test/adjoint.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
// SPDX-License-Identifier: MPL-2.0

#include "main.h"
#include "fp_control.h"
#include "CustomComplex.h"

// |value|, promoted so that the error bounds below accumulate in the widest available type.
template <typename T>
long double wide_abs(const T& value) {
  using std::abs;
  return static_cast<long double>(abs(value));
}

template <bool IsInteger>
struct adjoint_specific;

template <>
struct adjoint_specific<true> {
  template <typename Vec, typename Mat, typename Scalar>
  static void run(const Vec& v1, const Vec& v2, Vec& v3, const Mat& square, Scalar s1, Scalar s2) {
    VERIFY_IS_EQUAL((s1 * v1 + s2 * v2).dot(v3), numext::conj(s1) * v1.dot(v3) + numext::conj(s2) * v2.dot(v3));
    VERIFY_IS_EQUAL(v3.dot(s1 * v1 + s2 * v2), s1 * v3.dot(v1) + s2 * v3.dot(v2));

    // check compatibility of dot and adjoint
    VERIFY_IS_EQUAL(v1.dot(square * v2), (square.adjoint() * v1).dot(v2));
  }
};

template <>
struct adjoint_specific<false> {
  template <typename Vec, typename Mat, typename Scalar>
  static void run(const Vec& v1, const Vec& v2, Vec& v3, const Mat& square, Scalar s1, Scalar s2) {
    using RealScalar = typename NumTraits<Scalar>::Real;

    const long double eps = static_cast<long double>(NumTraits<RealScalar>::epsilon());
    const long double absS1 = wide_abs(s1), absS2 = wide_abs(s2);
    long double linearityScale = 0;
    for (Index i = 0; i < v1.size(); ++i)
      linearityScale += (absS1 * wide_abs(v1(i)) + absS2 * wide_abs(v2(i))) * wide_abs(v3(i));
    // Two evaluation orders, including complex multiply-adds and scalar-vector products.
    const long double linearityBound = 8 * (v1.size() + 4) * eps * linearityScale;
    VERIFY((numext::isfinite)(linearityBound));
    VERIFY(wide_abs((s1 * v1 + s2 * v2).dot(v3) - (numext::conj(s1) * v1.dot(v3) + numext::conj(s2) * v2.dot(v3))) <=
           linearityBound);
    VERIFY(wide_abs(v3.dot(s1 * v1 + s2 * v2) - (s1 * v3.dot(v1) + s2 * v3.dot(v2))) <= linearityBound);

    VERIFY_IS_APPROX(v1.squaredNorm(), v1.norm() * v1.norm());
    // check normalized() and normalize()
    VERIFY_IS_APPROX(v1, v1.norm() * v1.normalized());
    v3 = v1;
    v3.normalize();
    VERIFY_IS_APPROX(v1, v1.norm() * v3);
    VERIFY_IS_APPROX(v3, v1.normalized());
    VERIFY_IS_APPROX(v3.norm(), RealScalar(1));

    // check null inputs
    VERIFY_IS_APPROX((v1 * 0).normalized(), (v1 * 0));
#if (!EIGEN_ARCH_i386) || defined(EIGEN_VECTORIZE)
    RealScalar very_small = (std::numeric_limits<RealScalar>::min)();
    // Materialized once: ARMv7 NEON flushes the subnormal products that a coefficient-wise read of the lazy
    // expression keeps.
    const Vec tiny = v1 * very_small;
    VERIFY(numext::is_exactly_zero(tiny.norm()));
    VERIFY_IS_APPROX(tiny.normalized(), tiny);
    v3 = tiny;
    v3.normalize();
    VERIFY_IS_APPROX(v3, tiny);
#endif

    // check compatibility of dot and adjoint
    long double adjointScale = 0;
    for (Index i = 0; i < square.rows(); ++i)
      for (Index j = 0; j < square.cols(); ++j)
        adjointScale += wide_abs(v1(i)) * wide_abs(square(i, j)) * wide_abs(v2(j));
    // Each order has two reductions; allow 8*n*eps per complex product and its accumulation.
    const long double adjointBound = 16 * (v1.size() + 1) * eps * adjointScale;
    VERIFY((numext::isfinite)(adjointBound));
    VERIFY(wide_abs(v1.dot(square * v2) - (square.adjoint() * v1).dot(v2)) <= adjointBound);

    // check that Random().normalized() works: tricky as the random xpr must be evaluated by
    // normalized() in order to produce a consistent result.
    VERIFY_IS_APPROX(Vec::Random(v1.size()).normalized().norm(), RealScalar(1));
  }
};

template <typename MatrixType>
void adjoint(const MatrixType& m) {
  /* this test covers the following files:
     Transpose.h Conjugate.h Dot.h
  */
  using std::abs;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> SquareMatrixType;
  const Index PacketSize = internal::packet_traits<Scalar>::size;

  Index rows = m.rows();
  Index cols = m.cols();

  // Avoid integer overflow by limiting input values.
  RealScalar rmin = static_cast<RealScalar>(NumTraits<Scalar>::IsInteger ? NumTraits<Scalar>::IsSigned ? -100 : 0 : -1);
  RealScalar rmax = static_cast<RealScalar>(NumTraits<Scalar>::IsInteger ? 100 : 1);

