decltype(std::cin)& operator >> (decltype(std::cin) &in, modint &x) {
typename modint::UInt r;
auto &res = in >> r;
x.setr(r);
return res;
}
template<modint_type modint>
decltype(std::cout)& operator << (decltype(std::cout) &out, modint const& x) {
return out << x.getr();
}
template<auto m>
struct modint: modint_base<modint<m>, decltype(m)> {
using Base = modint_base<modint<m>, decltype(m)>;
using Base::Base;
static constexpr Base::Int mod() {return m;}
static constexpr Base::UInt remod() {return m;}
auto getr() const {return Base::r;}
};
template<typename Int = int>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
static Base::UInt m_reduce(Base::UInt2 ab) {
if(mod() % 2 == 0) [[unlikely]] {
return typename Base::UInt(ab % mod());
} else {
typename Base::UInt2 m = typename Base::UInt(ab) * imod();
return typename Base::UInt((ab + m * mod()) >> Base::bits);
}
}
static Base::UInt m_transform(Base::UInt a) {
if(mod() % 2 == 0) [[unlikely]] {
return a;
} else {
return m_reduce(a * pw128());
}
}
dynamic_modint& operator *= (const dynamic_modint &t) {
Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
return *this;
}
void setr(Base::UInt rr) {
Base::r = m_transform(rr);
}
Base::UInt getr() const {
typename Base::UInt res = m_reduce(Base::r);
return std::min(res, res - mod());
}
static Int mod() {return m;}
static Int remod() {return 2 * m;}
static Base::UInt imod() {return im;}
static Base::UInt2 pw128() {return r2;}
static void switch_mod(Int nm) {
m = nm;
im = m % 2 ? inv2(-m) : 0;
r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
}
// Wrapper for temp switching
auto static with_mod(Int tmp, auto callback) {
struct scoped {
Int prev = mod();
~scoped() {switch_mod(prev);}
} _;
switch_mod(tmp);
return callback();
}
private:
static thread_local Int m;
static thread_local Base::UInt im, r2;
};
template<typename Int>
Int thread_local dynamic_modint<Int>::m = 1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}
#line 1 "cp-algo/util/checkpoint.hpp"
#line 7 "cp-algo/util/checkpoint.hpp"
namespace cp_algo {
std::map<std::string, double> checkpoints;
template<bool final = false>
void checkpoint([[maybe_unused]] std::string const& msg = "") {
#ifdef CP_ALGO_CHECKPOINT
static double last = 0;
double now = (double)clock() / CLOCKS_PER_SEC;
double delta = now - last;
last = now;
if(msg.size() && !final) {
checkpoints[msg] += delta;
}
if(final) {
for(auto const& [key, value] : checkpoints) {
std::cerr << key << ": " << value * 1000 << " ms\n";
}
std::cerr << "Total: " << now * 1000 << " ms\n";
}
#endif
}
}
#line 1 "cp-algo/random/rng.hpp"
#line 5 "cp-algo/random/rng.hpp"
namespace cp_algo::random {
uint64_t rng() {
static std::mt19937_64 rng(
std::chrono::steady_clock::now().time_since_epoch().count()
);
return rng();
}
}
#line 1 "cp-algo/math/cvector.hpp"
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 7 "cp-algo/util/simd.hpp"
namespace cp_algo {
template<typename T, size_t len>
using simd [[gnu::vector_size(len * sizeof(T))]] = T;
using i64x4 = simd<int64_t, 4>;
using u64x4 = simd<uint64_t, 4>;
using u32x8 = simd<uint32_t, 8>;
using i32x4 = simd<int32_t, 4>;
using u32x4 = simd<uint32_t, 4>;
using i16x4 = simd<int16_t, 4>;
using u8x32 = simd<uint8_t, 32>;
using dx4 = simd<double, 4>;
[[gnu::target("avx2")]] inline dx4 abs(dx4 a) {
return a < 0 ? -a : a;
}
// https://stackoverflow.com/a/77376595
// works for ints in (-2^51, 2^51)
static constexpr dx4 magic = dx4() + (3ULL << 51);
[[gnu::target("avx2")]] inline i64x4 lround(dx4 x) {
return i64x4(x + magic) - i64x4(magic);
}
[[gnu::target("avx2")]] inline dx4 to_double(i64x4 x) {
return dx4(x + i64x4(magic)) - magic;
}
[[gnu::target("avx2")]] inline dx4 round(dx4 a) {
return dx4{
std::nearbyint(a[0]),
std::nearbyint(a[1]),
std::nearbyint(a[2]),
std::nearbyint(a[3])
};
}
[[gnu::target("avx2")]] inline u64x4 low32(u64x4 x) {
return x & uint32_t(-1);
}
[[gnu::target("avx2")]] inline auto swap_bytes(auto x) {
return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
}
[[gnu::target("avx2")]] inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
return swap_bytes(x);
}
[[gnu::target("avx2")]] inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t imod) {
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
}
[[gnu::target("avx2")]] inline u32x8 montgomery_mul(u32x8 x, u32x8 y, uint32_t mod, uint32_t imod) {
return u32x8(montgomery_mul(u64x4(x), u64x4(y), mod, imod)) |
u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)), u64x4(swap_bytes(y)), mod, imod)));
}
[[gnu::target("avx2")]] inline dx4 rotate_right(dx4 x) {
static constexpr u64x4 shuffler = {3, 0, 1, 2};
return __builtin_shuffle(x, shuffler);
}
template<std::size_t Align = 32>
[[gnu::target("avx2")]] inline bool is_aligned(const auto* p) noexcept {
return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
}
template<class Target>
[[gnu::target("avx2")]] inline Target& vector_cast(auto &&p) {
return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
}
}
#line 1 "cp-algo/util/complex.hpp"
#line 5 "cp-algo/util/complex.hpp"
namespace cp_algo {
// Custom implementation, since std::complex is UB on non-floating types
template<typename T>
struct complex {
using value_type = T;
T x, y;
constexpr complex(): x(), y() {}
constexpr complex(T x): x(x), y() {}
constexpr complex(T x, T y): x(x), y(y) {}
complex& operator *= (T t) {x *= t; y *= t; return *this;}
complex& operator /= (T t) {x /= t; y /= t; return *this;}
complex operator * (T t) const {return complex(*this) *= t;}
complex operator / (T t) const {return complex(*this) /= t;}
complex& operator += (complex t) {x += t.x; y += t.y; return *this;}
complex& operator -= (complex t) {x -= t.x; y -= t.y; return *this;}
complex operator * (complex t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
complex operator / (complex t) const {return *this * t.conj() / t.norm();}
complex operator + (complex t) const {return complex(*this) += t;}
complex operator - (complex t) const {return complex(*this) -= t;}
complex& operator *= (complex t) {return *this = *this * t;}
complex& operator /= (complex t) {return *this = *this / t;}
complex operator - () const {return {-x, -y};}
complex conj() const {return {x, -y};}
T norm() const {return x * x + y * y;}
T abs() const {return std::sqrt(norm());}
T const real() const {return x;}
T const imag() const {return y;}
T& real() {return x;}
T& imag() {return y;}
static constexpr complex polar(T r, T theta) {return {T(r * cos(theta)), T(r * sin(theta))};}
auto operator <=> (complex const& t) const = default;
};
template<typename T>
complex<T> operator * (auto x, complex<T> y) {return y *= x;}
template<typename T> complex<T> conj(complex<T> x) {return x.conj();}
template<typename T> T norm(complex<T> x) {return x.norm();}
template<typename T> T abs(complex<T> x) {return x.abs();}
template<typename T> T& real(complex<T> &x) {return x.real();}
template<typename T> T& imag(complex<T> &x) {return x.imag();}
template<typename T> T const real(complex<T> const& x) {return x.real();}
template<typename T> T const imag(complex<T> const& x) {return x.imag();}
template<typename T>
constexpr complex<T> polar(T r, T theta) {
return complex<T>::polar(r, theta);
}
template<typename T>
std::ostream& operator << (std::ostream &out, complex<T> x) {
return out << x.real() << ' ' << x.imag();
}
}
#line 1 "cp-algo/util/big_alloc.hpp"
#line 6 "cp-algo/util/big_alloc.hpp"
// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
# define CP_ALGO_USE_MMAP 1
# include <sys/mman.h>
#else
# define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo {
template <typename T, std::size_t Align = 32>
class big_alloc {
static_assert( Align >= alignof(void*), "Align must be at least pointer-size");
static_assert(std::popcount(Align) == 1, "Align must be a power of two");
public:
using value_type = T;
template <class U> struct rebind { using other = big_alloc<U, Align>; };
constexpr bool operator==(const big_alloc&) const = default;
constexpr bool operator!=(const big_alloc&) const = default;
big_alloc() noexcept = default;
template <typename U, std::size_t A>
big_alloc(const big_alloc<U, A>&) noexcept {}
[[nodiscard]] T* allocate(std::size_t n) {
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if (padded >= MEGABYTE) {
