test_matpack.cc

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/* Copyright (C) 2001-2012 Stefan Buehler <sbuehler@ltu.se>
 
   This program is free software; you can redistribute it and/or modify it
   under the terms of the GNU General Public License as published by the
   Free Software Foundation; either version 2, or (at your option) any
   later version.
 
   This program is distributed in the hope that it will be useful,
   but WITHOUT ANY WARRANTY; without even the implied warranty of
   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
   GNU General Public License for more details.
 
   You should have received a copy of the GNU General Public License
   along with this program; if not, write to the Free Software
   Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307,
   USA. */
 
#include <algorithm>
#include <cmath>
#include <cstdlib>
#include <iostream>
#include "array.h"
#include "describe.h"
#include "exceptions.h"
#include "logic.h"
#include "math_funcs.h"
#include "matpackII.h"
#include "matpackVII.h"
#include "mystring.h"
#include "rational.h"
#include "test_utils.h"
#include "time.h"
#include "wigner_functions.h"
 
using std::cout;
using std::endl;
using std::runtime_error;
 
Numeric by_reference(const Numeric& x) { return x + 1; }
 
Numeric by_value(Numeric x) { return x + 1; }
 
void fill_with_junk(VectorView x) { x = 999; }
 
void fill_with_junk(MatrixView x) { x = 888; }
 
int test1() {
  Vector v(20);
 
  cout << "v.nelem() = " << v.nelem() << "\n";
 
  for (Index i = 0; i < v.nelem(); ++i) v[i] = (Numeric)i;
 
  cout << "v.begin() = " << *v.begin() << "\n";
 
  cout << "v = \n" << v << "\n";
 
  fill_with_junk(v[Range(1, 8, 2)][Range(2, joker)]);
  //  fill_with_junk(v);
 
  Vector v2 = v[Range(2, 4)];
 
  cout << "v2 = \n" << v2 << "\n";
 
  for (Index i = 0; i < 1000; ++i) {
    Vector v3(1000);
    v3 = (Numeric)i;
  }
 
  v2[Range(joker)] = 88;
 
  v2[Range(0, 2)] = 77;
 
  cout << "v = \n" << v << "\n";
  cout << "v2 = \n" << v2 << "\n";
  cout << "v2.nelem() = \n" << v2.nelem() << "\n";
 
  Vector v3;
  v3.resize(v2.nelem());
  v3 = v2;
 
  cout << "\nv3 = \n" << v3 << "\n";
  fill_with_junk((VectorView)v2);
  cout << "\nv3 after junking v2 = \n" << v3 << "\n";
  v3 *= 2;
  cout << "\nv3 after *2 = \n" << v3 << "\n";
 
  Matrix M(10, 15);
  {
    Numeric n = 0;
    for (Index i = 0; i < M.nrows(); ++i)
      for (Index j = 0; j < M.ncols(); ++j) M(i, j) = ++n;
  }
 
  cout << "\nM =\n" << M << "\n";
 
  cout << "\nM(Range(2,4),Range(2,4)) =\n"
       << M(Range(2, 4), Range(2, 4)) << "\n";
 
  cout << "\nM(Range(2,4),Range(2,4))(Range(1,2),Range(1,2)) =\n"
       << M(Range(2, 4), Range(2, 4))(Range(1, 2), Range(1, 2)) << "\n";
 
  cout << "\nM(1,Range(joker)) =\n" << M(1, Range(joker)) << "\n";
 
  cout << "\nFilling M(1,Range(1,2)) with junk.\n";
  fill_with_junk(M(1, Range(1, 2)));
 
  cout << "\nM(Range(0,4),Range(0,4)) =\n"
       << M(Range(0, 4), Range(0, 4)) << "\n";
 
  cout << "\nFilling M(Range(4,2,2),Range(6,3)) with junk.\n";
 
  MatrixView s = M(Range(4, 2, 2), Range(6, 3));
  fill_with_junk(s);
 
  cout << "\nM =\n" << M << "\n";
 
  const Matrix C = M;
 
  cout << "\nC(Range(3,4,2),Range(2,3,3)) =\n"
       << C(Range(3, 4, 2), Range(2, 3, 3)) << "\n";
 
  cout << "\nC(Range(3,4,2),Range(2,3,3)).transpose() =\n"
       << transpose(C(Range(3, 4, 2), Range(2, 3, 3))) << "\n";
 
  return 0;
}
 
void test2() {
  Vector v(50000000);
 
  cout << "v.nelem() = " << v.nelem() << "\n";
 
  cout << "Filling\n";
  //   for (Index i=0; i<v.nelem(); ++i )
  //     v[i] = sqrt(i);
  v = 1.;
  cout << "Done\n";
}
 
void test4() {
  Vector a(10);
  Vector b(a.nelem());
 
  for (Index i = 0; i < a.nelem(); ++i) {
    a[i] = (Numeric)(i + 1);
    b[i] = (Numeric)(a.nelem() - i);
  }
 
  cout << "a = \n" << a << "\n";
  cout << "b = \n" << b << "\n";
  cout << "a*b \n= " << a * b << "\n";
 
  Matrix A(11, 6);
  Matrix B(10, 20);
  Matrix C(20, 5);
 
  B = 2;
  C = 3;
  mult(A(Range(1, joker), Range(1, joker)), B, C);
 
  //  cout << "\nB =\n" << B << "\n";
  //  cout << "\nC =\n" << C << "\n";
  cout << "\nB*C =\n" << A << "\n";
}
 
void test5() {
  Vector a(10);
  Vector b(20);
  Matrix M(10, 20);
 
  // Fill b and M with a constant number:
  b = 1;
  M = 2;
 
  cout << "b = \n" << b << "\n";
  cout << "M =\n" << M << "\n";
 
  mult(a, M, b);  // a = M*b
  cout << "\na = M*b = \n" << a << "\n";
 
  mult(transpose((MatrixView)b), transpose((MatrixView)a), M);  // b^t = a^t * M
  cout << "\nb^t = a^t * M = \n" << transpose((MatrixView)b) << "\n";
}
 
void test6() {
  Index n = 5000;
  Vector x(1, n, 1), y(n);
  Matrix M(n, n);
  M = 1;
  //  cout << "x = \n" << x << "\n";
 
  cout << "Transforming.\n";
  //  transform(x,sin,x);
  // transform(transpose(y),sin,transpose(x));
  //  cout << "sin(x) =\n" << y << "\n";
  for (Index i = 0; i < 1000; ++i) {
    //      mult(y,M,x);
    transform(y, sin, static_cast<MatrixView>(x));
    x += 1;
  }
  //  cout << "y =\n" << y << "\n";
 
  cout << "Done.\n";
}
 
void test7() {
  Vector x(1, 20000000, 1);
  Vector y(x.nelem());
  transform(y, sin, x);
  cout << "min(sin(x)), max(sin(x)) = " << min(y) << ", " << max(y) << "\n";
}
 
void test8() {
  Vector x(80000000);
  for (Index i = 0; i < x.nelem(); ++i) x[i] = (Numeric)i;
  cout << "Done."
       << "\n";
}
 
void test9() {
  // Initialization of Matrix with view of other Matrix:
  Matrix A(4, 8);
  Matrix B(A(Range(joker), Range(0, 3)));
  cout << "B = " << B << "\n";
}
 
void test10() {
  // Initialization of Matrix with a vector (giving a 1 column Matrix).
 
