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wedge.cc
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wedge.cc
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// -*- mode: c++; coding: utf-8 -*-
// ra-ra/test - Test generic wedge product with compile-time dimensions.
// (c) Daniel Llorens - 2008-2010, 2015
// This library is free software; you can redistribute it and/or modify it under
// the terms of the GNU Lesser General Public License as published by the Free
// Software Foundation; either version 3 of the License, or (at your option) any
// later version.
#include <iostream>
#include "mpdebug.hh"
#include "ra/test.hh"
using std::cout, std::endl, std::flush, ra::TestRecorder;
using ra::mp::Wedge, ra::mp::hodgex, ra::mp::int_list;
using real = double;
using complex = std::complex<double>;
real const GARBAGE(99);
template <class T, ra::dim_t N> using vec = ra::Small<T, N>;
using real1 = vec<real, 1>;
using real2 = vec<real, 2>;
using real3 = vec<real, 3>;
using real4 = vec<real, 4>;
using real6 = vec<real, 6>;
using complex1 = vec<complex, 1>;
using complex2 = vec<complex, 2>;
using complex3 = vec<complex, 3>;
template <class P, class Plist, int w, int s>
struct FindCombinationTester
{
using finder = ra::mp::FindCombination<P, Plist>;
static_assert(finder::where==w && finder::sign==s, "bad");
static void check() {};
};
template <int N, int O>
void
test_optimized_hodge_aux(TestRecorder & tr)
{
if constexpr (O<=N) {
tr.section(ra::format("hodge() vs hodgex() with N=", N, " O=", O));
static_assert(N>=O, "bad_N_or_bad_O");
using Va = vec<real, Wedge<N, O, N-O>::Na>;
using Vb = vec<real, Wedge<N, O, N-O>::Nb>;
Va u = ra::iota(u.size(), 1);
Vb w(GARBAGE);
hodge<N, O>(u, w);
cout << "-> " << u << " hodge " << w << endl;
// this is the property that u^(*u) = dot(u, u)*vol form.
if (O==1) {
real S = sum(sqr(u));
// since the volume form and the 1-forms are always ordered lexicographically (0 1 2...) vs (0) (1) (2) ...
tr.info("with O=1, S: ", S, " vs wedge(u, w): ", ra::wedge<N, O, N-O>(u, w))
.test_eq(S, ra::wedge<N, O, N-O>(u, w));
} else if (O+1==N) {
real S = sum(sqr(w));
// compare with the case above, this is the sign of the (anti)commutativity of the exterior product.
S *= ra::odd(O*(N-O)) ? -1 : +1;
tr.info("with O=N-1, S: ", S, " vs wedge(u, w): ", ra::wedge<N, N-O, O>(u, w))
.test_eq(S, ra::wedge<N, N-O, O>(u, w));
}
// test that it does the same as hodgex().
Vb x(GARBAGE);
hodgex<N, O>(u, x);
if (2*O==N) {
tr.info("-> ", u, " hodgex ", x).test_eq(ra::wedge<N, O, N-O>(u, w), ra::wedge<N, O, N-O>(u, x));
}
// test basic duality property, **w = (-1)^{o(n-o)} w.