  MatrixType m1 = RandomMatrix<MatrixType>(rows, cols, rmin, rmax),
             m2 = RandomMatrix<MatrixType>(rows, cols, rmin, rmax), m3(rows, cols),
             square = RandomMatrix<SquareMatrixType>(rows, rows, rmin, rmax);
  VectorType v1 = RandomMatrix<VectorType>(rows, 1, rmin, rmax), v2 = RandomMatrix<VectorType>(rows, 1, rmin, rmax),
             v3 = RandomMatrix<VectorType>(rows, 1, rmin, rmax), vzero = VectorType::Zero(rows);

  Scalar s1 = internal::random<Scalar>(rmin, rmax), s2 = internal::random<Scalar>(rmin, rmax);

  // check basic compatibility of adjoint, transpose, conjugate
  VERIFY_IS_APPROX(m1.transpose().conjugate().adjoint(), m1);
  VERIFY_IS_APPROX(m1.adjoint().conjugate().transpose(), m1);

  // check multiplicative behavior
  VERIFY_IS_APPROX((m1.adjoint() * m2).adjoint(), m2.adjoint() * m1);
  VERIFY_IS_APPROX((s1 * m1).adjoint(), numext::conj(s1) * m1.adjoint());

  // check basic properties of dot, squaredNorm
  VERIFY_IS_APPROX(numext::conj(v1.dot(v2)), v2.dot(v1));
  VERIFY_IS_APPROX(numext::real(v1.dot(v1)), v1.squaredNorm());

  adjoint_specific<NumTraits<Scalar>::IsInteger>::run(v1, v2, v3, square, s1, s2);

  VERIFY_IS_MUCH_SMALLER_THAN(abs(vzero.dot(v1)), static_cast<RealScalar>(1));

  // like in testBasicStuff, test operator() to check const-qualification
  Index r = internal::random<Index>(0, rows - 1), c = internal::random<Index>(0, cols - 1);
  VERIFY_IS_APPROX(m1.conjugate()(r, c), numext::conj(m1(r, c)));
  VERIFY_IS_APPROX(m1.adjoint()(c, r), numext::conj(m1(r, c)));

  // check inplace transpose
  m3 = m1;
  m3.transposeInPlace();
  VERIFY_IS_APPROX(m3, m1.transpose());
  m3.transposeInPlace();
  VERIFY_IS_APPROX(m3, m1);

  if (PacketSize < m3.rows() && PacketSize < m3.cols()) {
    m3 = m1;
    Index i = internal::random<Index>(0, m3.rows() - PacketSize);
    Index j = internal::random<Index>(0, m3.cols() - PacketSize);
    m3.template block<PacketSize, PacketSize>(i, j).transposeInPlace();
    VERIFY_IS_APPROX((m3.template block<PacketSize, PacketSize>(i, j)),
                     (m1.template block<PacketSize, PacketSize>(i, j).transpose()));
    m3.template block<PacketSize, PacketSize>(i, j).transposeInPlace();
    VERIFY_IS_APPROX(m3, m1);
  }

  // check inplace adjoint
  m3 = m1;
  m3.adjointInPlace();
  VERIFY_IS_APPROX(m3, m1.adjoint());
  m3.transposeInPlace();
  VERIFY_IS_APPROX(m3, m1.conjugate());

  // check mixed dot product
  typedef Matrix<RealScalar, MatrixType::RowsAtCompileTime, 1> RealVectorType;
  RealVectorType rv1 = RandomMatrix<RealVectorType>(rows, 1, rmin, rmax);

  VERIFY_IS_APPROX(v1.dot(rv1.template cast<Scalar>()), v1.dot(rv1));
  VERIFY_IS_APPROX(rv1.template cast<Scalar>().dot(v1), rv1.dot(v1));

  VERIFY(is_same_type(m1, m1.template conjugateIf<false>()));
  VERIFY(is_same_type(m1.conjugate(), m1.template conjugateIf<true>()));
}

template <int>
void adjoint_extra() {
  MatrixXcf a(10, 10), b(10, 10);
  VERIFY_RAISES_ASSERT(a = a.transpose());
  VERIFY_RAISES_ASSERT(a = a.transpose() + b);
  VERIFY_RAISES_ASSERT(a = b + a.transpose());
  VERIFY_RAISES_ASSERT(a = a.conjugate().transpose());
  VERIFY_RAISES_ASSERT(a = a.adjoint());
  VERIFY_RAISES_ASSERT(a = a.adjoint() + b);
  VERIFY_RAISES_ASSERT(a = b + a.adjoint());

  // no assertion should be triggered for these cases:
  a.transpose() = a.transpose();
  a.transpose() += a.transpose();
  a.transpose() += a.transpose() + b;
  a.transpose() = a.adjoint();
  a.transpose() += a.adjoint();
  a.transpose() += a.adjoint() + b;