  // At the moment doing this with a non-const Vector will result in a
  // warning message.
  Vector v(1, 8, 1);
  Matrix M((const Vector)v);
  cout << "M = " << M << "\n";
}
 
void test11() {
  // Assignment between Vector and Matrix:
 
  // At the moment doing this with a non-const Vector will result in a
  // warning message.
  Vector v(1, 8, 1);
  Matrix M(v.nelem(), 1);
  M = v;
  cout << "M = " << M << "\n";
}
 
void test12() {
  // Copying of Arrays
 
  Array<String> sa(3);
  sa[0] = "It's ";
  sa[1] = "a ";
  sa[2] = "test.";
 
  Array<String> sb(sa), sc(sa.nelem());
 
  cout << "sb = \n" << sb << "\n";
 
  sc = sa;
 
  cout << "sc = \n" << sc << "\n";
}
 
void test13() {
  // Mix vector and one-column matrix in += operator.
  const Vector v(1, 8, 1);  // The const is necessary here to
                            // avoid compiler warnings about
                            // different conversion paths.
  Matrix M(v);
  M += v;
  cout << "M = \n" << M << "\n";
}
 
void test14() {
  // Test explicit Array constructors.
  Array<String> a{"Test"};
  Array<Index> b{1, 2};
  Array<Numeric> c{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0};
  cout << "a = \n" << a << "\n";
  cout << "b = \n" << b << "\n";
  cout << "c = \n" << c << "\n";
}
 
void test15() {
  // Test String.
  String a = "Nur ein Test.";
  cout << "a = " << a << "\n";
  String b(a, 5, -1);
  cout << "b = " << b << "\n";
}
 
/*void test16()
{
  // Test interaction between Array<Numeric> and Vector.
  Vector a;
  Array<Numeric> b;
  b.push_back(1);
  b.push_back(2);
  b.push_back(3);
  a.resize(b.nelem());
  a = b;
  cout << "b =\n" << b << "\n";
  cout << "a =\n" << a << "\n";
}*/
 
void test17() {
  // Test Sum.
  Vector a(1, 10, 1);
  cout << "a.sum() = " << a.sum() << "\n";
}
 
void test18() {
  // Test elementvise square of a vector:
  Vector a(1, 10, 1);
  a *= a;
  cout << "a *= a =\n" << a << "\n";
}
 
void test19() {
  // There exists no explicit filling constructor of the form
  // Vector a(3,1.7).
  // But you can use the more general filling constructor with 3 arguments.
 
  Vector a(1, 10, 1);
  Vector b(5.3, 10, 0);
  cout << "a =\n" << a << "\n";
  cout << "b =\n" << b << "\n";
}
 
void test20() {
  // Test initialization list constructor:
  Vector a{1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
  cout << "a =\n" << a << "\n";
}
 
void test21() {
  Numeric s = 0;
  // Test speed of call by reference:
  cout << "By reference:\n";
  for (Index i = 0; i < (Index)1e8; ++i) {
    s += by_reference(s);
    s -= by_reference(s);
  }
  cout << "s = " << s << "\n";
}
 
void test22() {
  Numeric s = 0;
  // Test speed of call by value:
  cout << "By value:\n";
  for (Index i = 0; i < (Index)1e8; ++i) {
    s += by_value(s);
    s -= by_value(s);
  }
  cout << "s = " << s << "\n";
}
 
void test23() {
  // Test constructors that fills with constant:
  Vector a(10, 3.5);
  cout << "a =\n" << a << "\n";
  Matrix b(10, 10, 4.5);
  cout << "b =\n" << b << "\n";
}
 
void test24() {
  // Try element-vise multiplication of Matrix and Vector:
  Matrix a(5, 1, 2.5);
  Vector b(1, 5, 1);
  a *= b;
  cout << "a*=b =\n" << a << "\n";
  a /= b;
  cout << "a/=b =\n" << a << "\n";
  a += b;
  cout << "a+=b =\n" << a << "\n";
  a -= b;
  cout << "a-=b =\n" << a << "\n";
}
 
void test25() {
  // Test min and max for Array:
  Array<Index> a{1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1};
  cout << "min/max of a = " << min(a) << "/" << max(a) << "\n";
}
 
void test26() {
  cout << "Test filling constructor for Array:\n";
  Array<String> a(4, "Hello");
  cout << "a =\n" << a << "\n";
}
 
void test27() {
  cout << "Test Arrays of Vectors:\n";
  Array<Vector> a;
  a.push_back({1.0, 2.0});
  a.push_back(Vector(1.0, 10, 1.0));
  cout << "a =\n" << a << "\n";
}
 
void test28() {
  cout << "Test default constructor for Matrix:\n";
  Matrix a;
  Matrix b(a);
  cout << "b =\n" << b << "\n";
}
 
void test29() {
  cout << "Test Arrays of Matrix:\n";
  ArrayOfMatrix a;
  Matrix b;
 
  b.resize(2, 2);
  b(0, 0) = 1;
  b(0, 1) = 2;
  b(1, 0) = 3;
  b(1, 1) = 4;
  a.push_back(b);
  b *= 2;
  a.push_back(b);
 
  a[0].resize(2, 3);
  a[0] = 4;
 
  a.resize(3);
  a[2].resize(4, 5);
  a[2] = 5;
 
  cout << "a =\n" << a << "\n";
}
 
void test30() {
  cout << "Test Matrices of size 0:\n";
  Matrix a(0, 0);
  //  cout << "a(0,0) =\n" << a(0,0) << "\n";
  a.resize(2, 2);
  a = 1;
  cout << "a =\n" << a << "\n";
 
  Matrix b(3, 0);
  //  cout << "b(0,0) =\n" << b(0,0) << "\n";
  b.resize(b.nrows(), b.ncols() + 3);
  b = 2;
  cout << "b =\n" << b << "\n";
 
  Matrix c(0, 3);
  //  cout << "c(0,0) =\n" << c(0,0) << "\n";
  c.resize(c.nrows() + 3, c.ncols());
  c = 3;
  cout << "c =\n" << c << "\n";
}
 
void test31() {
  cout << "Test Tensor3:\n";
 
  Tensor3 a(2, 3, 4, 1.0);
 
  Index fill = 0;
 