void* raw = mmap(nullptr, padded,
PROT_READ | PROT_WRITE,
MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
madvise(raw, padded, MADV_HUGEPAGE);
madvise(raw, padded, MADV_POPULATE_WRITE);
return static_cast<T*>(raw);
}
#endif
return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
}
void deallocate(T* p, std::size_t n) noexcept {
if (!p) return;
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if (padded >= MEGABYTE) { munmap(p, padded); return; }
#endif
::operator delete(p, padded, std::align_val_t(align));
}
private:
static constexpr std::size_t MEGABYTE = 1 << 20;
static constexpr std::size_t round_up(std::size_t x) noexcept {
return (x + Align - 1) / Align * Align;
}
};
}
#line 7 "cp-algo/math/cvector.hpp"
#include <ranges>
#include <bit>
namespace stdx = std::experimental;
namespace cp_algo::math::fft {
static constexpr size_t flen = 4;
using ftype = double;
using vftype = dx4;
using point = complex<ftype>;
using vpoint = complex<vftype>;
static constexpr vftype vz = {};
vpoint vi(vpoint const& r) {
return {-imag(r), real(r)};
}
struct cvector {
std::vector<vpoint, big_alloc<vpoint>> r;
cvector(size_t n) {
n = std::max(flen, std::bit_ceil(n));
r.resize(n / flen);
checkpoint("cvector create");
}
vpoint& at(size_t k) {return r[k / flen];}
vpoint at(size_t k) const {return r[k / flen];}
template<class pt = point>
void set(size_t k, pt t) {
if constexpr(std::is_same_v<pt, point>) {
real(r[k / flen])[k % flen] = real(t);
imag(r[k / flen])[k % flen] = imag(t);
} else {
at(k) = t;
}
}
template<class pt = point>
pt get(size_t k) const {
if constexpr(std::is_same_v<pt, point>) {
return {real(r[k / flen])[k % flen], imag(r[k / flen])[k % flen]};
} else {
return at(k);
}
}
size_t size() const {
return flen * r.size();
}
static constexpr size_t eval_arg(size_t n) {
if(n < pre_evals) {
return eval_args[n];
} else {
return eval_arg(n / 2) | (n & 1) << (std::bit_width(n) - 1);
}
}
static constexpr point eval_point(size_t n) {
if(n % 2) {
return -eval_point(n - 1);
} else if(n % 4) {
return eval_point(n - 2) * point(0, 1);
} else if(n / 4 < pre_evals) {
return evalp[n / 4];
} else {
return polar<ftype>(1., std::numbers::pi / (ftype)std::bit_floor(n) * (ftype)eval_arg(n));
}
}
static constexpr std::array<point, 32> roots = []() {
std::array<point, 32> res;
for(size_t i = 2; i < 32; i++) {
res[i] = polar<ftype>(1., std::numbers::pi / (1ull << (i - 2)));
}
return res;
}();
static constexpr point root(size_t n) {
return roots[std::bit_width(n)];
}
template<int step>
static void exec_on_eval(size_t n, size_t k, auto &&callback) {
callback(k, root(4 * step * n) * eval_point(step * k));
}
template<int step>
static void exec_on_evals(size_t n, auto &&callback) {
point factor = root(4 * step * n);
for(size_t i = 0; i < n; i++) {
callback(i, factor * eval_point(step * i));
}
}
void dot(cvector const& t) {
size_t n = this->size();
exec_on_evals<1>(n / flen, [&](size_t k, point rt) {
k *= flen;
auto [Ax, Ay] = at(k);
auto Bv = t.at(k);
vpoint res = vz;
for (size_t i = 0; i < flen; i++) {
res += vpoint(vz + Ax[i], vz + Ay[i]) * Bv;
real(Bv) = rotate_right(real(Bv));
imag(Bv) = rotate_right(imag(Bv));
auto x = real(Bv)[0], y = imag(Bv)[0];
real(Bv)[0] = x * real(rt) - y * imag(rt);
imag(Bv)[0] = x * imag(rt) + y * real(rt);
}
set(k, res);
});
checkpoint("dot");
}
template<bool partial = true>
void ifft() {
size_t n = size();
if constexpr (!partial) {
point pi(0, 1);
exec_on_evals<4>(n / 4, [&](size_t k, point rt) {
k *= 4;
point v1 = conj(rt);
point v2 = v1 * v1;
point v3 = v1 * v2;
auto A = get(k);
auto B = get(k + 1);
auto C = get(k + 2);
auto D = get(k + 3);
set(k, (A + B) + (C + D));
set(k + 2, ((A + B) - (C + D)) * v2);
set(k + 1, ((A - B) - pi * (C - D)) * v1);
set(k + 3, ((A - B) + pi * (C - D)) * v3);
});
}
bool parity = std::countr_zero(n) % 2;
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&](size_t k, point rt) {
k *= 2 * flen;
vpoint cvrt = {vz + real(rt), vz - imag(rt)};
auto B = at(k) - at(k + flen);
at(k) += at(k + flen);
at(k + flen) = B * cvrt;
});
}
for(size_t leaf = 3 * flen; leaf < n; leaf += 4 * flen) {
size_t level = std::countr_one(leaf + 3);
for(size_t lvl = 4 + parity; lvl <= level; lvl += 2) {
size_t i = (1 << lvl) / 4;
exec_on_eval<4>(n >> lvl, leaf >> lvl, [&](size_t k, point rt) {
k <<= lvl;
vpoint v1 = {vz + real(rt), vz - imag(rt)};
vpoint v2 = v1 * v1;
vpoint v3 = v1 * v2;
for(size_t j = k; j < k + i; j += flen) {
auto A = at(j);
auto B = at(j + i);
auto C = at(j + 2 * i);
auto D = at(j + 3 * i);
at(j) = ((A + B) + (C + D));
at(j + 2 * i) = ((A + B) - (C + D)) * v2;
at(j + i) = ((A - B) - vi(C - D)) * v1;
at(j + 3 * i) = ((A - B) + vi(C - D)) * v3;
}
});
}
}
checkpoint("ifft");
for(size_t k = 0; k < n; k += flen) {
if constexpr (partial) {
set(k, get<vpoint>(k) /= vz + ftype(n / flen));
} else {
set(k, get<vpoint>(k) /= vz + ftype(n));
}
}
}
template<bool partial = true>
void fft() {
size_t n = size();
bool parity = std::countr_zero(n) % 2;
for(size_t leaf = 0; leaf < n; leaf += 4 * flen) {
size_t level = std::countr_zero(n + leaf);
level -= level % 2 != parity;
for(size_t lvl = level; lvl >= 4; lvl -= 2) {
size_t i = (1 << lvl) / 4;
exec_on_eval<4>(n >> lvl, leaf >> lvl, [&](size_t k, point rt) {
k <<= lvl;
vpoint v1 = {vz + real(rt), vz + imag(rt)};
vpoint v2 = v1 * v1;
vpoint v3 = v1 * v2;
for(size_t j = k; j < k + i; j += flen) {
auto A = at(j);
auto B = at(j + i) * v1;
auto C = at(j + 2 * i) * v2;
auto D = at(j + 3 * i) * v3;
at(j) = (A + C) + (B + D);
at(j + i) = (A + C) - (B + D);
at(j + 2 * i) = (A - C) + vi(B - D);
at(j + 3 * i) = (A - C) - vi(B - D);
}
});
}
}
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&](size_t k, point rt) {
k *= 2 * flen;
vpoint vrt = {vz + real(rt), vz + imag(rt)};
auto t = at(k + flen) * vrt;
at(k + flen) = at(k) - t;
at(k) += t;
});
}
if constexpr (!partial) {
point pi(0, 1);
exec_on_evals<4>(n / 4, [&](size_t k, point rt) {
k *= 4;
point v1 = rt;
point v2 = v1 * v1;
point v3 = v1 * v2;
auto A = get(k);
auto B = get(k + 1) * v1;
auto C = get(k + 2) * v2;
auto D = get(k + 3) * v3;
set(k, (A + C) + (B + D));
set(k + 1, (A + C) - (B + D));
set(k + 2, (A - C) + pi * (B - D));
set(k + 3, (A - C) - pi * (B - D));
});
}
checkpoint("fft");
}
static constexpr size_t pre_evals = 1 << 16;
static const std::array<size_t, pre_evals> eval_args;
static const std::array<point, pre_evals> evalp;
};
const std::array<size_t, cvector::pre_evals> cvector::eval_args = []() {
std::array<size_t, pre_evals> res = {};
for(size_t i = 1; i < pre_evals; i++) {
res[i] = res[i >> 1] | (i & 1) << (std::bit_width(i) - 1);
}
return res;
}();
const std::array<point, cvector::pre_evals> cvector::evalp = []() {
std::array<point, pre_evals> res = {};
res[0] = 1;
for(size_t n = 1; n < pre_evals; n++) {
res[n] = polar<ftype>(1., std::numbers::pi * ftype(eval_args[n]) / ftype(4 * std::bit_floor(n)));
}
return res;
}();
}
#line 9 "cp-algo/math/fft.hpp"
namespace cp_algo::math::fft {
template<modint_type base>
struct dft {
cvector A, B;
static base factor, ifactor;
using Int2 = base::Int2;
static bool _init;
static int split() {
static const int splt = int(std::sqrt(base::mod())) + 1;
return splt;
}
static uint32_t mod, imod;
static void init() {
if(!_init) {
factor = 1 + random::rng() % (base::mod() - 1);
ifactor = base(1) / factor;
mod = base::mod();
imod = -inv2<uint32_t>(base::mod());
_init = true;
}
}
dft(size_t n): A(n), B(n) {init();}
dft(auto const& a, size_t n, bool partial = true): A(n), B(n) {
init();
base b2x32 = bpow(base(2), 32);
u64x4 cur = {
(bpow(factor, 1) * b2x32).getr(),
(bpow(factor, 2) * b2x32).getr(),
(bpow(factor, 3) * b2x32).getr(),
(bpow(factor, 4) * b2x32).getr()
};
u64x4 step4 = u64x4{} + (bpow(factor, 4) * b2x32).getr();
u64x4 stepn = u64x4{} + (bpow(factor, n) * b2x32).getr();
for(size_t i = 0; i < std::min(n, std::size(a)); i += flen) {
auto splt = [&](size_t i, auto mul) {
if(i >= std::size(a)) {
return std::pair{vftype(), vftype()};
}
u64x4 au = {
i < std::size(a) ? a[i].getr() : 0,
i + 1 < std::size(a) ? a[i + 1].getr() : 0,
i + 2 < std::size(a) ? a[i + 2].getr() : 0,
i + 3 < std::size(a) ? a[i + 3].getr() : 0
};
au = montgomery_mul(au, mul, mod, imod);
au = au >= base::mod() ? au - base::mod() : au;