  // regression tests for check_for_aliasing
  MatrixXd c(10, 10);
  c = 1.0 * MatrixXd::Ones(10, 10) + c;
  c = MatrixXd::Ones(10, 10) * 1.0 + c;
  c = c + MatrixXd::Ones(10, 10).cwiseProduct(MatrixXd::Zero(10, 10));
  c = MatrixXd::Ones(10, 10) * MatrixXd::Zero(10, 10);

  // regression for bug 1646
  for (int j = 0; j < 10; ++j) {
    c.col(j).head(j) = c.row(j).head(j);
  }

  for (int j = 0; j < 10; ++j) {
    c.col(j) = c.row(j);
  }

  a.conservativeResize(1, 1);
  a = a.transpose();

  a.conservativeResize(0, 0);
  a = a.transpose();
}

template <typename Scalar>
void inner_product_boundary_sizes() {
  const Index PS = internal::packet_traits<Scalar>::size;
  // Sizes that exercise every branch in the 4-way unrolled vectorized inner product:
  // scalar fallback (< PS), 1-3 packets, quad loop entry/exit, remainder packets, scalar cleanup
  const Index sizes[] = {0,
                         1,
                         PS - 1,
                         PS,
                         PS + 1,
                         2 * PS - 1,
                         2 * PS,
                         2 * PS + 1,
                         3 * PS - 1,
                         3 * PS,
                         3 * PS + 1,
                         4 * PS - 1,
                         4 * PS,
                         4 * PS + 1,
                         8 * PS,
                         8 * PS + 1,
                         8 * PS + PS,
                         8 * PS + 2 * PS,
                         8 * PS + 3 * PS,
                         8 * PS + 3 * PS + 1};
  for (int si = 0; si < 20; ++si) {
    const Index n = sizes[si];
    if (n <= 0) continue;
    typedef Matrix<Scalar, Dynamic, 1> Vec;
    Vec v1 = Vec::Random(n);
    Vec v2 = Vec::Random(n);
    // Reference: scalar loop
    Scalar expected(0);
    for (Index k = 0; k < n; ++k) expected += numext::conj(v1(k)) * v2(k);
    VERIFY_IS_APPROX(v1.dot(v2), expected);
    // Also test squaredNorm
    Scalar sq_expected(0);
    for (Index k = 0; k < n; ++k) sq_expected += numext::conj(v1(k)) * v1(k);
    VERIFY_IS_APPROX(v1.squaredNorm(), numext::real(sq_expected));
  }
}

// dot() maps operands with a runtime unit stride to contiguous vectors and rewraps their unary ops around the map.
// adjoint() nests a non-const Transpose, so the rewrap must produce the const-nested type unaryExpr() returns.
template <typename Scalar>
void inner_product_adjoint_rewrap() {
  // Column-major so that row(r) has a compile-time non-unit stride and the raw maps below walk a row and a column.
  using Mat = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
  using Vec = Matrix<Scalar, Dynamic, 1>;
  using StridedMap = Map<const Vec, Unaligned, InnerStride<Dynamic>>;
  const Index n = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);
  const Index r = internal::random<Index>(0, n - 1);
  Mat m = Mat::Random(n, n);
  Vec v = Vec::Random(n);
  Scalar expected(0);
  for (Index k = 0; k < n; ++k) expected += m(r, k) * v(k);

  // Compile-time non-unit stride: the map path is instantiated but not taken.
  VERIFY_IS_APPROX(m.row(r).adjoint().dot(v), expected);
  VERIFY_IS_APPROX(v.dot(m.row(r).adjoint()), numext::conj(expected));
  // Runtime strides: n skips the map, 1 takes it.
  StridedMap row(m.data() + r, n, InnerStride<Dynamic>(n));
  StridedMap column(m.data() + r * n, n, InnerStride<Dynamic>(1));
  VERIFY_IS_APPROX(row.adjoint().dot(v), expected);
  VERIFY_IS_APPROX(v.dot(row.adjoint()), numext::conj(expected));
  VERIFY_IS_APPROX(column.adjoint().dot(v), (m.col(r).transpose() * v).value());
  VERIFY_IS_APPROX(v.dot(column.adjoint()), numext::conj((m.col(r).transpose() * v).value()));
}

template <typename Lhs, typename Rhs>
void check_inner_product(const MatrixBase<Lhs>& lhs, const MatrixBase<Rhs>& rhs) {
  using Scalar = typename ScalarBinaryOpTraits<typename Lhs::Scalar, typename Rhs::Scalar>::ReturnType;
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Wide = std::complex<long double>;
  const auto widen = [](const auto& value) {
    return Wide(static_cast<long double>(numext::real(value)), static_cast<long double>(numext::imag(value)));
  };
  Wide dot(0), product(0);
  long double magnitude = 0;
  for (Index i = 0; i < lhs.size(); ++i) {
    const Wide a = widen(lhs.coeff(i)), b = widen(rhs.coeff(i));
    dot += std::conj(a) * b;
    product += a * b;
    magnitude += std::abs(a) * std::abs(b);
  }
  // Complex multiply/add and the reference reduction fit within 8*n*eps*sum(|a_i|*|b_i|).
  const long double bound = 8 * lhs.size() * static_cast<long double>(NumTraits<RealScalar>::epsilon()) * magnitude;
  VERIFY(std::abs(widen(lhs.dot(rhs)) - dot) <= bound);
  VERIFY(std::abs(widen((lhs.adjoint() * rhs).value()) - dot) <= bound);
  VERIFY(std::abs(widen((lhs.transpose() * rhs).value()) - product) <= bound);
  Matrix<Scalar, 1, 1> result;
  result.noalias() = lhs.transpose() * rhs;
  VERIFY(std::abs(widen(result.value()) - product) <= bound);
  result.noalias() += lhs.transpose() * rhs;
  VERIFY(std::abs(widen(result.value()) - 2.0L * product) <= 2 * bound);
  result.noalias() -= lhs.transpose() * rhs;
  VERIFY(std::abs(widen(result.value()) - product) <= 3 * bound);
}