{
Va b(GARBAGE);
hodgex<N, N-O>(x, b);
tr.info("duality test with hodgex() (N ", N, " O ", O, ") -> ", u, " hodge ", x, " hodge(hodge) ", b)
.test_eq((ra::odd(O*(N-O)) ? -1 : +1)*u, b);
}
{
Va a(GARBAGE);
hodge<N, N-O>(w, a);
tr.info("duality test with hodge() (N ", N, " O ", O, ") -> ", u, " hodge ", w, " hodge(hodge) ", a)
.test_eq((ra::odd(O*(N-O)) ? -1 : +1)*u, a);
}
test_optimized_hodge_aux<N, O+1>(tr);
}
}
template <int N>
void
test_optimized_hodge(TestRecorder & tr)
{
static_assert(N>=0, "bad_N");
test_optimized_hodge_aux<N, 0>(tr);
test_optimized_hodge<N-1>(tr);
}
template <>
void
test_optimized_hodge<-1>(TestRecorder & tr)
{
}
template <int D, class R, class A, class B>
R
test_scalar_case(A const & a, B const & b)
{
R r = ra::wedge<D, 0, 0>(a, b);
cout << "[" << D << "/0/0] " << a << " ^ " << b << " -> " << r << endl;
return r;
}
template <int D, int OA, int OB, class R, class A, class B>
R
test_one_one_case(TestRecorder & tr, A const & a, B const & b)
{
R r1(GARBAGE);
Wedge<D, OA, OB>::product(a, b, r1);
cout << "[" << D << "/" << OA << "/" << OB << "] " << a << " ^ " << b << " -> " << r1 << endl;
R r2(ra::wedge<D, OA, OB>(a, b));
cout << "[" << D << "/" << OA << "/" << OB << "] " << a << " ^ " << b << " -> " << r2 << endl;
tr.test_eq(r1, r2);
return r1;
}
template <int D, int OA, int OB, class R, class A, class B>
R
test_one_scalar_case(A const & a, B const & b)
{
R r2(ra::wedge<D, OA, OB>(a, b));
cout << "[" << D << "/" << OA << "/" << OB << "] " << a << " ^ " << b << " -> " << r2 << endl;
return r2;
}
int
main()
{
TestRecorder tr(std::cout);
static_assert(ra::mp::binom(0, 0)==1, "");
if constexpr (sizeof(size_t)>=4) {
tr.test_eq(size_t(265182525), ra::mp::binom(31, 14));
}
if constexpr (sizeof(size_t)>=8) {
tr.test_eq(size_t(629308289804197437), ra::mp::binom(63, 28));
}
tr.section("Testing FindCombination");
{
using la = ra::mp::iota<3>;
using ca = ra::mp::combinations<la, 2>;
FindCombinationTester<int_list<0, 1>, ca, 0, +1>::check();
FindCombinationTester<int_list<1, 0>, ca, 0, -1>::check();
FindCombinationTester<int_list<0, 2>, ca, 1, +1>::check();
FindCombinationTester<int_list<2, 0>, ca, 1, -1>::check();
FindCombinationTester<int_list<1, 2>, ca, 2, +1>::check();
FindCombinationTester<int_list<2, 1>, ca, 2, -1>::check();
FindCombinationTester<int_list<0, 0>, ca, -1, 0>::check();
FindCombinationTester<int_list<1, 1>, ca, -1, 0>::check();
FindCombinationTester<int_list<2, 2>, ca, -1, 0>::check();
FindCombinationTester<int_list<3, 0>, ca, -1, 0>::check();
}
tr.section("Testing AntiCombination");
{
using la = ra::mp::iota<3>;
using ca = ra::mp::combinations<la, 1>;
using cc0 = ra::mp::AntiCombination<ra::mp::ref<ca, 0>, 3>::type;
static_assert(ra::mp::check_idx<cc0, 1, 2>::value, "bad");
using cc1 = ra::mp::AntiCombination<ra::mp::ref<ca, 1>, 3>::type;
static_assert(ra::mp::check_idx<cc1, 2, 0>::value, "bad");
using cc2 = ra::mp::AntiCombination<ra::mp::ref<ca, 2>, 3>::type;
static_assert(ra::mp::check_idx<cc2, 0, 1>::value, "bad");
}
tr.section("Testing ChooseComponents");
{
using c1 = ra::mp::ChooseComponents<3, 1>;
static_assert(ra::mp::len<c1> == 3, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c1, 0>, 0>::value, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c1, 1>, 1>::value, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c1, 2>, 2>::value, "bad");
using c2 = ra::mp::ChooseComponents<3, 2>;
static_assert(ra::mp::len<c2> == 3, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c2, 0>, 1, 2>::value, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c2, 1>, 2, 0>::value, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c2, 2>, 0, 1>::value, "bad");
using c3 = ra::mp::ChooseComponents<3, 3>;
static_assert(ra::mp::len<c3> == 1, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c3, 0>, 0, 1, 2>::value, "bad");
}
{
using c0 = ra::mp::ChooseComponents<1, 0>;
static_assert(ra::mp::len<c0> == 1, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c0, 0>>::value, "bad");
using c1 = ra::mp::ChooseComponents<1, 1>;
static_assert(ra::mp::len<c1> == 1, "bad");
static_assert(ra::mp::check_idx<ra::mp::ref<c1, 0>, 0>::value, "bad");
}
tr.section("Testing Wedge<>::product()");
{
real1 a(1);
real1 b(3);
real1 r(GARBAGE);
Wedge<1, 0, 0>::product(a, b, r);
tr.info("[1/0/0] ", a, " ^ ", b, " -> ", r).test_eq(3, r[0]);
real1 h(GARBAGE);
hodgex<1, 0>(r, h);
tr.info("thodge-star: ", h).test_eq(3, h[0]);
}
tr.section("change order changes sign");
{
real3 a {1, 0, 0};
real3 b {0, 1, 0};
real3 r(GARBAGE);
Wedge<3, 1, 1>::product(a, b, r);
tr.info("[3/1/1] ", a, " ^ ", b, " -> ", r).test_eq(real3{0, 0, +1}, r); // +1, 0, 0 in lex. order.