auto ai = to_double(i64x4(au >= base::mod() / 2 ? au - base::mod() : au));
auto quo = round(ai / split());
return std::pair{ai - quo * split(), quo};
};
auto [rai, qai] = splt(i, cur);
auto [rani, qani] = splt(n + i, montgomery_mul(cur, stepn, mod, imod));
A.at(i) = vpoint(rai, rani);
B.at(i) = vpoint(qai, qani);
cur = montgomery_mul(cur, step4, mod, imod);
}
checkpoint("dft init");
if(n) {
if(partial) {
A.fft();
B.fft();
} else {
A.template fft<false>();
B.template fft<false>();
}
}
}
template<bool overwrite = true, bool partial = true>
void dot(auto const& C, auto const& D, auto &Aout, auto &Bout, auto &Cout) const {
cvector::exec_on_evals<1>(A.size() / flen, [&](size_t k, point rt) {
k *= flen;
vpoint AC, AD, BC, BD;
AC = AD = BC = BD = vz;
auto Cv = C.at(k), Dv = D.at(k);
if constexpr(partial) {
auto [Ax, Ay] = A.at(k);
auto [Bx, By] = B.at(k);
for (size_t i = 0; i < flen; i++) {
vpoint Av = {vz + Ax[i], vz + Ay[i]}, Bv = {vz + Bx[i], vz + By[i]};
AC += Av * Cv; AD += Av * Dv;
BC += Bv * Cv; BD += Bv * Dv;
real(Cv) = rotate_right(real(Cv));
imag(Cv) = rotate_right(imag(Cv));
real(Dv) = rotate_right(real(Dv));
imag(Dv) = rotate_right(imag(Dv));
auto cx = real(Cv)[0], cy = imag(Cv)[0];
auto dx = real(Dv)[0], dy = imag(Dv)[0];
real(Cv)[0] = cx * real(rt) - cy * imag(rt);
imag(Cv)[0] = cx * imag(rt) + cy * real(rt);
real(Dv)[0] = dx * real(rt) - dy * imag(rt);
imag(Dv)[0] = dx * imag(rt) + dy * real(rt);
}
} else {
AC = A.at(k) * Cv;
AD = A.at(k) * Dv;
BC = B.at(k) * Cv;
BD = B.at(k) * Dv;
}
if constexpr (overwrite) {
Aout.at(k) = AC;
Cout.at(k) = AD + BC;
Bout.at(k) = BD;
} else {
Aout.at(k) += AC;
Cout.at(k) += AD + BC;
Bout.at(k) += BD;
}
});
checkpoint("dot");
}
void dot(auto &&C, auto const& D) {
dot(C, D, A, B, C);
}
void recover_mod(auto &&C, auto &res, size_t k) {
size_t check = (k + flen - 1) / flen * flen;
assert(res.size() >= check);
size_t n = A.size();
auto const splitsplit = base(split() * split()).getr();
base b2x32 = bpow(base(2), 32);
base b2x64 = bpow(base(2), 64);
u64x4 cur = {
(bpow(ifactor, 2) * b2x64).getr(),
(bpow(ifactor, 3) * b2x64).getr(),
(bpow(ifactor, 4) * b2x64).getr(),
(bpow(ifactor, 5) * b2x64).getr()
};
u64x4 step4 = u64x4{} + (bpow(ifactor, 4) * b2x32).getr();
u64x4 stepn = u64x4{} + (bpow(ifactor, n) * b2x32).getr();
for(size_t i = 0; i < std::min(n, k); i += flen) {
auto [Ax, Ay] = A.at(i);
auto [Bx, By] = B.at(i);
auto [Cx, Cy] = C.at(i);
auto set_i = [&](size_t i, auto A, auto B, auto C, auto mul) {
auto A0 = lround(A), A1 = lround(C), A2 = lround(B);
auto Ai = A0 + A1 * split() + A2 * splitsplit + uint64_t(base::modmod());
auto Au = montgomery_reduce(u64x4(Ai), mod, imod);
Au = montgomery_mul(Au, mul, mod, imod);
Au = Au >= base::mod() ? Au - base::mod() : Au;
for(size_t j = 0; j < flen; j++) {
res[i + j].setr(typename base::UInt(Au[j]));
}
};
set_i(i, Ax, Bx, Cx, cur);
if(i + n < k) {
set_i(i + n, Ay, By, Cy, montgomery_mul(cur, stepn, mod, imod));
}
cur = montgomery_mul(cur, step4, mod, imod);
}
checkpoint("recover mod");
}
void mul(auto &&C, auto const& D, auto &res, size_t k) {
assert(A.size() == C.size());
size_t n = A.size();
if(!n) {
res = {};
return;
}
dot(C, D);
A.ifft();
B.ifft();
C.ifft();
recover_mod(C, res, k);
}
void mul_inplace(auto &&B, auto& res, size_t k) {
mul(B.A, B.B, res, k);
}
void mul(auto const& B, auto& res, size_t k) {
mul(cvector(B.A), B.B, res, k);
}
std::vector<base, big_alloc<base>> operator *= (dft &B) {
std::vector<base, big_alloc<base>> res(2 * A.size());
mul_inplace(B, res, 2 * A.size());
return res;
}
std::vector<base, big_alloc<base>> operator *= (dft const& B) {
std::vector<base, big_alloc<base>> res(2 * A.size());
mul(B, res, 2 * A.size());
return res;
}
auto operator * (dft const& B) const {
return dft(*this) *= B;
}
point operator [](int i) const {return A.get(i);}
};
template<modint_type base> base dft<base>::factor = 1;
template<modint_type base> base dft<base>::ifactor = 1;
template<modint_type base> bool dft<base>::_init = false;
template<modint_type base> uint32_t dft<base>::mod = {};
template<modint_type base> uint32_t dft<base>::imod = {};
void mul_slow(auto &a, auto const& b, size_t k) {
if(std::empty(a) || std::empty(b)) {
a.clear();
} else {
size_t n = std::min(k, std::size(a));
size_t m = std::min(k, std::size(b));
a.resize(k);
for(int j = int(k - 1); j >= 0; j--) {
a[j] *= b[0];
for(int i = std::max(j - (int)n, 0) + 1; i < std::min(j + 1, (int)m); i++) {
a[j] += a[j - i] * b[i];
}
}
}
}
size_t com_size(size_t as, size_t bs) {
if(!as || !bs) {
return 0;
}
return std::max(flen, std::bit_ceil(as + bs - 1) / 2);
}
void mul_truncate(auto &a, auto const& b, size_t k) {
using base = std::decay_t<decltype(a[0])>;
if(std::min({k, std::size(a), std::size(b)}) < magic) {
mul_slow(a, b, k);
return;
}
auto n = std::max(flen, std::bit_ceil(
std::min(k, std::size(a)) + std::min(k, std::size(b)) - 1
) / 2);
auto A = dft<base>(a | std::views::take(k), n);
auto B = dft<base>(b | std::views::take(k), n);
a.resize((k + flen - 1) / flen * flen);
A.mul_inplace(B, a, k);
a.resize(k);
}
// store mod x^n-k in first half, x^n+k in second half
void mod_split(auto &&x, size_t n, auto k) {
using base = std::decay_t<decltype(k)>;
dft<base>::init();
assert(std::size(x) == 2 * n);
u64x4 cur = u64x4{} + (k * bpow(base(2), 32)).getr();
for(size_t i = 0; i < n; i += flen) {
u64x4 xl = {
x[i].getr(),
x[i + 1].getr(),
x[i + 2].getr(),
x[i + 3].getr()
};
u64x4 xr = {
x[n + i].getr(),
x[n + i + 1].getr(),
x[n + i + 2].getr(),
x[n + i + 3].getr()
};
xr = montgomery_mul(xr, cur, dft<base>::mod, dft<base>::imod);
xr = xr >= base::mod() ? xr - base::mod() : xr;
auto t = xr;
xr = xl - t;
xl += t;
xl = xl >= base::mod() ? xl - base::mod() : xl;
xr = xr >= base::mod() ? xr + base::mod() : xr;
for(size_t k = 0; k < flen; k++) {
x[i + k].setr(typename base::UInt(xl[k]));
x[n + i + k].setr(typename base::UInt(xr[k]));
}
}
cp_algo::checkpoint("mod split");
}
void cyclic_mul(auto &a, auto &&b, size_t k) {
assert(std::popcount(k) == 1);
assert(std::size(a) == std::size(b) && std::size(a) == k);
using base = std::decay_t<decltype(a[0])>;
dft<base>::init();
if(k <= (1 << 16)) {
auto ap = std::ranges::to<std::vector<base, big_alloc<base>>>(a);
mul_truncate(ap, b, 2 * k);
mod_split(ap, k, bpow(dft<base>::factor, k));
std::ranges::copy(ap | std::views::take(k), begin(a));
return;
}
k /= 2;
auto factor = bpow(dft<base>::factor, k);
mod_split(a, k, factor);
mod_split(b, k, factor);
auto la = std::span(a).first(k);
auto lb = std::span(b).first(k);
auto ra = std::span(a).last(k);
auto rb = std::span(b).last(k);
cyclic_mul(la, lb, k);
auto A = dft<base>(ra, k / 2);
auto B = dft<base>(rb, k / 2);
A.mul_inplace(B, ra, k);
base i2 = base(2).inv();
factor = factor.inv() * i2;
for(size_t i = 0; i < k; i++) {
auto t = (a[i] + a[i + k]) * i2;
a[i + k] = (a[i] - a[i + k]) * factor;
a[i] = t;
}
cp_algo::checkpoint("mod join");
}
auto make_copy(auto &&x) {
return x;
}
void cyclic_mul(auto &a, auto const& b, size_t k) {
return cyclic_mul(a, make_copy(b), k);
}
void mul(auto &a, auto &&b) {
size_t N = size(a) + size(b);
if(N > (1 << 20)) {
N--;
size_t NN = std::bit_ceil(N);
a.resize(NN);
b.resize(NN);
cyclic_mul(a, b, NN);
a.resize(N);
} else {
mul_truncate(a, b, N - 1);
}
}
void mul(auto &a, auto const& b) {
size_t N = size(a) + size(b);
if(N > (1 << 20)) {
mul(a, make_copy(b));
} else {
mul_truncate(a, b, N - 1);
}
}
}
#line 12 "cp-algo/math/poly/impl/euclid.hpp"
// operations related to gcd and Euclidean algo
namespace cp_algo::math::poly::impl {
template<typename poly>
using gcd_result = std::pair<
std::list<std::decay_t<poly>>,
linfrac<std::decay_t<poly>>>;
template<typename poly>
gcd_result<poly> half_gcd(poly &&A, poly &&B) {
assert(A.deg() >= B.deg());
size_t m = size(A.a) / 2;
if(B.deg() < (int)m) {
return {};
}
auto [ai, R] = A.divmod(B);
std::tie(A, B) = {B, R};
std::list a = {ai};
auto T = -linfrac(ai).adj();
auto advance = [&](size_t k) {
auto [ak, Tk] = half_gcd(A.div_xk(k), B.div_xk(k));
a.splice(end(a), ak);
T.prepend(Tk);
return Tk;
};
advance(m).apply(A, B);
if constexpr (std::is_reference_v<poly>) {
advance(2 * m - A.deg()).apply(A, B);
} else {
advance(2 * m - A.deg());
}
return {std::move(a), std::move(T)};
}