template <typename Scalar>
void inner_product_runtime_stride() {
  using Vec = Matrix<Scalar, Dynamic, 1>;
  using StridedMap = Map<const Vec, Unaligned, InnerStride<Dynamic>>;
  const Index packetSize = internal::packet_traits<Scalar>::size;
  const Index sizes[] = {
      0, 1, 2, 3, 4, 5, packetSize - 1, packetSize, packetSize + 1, 4 * packetSize, 9 * packetSize + 1};
  for (Index n : sizes) {
    Vec a = Vec::Random(2 * n + 2), b = Vec::Random(2 * n + 2);
    for (Index lhsStride : {1, 2, -2}) {
      for (Index rhsStride : {1, 2, -2}) {
        const StridedMap lhs(a.data() + (lhsStride < 0 && n > 0 ? 2 * n : 1), n, InnerStride<Dynamic>(lhsStride));
        const StridedMap rhs(b.data() + (rhsStride < 0 && n > 0 ? 2 * n : 1), n, InnerStride<Dynamic>(rhsStride));
        check_inner_product(lhs, rhs);
        check_inner_product(-lhs.conjugate(), -rhs);
        // A functor with state must survive rewrapping, including when it has no packet support.
        const Scalar shift = internal::random<Scalar>();
        check_inner_product(lhs.unaryExpr([=](Scalar x) { return x + shift; }), rhs);
        check_inner_product(lhs.real(), rhs);
        check_inner_product(lhs, rhs.real());
      }
    }
    Map<const Vec, AlignedMax, InnerStride<Dynamic>> aligned(a.data(), n, InnerStride<Dynamic>(1));
    Ref<const Vec, Unaligned, InnerStride<Dynamic>> ref(aligned);
    check_inner_product(aligned, ref);
    check_inner_product(ref, b.head(n));
    check_inner_product(a.head(n), ref);
    // Rows and columns can both become contiguous through their runtime outer dimension.
    Matrix<Scalar, Dynamic, Dynamic, ColMajor> row = a.head(n).transpose();
    Matrix<Scalar, Dynamic, Dynamic, RowMajor> col = b.head(n);
    check_inner_product(row.row(0).transpose(), col.col(0));
    VERIFY_RAISES_ASSERT(ref.dot(b));
  }
}

template <typename Scalar>
void inner_product_exact_values() {
  using Vec = Matrix<Scalar, Dynamic, 1>;
  for (Index n : {0, 1, 2, 3, 4, 5, 17, 65}) {
    Vec a(2 * n), b(2 * n);
    for (Index i = 0; i < 2 * n; ++i) {
      a(i) = Scalar(i % 3 == 0);
      b(i) = Scalar(i % 2 == 0);
    }
    for (Index stride : {1, 2}) {
      const Map<const Vec, Unaligned, InnerStride<Dynamic>> lhs(a.data(), n, InnerStride<Dynamic>(stride));
      const Map<const Vec, Unaligned, InnerStride<Dynamic>> rhs(b.data(), n, InnerStride<Dynamic>(stride));
      Index count = 0;
      for (Index i = 0; i < n; ++i) count += (i * stride) % 6 == 0;
      VERIFY_IS_EQUAL(lhs.dot(rhs), Scalar(count));
      VERIFY_IS_EQUAL((lhs.transpose() * rhs).value(), Scalar(count));
    }
  }
}

// isOrthogonal() must hold at every representable scale, including scales whose squares or fourth powers overflow or
// underflow.
template <typename Scalar>
void is_orthogonal_scaling() {
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Vec = Matrix<Scalar, Dynamic, 1>;
  Vec v = Vec::Zero(5), w = Vec::Zero(5);
  v(1) = Scalar(1);
  v(3) = Scalar(2);
  w(0) = Scalar(3);
  w(4) = Scalar(-1);
  const RealScalar scales[] = {RealScalar(1), RealScalar(1) / NumTraits<RealScalar>::epsilon(),
                               NumTraits<RealScalar>::highest() / RealScalar(4),
                               NumTraits<RealScalar>::epsilon() * NumTraits<RealScalar>::epsilon(),
                               (std::numeric_limits<RealScalar>::min)() * RealScalar(4)};
  for (RealScalar sv : scales) {
    for (RealScalar sw : scales) {
      const Vec a = v * sv, b = w * sw, c = (v + w) * sw;
      VERIFY(a.isOrthogonal(b));
      VERIFY(!a.isOrthogonal(a));
      VERIFY(!a.isOrthogonal(Vec(v * sw)));
      VERIFY(!a.isOrthogonal(c));
    }
  }
  VERIFY(Vec::Zero(5).isOrthogonal(v));
}