real3 h(GARBAGE);
hodgex<3, 2>(r, h);
tr.info("hodge-star: ", h).test_eq(real3{0, 0, 1}, h);
}
{
real3 a {0, 1, 0};
real3 b {1, 0, 0};
real3 r(GARBAGE);
Wedge<3, 1, 1>::product(a, b, r);
tr.info("[3/1/1] ", a, " ^ ", b, " -> ", r).test_eq(real3{0, 0, -1}, r); // -1, 0, 0 in lex order.
real3 h(GARBAGE);
hodgex<3, 2>(r, h);
tr.info("hodge-star: ", h).test_eq(real3{0, 0, -1}, h);
}
tr.section("check type promotion");
{
complex3 a {complex(0, 1), 0, 0};
real3 b{0, 1, 0};
complex3 r(GARBAGE);
Wedge<3, 1, 1>::product(a, b, r);
tr.info("[3/1/1] ", a, " ^ ", b, " -> ", r).test_eq(complex3{0, 0, complex(0, 1)}, r); // +j, 0, 0 in lex. o.
complex3 h(GARBAGE);
hodgex<3, 2>(r, h);
tr.info("hodge-star: ", h).test_eq(complex3{0, 0, complex(0, 1)}, h);
}
tr.section("sign change in going from lexicographic -> our peculiar order");
{
real3 a {1, 0, 0};
real3 b {0, 0, 2};
real3 r(GARBAGE);
Wedge<3, 1, 1>::product(a, b, r);
tr.info("[3/1/1] ", a, " ^ ", b, " -> ", r).test_eq(real3{0, -2, 0}, r); // 0, 2, 0 in lex order.
real3 h(GARBAGE);
hodgex<3, 2>(r, h);
tr.info("hodge-star: ", h).test_eq(real3{0, -2, 0}, h);
}
{
real3 a {1, 0, 2};
real3 b {1, 0, 2};
real3 r(GARBAGE);
Wedge<3, 1, 1>::product(a, b, r);
tr.info("[3/1/1] ", a, " ^ ", b, " -> ", r).test_eq(0., r);
real3 h(GARBAGE);
hodgex<3, 2>(r, h);
tr.info("hodge-star: ", h).test_eq(0., h);
}
{
real3 a {0, 1, 0};
real3 b {0, -1, 0}; // 0, 1, 0 in lex order.
real1 r(GARBAGE);
Wedge<3, 1, 2>::product(a, b, r);
tr.info("[3/1/2] ", a, " ^ ", b, " -> ", r).test_eq(-1, r[0]);
real1 h(GARBAGE);
hodgex<3, 3>(r, h);
tr.info("\thodge-star: ", h).test_eq(-1, h[0]);
// this is not forced for hodgex (depends on vec::ChooseComponents<> as used in Wedge<>) so if you change that, change this too.