template<typename poly>
gcd_result<poly> full_gcd(poly &&A, poly &&B) {
using poly_t = std::decay_t<poly>;
std::list<poly_t> ak;
std::vector<linfrac<poly_t>> trs;
while(!B.is_zero()) {
auto [a0, R] = A.divmod(B);
ak.push_back(a0);
trs.push_back(-linfrac(a0).adj());
std::tie(A, B) = {B, R};
auto [a, Tr] = half_gcd(A, B);
ak.splice(end(ak), a);
trs.push_back(Tr);
}
return {ak, std::accumulate(rbegin(trs), rend(trs), linfrac<poly_t>{}, std::multiplies{})};
}
// computes product of linfrac on [L, R)
auto convergent(auto L, auto R) {
using poly = decltype(L)::value_type;
if(R == next(L)) {
return linfrac(*L);
} else {
int s = std::transform_reduce(L, R, 0, std::plus{}, std::mem_fn(&poly::deg));
auto M = L;
for(int c = M->deg(); 2 * c <= s; M++) {
c += next(M)->deg();
}
return convergent(L, M) * convergent(M, R);
}
}
template<typename poly>
poly min_rec(poly const& p, size_t d) {
auto R2 = p.mod_xk(d).reversed(d), R1 = poly::xk(d);
if(R2.is_zero()) {
return poly(1);
}
auto [a, Tr] = full_gcd(R1, R2);
a.emplace_back();
auto pref = begin(a);
for(int delta = (int)d - a.front().deg(); delta >= 0; pref++) {
delta -= pref->deg() + next(pref)->deg();
}
return convergent(begin(a), pref).a;
}
template<typename poly>
std::optional<poly> inv_mod(poly p, poly q) {
assert(!q.is_zero());
auto [a, Tr] = full_gcd(q, p);
if(q.deg() != 0) {
return std::nullopt;
}
return Tr.b / q[0];
}
}
#line 1 "cp-algo/math/poly/impl/div.hpp"
#line 6 "cp-algo/math/poly/impl/div.hpp"
// operations related to polynomial division
namespace cp_algo::math::poly::impl {
auto divmod_slow(auto const& p, auto const& q) {
auto R = p;
auto D = decltype(p){};
auto q_lead_inv = q.lead().inv();
while(R.deg() >= q.deg()) {
D.a.push_back(R.lead() * q_lead_inv);
if(D.lead() != 0) {
for(size_t i = 1; i <= q.a.size(); i++) {
R.a[R.a.size() - i] -= D.lead() * q.a[q.a.size() - i];
}
}
R.a.pop_back();
}
std::ranges::reverse(D.a);
R.normalize();
return std::array{D, R};
}
template<typename poly>
auto divmod_hint(poly const& p, poly const& q, poly const& qri) {
assert(!q.is_zero());
int d = p.deg() - q.deg();
if(std::min(d, q.deg()) < magic) {
return divmod_slow(p, q);
}
poly D;
if(d >= 0) {
D = (p.reversed().mod_xk(d + 1) * qri.mod_xk(d + 1)).mod_xk(d + 1).reversed(d + 1);
}
return std::array{D, p - D * q};
}
auto divmod(auto const& p, auto const& q) {
assert(!q.is_zero());
int d = p.deg() - q.deg();
if(std::min(d, q.deg()) < magic) {
return divmod_slow(p, q);
}
return divmod_hint(p, q, q.reversed().inv(d + 1));
}
template<typename poly>
poly powmod_hint(poly const& p, int64_t k, poly const& md, poly const& mdri) {
return bpow(p, k, poly(1), [&](auto const& p, auto const& q){
return divmod_hint(p * q, md, mdri)[1];
});
}
template<typename poly>
auto powmod(poly const& p, int64_t k, poly const& md) {
int d = md.deg();
if(p == poly::xk(1) && false) { // does it actually speed anything up?..
if(k < md.deg()) {
return poly::xk(k);
} else {
auto mdr = md.reversed();
return (mdr.inv(k - md.deg() + 1, md.deg()) * mdr).reversed(md.deg());
}
}
if(md == poly::xk(d)) {
return p.pow(k, d);
}
if(md == poly::xk(d) - poly(1)) {
return p.powmod_circular(k, d);
}
return powmod_hint(p, k, md, md.reversed().inv(md.deg() + 1));
}
template<typename poly>
poly& inv_inplace(poly& q, int64_t k, size_t n) {
using poly_t = std::decay_t<poly>;
using base = poly_t::base;
if(k <= std::max<int64_t>(n, size(q.a))) {
return q.inv_inplace(k + n).div_xk_inplace(k);
}
if(k % 2) {
return inv_inplace(q, k - 1, n + 1).div_xk_inplace(1);
}
auto [q0, q1] = q.bisect();
auto qq = q0 * q0 - (q1 * q1).mul_xk_inplace(1);
inv_inplace(qq, k / 2 - q.deg() / 2, (n + 1) / 2 + q.deg() / 2);
size_t N = fft::com_size(size(q0.a), size(qq.a));
auto q0f = fft::dft<base>(q0.a, N);
auto q1f = fft::dft<base>(q1.a, N);
auto qqf = fft::dft<base>(qq.a, N);
size_t M = q0.deg() + (n + 1) / 2;
typename poly::Vector A, B;
A.resize((M + fft::flen - 1) / fft::flen * fft::flen);
B.resize((M + fft::flen - 1) / fft::flen * fft::flen);
q0f.mul(qqf, A, M);
q1f.mul_inplace(qqf, B, M);
q.a.resize(n + 1);
for(size_t i = 0; i < n; i += 2) {
q.a[i] = A[q0.deg() + i / 2];
q.a[i + 1] = -B[q0.deg() + i / 2];
}
q.a.pop_back();
q.normalize();
return q;
}
template<typename poly>
poly& inv_inplace(poly& p, size_t n) {
using poly_t = std::decay_t<poly>;
using base = poly_t::base;
if(n == 1) {
return p = base(1) / p[0];
}
// Q(-x) = P0(x^2) + xP1(x^2)
auto [q0, q1] = p.bisect(n);
size_t N = fft::com_size(size(q0.a), (n + 1) / 2);
auto q0f = fft::dft<base>(q0.a, N);
auto q1f = fft::dft<base>(q1.a, N);
// Q(x)*Q(-x) = Q0(x^2)^2 - x^2 Q1(x^2)^2
auto qq = poly_t(q0f * q0f) - poly_t(q1f * q1f).mul_xk_inplace(1);
inv_inplace(qq, (n + 1) / 2);
auto qqf = fft::dft<base>(qq.a, N);
typename poly::Vector A, B;
A.resize(((n + 1) / 2 + fft::flen - 1) / fft::flen * fft::flen);
B.resize(((n + 1) / 2 + fft::flen - 1) / fft::flen * fft::flen);
q0f.mul(qqf, A, (n + 1) / 2);
q1f.mul_inplace(qqf, B, (n + 1) / 2);
p.a.resize(n + 1);
for(size_t i = 0; i < n; i += 2) {
p.a[i] = A[i / 2];
p.a[i + 1] = -B[i / 2];
}
p.a.pop_back();
p.normalize();
return p;
}
}
#line 1 "cp-algo/math/combinatorics.hpp"
#line 6 "cp-algo/math/combinatorics.hpp"
namespace cp_algo::math {
// fact/rfact/small_inv are caching
// Beware of usage with dynamic mod
template<typename T>
T fact(auto n) {
static std::vector<T> F(maxn);
static bool init = false;
if(!init) {
F[0] = T(1);
for(int i = 1; i < maxn; i++) {
F[i] = F[i - 1] * T(i);
}
init = true;
}
return F[n];
}
// Only works for modint types
template<typename T>
T rfact(auto n) {
static std::vector<T> F(maxn);
static bool init = false;
if(!init) {
int t = (int)std::min<int64_t>(T::mod(), maxn) - 1;
F[t] = T(1) / fact<T>(t);
for(int i = t - 1; i >= 0; i--) {
F[i] = F[i + 1] * T(i + 1);
}
init = true;
}
return F[n];
}
template<typename T, int base>
T pow_fixed(int n) {
static std::vector<T> prec_low(1 << 16);
static std::vector<T> prec_high(1 << 16);
static bool init = false;
if(!init) {
init = true;
prec_low[0] = prec_high[0] = T(1);
T step_low = T(base);
T step_high = bpow(T(base), 1 << 16);
for(int i = 1; i < (1 << 16); i++) {
prec_low[i] = prec_low[i - 1] * step_low;
prec_high[i] = prec_high[i - 1] * step_high;
}
}
return prec_low[n & 0xFFFF] * prec_high[n >> 16];
}
template<typename T>
std::vector<T> bulk_invs(auto const& args) {
std::vector<T> res(std::size(args), args[0]);
for(size_t i = 1; i < std::size(args); i++) {
res[i] = res[i - 1] * args[i];
}
auto all_invs = T(1) / res.back();
for(size_t i = std::size(args) - 1; i > 0; i--) {
res[i] = all_invs * res[i - 1];
all_invs *= args[i];
}
res[0] = all_invs;
return res;
}
template<typename T>
T small_inv(auto n) {
static auto F = bulk_invs<T>(std::views::iota(1, maxn));
return F[n - 1];
}
template<typename T>
T binom_large(T n, auto r) {
assert(r < maxn);
T ans = 1;
for(decltype(r) i = 0; i < r; i++) {
ans = ans * T(n - i) * small_inv<T>(i + 1);
}
return ans;
}
template<typename T>
T binom(auto n, auto r) {
if(r < 0 || r > n) {
return T(0);
} else if(n >= maxn) {
return binom_large(T(n), r);
} else {
return fact<T>(n) * rfact<T>(r) * rfact<T>(n - r);
}
}
}
#line 1 "cp-algo/number_theory/discrete_sqrt.hpp"
#line 6 "cp-algo/number_theory/discrete_sqrt.hpp"
namespace cp_algo::math {
// https://en.wikipedia.org/wiki/Berlekamp-Rabin_algorithm
template<modint_type base>
std::optional<base> sqrt(base b) {
if(b == base(0)) {
return base(0);
} else if(bpow(b, (b.mod() - 1) / 2) != base(1)) {
return std::nullopt;
} else {
while(true) {
base z = random::rng();
if(z * z == b) {
return z;
}
lin<base> x(1, z, b); // x + z (mod x^2 - b)
x = bpow(x, (b.mod() - 1) / 2, lin<base>(0, 1, b));
if(x.a != base(0)) {
return x.a.inv();
}
}
}
}
}
#line 15 "cp-algo/math/poly.hpp"
namespace cp_algo::math {
template<typename T, class Alloc = big_alloc<T>>
struct poly_t {
using Vector = std::vector<T, Alloc>;
using base = T;
Vector a;
poly_t& normalize() {
while(deg() >= 0 && lead() == base(0)) {
a.pop_back();
}
return *this;
}
poly_t(){}
poly_t(T a0): a{a0} {normalize();}
poly_t(Vector const& t): a(t) {normalize();}
poly_t(Vector &&t): a(std::move(t)) {normalize();}
poly_t& negate_inplace() {
std::ranges::transform(a, begin(a), std::negate{});
return *this;
}
poly_t operator -() const {