// A tight prec must not be squared away: <a/|a|, b/|b|> = t, where t^2 rounds to zero. This holds at every
// power-of-two scale that keeps t |a| normal, including scales where the product in <a, b> underflows.
template <typename RealScalar>
void is_orthogonal_tight_prec() {
  using Vec = Matrix<RealScalar, 2, 1>;
  const RealScalar t = numext::sqrt(std::numeric_limits<RealScalar>::denorm_min()) / RealScalar(2);
  const int minExponent = std::numeric_limits<RealScalar>::min_exponent;
  const int maxExponent = std::numeric_limits<RealScalar>::max_exponent;
  for (int ea = minExponent; ea < maxExponent; ea += 7) {
    const RealScalar sa = std::ldexp(RealScalar(1), ea);
    if (!(t * sa >= (std::numeric_limits<RealScalar>::min)())) continue;
    for (int eb = minExponent; eb < maxExponent; eb += 5) {
      const Vec a(sa, t * sa), b(RealScalar(0), std::ldexp(RealScalar(1), eb));
      VERIFY(!a.isOrthogonal(b, RealScalar(0)));
      VERIFY(!a.isOrthogonal(b, t / RealScalar(2)));
      VERIFY(a.isOrthogonal(b, RealScalar(2) * t));
    }
  }
}

// <a, b> = 0 exactly must stay zero for prec = 0 and tiny positive prec at power-of-two scales near both ends of the
// exponent range; for float and double, their squares overflow or underflow.
template <typename Scalar>
void is_orthogonal_exact_cancellation(const Scalar& phaseA = Scalar(1), const Scalar& phaseB = Scalar(1)) {
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Vec = Matrix<Scalar, 3, 1>;
  const Vec a = Vec(Scalar(1), Scalar(3), Scalar(1)) * phaseA;
  const Vec b = Vec(Scalar(3), Scalar(-1), Scalar(0)) * phaseB;
  const RealScalar scales[] = {RealScalar(1),
                               RealScalar(std::ldexp(1.0, std::numeric_limits<RealScalar>::max_exponent - 4)),
                               RealScalar(std::ldexp(1.0, std::numeric_limits<RealScalar>::min_exponent + 1))};
  const RealScalar eps = NumTraits<RealScalar>::epsilon();
  const RealScalar precs[] = {RealScalar(0), std::numeric_limits<RealScalar>::denorm_min(), eps * eps};
  for (RealScalar sa : scales) {
    for (RealScalar sb : scales) {
      const Vec x = a * sa, y = b * sb;
      for (RealScalar prec : precs) {
        VERIFY(x.isOrthogonal(y, prec));
        VERIFY(y.isOrthogonal(x, prec));
        VERIFY(!x.isOrthogonal(x, prec));
      }
    }
  }
}

// <v, w> = 0 exactly although v spans more exponents than scaling its maximum to 1 keeps normal. In float,
// v = (2^63, 16393 * 2^-90) and w = (-245895 * 2^-126, 15 * 2^27), with 16393 * 15 = 245895.
template <typename RealScalar>
void is_orthogonal_wide_range_cancellation() {
  using Vec = Matrix<RealScalar, 2, 1>;
  const int p = std::numeric_limits<RealScalar>::max_exponent / 2 - 1;
  const int q = p + std::numeric_limits<RealScalar>::min_exponent - std::numeric_limits<RealScalar>::digits - 4;
  const int s = std::numeric_limits<RealScalar>::digits + 3;
  const Vec v(std::ldexp(RealScalar(1), p), std::ldexp(RealScalar(16393), q));
  const Vec w(std::ldexp(RealScalar(-245895), q + s - p), std::ldexp(RealScalar(15), s));
  VERIFY_IS_EQUAL(v.dot(w), RealScalar(0));
  const RealScalar eps = NumTraits<RealScalar>::epsilon();
  for (RealScalar prec : {RealScalar(0), std::numeric_limits<RealScalar>::denorm_min(), eps * eps, eps}) {
    VERIFY(v.isOrthogonal(w, prec));
    VERIFY(w.isOrthogonal(v, prec));
  }
}