  // Fill with some numbers
  for (Index i = 0; i < a.npages(); ++i)
    for (Index j = 0; j < a.nrows(); ++j)
      for (Index k = 0; k < a.ncols(); ++k) a(i, j, k) = (Numeric)(++fill);
 
  cout << "a =\n" << a << "\n";
 
  cout << "Taking out first row of first page:\n"
       << a(0, 0, Range(joker)) << "\n";
 
  cout << "Taking out last column of second page:\n"
       << a(1, Range(joker), a.ncols() - 1) << "\n";
 
  cout << "Taking out the first letter on every page:\n"
       << a(Range(joker), 0, 0) << "\n";
 
  cout << "Taking out first page:\n"
       << a(0, Range(joker), Range(joker)) << "\n";
 
  cout << "Taking out last row of all pages:\n"
       << a(Range(joker), a.nrows() - 1, Range(joker)) << "\n";
 
  cout << "Taking out second column of all pages:\n"
       << a(Range(joker), Range(joker), 1) << "\n";
 
  a *= 2;
 
  cout << "After element-vise multiplication with 2:\n" << a << "\n";
 
  transform(a, sqrt, a);
 
  cout << "After taking the square-root:\n" << a << "\n";
 
  Index s = 200;
  cout << "Let's allocate a large tensor, "
       << (Numeric)(s * s * s * 8) / 1024. / 1024. << " MB...\n";
 
  a.resize(s, s, s);
 
  cout << "Set it to 1...\n";
 
  a = 1;
 
  cout << "a(900,900,900) = " << a(90, 90, 90) << "\n";
 
  fill = 0;
 
  cout << "Fill with running numbers, using for loops...\n";
  for (Index i = 0; i < a.npages(); ++i)
    for (Index j = 0; j < a.nrows(); ++j)
      for (Index k = 0; k < a.ncols(); ++k) a(i, j, k) = (Numeric)(++fill);
 
  cout << "Max(a) = ...\n";
 
  cout << max(a) << "\n";
}
 
void test32() {
  cout << "Test von X = A*X:\n";
  Matrix X(3, 3), A(3, 3), B(3, 3);
 
  for (Index j = 0; j < A.nrows(); ++j)
    for (Index k = 0; k < A.ncols(); ++k) {
      X(j, k) = 1;
      A(j, k) = (Numeric)(j + k);
    }
  cout << "A:\n" << A << "\n";
  cout << "X:\n" << X << "\n";
 
  mult(B, A, X);
  cout << "B = A*X:\n" << B << "\n";
  mult(X, A, X);
  cout << "X = A*X:\n" << X << "\n";
 
  cout << "This is not the same, and should not be, because you\n"
       << "are not allowed to use the same matrix as input and output\n"
       << "for mult!\n";
}
 
void test33() {
  cout << "Making things look bigger than they are...\n";
 
  {
    cout << "1. Make a scalar look like a vector:\n";
    Numeric a = 3.1415;  // Just any number here.
    VectorView av(a);
    cout << "a, viewed as a vector: " << av << "\n";
    cout << "Describe a: " << describe(a) << "\n";
    av[0] += 1;
    cout << "a, after the first element\n"
         << "of the vector has been increased by 1: " << a << "\n";
  }
 
  {
    cout
        << "\n2. Make a vector look like a matrix:\n"
        << "This is an exception, because the new dimension is added at the end.\n";
    Vector a{1, 2, 3, 4, 5};
    MatrixView am = a;
    cout << "a, viewed as a matrix:\n" << am << "\n";
    cout << "Trasnpose view:\n" << transpose(am) << "\n";
    cout << "Describe transpose(am): " << describe(transpose(am)) << "\n";
 
    cout << "\n3. Make a matrix look like a tensor3:\n";
    Tensor3View at3 = am;
    cout << "at3 = \n" << at3 << "\n";
    cout << "Describe at3: " << describe(at3) << "\n";
    at3(0, 2, 0) += 1;
    cout << "a after Increasing element at3(0,2,0) by 1: \n" << a << "\n\n";
 
    Tensor4View at4 = at3;
    cout << "at4 = \n" << at4 << "\n";
    cout << "Describe at4: " << describe(at4) << "\n";
 
    Tensor5View at5 = at4;
    cout << "at5 = \n" << at5 << "\n";
    cout << "Describe at5: " << describe(at5) << "\n";
 
    Tensor6View at6 = at5;
    cout << "at6 = \n" << at6 << "\n";
    cout << "Describe at6: " << describe(at6) << "\n";
 
    Tensor7View at7 = at6;
    cout << "at7 = \n" << at7 << "\n";
    cout << "Describe at7: " << describe(at7) << "\n";
 
    at7(0, 0, 0, 0, 0, 2, 0) -= 1;
 
    cout << "After subtracting 1 from at7(0,0,0,0,0,2,0)\n"
         << "a = " << a << "\n";
 
    cout << "\nAll in one go:\n";
    Numeric b = 3.1415;  // Just any number here.
    Tensor7View bt7 = Tensor6View(
        Tensor5View(Tensor4View(Tensor3View(MatrixView(VectorView(b))))));
    cout << "bt7:\n" << bt7 << "\n";
    cout << "Describe bt7: " << describe(bt7) << "\n";
  }
}
 
void junk4(Tensor4View a) { cout << "Describe a: " << describe(a) << "\n"; }
 
void junk2(ConstVectorView a) { cout << "Describe a: " << describe(a) << "\n"; }
 
void test34() {
  cout << "Test, if dimension expansion works implicitly.\n";
 
  Tensor3 t3(2, 3, 4);
  junk4(t3);
 
  Numeric x;
  junk2(ConstVectorView(x));
}
 
void test35() {
  cout << "Test the new copy semantics.\n";
  Vector a(1, 4, 1);
  Vector b;
 
  b = a;
  cout << "b = " << b << "\n";
 
  Vector aa(1, 5, 1);
  ConstVectorView c = aa;
  b = c;
  cout << "b = " << b << "\n";
 
  Vector aaa(1, 6, 1);
  VectorView d = aaa;
  b = d;
  cout << "b = " << b << "\n";
}
 
void test36() {
  cout << "Test using naked joker on Vector.\n";
  Vector a(1, 4, 1);
  VectorView b = a[joker];
  cout << "a = " << a << "\n";
  cout << "b = " << b << "\n";
}
 
void test37(const Index& i) {
  Vector v1(5e-15, 10, 0.42e-15 / 11);
  Vector v2 = v1;
  //  Index i = 10000000;
 
  v1 /= (Numeric)i;
  v2 /= (Numeric)i;
  cout.precision(12);
  //  cout.setf(ios_base::scientific,ios_base::floatfield);
  v1 *= v1;
  v2 *= v2;
  cout << v1 << endl;
  cout << v2 << endl;
}
 
void test38() {
  Vector v(5, 0.);
  Numeric* const a = v.get_c_array();
 
  a[4] = 5.;
 
  cout << v << endl;
  cout << endl << "========================" << endl << endl;
 
  Matrix m(5, 5, 0.);
  Numeric* const b = m.get_c_array();
 
  b[4] = 5.;
 
  cout << m << endl;
  cout << endl << "========================" << endl << endl;
 