real3 c;
hodgex<3, 1>(b, c);
tr.info("hodge<3, 1>(", b, "): ", c).test_eq(real3{0, -1, 0}, b);
hodgex<3, 2>(b, c);
tr.info("hodge<3, 2>(", b, "): ", c).test_eq(real3{0, -1, 0}, b);
}
{
real4 a {1, 0, 0, 0};
real4 b {0, 0, 1, 0};
real6 r(GARBAGE);
Wedge<4, 1, 1>::product(a, b, r);
tr.info("[4/1/1] ", a, " ^ ", b, " -> ", r).test_eq(real6{0, 1, 0, 0, 0, 0}, r);
real6 h(GARBAGE);
hodgex<4, 2>(r, h);
tr.info("hodge-star: ", h).test_eq(real6{0, 0, 0, 0, -1, 0}, h);
r = GARBAGE;
hodgex<4, 2>(h, r);
tr.info("hodge-star(hodge-star()): ", r).test_eq(real6{0, 1, 0, 0, 0, 0}, r);
}
{
real4 a {0, 0, 1, 0};
real4 b {1, 0, 0, 0};
real6 r(GARBAGE);
Wedge<4, 1, 1>::product(a, b, r);
tr.info("[4/1/1] ", a, " ^ ", b, " -> ", r).test_eq(real6{0, -1, 0, 0, 0, 0}, r);
}
{
real6 r {1, 0, 0, 0, 0, 0};
real6 h(GARBAGE);
hodgex<4, 2>(r, h);
tr.info("r: ", r, " -> hodge-star: ", h).test_eq(real6{0, 0, 0, 0, 0, 1}, h);
}
tr.section("important as a case where a^b==b^a");
{
real6 a {1, 0, 0, 0, 0, 0};
real6 b {0, 0, 0, 0, 0, 1};
real1 r(GARBAGE);
Wedge<4, 2, 2>::product(a, b, r);
tr.info("[4/2/2] ", a, " ^ ", b, " -> ", r).test_eq(1, r[0]);
Wedge<4, 2, 2>::product(b, a, r);
tr.info("[4/2/2] ", a, " ^ ", b, " -> ", r).test_eq(1, r[0]);
}
tr.section("important as a case where a^a!=0, see DoCarmo1994, Ch. 1 after Prop. 2.");
{
real6 a {1, 0, 0, 0, 0, 1};
real6 b {1, 0, 0, 0, 0, 1};
real1 r(GARBAGE);
Wedge<4, 2, 2>::product(a, b, r);
tr.info("[4/2/2] ", a, " ^ ", b, " -> ", r).test_eq(2, r[0]);
}
tr.section("important as a case where a^b is not dot(a, b) even though O(a)=D-O(b). This happens when O(a)==O(b), i.e. they have the same components");
{
real2 a {1, 0};
real2 b {0, 1};
real1 r(GARBAGE);
Wedge<2, 1, 1>::product(a, b, r);
tr.info("[2/1/1] ", a, " ^ ", b, " -> ", r).test_eq(1, r[0]);
real2 p{1, 2};
real2 q(GARBAGE);
hodgex<2, 1>(p, q);
tr.info("p: ", p, " -> hodge-star: ", q).test_eq(real2{-2, 1}, q);
}
tr.section("test the specializations in cross(), wedge<>()");
{
real2 a {1, 0};
real2 b {0, 1};
real c(cross(a, b));
tr.info("a cross b: ", c).test_eq(1, c);
c = cross(b, a);
tr.test_eq(-1, c);
// accepts expr arguments.