// <x, y> != 0 is not orthogonal at prec = 0, although |<x, y>| / (|x| |y|) < denorm_min. For a = (2^e, 2^-e, 0) and
// b = (0, 2^e, 2^e), <a, b> = 1 while |a|^2 and |b|^2 overflow and a(1) 2^-e underflows. For c = (1, 2^-f, 0) and
// d = (0, 2^-f, 1), c(1) d(1) < denorm_min / 2 underflows although |c|^2 and |d|^2 are in range.
template <typename Scalar>
void is_orthogonal_nonzero_overlap(const Scalar& phaseA = Scalar(1), const Scalar& phaseB = Scalar(1)) {
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Vec = Matrix<Scalar, 3, 1>;
  const int e = 3 * std::numeric_limits<RealScalar>::max_exponent / 4;
  const int f = (std::numeric_limits<RealScalar>::digits - std::numeric_limits<RealScalar>::min_exponent) / 2 + 2;
  const RealScalar large = std::ldexp(RealScalar(1), e), small = std::ldexp(RealScalar(1), -e);
  const RealScalar tiny = std::ldexp(RealScalar(1), -f);
  const Vec a = Vec(Scalar(large), Scalar(small), Scalar(0)) * phaseA;
  const Vec b = Vec(Scalar(0), Scalar(large), Scalar(large)) * phaseB;
  const Vec c = Vec(Scalar(1), Scalar(tiny), Scalar(0)) * phaseA;
  const Vec d = Vec(Scalar(0), Scalar(tiny), Scalar(1)) * phaseB;
  VERIFY_IS_EQUAL(a.dot(b), numext::conj(phaseA) * phaseB);
  for (const auto& pair : {std::make_pair(a, b), std::make_pair(c, d)}) {
    VERIFY(!pair.first.isOrthogonal(pair.second, RealScalar(0)));
    VERIFY(!pair.second.isOrthogonal(pair.first, RealScalar(0)));
    VERIFY(pair.first.isOrthogonal(pair.second, std::numeric_limits<RealScalar>::denorm_min()));
    VERIFY(pair.first.isOrthogonal(pair.second));
  }
}

// A non-finite coefficient is orthogonal to nothing; a zero or empty vector is orthogonal to everything.
template <typename Scalar>
void is_orthogonal_special_values() {
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Vec = Matrix<Scalar, Dynamic, 1>;
  Vec v(3), w(3);
  v << Scalar(1), Scalar(3), Scalar(1);
  w << Scalar(3), Scalar(-1), Scalar(0);
  for (const RealScalar special :
       {NumTraits<RealScalar>::quiet_NaN(), NumTraits<RealScalar>::infinity(), -NumTraits<RealScalar>::infinity()}) {
    for (Index i = 0; i < 3; ++i) {
      Vec x = v;
      x(i) = Scalar(special);
      VERIFY(!x.isOrthogonal(w));
      VERIFY(!w.isOrthogonal(x));
      VERIFY(!x.isOrthogonal(Vec::Zero(3)));
      VERIFY(!x.isOrthogonal(x));
    }
  }
  VERIFY(Vec::Zero(3).isOrthogonal(v, RealScalar(0)));
  VERIFY(v.isOrthogonal(Vec::Zero(3), RealScalar(0)));
  VERIFY(Vec().isOrthogonal(Vec(), RealScalar(0)));
}

// Under FTZ and DAZ, a vector whose largest coefficient is subnormal is still nonzero.
template <typename Scalar>
void is_orthogonal_flush_to_zero() {
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Bits = typename internal::binary_floating_point_traits<RealScalar>::Bits;
  using Vec = Matrix<Scalar, 2, 1>;
  // A runtime bit pattern keeps the subnormal out of constant-folded floating-point arithmetic.
  volatile Bits subnormalBits = Bits(1);
  const Scalar tiny(numext::bit_cast<RealScalar>(Bits(subnormalBits)));
  const Vec v(tiny, Scalar(0)), w(Scalar(0), tiny), u(tiny, tiny);
  ScopedFlushToZero flushToZero(FlushToZeroMode::Both);
  if (!flushToZero.isSupported()) return;
  VERIFY(!v.isOrthogonal(v));
  VERIFY(!v.isOrthogonal(v, RealScalar(0)));
  VERIFY(!v.isOrthogonal(u));
  VERIFY(v.isOrthogonal(w));
}

// A custom complex scalar keeps the squared form, which needs no conversion to std::complex.
void is_orthogonal_custom_complex() {
  using Complex = CustomComplex<double>;
  Matrix<Complex, 2, 1> a, b;
  a << Complex(1, 2), Complex(3, 0);
  b << Complex(3, 0), Complex(-1, 2);
  VERIFY(a.isOrthogonal(b, 0.0));
  VERIFY(b.isOrthogonal(a, 0.0));
  VERIFY(!a.isOrthogonal(a));
}

// Test transposeInPlace at vectorization boundary sizes.
// BlockedInPlaceTranspose uses PacketSize-blocked loops with a scalar remainder (line 273),
// exercising off-by-one-prone transitions.
template <typename Scalar>
void transposeInPlace_boundary() {
  const Index PS = internal::packet_traits<Scalar>::size;
  // Sizes around packet boundaries where the blocked path's remainder handling is exercised.
  const Index sizes[] = {1,      2,          3,      PS - 1,     PS,     PS + 1,    2 * PS - 1,
                         2 * PS, 2 * PS + 1, 3 * PS, 3 * PS + 1, 4 * PS, 4 * PS + 1};
  for (int si = 0; si < 13; ++si) {
    Index n = sizes[si];
    if (n <= 0) continue;
    typedef Matrix<Scalar, Dynamic, Dynamic> Mat;