  Tensor3 t3(5, 6, 7, 0.);
  Numeric* const c = t3.get_c_array();
 
  c[6] = 5.;
 
  cout << t3 << endl;
}
 
void test39() {
  Vector v1(1, 5, 1), v2(5);
 
  v2 = v1 * 2;
  // Unfortunately, this thing compiles, but at least it gives an
  // assertion failure at runtime. I think what happens is that it
  // implicitly creates a one element vector out of the "2", then
  // tries to do a scalar product with v1.
 
  cout << v2 << endl;
}
 
void test40() {
  Vector v(100);
 
  v = 5;
 
  cout << v << endl;
}
 
void test41() {
  const Vector v1(10, 1);
 
  ConstVectorView vv = v1[Range(0, 5)];
 
  cout << "Vector:     " << v1 << endl;
  cout << "VectorView: " << vv << endl;
 
  try {
    vv = 2;
  } catch (const std::runtime_error& e) {
    std::cerr << e.what() << endl;
    exit(EXIT_FAILURE);
  }
 
  cout << "VectorView: " << vv << endl;
  cout << "Vector:     " << v1 << endl;
}
 
// Test behaviour of VectorView::operator MatrixView, for which I have fixed
// a bug today.
// SAB 2013-01-18
void test42() {
  cout << "test42\n";
  Vector x{1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
  cout << "x: " << x << endl;
 
  VectorView y = x[Range(2, 4, 2)];
  cout << "y: " << y << endl;
 
  ConstMatrixView z((MatrixView)y);
  cout << "z:\n" << z << endl;
 
  cout << "Every other z:\n" << z(Range(1, 2, 2), joker) << endl;
 
  // Try write access also:
  MatrixView zz(y);
  zz(Range(1, 2, 2), joker) = 0;
  cout << "zz:\n" << zz << endl;
  cout << "New x: " << x << endl;
}
 
 
// Test function for internal use, must return 2 as last n2r
constexpr Rational test_numeric2rational(const Index i, const Index maxi, const Rational r=0, const Rational n2r=0) {
  if (i > maxi)
    return n2r;
  else {
    return 
    (r == n2r) ?
    test_numeric2rational(i+1, maxi, Rational(maxi + i, maxi), numeric2rational(1 + Numeric(i)/Numeric(maxi), 12)) :
    throw std::logic_error("Fail to initialize");;
  }
}
 
 
void test43() {
  // Simple construction compile-time test
  constexpr Rational r=3_2;  // should be 3/2
  static_assert(r.Nom() == 3, "Rational fail to initialize properly");
  static_assert(r.Denom() == 2, "Rational fail to initialize properly");
 
  // Simple expression compile-time test
  constexpr Rational r2 = 1 + r;  // should be 5/2
  static_assert(r2.Nom() == 5, "Rational fail to initialize properly");
  static_assert(r2.Denom() == 2, "Rational fail to initialize properly");
 
  // Div-zero by index generates an undefined rational:  0/0
  constexpr Rational r3 = r / 0;  // should be undefined
  static_assert(r3.Nom() == 0, "Rational fail to initialize properly");
  static_assert(r3.Denom() == 0, "Rational fail to initialize properly");
  static_assert(r3 not_eq r3, "Undefined rational does not equal self");
  static_assert(r3.isUndefined(), "Rational fail to initialize properly");
  
  // Mul-zero gives 0/1
  constexpr Rational r4 = r * 0;  // should be undefined
  static_assert(r4.Nom() == 0, "Rational fail to initialize properly");
  static_assert(r4.Denom() == 1, "Rational fail to initialize properly");
 
  // Complicated expression compile-time test
  constexpr Rational r5 = (Rational(1, 9) * r) % 3;  // should be 1/6
  static_assert(r5.Nom() == 1, "Rational fail to initialize properly");
  static_assert(r5.Denom() == 6, "Rational fail to initialize properly");
 
  // The simplify operation simplifies expressions
  constexpr Rational r6 = 10 % r;  // should be 1
  static_assert(r6.Nom() == 1, "Rational fail to initialize properly");
  static_assert(r6.Denom() == 1, "Rational fail to initialize properly");
  static_assert(r6.toInt() == 1, "Rational fail to initialize properly");
  static_assert(r6.toIndex() == 1, "Rational fail to initialize properly");
  static_assert(r6.toNumeric() == 1e0, "Rational fail to initialize properly");
  
  constexpr Index rattest=1<<8;
  constexpr Rational r7 = test_numeric2rational(0, rattest);
  static_assert(r7 == 2, "Rational fail to initialize properly");
}
 
void test44() {
#define docheck(fn, val, expect)                                           \
  cout << #fn << "(" << val << ") = " << fn(x) << " (expected " << #expect \
       << ")" << std::endl;
 
  Vector x{1, 2, 3};
  docheck(is_increasing, x, 1) docheck(is_decreasing, x, 0)
      docheck(is_sorted, x, 1)
 
          x = {3, 2, 1};
  docheck(is_increasing, x, 0) docheck(is_decreasing, x, 1)
      docheck(is_sorted, x, 0)
 
          x = {1, 2, 2};
  docheck(is_increasing, x, 0) docheck(is_decreasing, x, 0)
      docheck(is_sorted, x, 1)
 
          x = {2, 2, 1};
  docheck(is_increasing, x, 0) docheck(is_decreasing, x, 0)
      docheck(is_sorted, x, 0)
 
          x = {1, NAN, 2};
  docheck(is_increasing, x, 0) docheck(is_decreasing, x, 0)
      docheck(is_sorted, x, 0)
 
          x = {2, NAN, 1};
  docheck(is_increasing, x, 0) docheck(is_decreasing, x, 0)
      docheck(is_sorted, x, 0)
 
          x = {NAN, NAN, NAN};
  docheck(is_increasing, x, 0) docheck(is_decreasing, x, 0)
      docheck(is_sorted, x, 0)
 
#undef docheck
}
 
void test46() {
  Vector v(5, 0.);
  nlinspace(v, 1, 10, 10);
  VectorView v1 = v;
  auto compare_func = [](Numeric n) { return n != 0; };
  cout << std::any_of(v1.begin(), v1.end(), compare_func) << endl;
  v1 = 0.;
  cout << std::any_of(v1.begin(), v1.end(), compare_func) << endl;
}
 
 
bool test_diagonal(Index ntests) {
  Rand<Index> rand(1, 100);
 
  Index n_diag;
 
  bool pass = true;
 
  for (Index i = 0; i < ntests; i++) {
    Matrix A(rand(), rand());
    n_diag = std::min(A.ncols(), A.nrows());
 
    ConstMatrixView AT = transpose(A);
    ConstMatrixView B = A;
    if (n_diag > 3) {
      B = A(Range(1, Joker(), 2), Range(1, Joker(), 2));
    }
 