return poly_t(*this).negate_inplace();
}
poly_t& operator += (poly_t const& t) {
a.resize(std::max(size(a), size(t.a)));
std::ranges::transform(a, t.a, begin(a), std::plus{});
return normalize();
}
poly_t& operator -= (poly_t const& t) {
a.resize(std::max(size(a), size(t.a)));
std::ranges::transform(a, t.a, begin(a), std::minus{});
return normalize();
}
poly_t operator + (poly_t const& t) const {return poly_t(*this) += t;}
poly_t operator - (poly_t const& t) const {return poly_t(*this) -= t;}
poly_t& mod_xk_inplace(size_t k) {
a.resize(std::min(size(a), k));
return normalize();
}
poly_t& mul_xk_inplace(size_t k) {
a.insert(begin(a), k, T(0));
return normalize();
}
poly_t& div_xk_inplace(int64_t k) {
if(k < 0) {
return mul_xk_inplace(-k);
}
a.erase(begin(a), begin(a) + std::min<size_t>(k, size(a)));
return normalize();
}
poly_t &substr_inplace(size_t l, size_t k) {
return mod_xk_inplace(l + k).div_xk_inplace(l);
}
poly_t mod_xk(size_t k) const {return poly_t(*this).mod_xk_inplace(k);}
poly_t mul_xk(size_t k) const {return poly_t(*this).mul_xk_inplace(k);}
poly_t div_xk(int64_t k) const {return poly_t(*this).div_xk_inplace(k);}
poly_t substr(size_t l, size_t k) const {return poly_t(*this).substr_inplace(l, k);}
poly_t& operator *= (const poly_t &t) {fft::mul(a, t.a); normalize(); return *this;}
poly_t operator * (const poly_t &t) const {return poly_t(*this) *= t;}
poly_t& operator /= (const poly_t &t) {return *this = divmod(t)[0];}
poly_t& operator %= (const poly_t &t) {return *this = divmod(t)[1];}
poly_t operator / (poly_t const& t) const {return poly_t(*this) /= t;}
poly_t operator % (poly_t const& t) const {return poly_t(*this) %= t;}
poly_t& operator *= (T const& x) {
for(auto &it: a) {
it *= x;
}
return normalize();
}
poly_t& operator /= (T const& x) {return *this *= x.inv();}
poly_t operator * (T const& x) const {return poly_t(*this) *= x;}
poly_t operator / (T const& x) const {return poly_t(*this) /= x;}
poly_t& reverse(size_t n) {
a.resize(n);
std::ranges::reverse(a);
return normalize();
}
poly_t& reverse() {return reverse(size(a));}
poly_t reversed(size_t n) const {return poly_t(*this).reverse(n);}
poly_t reversed() const {return poly_t(*this).reverse();}
std::array<poly_t, 2> divmod(poly_t const& b) const {
return poly::impl::divmod(*this, b);
}
// reduces A/B to A'/B' such that
// deg B' < deg A / 2
static std::pair<std::list<poly_t>, linfrac<poly_t>> half_gcd(auto &&A, auto &&B) {
return poly::impl::half_gcd(A, B);
}
// reduces A / B to gcd(A, B) / 0
static std::pair<std::list<poly_t>, linfrac<poly_t>> full_gcd(auto &&A, auto &&B) {
return poly::impl::full_gcd(A, B);
}
static poly_t gcd(poly_t &&A, poly_t &&B) {
full_gcd(A, B);
return A;
}
// Returns a (non-monic) characteristic polynomial
// of the minimum linear recurrence for the sequence
poly_t min_rec(size_t d) const {
return poly::impl::min_rec(*this, d);
}
// calculate inv to *this modulo t
std::optional<poly_t> inv_mod(poly_t const& t) const {
return poly::impl::inv_mod(*this, t);
};
poly_t negx() const { // A(x) -> A(-x)
auto res = *this;
for(int i = 1; i <= deg(); i += 2) {
res.a[i] = -res[i];
}
return res;
}
void print(int n) const {
for(int i = 0; i < n; i++) {
std::cout << (*this)[i] << ' ';
}
std::cout << "\n";
}
void print() const {
print(deg() + 1);
}
T eval(T x) const { // evaluates in single point x
T res(0);
for(int i = deg(); i >= 0; i--) {
res *= x;
res += a[i];
}
return res;
}
T lead() const { // leading coefficient
assert(!is_zero());
return a.back();
}
int deg() const { // degree, -1 for P(x) = 0
return (int)a.size() - 1;
}
bool is_zero() const {
return a.empty();
}
T operator [](int idx) const {
return idx < 0 || idx > deg() ? T(0) : a[idx];
}
T& coef(size_t idx) { // mutable reference at coefficient
return a[idx];
}
bool operator == (const poly_t &t) const {return a == t.a;}
bool operator != (const poly_t &t) const {return a != t.a;}
poly_t& deriv_inplace(int k = 1) {
if(deg() + 1 < k) {
return *this = poly_t{};
}
for(int i = k; i <= deg(); i++) {
a[i - k] = fact<T>(i) * rfact<T>(i - k) * a[i];
}
a.resize(deg() + 1 - k);
return *this;
}
poly_t deriv(int k = 1) const { // calculate derivative
return poly_t(*this).deriv_inplace(k);
}
poly_t& integr_inplace() {
a.push_back(0);
for(int i = deg() - 1; i >= 0; i--) {
a[i + 1] = a[i] * small_inv<T>(i + 1);
}
a[0] = 0;
return *this;
}
poly_t integr() const { // calculate integral with C = 0
Vector res(deg() + 2);
for(int i = 0; i <= deg(); i++) {
res[i + 1] = a[i] * small_inv<T>(i + 1);
}
return res;
}
size_t trailing_xk() const { // Let p(x) = x^k * t(x), return k
if(is_zero()) {
return -1;
}
int res = 0;
while(a[res] == T(0)) {
res++;
}
return res;
}
// calculate log p(x) mod x^n
poly_t& log_inplace(size_t n) {
assert(a[0] == T(1));
mod_xk_inplace(n);
return (inv_inplace(n) *= mod_xk_inplace(n).deriv()).mod_xk_inplace(n - 1).integr_inplace();
}
poly_t log(size_t n) const {
return poly_t(*this).log_inplace(n);
}
poly_t& mul_truncate(poly_t const& t, size_t k) {
fft::mul_truncate(a, t.a, k);
return normalize();
}
poly_t& exp_inplace(size_t n) {
if(is_zero()) {
return *this = T(1);
}
assert(a[0] == T(0));
a[0] = 1;
size_t a = 1;
while(a < n) {
poly_t C = log(2 * a).div_xk_inplace(a) - substr(a, 2 * a);
*this -= C.mul_truncate(*this, a).mul_xk_inplace(a);
a *= 2;
}
return mod_xk_inplace(n);
}
poly_t exp(size_t n) const { // calculate exp p(x) mod x^n
return poly_t(*this).exp_inplace(n);
}
poly_t pow_bin(int64_t k, size_t n) const { // O(n log n log k)
if(k == 0) {
return poly_t(1).mod_xk(n);
} else {
auto t = pow(k / 2, n);
t = (t * t).mod_xk(n);
return (k % 2 ? *this * t : t).mod_xk(n);
}
}
poly_t circular_closure(size_t m) const {
if(deg() == -1) {
return *this;
}
auto t = *this;
for(size_t i = t.deg(); i >= m; i--) {
t.a[i - m] += t.a[i];
}
t.a.resize(std::min(t.a.size(), m));
return t;
}
static poly_t mul_circular(poly_t const& a, poly_t const& b, size_t m) {
return (a.circular_closure(m) * b.circular_closure(m)).circular_closure(m);
}
poly_t powmod_circular(int64_t k, size_t m) const {
if(k == 0) {
return poly_t(1);
} else {
auto t = powmod_circular(k / 2, m);
t = mul_circular(t, t, m);
if(k % 2) {
t = mul_circular(t, *this, m);
}
return t;
}
}
poly_t powmod(int64_t k, poly_t const& md) const {
return poly::impl::powmod(*this, k, md);
}
// O(d * n) with the derivative trick from
// https://codeforces.com/blog/entry/73947?#comment-581173
poly_t pow_dn(int64_t k, size_t n) const {
if(n == 0) {
return poly_t(T(0));
}
assert((*this)[0] != T(0));
Vector Q(n);
Q[0] = bpow(a[0], k);
auto a0inv = a[0].inv();
for(int i = 1; i < (int)n; i++) {
for(int j = 1; j <= std::min(deg(), i); j++) {
Q[i] += a[j] * Q[i - j] * (T(k) * T(j) - T(i - j));
}
Q[i] *= small_inv<T>(i) * a0inv;
}
return Q;
}
// calculate p^k(n) mod x^n in O(n log n)
// might be quite slow due to high constant
poly_t pow(int64_t k, size_t n) const {
if(is_zero()) {
return k ? *this : poly_t(1);
}
size_t i = trailing_xk();
if(i > 0) {
return k >= int64_t(n + i - 1) / (int64_t)i ? poly_t(T(0)) : div_xk(i).pow(k, n - i * k).mul_xk(i * k);
}
if(std::min(deg(), (int)n) <= magic) {
return pow_dn(k, n);
}
if(k <= magic) {
return pow_bin(k, n);
}
T j = a[i];
poly_t t = *this / j;
return bpow(j, k) * (t.log(n) * T(k)).exp(n).mod_xk(n);
}
// returns std::nullopt if undefined
std::optional<poly_t> sqrt(size_t n) const {
if(is_zero()) {
return *this;
}
size_t i = trailing_xk();
if(i % 2) {
return std::nullopt;
} else if(i > 0) {
auto ans = div_xk(i).sqrt(n - i / 2);
return ans ? ans->mul_xk(i / 2) : ans;
}
auto st = math::sqrt((*this)[0]);
if(st) {
poly_t ans = *st;
size_t a = 1;
while(a < n) {
a *= 2;
ans -= (ans - mod_xk(a) * ans.inv(a)).mod_xk(a) / 2;
}
return ans.mod_xk(n);
}
return std::nullopt;
}
poly_t mulx(T a) const { // component-wise multiplication with a^k
T cur = 1;
poly_t res(*this);
for(int i = 0; i <= deg(); i++) {
res.coef(i) *= cur;
cur *= a;
}
return res;
}
poly_t mulx_sq(T a) const { // component-wise multiplication with a^{k choose 2}
T cur = 1, total = 1;
poly_t res(*this);
for(int i = 0; i <= deg(); i++) {
res.coef(i) *= total;
cur *= a;
total *= cur;
}
return res;
}
// be mindful of maxn, as the function
// requires multiplying polynomials of size deg() and n+deg()!