    // Square transposeInPlace
    Mat m1 = Mat::Random(n, n);
    Mat m2 = m1;
    m2.transposeInPlace();
    VERIFY_IS_APPROX(m2, m1.transpose());
    // Double transpose should return to original
    m2.transposeInPlace();
    VERIFY_IS_APPROX(m2, m1);
  }

  // Non-square transposeInPlace (resizable dynamic matrices)
  const Index rect_sizes[][2] = {{2, 5}, {PS, 2 * PS + 1}, {3, 1}, {1, 7}, {2 * PS, PS + 1}};
  for (int si = 0; si < 5; ++si) {
    Index r = rect_sizes[si][0], c = rect_sizes[si][1];
    if (r <= 0 || c <= 0) continue;
    typedef Matrix<Scalar, Dynamic, Dynamic> Mat;
    Mat m1 = Mat::Random(r, c);
    Mat expected = m1.transpose();
    Mat m2 = m1;
    m2.transposeInPlace();
    VERIFY_IS_APPROX(m2, expected);
    VERIFY(m2.rows() == c && m2.cols() == r);
  }
}

template <typename Scalar, int Order, int InnerStride>
void transposeInPlace_strided() {
  using MatrixType = Matrix<Scalar, Dynamic, Dynamic, Order>;
  using MapType = Map<MatrixType, Unaligned, Stride<Dynamic, InnerStride>>;
  STATIC_CHECK(!(internal::evaluator<MapType>::Flags & PacketAccessBit));
  const Index packetSize = internal::packet_traits<Scalar>::size;
  const Index sizes[] = {1, packetSize, packetSize + 1, 2 * packetSize, 2 * packetSize + 1};
  for (Index size : sizes) {
    for (Index innerStride : {1, 2, 3}) {
      if (InnerStride != Dynamic && innerStride != InnerStride) continue;
      const Index outerStride = size * innerStride + 3;
      Matrix<Scalar, Dynamic, 1> storage(size * outerStride + 2);
      for (Index i = 0; i < storage.size(); ++i) storage(i) = Scalar(i + 1);
      const Matrix<Scalar, Dynamic, 1> original = storage;
      Matrix<Scalar, Dynamic, 1> expected = original;
      // Include padding and interleaved coefficients in the exact comparison.
      for (Index outer = 0; outer < size; ++outer)
        for (Index inner = 0; inner < size; ++inner)
          expected(1 + outer * outerStride + inner * innerStride) =
              original(1 + inner * outerStride + outer * innerStride);
      MapType mapped(storage.data() + 1, size, size, Stride<Dynamic, InnerStride>(outerStride, innerStride));
      mapped.transposeInPlace();
      VERIFY(storage == expected);
      mapped.transposeInPlace();
      VERIFY(storage == original);
    }
  }
}

// An aligned start does not make every column aligned when the outer stride is not a multiple of the packet size.
template <typename Scalar>
void transposeInPlace_aligned_map_outer_stride() {
  constexpr int P = internal::packet_traits<Scalar>::size;
  Matrix<Scalar, Dynamic, 1> storage =
      Matrix<Scalar, Dynamic, 1>::LinSpaced(2 * P * (2 * P + 1), 1, 2 * P * (2 * P + 1));
  Map<Matrix<Scalar, Dynamic, Dynamic>, AlignedMax, OuterStride<>> dynamicMap(storage.data(), 2 * P, 2 * P,
                                                                              OuterStride<>(2 * P + 1));
  Matrix<Scalar, Dynamic, Dynamic> expected = dynamicMap.transpose();
  dynamicMap.transposeInPlace();
  VERIFY_IS_EQUAL(dynamicMap, expected);
  Map<Matrix<Scalar, P, P>, AlignedMax, OuterStride<>> fixedMap(storage.data(), OuterStride<>(P + 1));
  Matrix<Scalar, P, P> fixedExpected = fixedMap.transpose();
  fixedMap.transposeInPlace();
  VERIFY_IS_EQUAL(fixedMap, fixedExpected);
}

EIGEN_DECLARE_TEST(adjoint) {
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(adjoint(Matrix<float, 1, 1>()));
    CALL_SUBTEST_2(adjoint(Matrix3d()));
    CALL_SUBTEST_3(adjoint(Matrix4f()));

    CALL_SUBTEST_4(adjoint(MatrixXcf(internal::random<int>(1, EIGEN_TEST_MAX_SIZE / 2),
                                     internal::random<int>(1, EIGEN_TEST_MAX_SIZE / 2))));
    CALL_SUBTEST_5(adjoint(
        MatrixXi(internal::random<int>(1, EIGEN_TEST_MAX_SIZE), internal::random<int>(1, EIGEN_TEST_MAX_SIZE))));
    CALL_SUBTEST_6(adjoint(
        MatrixXf(internal::random<int>(1, EIGEN_TEST_MAX_SIZE), internal::random<int>(1, EIGEN_TEST_MAX_SIZE))));

    // Complement for 128 bits vectorization:
    CALL_SUBTEST_8(adjoint(Matrix2d()));
    CALL_SUBTEST_9(adjoint(Matrix<int, 4, 4>()));