    ConstVectorView v(A.diagonal());
    ConstVectorView vt(AT.diagonal());
    ConstVectorView vb(B.diagonal());
 
    random_fill_matrix(A, 10, true);
 
    for (Index j = 0; j < n_diag; j++) {
      pass = pass && (v[j] == A(j, j));
      pass = pass && (vt[j] == AT(j, j));
 
      if (j < vb.nelem()) pass = pass && (vb[j] == B(j, j));
    }
    cout << endl;
  }
 
  if (pass)
    cout << "test diagonal: Passed all tests." << endl;
  else
    cout << "test diagonal: Failed." << endl;
 
  return pass;
}
 
Numeric matrix_vector_mult(Index m, Index n, Index ntests, bool verbose) {
  Numeric max_err = 0;
  Matrix A(m, n);
  Vector x(n), xt, x_ref(n), y(m), y_ref(m);
 
  Rand<Index> random_row_stride(1, n / 4);
  Rand<Index> random_column_stride(1, m / 4);
 
  for (Index i = 0; i < ntests; i++) {
    // A * x
 
    random_fill_matrix(A, 1000, false);
    random_fill_vector(x, 1000, false);
 
    mult(y, A, x);
    mult_general(y_ref, A, x);
 
    Numeric err_mul = get_maximum_error(y, y_ref, true);
    if (err_mul > max_err) max_err = err_mul;
 
    if (verbose) {
      cout << "\t A * x: max. rel. error = " << err_mul << endl;
    }
 
    // A^T  * y
 
    mult(x, transpose(A), y);
    mult_general(x_ref, transpose(A), y);
 
    err_mul = get_maximum_error(x, x_ref, true);
    if (err_mul > max_err) max_err = err_mul;
 
    if (verbose) {
      cout << "\t A^T * x: max. rel. error = " << err_mul << endl;
    }
 
    // Random stride
    Index column_stride(random_column_stride()),
        row_stride(random_row_stride());
 
    Index m_sub, n_sub;
    m_sub = (m - 1) / row_stride + 1;
    n_sub = (n - 1) / column_stride + 1;
 
    mult(y[Range(0, joker, row_stride)],
         A(Range(0, m_sub), Range(0, n_sub)),
         x[Range(0, joker, column_stride)]);
    mult_general(y_ref[Range(0, joker, row_stride)],
                 A(Range(0, m_sub), Range(0, n_sub)),
                 x[Range(0, joker, column_stride)]);
 
    err_mul = get_maximum_error(y[Range(0, joker, row_stride)],
                                y_ref[Range(0, joker, row_stride)],
                                true);
    if (err_mul > max_err) max_err = err_mul;
 
    if (verbose) {
      cout << "\t Random stride: max. rel. error = " << err_mul << endl << endl;
    }
 
    // Random offset
    if ((m > 1) && (n > 1)) {
      Index y_offset = rand() % (m - 1);
      Index x_offset = rand() % (n - 1);
 
      m_sub = m - y_offset - 1;
      n_sub = n - x_offset - 1;
 
      mult(y[Range(y_offset, m_sub)],
           A(Range(y_offset, m_sub), Range(x_offset, n_sub)),
           x[Range(x_offset, n_sub)]);
      mult_general(y_ref[Range(y_offset, m_sub)],
                   A(Range(y_offset, m_sub), Range(x_offset, n_sub)),
                   x[Range(x_offset, n_sub)]);
 
      err_mul = get_maximum_error(
          y[Range(y_offset, m_sub)], y_ref[Range(y_offset, m_sub)], true);
      if (err_mul > max_err) max_err = err_mul;
 
      if (verbose) {
        cout << "\t Random offset: max. rel. error = " << err_mul << endl
             << endl;
      }
    }
  }
 
  return max_err;
}
 
Numeric test_matrix_vector_multiplication(bool verbose) {
  Numeric err, max_err;
 
  if (verbose)
    cout << "Matrix-Vector Multiplication: n = m = 100, ntests = 100" << endl;
 
  max_err = matrix_vector_mult(100, 100, 100, verbose);
  if (verbose) {
    cout << endl;
    cout << "Matrix-Vector Multiplication: n = 100, m = 20, ntests = 100"
         << endl;
  }
 
  err = matrix_vector_mult(100, 20, 100, verbose);
  if (err > max_err) max_err = err;
  if (verbose) {
    cout << endl;
    cout << "Matrix-Vector Multiplication: n = 20, m = 100, ntests = 100"
         << endl;
  }
 
  err = matrix_vector_mult(20, 100, 100, verbose);
  if (err > max_err) max_err = err;
  if (verbose) {
    if (max_err < 1e-9)
      cout << endl << "Matrix Vector Multiplication: PASSED" << endl;
    else
      cout << endl << "Matrix Vector Multiplication: FAILED" << endl;
  }
 
  err = matrix_vector_mult(100, 1, 100, verbose);
  if (err > max_err) max_err = err;
  if (verbose) {
    cout << endl;
    cout << "Matrix-Vector Multiplication: n = 20, m = 100, ntests = 100"
         << endl;
  }
 
  err = matrix_vector_mult(1, 100, 100, verbose);
  if (err > max_err) max_err = err;
  if (verbose) {
    cout << endl;
    cout << "Matrix-Vector Multiplication: n = 20, m = 100, ntests = 100"
         << endl;
  }
 
  if (verbose) {
    if (max_err < 1e-9)
      cout << endl << "Matrix Vector Multiplication: PASSED" << endl;
    else
      cout << endl << "Matrix Vector Multiplication: FAILED" << endl;
  }
  return max_err;
}
 
 
Numeric matrix_mult(
    Index k, Index m, Index n, Index ntests, Index nsubtests, bool verbose) {
  Numeric max_err = 0;
  Matrix A(m, k), B(k, n), C(m, n), C_ref(m, n);
 
  for (Index i = 0; i < ntests; i++) {
    random_fill_matrix(A, 1000, false);
    random_fill_matrix(B, 1000, false);
 
    if (verbose) {
      cout.precision(15);
      cout << "MATRIX MULTIPLICATION: m = " << m << ", k = " << k;
      cout << ", n = " << n << endl;
    }
 
    // A * B
 
    random_fill_matrix(C, 10, false);
    random_fill_matrix(C_ref, 10, false);
 
    mult(C, A, B);
    mult_general(C_ref, A, B);
 
    Numeric err_mul = get_maximum_error(C, C_ref, true);
    if (err_mul > max_err) max_err = err_mul;
 
    if (verbose) {
      cout << "\t"
           << "A * B: max. rel. error = " << err_mul << endl;
    }
 
    // A^T * B
 
    Matrix AT(k, m);
    random_fill_matrix(AT, 100.0, false);
 
    mult(C, transpose(AT), B);
    mult_general(C_ref, transpose(AT), B);
    Numeric err_trans1 = get_maximum_error(C, C_ref, true);
    if (err_trans1 > max_err) max_err = err_trans1;
 
    if (verbose) {
      cout << "\t"
           << "A^T * B: max. rel. err = " << err_trans1 << endl;
    }
 