poly_t chirpz(T z, int n) const { // P(1), P(z), P(z^2), ..., P(z^(n-1))
if(is_zero()) {
return Vector(n, 0);
}
if(z == T(0)) {
Vector ans(n, (*this)[0]);
if(n > 0) {
ans[0] = accumulate(begin(a), end(a), T(0));
}
return ans;
}
auto A = mulx_sq(z.inv());
auto B = ones(n+deg()).mulx_sq(z);
return semicorr(B, A).mod_xk(n).mulx_sq(z.inv());
}
// res[i] = prod_{1 <= j <= i} 1/(1 - z^j)
static auto _1mzk_prod_inv(T z, int n) {
Vector res(n, 1), zk(n);
zk[0] = 1;
for(int i = 1; i < n; i++) {
zk[i] = zk[i - 1] * z;
res[i] = res[i - 1] * (T(1) - zk[i]);
}
res.back() = res.back().inv();
for(int i = n - 2; i >= 0; i--) {
res[i] = (T(1) - zk[i+1]) * res[i+1];
}
return res;
}
// prod_{0 <= j < n} (1 - z^j x)
static auto _1mzkx_prod(T z, int n) {
if(n == 1) {
return poly_t(Vector{1, -1});
} else {
auto t = _1mzkx_prod(z, n / 2);
t *= t.mulx(bpow(z, n / 2));
if(n % 2) {
t *= poly_t(Vector{1, -bpow(z, n - 1)});
}
return t;
}
}
poly_t chirpz_inverse(T z, int n) const { // P(1), P(z), P(z^2), ..., P(z^(n-1))
if(is_zero()) {
return {};
}
if(z == T(0)) {
if(n == 1) {
return *this;
} else {
return Vector{(*this)[1], (*this)[0] - (*this)[1]};
}
}
Vector y(n);
for(int i = 0; i < n; i++) {
y[i] = (*this)[i];
}
auto prods_pos = _1mzk_prod_inv(z, n);
auto prods_neg = _1mzk_prod_inv(z.inv(), n);
T zn = bpow(z, n-1).inv();
T znk = 1;
for(int i = 0; i < n; i++) {
y[i] *= znk * prods_neg[i] * prods_pos[(n - 1) - i];
znk *= zn;
}
poly_t p_over_q = poly_t(y).chirpz(z, n);
poly_t q = _1mzkx_prod(z, n);
return (p_over_q * q).mod_xk_inplace(n).reverse(n);
}
static poly_t build(std::vector<poly_t> &res, int v, auto L, auto R) { // builds evaluation tree for (x-a1)(x-a2)...(x-an)
if(R - L == 1) {
return res[v] = Vector{-*L, 1};
} else {
auto M = L + (R - L) / 2;
return res[v] = build(res, 2 * v, L, M) * build(res, 2 * v + 1, M, R);
}
}
poly_t to_newton(std::vector<poly_t> &tree, int v, auto l, auto r) {
if(r - l == 1) {
return *this;
} else {
auto m = l + (r - l) / 2;
auto A = (*this % tree[2 * v]).to_newton(tree, 2 * v, l, m);
auto B = (*this / tree[2 * v]).to_newton(tree, 2 * v + 1, m, r);
return A + B.mul_xk(m - l);
}
}
poly_t to_newton(Vector p) {
if(is_zero()) {
return *this;
}
size_t n = p.size();
std::vector<poly_t> tree(4 * n);
build(tree, 1, begin(p), end(p));
return to_newton(tree, 1, begin(p), end(p));
}
Vector eval(std::vector<poly_t> &tree, int v, auto l, auto r) { // auxiliary evaluation function
if(r - l == 1) {
return {eval(*l)};
} else {
auto m = l + (r - l) / 2;
auto A = (*this % tree[2 * v]).eval(tree, 2 * v, l, m);
auto B = (*this % tree[2 * v + 1]).eval(tree, 2 * v + 1, m, r);
A.insert(end(A), begin(B), end(B));
return A;
}
}
Vector eval(Vector x) { // evaluate polynomial in (x1, ..., xn)
size_t n = x.size();
if(is_zero()) {
return Vector(n, T(0));
}
std::vector<poly_t> tree(4 * n);
build(tree, 1, begin(x), end(x));
return eval(tree, 1, begin(x), end(x));
}
poly_t inter(std::vector<poly_t> &tree, int v, auto ly, auto ry) { // auxiliary interpolation function
if(ry - ly == 1) {
return {*ly / a[0]};
} else {
auto my = ly + (ry - ly) / 2;
auto A = (*this % tree[2 * v]).inter(tree, 2 * v, ly, my);
auto B = (*this % tree[2 * v + 1]).inter(tree, 2 * v + 1, my, ry);
return A * tree[2 * v + 1] + B * tree[2 * v];
}
}
static auto inter(Vector x, Vector y) { // interpolates minimum polynomial from (xi, yi) pairs
size_t n = x.size();
std::vector<poly_t> tree(4 * n);
return build(tree, 1, begin(x), end(x)).deriv().inter(tree, 1, begin(y), end(y));
}
static auto resultant(poly_t a, poly_t b) { // computes resultant of a and b
if(b.is_zero()) {
return 0;
} else if(b.deg() == 0) {
return bpow(b.lead(), a.deg());
} else {
int pw = a.deg();
a %= b;
pw -= a.deg();
auto mul = bpow(b.lead(), pw) * T((b.deg() & a.deg() & 1) ? -1 : 1);
auto ans = resultant(b, a);
return ans * mul;
}
}
static poly_t xk(size_t n) { // P(x) = x^n
return poly_t(T(1)).mul_xk(n);
}
static poly_t ones(size_t n) { // P(x) = 1 + x + ... + x^{n-1}
return Vector(n, 1);
}
static poly_t expx(size_t n) { // P(x) = e^x (mod x^n)
return ones(n).borel();
}
static poly_t log1px(size_t n) { // P(x) = log(1+x) (mod x^n)
Vector coeffs(n, 0);
for(size_t i = 1; i < n; i++) {
coeffs[i] = (i & 1 ? T(i).inv() : -T(i).inv());
}
return coeffs;
}
static poly_t log1mx(size_t n) { // P(x) = log(1-x) (mod x^n)
return -ones(n).integr();
}
// [x^k] (a corr b) = sum_{i} a{(k-m)+i}*bi
static poly_t corr(poly_t const& a, poly_t const& b) { // cross-correlation
return a * b.reversed();
}
// [x^k] (a semicorr b) = sum_i a{i+k} * b{i}
static poly_t semicorr(poly_t const& a, poly_t const& b) {
return corr(a, b).div_xk(b.deg());
}
poly_t invborel() const { // ak *= k!
auto res = *this;
for(int i = 0; i <= deg(); i++) {
res.coef(i) *= fact<T>(i);
}
return res;
}
poly_t borel() const { // ak /= k!