c = cross(a, b+1.);
tr.test_eq(2, c);
}
tr.section("test the cross product some more");
{
real3 x3 {1., 0. ,0.};
real3 y3 {0., 1., 0.};
real3 z3 {0., 0., 1.};
tr.test_eq(z3, cross(x3, y3));
tr.test_eq(x3, cross(y3, z3));
tr.test_eq(y3, cross(z3, x3));
tr.test_eq(-z3, cross(y3, x3));
tr.test_eq(-x3, cross(z3, y3));
tr.test_eq(-y3, cross(x3, z3));
real2 x2 {1., 0.};
real2 y2 {0., 1.};
tr.test_eq(1., cross(x2, y2));
tr.test_eq(-1., cross(y2, x2));
complex2 cy2{0., 1.};
tr.test_eq(complex(1., 0.), cross(x2, cy2));
}
tr.section("verify that wedge<>() returns an expression where appropriate");
{
real3 u {1., 2., 3.};
real3 v {3., 2., 1.};
tr.test_eq(10., ra::wedge<3, 1, 2>(u, v));
tr.test_eq(cross(u, v), ra::wedge<3, 1, 1>(u, v));
tr.test_eq(10., ra::wedge<3, 1, 2>(u, v));
}
tr.section("verify that we are allowed to choose our return type to wedge<>(a, b, r)");
{
real a(GARBAGE);
real1 b(GARBAGE);
ra::wedge<2, 1, 1>(real2 {1, 0}, real2 {0, 1}, a);
ra::wedge<2, 1, 1>(real2 {1, 0}, real2 {0, 1}, b);
tr.test_eq(1, a);
tr.test_eq(1, b[0]);
}
tr.section("check the optimization of hodgex() that relies on a complementary order of bases in the 2*O>D forms");
{
test_optimized_hodge<6>(tr);
}
tr.section("Test scalar arg cases");
{
tr.test_eq(6, test_scalar_case<0, real>(real1(2), real(3)));
tr.test_eq(6, test_scalar_case<1, real>(real1(2), real(3)));
tr.test_eq(6, test_scalar_case<0, real>(real(2), real(3)));
tr.test_eq(6, test_scalar_case<1, real>(real(2), real(3)));
tr.test_eq(6, test_scalar_case<0, real>(real(2), real1(3)));
tr.test_eq(6, test_scalar_case<1, real>(real(2), real1(3)));
tr.test_eq(6, test_scalar_case<0, real>(real1(2), real1(3)));
tr.test_eq(6, test_scalar_case<1, real>(real1(2), real1(3)));
tr.test_eq(6, test_scalar_case<0, real1>(real(2), real(3)));
tr.test_eq(6, test_scalar_case<1, real1>(real(2), real(3)));
tr.test_eq(6, test_scalar_case<0, real1>(real1(2), real(3)));
tr.test_eq(6, test_scalar_case<1, real1>(real1(2), real(3)));
tr.test_eq(6, test_scalar_case<0, real1>(real(2), real1(3)));
tr.test_eq(6, test_scalar_case<1, real1>(real(2), real1(3)));
tr.test_eq(6, test_scalar_case<0, real1>(real1(2), real1(3)));
tr.test_eq(6, test_scalar_case<1, real1>(real1(2), real1(3)));
}
tr.section("Test scalar x nonscalar arg cases.");
{
tr.test_eq(real2{6, 10}, test_one_one_case<2, 0, 1, real2>(tr, real1(2), real2{3, 5}));
tr.test_eq(real2{6, 10}, test_one_one_case<2, 1, 0, real2>(tr, real2{3, 5}, real1(2)));
tr.test_eq(real3{2, 6, 10}, test_one_one_case<3, 0, 1, real3>(tr, real1(2), real3{1, 3, 5}));
tr.test_eq(real3{2, 6, 10}, test_one_one_case<3, 1, 0, real3>(tr, real3{1, 3, 5}, real1(2)));
}
{
tr.test_eq(real2{6, 10}, test_one_scalar_case<2, 0, 1, real2>(real1(2), real2{3, 5}));
tr.test_eq(real2{6, 10}, test_one_scalar_case<2, 1, 0, real2>(real2{3, 5}, real1(2)));
tr.test_eq(real3{2, 6, 10}, test_one_scalar_case<3, 0, 1, real3>(real1(2), real3{1, 3, 5}));
tr.test_eq(real3{2, 6, 10}, test_one_scalar_case<3, 1, 0, real3>(real3{1, 3, 5}, real1(2)));
}
tr.section("Test scalar x ~scalar arg cases.");
{
tr.test_eq(6., ra::wedge<1, 0, 1>(3., complex1(2.)));
}
return tr.summary();
}