    // A * B^T
 
    Matrix BT(n, k);
    random_fill_matrix(BT, 100.0, false);
 
    mult(C, A, transpose(BT));
    mult_general(C_ref, A, transpose(BT));
    Numeric err_trans2 = get_maximum_error(C, C_ref, true);
    if (err_trans2 > max_err) max_err = err_trans2;
 
    if (verbose) {
      cout << "\t"
           << "A * B^T: max. rel. err = " << err_trans2 << endl;
    }
 
    // A^T * B^T
 
    mult(C, transpose(AT), transpose(BT));
    mult_general(C_ref, transpose(AT), transpose(BT));
    Numeric err_trans3 = get_maximum_error(C, C_ref, true);
    if (err_trans3 > max_err) max_err = err_trans3;
 
    if (verbose) {
      cout << "\t"
           << "A^T * B^T: max. rel. err = " << err_trans3 << endl;
      cout << endl;
    }
  }
 
  // Multiplication of submatrices.
  Rand<Index> k_rand(1, k - 1), m_rand(1, m - 1), n_rand(1, n - 1);
 
  for (Index i = 0; i < nsubtests - 1; i++) {
    Index k1(k_rand()), m1(m_rand()), n1(n_rand());
 
    Range r1(random_range(m1));
    Range r2(random_range(k1));
    Range r3(random_range(n1));
 
    MatrixView C_sub(C(r1, r3));
    MatrixView C_sub_ref(C_ref(r1, r3));
 
    ConstMatrixView A_sub(A(r1, r2));
    ConstMatrixView B_sub(B(r2, r3));
 
    mult(C_sub, A_sub, B_sub);
    mult_general(C_sub_ref, A_sub, B_sub);
 
    Numeric err = get_maximum_error(C_sub, C_sub_ref, true);
    if (err > max_err) max_err = err;
 
    if (verbose) {
      cout << "\t"
           << "Submatrix multiplication: max. rel. err = " << err;
      cout << endl;
    }
  }
 
  if (verbose) {
    cout << endl;
  }
 
  return max_err;
}
 
 
Numeric test_matrix_multiplication(bool verbose) {
  Numeric max_err, err;
 
  // Trivial case k = m = n = 0.
  max_err = matrix_mult(0, 0, 0, 10, 0, verbose);
 
  // k = 1, m = 1, n = 1.
  err = matrix_mult(1, 1, 1, 10, 0, verbose);
  if (err > max_err) max_err = err;
 
  // k = 10, m = 1, n = 10.
  err = matrix_mult(10, 1, 10, 20, 20, verbose);
  if (err > max_err) max_err = err;
 
  // k = 10, m = 1, n = 10.
  err = matrix_mult(10, 1, 10, 20, 20, verbose);
  if (err > max_err) max_err = err;
 
  // k = 10, m = 1, n = 1.
  err = matrix_mult(10, 1, 1, 20, 20, verbose);
  if (err > max_err) max_err = err;
 
  // k = 100, m = 100, n = 100.
  err = matrix_mult(200, 200, 200, 20, 20, verbose);
  if (err > max_err) max_err = err;
 
  // k = 10, m = 100, n = 10.
  err = matrix_mult(10, 100, 10, 20, 20, verbose);
  if (err > max_err) max_err = err;
 
  // k = 10, m = 100, n = 100.
  err = matrix_mult(10, 100, 100, 20, 20, verbose);
  if (err > max_err) max_err = err;
 
  // k = 100, m = 100, n = 100, 100 submatrix multiplications.
  err = matrix_mult(100, 100, 100, 0, 100, verbose);
  if (err > max_err) max_err = err;
 
  return max_err;
}
 
void test_empty() {
  {
    cout << "Array" << endl;
    ArrayOfIndex v;
    v.resize(0);
    cout << v.empty() << endl;
    v.resize(1);
    cout << v.empty() << endl;
  }
  {
    cout << "Vector" << endl;
    Vector v;
    v.resize(0);
    cout << v.empty() << endl;
    v.resize(1);
    cout << v.empty() << endl;
  }
  {
    cout << "Matrix" << endl;
    Matrix v;
    v.resize(0, 1);
    cout << v.empty() << endl;
    v.resize(1, 0);
    cout << v.empty() << endl;
    v.resize(1, 1);
    cout << v.empty() << endl;
  }
  {
    cout << "Tensor3" << endl;
    Tensor3 v;
    v.resize(0, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 0, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 0);
    cout << v.empty() << endl;
    v.resize(1, 1, 1);
    cout << v.empty() << endl;
  }
  {
    cout << "Tensor4" << endl;
    Tensor4 v;
    v.resize(0, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 0, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 0, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 0);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1);
    cout << v.empty() << endl;
  }
  {
    cout << "Tensor5" << endl;
    Tensor5 v;
    v.resize(0, 1, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 0, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 0, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 0, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 0);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 1);
    cout << v.empty() << endl;
  }
  {
    cout << "Tensor6" << endl;
    Tensor6 v;
    v.resize(0, 1, 1, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 0, 1, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 0, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 0, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 0, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 1, 0);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 1, 1);
    cout << v.empty() << endl;
  }
  {
    cout << "Tensor7" << endl;
    Tensor7 v;
    v.resize(0, 1, 1, 1, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 0, 1, 1, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 0, 1, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 0, 1, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 0, 1, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 1, 0, 1);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 1, 1, 0);
    cout << v.empty() << endl;
    v.resize(1, 1, 1, 1, 1, 1, 1);
    cout << v.empty() << endl;
  }
}
 
void nlinspace(Vector& x,
               const Numeric start,
               const Numeric stop,
               const Index n) {
  ARTS_ASSERT(1 < n);  // Number of points must be greater 1.
  x.resize(n);
  Numeric step = (stop - start) / ((double)n - 1);
  for (Index i = 0; i < n - 1; i++) x[i] = start + (double)i * step;
  x[n - 1] = stop;
}
 
void test47() {
  // Selecting empty matpack dimensions with Joker shouldn't fail
  Vector v;
  Matrix m;
 
  VectorView vv = v[joker];
  MatrixView mv = m(joker, joker);
 
  std::cout << "vv.nelem: " << vv.nelem() << std::endl;
  std::cout << "mv.nrows: " << mv.nrows() << " ";
  std::cout << "mv.ncols: " << mv.ncols() << std::endl;
 