auto res = *this;
for(int i = 0; i <= deg(); i++) {
res.coef(i) *= rfact<T>(i);
}
return res;
}
poly_t shift(T a) const { // P(x + a)
return semicorr(invborel(), expx(deg() + 1).mulx(a)).borel();
}
poly_t x2() { // P(x) -> P(x^2)
Vector res(2 * a.size());
for(size_t i = 0; i < a.size(); i++) {
res[2 * i] = a[i];
}
return res;
}
// Return {P0, P1}, where P(x) = P0(x) + xP1(x)
std::array<poly_t, 2> bisect(size_t n) const {
n = std::min(n, size(a));
Vector res[2];
for(size_t i = 0; i < n; i++) {
res[i % 2].push_back(a[i]);
}
return {res[0], res[1]};
}
std::array<poly_t, 2> bisect() const {
return bisect(size(a));
}
// Find [x^k] P / Q
static T kth_rec_inplace(poly_t &P, poly_t &Q, int64_t k) {
while(k > Q.deg()) {
size_t n = Q.a.size();
auto [Q0, Q1] = Q.bisect();
auto [P0, P1] = P.bisect();
size_t N = fft::com_size((n + 1) / 2, (n + 1) / 2);
auto Q0f = fft::dft<T>(Q0.a, N);
auto Q1f = fft::dft<T>(Q1.a, N);
auto P0f = fft::dft<T>(P0.a, N);
auto P1f = fft::dft<T>(P1.a, N);
Q = poly_t(Q0f * Q0f) -= poly_t(Q1f * Q1f).mul_xk_inplace(1);
if(k % 2) {
P = poly_t(Q0f *= P1f) -= poly_t(Q1f *= P0f);
} else {
P = poly_t(Q0f *= P0f) -= poly_t(Q1f *= P1f).mul_xk_inplace(1);
}
k /= 2;
}
return (P *= Q.inv_inplace(Q.deg() + 1))[(int)k];
}
static T kth_rec(poly_t const& P, poly_t const& Q, int64_t k) {
return kth_rec_inplace(poly_t(P), poly_t(Q), k);
}
// inverse series mod x^n
poly_t& inv_inplace(size_t n) {
return poly::impl::inv_inplace(*this, n);
}
poly_t inv(size_t n) const {
return poly_t(*this).inv_inplace(n);
}
// [x^k]..[x^{k+n-1}] of inv()
// supports negative k if k+n >= 0
poly_t& inv_inplace(int64_t k, size_t n) {
return poly::impl::inv_inplace(*this, k, n);
}
poly_t inv(int64_t k, size_t n) const {
return poly_t(*this).inv_inplace(k, n);
}
// compute A(B(x)) mod x^n in O(n^2)
static poly_t compose(poly_t A, poly_t B, int n) {
int q = std::sqrt(n);
std::vector<poly_t> Bk(q);
auto Bq = B.pow(q, n);
Bk[0] = poly_t(T(1));
for(int i = 1; i < q; i++) {
Bk[i] = (Bk[i - 1] * B).mod_xk(n);
}
poly_t Bqk(1);
poly_t ans;
for(int i = 0; i <= n / q; i++) {
poly_t cur;
for(int j = 0; j < q; j++) {
cur += Bk[j] * A[i * q + j];
}
ans += (Bqk * cur).mod_xk(n);
Bqk = (Bqk * Bq).mod_xk(n);
}
return ans;
}
// compute A(B(x)) mod x^n in O(sqrt(pqn log^3 n))
// preferrable when p = deg A and q = deg B
// are much less than n
static poly_t compose_large(poly_t A, poly_t B, int n) {
if(B[0] != T(0)) {
return compose_large(A.shift(B[0]), B - B[0], n);
}
int q = std::sqrt(n);
auto [B0, B1] = std::make_pair(B.mod_xk(q), B.div_xk(q));
B0 = B0.div_xk(1);
std::vector<poly_t> pw(A.deg() + 1);
auto getpow = [&](int k) {
return pw[k].is_zero() ? pw[k] = B0.pow(k, n - k) : pw[k];
};
std::function<poly_t(poly_t const&, int, int)> compose_dac = [&getpow, &compose_dac](poly_t const& f, int m, int N) {
if(f.deg() <= 0) {
return f;
}
int k = m / 2;
auto [f0, f1] = std::make_pair(f.mod_xk(k), f.div_xk(k));
auto [A, B] = std::make_pair(compose_dac(f0, k, N), compose_dac(f1, m - k, N - k));
return (A + (B.mod_xk(N - k) * getpow(k).mod_xk(N - k)).mul_xk(k)).mod_xk(N);
};
int r = n / q;
auto Ar = A.deriv(r);
auto AB0 = compose_dac(Ar, Ar.deg() + 1, n);
auto Bd = B0.mul_xk(1).deriv();
poly_t ans = T(0);
std::vector<poly_t> B1p(r + 1);
B1p[0] = poly_t(T(1));
for(int i = 1; i <= r; i++) {
B1p[i] = (B1p[i - 1] * B1.mod_xk(n - i * q)).mod_xk(n - i * q);
}
while(r >= 0) {
ans += (AB0.mod_xk(n - r * q) * rfact<T>(r) * B1p[r]).mul_xk(r * q).mod_xk(n);
r--;
if(r >= 0) {
AB0 = ((AB0 * Bd).integr() + A[r] * fact<T>(r)).mod_xk(n);
}
}
return ans;
}
};
template<typename base>
static auto operator * (const auto& a, const poly_t<base>& b) {
return b * a;
}
};
#line 1 "cp-algo/linalg/matrix.hpp"
#line 1 "cp-algo/linalg/vector.hpp"
#line 15 "cp-algo/linalg/vector.hpp"
namespace cp_algo::linalg {
template<typename base, class Alloc = big_alloc<base>>
struct vec: std::basic_string<base, std::char_traits<base>, Alloc> {
using Base = std::basic_string<base, std::char_traits<base>, Alloc>;
using Base::Base;
vec(Base const& t): Base(t) {}
vec(Base &&t): Base(std::move(t)) {}
vec(size_t n): Base(n, base()) {}
vec(auto &&r): Base(std::ranges::to<Base>(r)) {}
static vec ei(size_t n, size_t i) {
vec res(n);
res[i] = 1;
return res;
}
auto operator-() const {
return *this | std::views::transform([](auto x) {return -x;});
}
auto operator *(base t) const {
return *this | std::views::transform([t](auto x) {return x * t;});
}
auto operator *=(base t) {
for(auto &it: *this) {
it *= t;
}
return *this;
}
virtual void add_scaled(vec const& b, base scale, size_t i = 0) {
if(scale != base(0)) {
for(; i < size(*this); i++) {
(*this)[i] += scale * b[i];
}
}
}
virtual vec const& normalize() {
return static_cast<vec&>(*this);
}
virtual base normalize(size_t i) {
return (*this)[i];
}
void read() {
for(auto &it: *this) {
std::cin >> it;
}
}
void print() const {
for(auto &it: *this) {
std::cout << it << " ";
}
std::cout << "\n";
}
static vec random(size_t n) {
vec res(n);
std::ranges::generate(res, random::rng);
return res;
}
// Concatenate vectors
vec operator |(vec const& t) const {
return std::views::join(std::array{
std::views::all(*this),
std::views::all(t)
});
}
// Generally, vec shouldn't be modified
// after its pivot index is set
std::pair<size_t, base> find_pivot() {
if(pivot == size_t(-1)) {
pivot = 0;
while(pivot < size(*this) && normalize(pivot) == base(0)) {
pivot++;
}
if(pivot < size(*this)) {
pivot_inv = base(1) / (*this)[pivot];
}
}
return {pivot, pivot_inv};
}
void reduce_by(vec &t) {
auto [pivot, pinv] = t.find_pivot();
if(pivot < size(*this)) {
add_scaled(t, -normalize(pivot) * pinv, pivot);
}
}
private:
size_t pivot = -1;
base pivot_inv;
};
template<math::modint_type base, class Alloc = big_alloc<base>>
struct modint_vec: vec<base, Alloc> {
using Base = vec<base, Alloc>;
using Base::Base;
modint_vec(Base const& t): Base(t) {}
modint_vec(Base &&t): Base(std::move(t)) {}
void add_scaled(Base const& b, base scale, size_t i = 0) override {
static_assert(base::bits >= 64, "Only wide modint types for linalg");
if(scale != base(0)) {
assert(Base::size() == b.size());
size_t n = size(*this);
u64x4 scaler = u64x4() + scale.getr();
if (is_aligned(&(*this)[0]) && is_aligned(&b[0])) // verify we're not in SSO
for(i -= i % 4; i + 3 < n; i += 4) {
auto &ai = vector_cast<u64x4>((*this)[i]);
auto bi = vector_cast<u64x4 const>(b[i]);
#ifdef __AVX2__
ai += u64x4(_mm256_mul_epu32(__m256i(scaler), __m256i(bi)));
#else
ai += scaler * bi;
#endif
}
for(; i < n; i++) {
(*this)[i].add_unsafe(b[i].getr_direct() * scale.getr());
}
if(++counter == 4) {
for(auto &it: *this) {
it.pseudonormalize();
}
counter = 0;
}
}
}
Base const& normalize() override {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
base normalize(size_t i) override {
return (*this)[i].normalize();
}
private:
size_t counter = 0;
};
}
#line 11 "cp-algo/linalg/matrix.hpp"
namespace cp_algo::linalg {
enum gauss_mode {normal, reverse};
template<typename base_t, class _vec_t = std::conditional_t<
math::modint_type<base_t>,
modint_vec<base_t>,
vec<base_t>>>
struct matrix: std::vector<_vec_t> {
using vec_t = _vec_t;
using base = base_t;
using Base = std::vector<vec_t>;
using Base::Base;
matrix(size_t n): Base(n, vec_t(n)) {}
matrix(size_t n, size_t m): Base(n, vec_t(m)) {}
matrix(Base const& t): Base(t) {}
matrix(Base &&t): Base(std::move(t)) {}
template<std::ranges::input_range R>
matrix(R &&r): Base(std::ranges::to<Base>(std::forward<R>(r))) {}
size_t n() const {return size(*this);}
size_t m() const {return n() ? size(row(0)) : 0;}
void resize(size_t n, size_t m) {
Base::resize(n);
for(auto &it: *this) {
it.resize(m);
}
}
auto& row(size_t i) {return (*this)[i];}
auto const& row(size_t i) const {return (*this)[i];}
auto elements() {return *this | std::views::join;}
auto elements() const {return *this | std::views::join;}
matrix operator-() const {
return *this | std::views::transform([](auto x) {return vec_t(-x);});
}
matrix& operator+=(matrix const& t) {
for(auto [a, b]: std::views::zip(elements(), t.elements())) {
a += b;
}
return *this;
}
matrix& operator -=(matrix const& t) {
for(auto [a, b]: std::views::zip(elements(), t.elements())) {
a -= b;
}
return *this;
}
matrix operator+(matrix const& t) const {return matrix(*this) += t;}
matrix operator-(matrix const& t) const {return matrix(*this) -= t;}
matrix& operator *=(base t) {for(auto &it: *this) it *= t; return *this;}
matrix operator *(base t) const {return matrix(*this) *= t;}
matrix& operator /=(base t) {return *this *= base(1) / t;}
matrix operator /(base t) const {return matrix(*this) /= t;}
// Make sure the result is matrix, not Base
matrix& operator *=(matrix const& t) {return *this = *this * t;}
void read_transposed() {
for(size_t j = 0; j < m(); j++) {
for(size_t i = 0; i < n(); i++) {
std::cin >> (*this)[i][j];
}
}
}
void read() {
for(auto &it: *this) {
it.read();
}
}
void print() const {
for(auto const& it: *this) {
it.print();
}
}
static matrix block_diagonal(std::vector<matrix> const& blocks) {