  Matrix m2(0, 5);
  MatrixView mv2 = m2(joker, 3);
  std::cout << "mv2.nrows: " << mv2.nrows() << " ";
  std::cout << "mv2.ncols: " << mv2.ncols() << std::endl;
}
 
void test48() {
  
  // Test each simple reduction
  std::cout << Tensor7(2, 2, 2, 2, 2, 2, 1, 8).reduce_rank<0, 1, 2, 3, 4, 5>() << '\n';
  std::cout << Tensor7(2, 2, 2, 2, 2, 1, 1, 9).reduce_rank<0, 1, 2, 3, 4>() << '\n';
  std::cout << Tensor7(2, 2, 2, 2, 1, 1, 1, 10).reduce_rank<0, 1, 2, 3>() << '\n';
  std::cout << Tensor7(2, 2, 2, 1, 1, 1, 1, 11).reduce_rank<0, 1, 2>() << '\n';
  std::cout << Tensor7(2, 2, 1, 1, 1, 1, 1, 12).reduce_rank<0, 1>() << '\n';
  std::cout << Tensor7(2, 1, 1, 1, 1, 1, 1, 13).reduce_rank<0>() << '\n';
  std::cout << Tensor7(1, 1, 1, 1, 1, 1, 1, 14) << '\n';
  std::cout << Tensor6(2, 2, 2, 2, 2, 1, 15).reduce_rank<0, 1, 2, 3, 4>() << '\n';
  std::cout << Tensor6(2, 2, 2, 2, 1, 1, 16).reduce_rank<0, 1, 2, 3>() << '\n';
  std::cout << Tensor6(2, 2, 2, 1, 1, 1, 17).reduce_rank<0, 1, 2>() << '\n';
  std::cout << Tensor6(2, 2, 1, 1, 1, 1, 18).reduce_rank<0, 1>() << '\n';
  std::cout << Tensor6(2, 1, 1, 1, 1, 1, 19).reduce_rank<0>() << '\n';
  std::cout << Tensor6(1, 1, 1, 1, 1, 1, 20) << '\n';
  std::cout << Tensor5(2, 2, 2, 2, 1, 21).reduce_rank<0, 1, 2, 3>() << '\n';
  std::cout << Tensor5(2, 2, 2, 1, 1, 22).reduce_rank<0, 1, 2>() << '\n';
  std::cout << Tensor5(2, 2, 1, 1, 1, 23).reduce_rank<0, 1>() << '\n';
  std::cout << Tensor5(2, 1, 1, 1, 1, 24).reduce_rank<0>() << '\n';
  std::cout << Tensor5(1, 1, 1, 1, 1, 25) << '\n';
  std::cout << Tensor4(2, 2, 2, 1, 26).reduce_rank<0, 1, 2>() << '\n';
  std::cout << Tensor4(2, 2, 1, 1, 27).reduce_rank<0, 1>() << '\n';
  std::cout << Tensor4(2, 1, 1, 1, 28).reduce_rank<0>() << '\n';
  std::cout << Tensor4(1, 1, 1, 1, 29) << '\n';
  std::cout << Tensor3(2, 2, 1, 30).reduce_rank<0, 1>() << '\n';
  std::cout << Tensor3(2, 1, 1, 31).reduce_rank<0>() << '\n';
  std::cout << Tensor3(1, 1, 1, 32) << '\n';
  std::cout << Matrix(2, 1, 33).reduce_rank<0>() << '\n';
  std::cout << Matrix(1, 1, 34) << '\n';
  
  // Test that the reductions work along different axis for Vector
  std::cout << MapToEigen(Tensor7(2, 1, 1, 1, 1, 1, 1, 35).reduce_rank<0, 1>()).transpose() << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 1, 1, 1, 1, 35).reduce_rank<0, 2>()).transpose() << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 1, 1, 1, 1, 35).reduce_rank<0, 3>()).transpose() << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 1, 1, 1, 1, 35).reduce_rank<0, 4>()).transpose() << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 1, 1, 1, 1, 35).reduce_rank<0, 5>()).transpose() << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 1, 1, 1, 1, 35).reduce_rank<0, 6>()).transpose() << '\n';
  
  // Test that the reductions work along different axis for Matrix
  std::cout << MapToEigen(Tensor7(2, 2, 1, 1, 1, 1, 1, 36).reduce_rank<0, 1>()) << '\n'
            << MapToEigen(Tensor7(2, 1, 2, 1, 1, 1, 1, 36).reduce_rank<0, 2>()) << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 2, 1, 1, 1, 36).reduce_rank<0, 3>()) << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 1, 2, 1, 1, 36).reduce_rank<0, 4>()) << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 1, 1, 2, 1, 36).reduce_rank<0, 5>()) << '\n'
            << MapToEigen(Tensor7(2, 1, 1, 1, 1, 1, 2, 36).reduce_rank<0, 6>()) << '\n';
            
  // Test that the right elements are accessed
  {
    Index val=1;
    Tensor7 testvar(9, 2, 4, 3, 5, 7, 11);for (Index i=0; i<9; i++)
    for (Index j=0; j<2; j++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              for (Index o=0; o<11; o++) {
                testvar(i, j, k, l, m, n, o) = Numeric(val++);
              }
            }
          }
        }
      }
    }
    
    //  Vector test
    for (Index j=0; j<2; j++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              for (Index o=0; o<11; o++) {
                std::cout << Tensor7(testvar(joker, Range(j, 1), Range(k, 1), Range(l, 1), Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<0>() << '\n';
                std::cout << testvar(joker, j, k, l, m, n, o) << '\n';
                std::cout << '\n';
              }
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              for (Index o=0; o<11; o++) {
                std::cout << Tensor7(testvar(Range(i, 1), joker, Range(k, 1), Range(l, 1), Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<1>() << '\n';
                std::cout << testvar(i, joker, k, l, m, n, o) << '\n';
                std::cout << '\n';
              }
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              for (Index o=0; o<11; o++) {
                std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), joker, Range(l, 1), Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<2>() << '\n';
                std::cout << testvar(i, j, joker, l, m, n, o) << '\n';
                std::cout << '\n';
              }
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              for (Index o=0; o<11; o++) {
                std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), joker, Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<3>() << '\n';
                std::cout << testvar(i, j, k, joker, m, n, o) << '\n';
                std::cout << '\n';
              }
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index l=0; l<3; l++) {
            for (Index n=0; n<7; n++) {
              for (Index o=0; o<11; o++) {
                std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), Range(l, 1), joker, Range(n, 1), Range(o, 1))).reduce_rank<4>() << '\n';
                std::cout << testvar(i, j, k, l, joker, n, o) << '\n';
                std::cout << '\n';
              }
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index l=0; l<3; l++) {
            for (Index m=0; m<5; m++) {
              for (Index o=0; o<11; o++) {
                std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), Range(l, 1), Range(m, 1), joker, Range(o, 1))).reduce_rank<5>() << '\n';
                std::cout << testvar(i, j, k, l, m, joker, o) << '\n';
                std::cout << '\n';
              }
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index l=0; l<3; l++) {
            for (Index m=0; m<5; m++) {
              for (Index n=0; n<7; n++) {
                std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), Range(l, 1), Range(m, 1), Range(n, 1), joker)).reduce_rank<6>() << '\n';
                std::cout << testvar(i, j, k, l, m, n, joker) << '\n';
                std::cout << '\n';
              }
            }
          }
        }
      }
    }
    
    // Matrix test
    for (Index k=0; k<4; k++) {
      for (Index l=0; l<3; l++) {
        for (Index m=0; m<5; m++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(joker, joker, Range(k, 1), Range(l, 1), Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<0, 1>() << '\n';
              std::cout << testvar(joker, joker, k, l, m, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index j=0; j<2; j++) {
      for (Index l=0; l<3; l++) {
        for (Index m=0; m<5; m++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(joker, Range(j, 1), joker, Range(l, 1), Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<0, 2>() << '\n';
              std::cout << testvar(joker, j, joker, l, m, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index j=0; j<2; j++) {
      for (Index k=0; k<4; k++) {
        for (Index m=0; m<5; m++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(joker, Range(j, 1), Range(k, 1), joker, Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<0, 3>() << '\n';
              std::cout << testvar(joker, j, k, joker, m, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index j=0; j<2; j++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(joker, Range(j, 1), Range(k, 1), Range(l, 1), joker, Range(n, 1), Range(o, 1))).reduce_rank<0, 4>() << '\n';
              std::cout << testvar(joker, j, k, l, joker, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index j=0; j<2; j++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(joker, Range(j, 1), Range(k, 1), Range(l, 1), Range(m, 1), joker, Range(o, 1))).reduce_rank<0, 5>() << '\n';
              std::cout << testvar(joker, j, k, l, m, joker, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index j=0; j<2; j++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              std::cout << Tensor7(testvar(joker, Range(j, 1), Range(k, 1), Range(l, 1), Range(m, 1), Range(n, 1), joker)).reduce_rank<0, 6>() << '\n';
              std::cout << testvar(joker, j, k, l, m, n, joker) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index l=0; l<3; l++) {
        for (Index m=0; m<5; m++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), joker, joker, Range(l, 1), Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<1, 2>() << '\n';
              std::cout << testvar(i, joker, joker, l, m, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index k=0; k<4; k++) {
        for (Index m=0; m<5; m++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), joker, Range(k, 1), joker, Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<1, 3>() << '\n';
              std::cout << testvar(i, joker, k, joker, m, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), joker, Range(k, 1), Range(l, 1), joker, Range(n, 1), Range(o, 1))).reduce_rank<1, 4>() << '\n';
              std::cout << testvar(i, joker, k, l, joker, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), joker, Range(k, 1), Range(l, 1), Range(m, 1), joker, Range(o, 1))).reduce_rank<1, 5>() << '\n';
              std::cout << testvar(i, joker, k, l, m, joker, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index k=0; k<4; k++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              std::cout << Tensor7(testvar(Range(i, 1), joker, Range(k, 1), Range(l, 1), Range(m, 1), Range(n, 1), joker)).reduce_rank<1, 6>() << '\n';
              std::cout << testvar(i, joker, k, l, m, n, joker) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index m=0; m<5; m++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), joker, joker, Range(m, 1), Range(n, 1), Range(o, 1))).reduce_rank<2, 3>() << '\n';
              std::cout << testvar(i, j, joker, joker, m, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index l=0; l<3; l++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), joker, Range(l, 1), joker, Range(n, 1), Range(o, 1))).reduce_rank<2, 4>() << '\n';
              std::cout << testvar(i, j, joker, l, joker, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), joker, Range(l, 1), Range(m, 1), joker, Range(o, 1))).reduce_rank<2, 5>() << '\n';
              std::cout << testvar(i, j, joker, l, m, joker, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index l=0; l<3; l++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), joker, Range(l, 1), Range(m, 1), Range(n, 1), joker)).reduce_rank<2, 6>() << '\n';
              std::cout << testvar(i, j, joker, l, m, n, joker) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index n=0; n<7; n++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), joker, joker, Range(n, 1), Range(o, 1))).reduce_rank<3, 4>() << '\n';
              std::cout << testvar(i, j, k, joker, joker, n, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index m=0; m<5; m++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), joker, Range(m, 1), joker, Range(o, 1))).reduce_rank<3, 5>() << '\n';
              std::cout << testvar(i, j, k, joker, m, joker, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index m=0; m<5; m++) {
            for (Index n=0; n<7; n++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), joker, Range(m, 1), Range(n, 1), joker)).reduce_rank<3, 6>() << '\n';
              std::cout << testvar(i, j, k, joker, m, n, joker) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index l=0; l<3; l++) {
            for (Index o=0; o<11; o++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), Range(l, 1), joker, joker, Range(o, 1))).reduce_rank<4, 5>() << '\n';
              std::cout << testvar(i, j, k, l, joker, joker, o) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index l=0; l<3; l++) {
            for (Index n=0; n<7; n++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), Range(l, 1), joker, Range(n, 1), joker)).reduce_rank<4, 6>() << '\n';
              std::cout << testvar(i, j, k, l, joker, n, joker) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
    for (Index i=0; i<9; i++) {
      for (Index j=0; j<2; j++) {
        for (Index k=0; k<4; k++) {
          for (Index l=0; l<3; l++) {
            for (Index m=0; m<5; m++) {
              std::cout << Tensor7(testvar(Range(i, 1), Range(j, 1), Range(k, 1), Range(l, 1), Range(m, 1), joker, joker)).reduce_rank<5, 6>() << '\n';
              std::cout << testvar(i, j, k, l, m, joker, joker) << '\n';
              std::cout << '\n';
            }
          }
        }
      }
    }
  }
}
 
int main() {
  //   test1();
  //   test2();
  //   test3();
  //   test4();
  //   test5();
  //   test6();
  //   test7();
  //   test8();
  //   test9();
  //   test10();
  //   test11();
  //   test12();
  //   test13();
  //   test14();
  //   test15();
  //   test16();
  //   test17();
  //   test18();
  //   test19();
  //   test20();
  //   test21();
  //   test22();
  //   test23();
  //   test24();
  //   test25();
  //   test26();
  //   test27();
  //   test28();
  //   test29();
  //   test30();
  //   test31();
  //   test32();
  //   test33();
  //   test34();
  //   test35();
  //   test36();
  //  Index i = 10000000;
  //  test37(i);
  //  test38();
  //  test39();
  //  test40();
  //  test41();
  //    test42();
  //   test43();
  //    test44();
  //    test45();
  //    test46();
  //  test47();
  test48();
 
  //    const double tolerance = 1e-9;
  //    double error;
  //
  //    // Matrix Vector Multiplication.
  //    error = test_matrix_vector_multiplication(false);
  //    cout << "Matrix Vector Multiplication: ";
  //    if (error > tolerance)
  //        cout << "FAILED, maximum error: " << error << endl;
  //    else
  //        cout << "PASSED." << endl;
  //
  //    // Matrix Matrix Multiplication.
  //    error = test_matrix_multiplication(false);
  //
  //    cout << "Matrix Matrix Multiplication: ";
  //    if (error > tolerance)
  //        cout << "FAILED, maximum error: " << error << endl;
  //    else
  //        cout << "PASSED." << endl;
 
  //test_diagonal( 100 );
  //test_empty();
 
  return 1;
}