size_t n = 0;
for(auto &it: blocks) {
assert(it.n() == it.m());
n += it.n();
}
matrix res(n);
n = 0;
for(auto &it: blocks) {
for(size_t i = 0; i < it.n(); i++) {
std::ranges::copy(it[i], begin(res[n + i]) + n);
}
n += it.n();
}
return res;
}
static matrix random(size_t n, size_t m) {
matrix res(n, m);
std::ranges::generate(res, std::bind(vec_t::random, m));
return res;
}
static matrix random(size_t n) {
return random(n, n);
}
static matrix eye(size_t n) {
matrix res(n);
for(size_t i = 0; i < n; i++) {
res[i][i] = 1;
}
return res;
}
// Concatenate matrices
matrix operator |(matrix const& b) const {
assert(n() == b.n());
matrix res(n(), m()+b.m());
for(size_t i = 0; i < n(); i++) {
res[i] = row(i) | b[i];
}
return res;
}
void assign_submatrix(auto viewx, auto viewy, matrix const& t) {
for(auto [a, b]: std::views::zip(*this | viewx, t)) {
std::ranges::copy(b, begin(a | viewy));
}
}
auto submatrix(auto viewx, auto viewy) const {
return *this | viewx | std::views::transform([viewy](auto const& y) {
return y | viewy;
});
}
matrix T() const {
matrix res(m(), n());
for(size_t i = 0; i < n(); i++) {
for(size_t j = 0; j < m(); j++) {
res[j][i] = row(i)[j];
}
}
return res;
}
matrix operator *(matrix const& b) const {
assert(m() == b.n());
matrix res(n(), b.m());
for(size_t i = 0; i < n(); i++) {
for(size_t j = 0; j < m(); j++) {
res[i].add_scaled(b[j], row(i)[j]);
}
}
return res.normalize();
}
vec_t apply(vec_t const& x) const {
return (matrix(1, x) * *this)[0];
}
matrix pow(uint64_t k) const {
assert(n() == m());
return bpow(*this, k, eye(n()));
}
matrix& normalize() {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
template<gauss_mode mode = normal>
void eliminate(size_t i, size_t k) {
auto kinv = base(1) / row(i).normalize()[k];
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).add_scaled(row(i), -row(j).normalize(k) * kinv);
}
}
}
template<gauss_mode mode = normal>
void eliminate(size_t i) {
row(i).normalize();
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).reduce_by(row(i));
}
}
}
template<gauss_mode mode = normal>
matrix& gauss() {
for(size_t i = 0; i < n(); i++) {
eliminate<mode>(i);
}
return normalize();
}
template<gauss_mode mode = normal>
auto echelonize(size_t lim) {
return gauss<mode>().sort_classify(lim);
}
template<gauss_mode mode = normal>
auto echelonize() {
return echelonize<mode>(m());
}
size_t rank() const {
if(n() > m()) {
return T().rank();
}
return size(matrix(*this).echelonize()[0]);
}
base det() const {
assert(n() == m());
matrix b = *this;
b.echelonize();
base res = 1;
for(size_t i = 0; i < n(); i++) {
res *= b[i][i];
}
return res;
}
std::pair<base, matrix> inv() const {
assert(n() == m());
matrix b = *this | eye(n());
if(size(b.echelonize<reverse>(n())[0]) < n()) {
return {0, {}};
}
base det = 1;
for(size_t i = 0; i < n(); i++) {
det *= b[i][i];
b[i] *= base(1) / b[i][i];
}
return {det, b.submatrix(std::views::all, std::views::drop(n()))};
}
// Can also just run gauss on T() | eye(m)
// but it would be slower :(
auto kernel() const {
auto A = *this;
auto [pivots, free] = A.template echelonize<reverse>();
matrix sols(size(free), m());
for(size_t j = 0; j < size(pivots); j++) {
base scale = base(1) / A[j][pivots[j]];
for(size_t i = 0; i < size(free); i++) {
sols[i][pivots[j]] = A[j][free[i]] * scale;
}
}
for(size_t i = 0; i < size(free); i++) {
sols[i][free[i]] = -1;
}
return sols;
}
// [solution, basis], transposed
std::optional<std::array<matrix, 2>> solve(matrix t) const {
matrix sols = (*this | t).kernel();
if(sols.n() < t.m() || matrix(sols.submatrix(
std::views::drop(sols.n() - t.m()),
std::views::drop(m())
)) != -eye(t.m())) {
return std::nullopt;
} else {
return std::array{
matrix(sols.submatrix(std::views::drop(sols.n() - t.m()), std::views::take(m()))),
matrix(sols.submatrix(std::views::take(sols.n() - t.m()), std::views::take(m())))
};
}
}
// To be called after a gaussian elimination run
// Sorts rows by pivots and classifies
// variables into pivots and free
auto sort_classify(size_t lim) {
size_t rk = 0;
std::vector<size_t> free, pivots;
for(size_t j = 0; j < lim; j++) {
for(size_t i = rk + 1; i < n() && row(rk)[j] == base(0); i++) {
if(row(i)[j] != base(0)) {
std::swap(row(i), row(rk));
row(rk) = -row(rk);
}
}
if(rk < n() && row(rk)[j] != base(0)) {
pivots.push_back(j);
rk++;
} else {
free.push_back(j);
}
}
return std::array{pivots, free};
}
};
template<typename base_t>
auto operator *(base_t t, matrix<base_t> const& A) {return A * t;}
}
#line 8 "cp-algo/linalg/frobenius.hpp"
namespace cp_algo::linalg {
enum frobenius_mode {blocks, full};
template<frobenius_mode mode = blocks>
auto frobenius_form(auto const& A) {
using matrix = std::decay_t<decltype(A)>;
using vec_t = matrix::vec_t;
using base = typename matrix::base;
using base = matrix::base;
using polyn = math::poly_t<base>;
assert(A.n() == A.m());
size_t n = A.n();
std::vector<polyn> charps;
std::vector<vec_t> basis, basis_init;
while(size(basis) < n) {
size_t start = size(basis);
auto generate_block = [&](auto x) {
while(true) {
vec_t y = x | vec_t::ei(n + 1, size(basis));
for(auto &it: basis) {
y.reduce_by(it);
}
y.normalize();
if(std::ranges::count(y | std::views::take(n), base(0)) == int(n)) {
return polyn(typename polyn::Vector(begin(y) + n, end(y)));
} else {
basis_init.push_back(x);
basis.push_back(y);
x = A.apply(x);
}
}
};
auto full_rec = generate_block(vec_t::random(n));
// Extra trimming to make it block-diagonal (expensive)
if constexpr (mode == full) {
if(full_rec.mod_xk(start) != polyn()) {
auto charp = full_rec.div_xk(start);
auto x = basis_init[start];
auto shift = full_rec / charp;
for(int j = 0; j < shift.deg(); j++) {
x.add_scaled(basis_init[j], shift[j]);
}
basis.resize(start);
basis_init.resize(start);
full_rec = generate_block(x.normalize());
}
}
charps.push_back(full_rec.div_xk(start));
}
// Find transform matrices while we're at it...
if constexpr (mode == full) {
for(size_t i = 0; i < n; i++) {
for(size_t j = i + 1; j < n; j++) {
basis[i].reduce_by(basis[j]);
}
basis[i].normalize();
}
auto T = matrix(basis_init);
auto Tinv = matrix(basis);
std::ignore = Tinv.sort_classify(n);
for(size_t i = 0; i < n; i++) {
Tinv[i] = vec_t(
Tinv[i] | std::views::drop(n) | std::views::take(n)
) * (base(1) / Tinv[i][i]);
}
return std::tuple{T, Tinv, charps};
} else {
return charps;
}
}
template<typename base>
auto with_frobenius(matrix<base> const& A, auto &&callback) {
auto [T, Tinv, charps] = frobenius_form<full>(A);
std::vector<matrix<base>> blocks;
for(auto charp: charps) {
matrix<base> block(charp.deg());
auto xk = callback(charp);
for(size_t i = 0; i < block.n(); i++) {
std::ranges::copy(xk.a, begin(block[i]));
xk = xk.mul_xk(1) % charp;
}
blocks.push_back(block);
}
auto S = matrix<base>::block_diagonal(blocks);
return Tinv * S * T;
}
template<typename base>
auto frobenius_pow(matrix<base> const& A, uint64_t k) {
return with_frobenius(A, [k](auto const& charp) {
return math::poly_t<base>::xk(1).powmod(k, charp);
});
}
};
#line 8 "verify/linalg/characteristic.test.cpp"
using namespace std;
using namespace cp_algo::math;
using namespace cp_algo::linalg;
const int64_t mod = 998244353;
using base = modint<mod>;
using polyn = poly_t<base>;
void solve() {
size_t n;
cin >> n;
matrix<base> A(n);
A.read();
auto blocks = frobenius_form(A);
reduce(begin(blocks), end(blocks), polyn(1), multiplies{}).print();
}
signed main() {
//freopen("input.txt", "r", stdin);
ios::sync_with_stdio(0);
cin.tie(0);
int t = 1;
//cin >> t;
while(t--) {
solve();
}
}
Test cases
| Env | Name | Status | Elapsed | Memory |
|---|---|---|---|---|
| g++ | example_00 |
AC |
5 ms | 6 MB |
| g++ | example_01 |
AC |
4 ms | 6 MB |
| g++ | example_02 |
AC |
5 ms | 6 MB |
| g++ | max_det_zero_random_00 |
AC |
58 ms | 18 MB |
| g++ | max_det_zero_random_01 |
AC |
57 ms | 18 MB |
| g++ | max_det_zero_random_02 |
AC |
56 ms | 18 MB |
| g++ | max_det_zero_random_03 |
AC |
55 ms | 18 MB |
| g++ | max_random_00 |
AC |
54 ms | 18 MB |
| g++ | max_random_01 |
AC |
53 ms | 18 MB |
| g++ | max_random_02 |
AC |
55 ms | 18 MB |
| g++ | max_random_03 |
AC |
56 ms | 18 MB |
| g++ | max_zero_00 |
AC |
60 ms | 16 MB |
| g++ | nontrivial_frobenius_form_00 |
AC |
77 ms | 16 MB |
| g++ | nontrivial_frobenius_form_01 |
AC |
74 ms | 17 MB |
| g++ | nontrivial_frobenius_form_02 |
AC |
68 ms | 17 MB |
| g++ | nontrivial_frobenius_form_03 |
AC |
65 ms | 17 MB |
| g++ | nontrivial_frobenius_form_04 |
AC |
62 ms | 17 MB |
| g++ | nontrivial_frobenius_form_05 |
AC |
58 ms | 18 MB |
| g++ | nontrivial_frobenius_form_06 |
AC |
59 ms | 16 MB |
| g++ | nontrivial_frobenius_form_07 |
AC |
59 ms | 16 MB |
| g++ | nontrivial_frobenius_form_08 |
AC |
60 ms | 16 MB |
| g++ | nontrivial_frobenius_form_09 |
AC |
56 ms | 16 MB |
| g++ | small_multiple_root_00 |
AC |
5 ms | 6 MB |
| g++ | small_multiple_root_01 |
AC |
5 ms | 6 MB |
| g++ | small_random_00 |
AC |
5 ms | 6 MB |
| g++ | small_random_01 |
AC |
4 ms | 6 MB |
| g++ | small_random_02 |
AC |
5 ms | 6 MB |
| g++ | small_random_03 |
AC |
4 ms | 6 MB |