    // 256 bits vectorization:
    CALL_SUBTEST_10(adjoint(Matrix<float, 8, 8>()));
    CALL_SUBTEST_11(adjoint(Matrix<double, 4, 4>()));
    CALL_SUBTEST_12(adjoint(Matrix<int, 8, 8>()));
  }
  // test a large static matrix only once
  CALL_SUBTEST_7(adjoint(Matrix<float, 100, 100>()));

  CALL_SUBTEST_13(adjoint_extra<0>());

  // Inner product vectorization boundary tests (deterministic, outside g_repeat)
  CALL_SUBTEST_14(inner_product_boundary_sizes<float>());
  CALL_SUBTEST_15(inner_product_boundary_sizes<double>());
  CALL_SUBTEST_16(inner_product_boundary_sizes<std::complex<float>>());
  CALL_SUBTEST_17(inner_product_boundary_sizes<std::complex<double>>());
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_19(inner_product_adjoint_rewrap<double>());
    CALL_SUBTEST_19(inner_product_adjoint_rewrap<std::complex<float>>());
    CALL_SUBTEST_19(inner_product_adjoint_rewrap<std::complex<double>>());
  }

  CALL_SUBTEST_20(inner_product_runtime_stride<float>());
  CALL_SUBTEST_21(inner_product_runtime_stride<double>());
  CALL_SUBTEST_22(inner_product_runtime_stride<std::complex<float>>());
  CALL_SUBTEST_23(inner_product_runtime_stride<std::complex<double>>());

  CALL_SUBTEST_24(inner_product_exact_values<int>());
  CALL_SUBTEST_24(inner_product_exact_values<bool>());
  CALL_SUBTEST_24(inner_product_exact_values<half>());
  CALL_SUBTEST_24(inner_product_exact_values<bfloat16>());
  CALL_SUBTEST_24(is_orthogonal_scaling<float>());
  CALL_SUBTEST_24(is_orthogonal_scaling<double>());
  CALL_SUBTEST_24(is_orthogonal_scaling<std::complex<float>>());
  CALL_SUBTEST_24(is_orthogonal_tight_prec<float>());
  CALL_SUBTEST_24(is_orthogonal_tight_prec<double>());
  CALL_SUBTEST_24(is_orthogonal_exact_cancellation<float>());
  CALL_SUBTEST_24(is_orthogonal_exact_cancellation<double>());
  CALL_SUBTEST_24(is_orthogonal_exact_cancellation<half>());
  CALL_SUBTEST_24(is_orthogonal_exact_cancellation<bfloat16>());
  // conj(1 + i) (2 - i) = 1 - 3i multiplies every product exactly.
  CALL_SUBTEST_24(is_orthogonal_exact_cancellation<std::complex<float>>({1, 1}, {2, -1}));
  CALL_SUBTEST_24(is_orthogonal_exact_cancellation<std::complex<double>>({1, 1}, {2, -1}));
  CALL_SUBTEST_24(is_orthogonal_wide_range_cancellation<float>());
  CALL_SUBTEST_24(is_orthogonal_wide_range_cancellation<double>());
  CALL_SUBTEST_24(is_orthogonal_nonzero_overlap<float>());
  CALL_SUBTEST_24(is_orthogonal_nonzero_overlap<double>());
  CALL_SUBTEST_24(is_orthogonal_nonzero_overlap<long double>());
  CALL_SUBTEST_24(is_orthogonal_nonzero_overlap<std::complex<float>>({1, 1}, {2, -1}));
  CALL_SUBTEST_24(is_orthogonal_nonzero_overlap<std::complex<double>>({1, 1}, {2, -1}));
  CALL_SUBTEST_24(is_orthogonal_special_values<float>());
  CALL_SUBTEST_24(is_orthogonal_special_values<double>());
  CALL_SUBTEST_24(is_orthogonal_special_values<std::complex<float>>());
  CALL_SUBTEST_24(is_orthogonal_special_values<half>());
  CALL_SUBTEST_24(is_orthogonal_flush_to_zero<float>());
  CALL_SUBTEST_24(is_orthogonal_flush_to_zero<double>());
  CALL_SUBTEST_24(is_orthogonal_flush_to_zero<std::complex<float>>());
  CALL_SUBTEST_24(is_orthogonal_custom_complex());

  // transposeInPlace at vectorization boundaries (deterministic, outside g_repeat).
  CALL_SUBTEST_18(transposeInPlace_boundary<float>());
  CALL_SUBTEST_18(transposeInPlace_boundary<double>());
  CALL_SUBTEST_18(transposeInPlace_boundary<std::complex<float>>());
  CALL_SUBTEST_18(transposeInPlace_aligned_map_outer_stride<float>());
  CALL_SUBTEST_18(transposeInPlace_aligned_map_outer_stride<double>());

  CALL_SUBTEST_19((transposeInPlace_strided<float, ColMajor, Dynamic>()));
  CALL_SUBTEST_19((transposeInPlace_strided<float, RowMajor, Dynamic>()));
  CALL_SUBTEST_19((transposeInPlace_strided<double, ColMajor, Dynamic>()));
  CALL_SUBTEST_19((transposeInPlace_strided<double, RowMajor, Dynamic>()));
  CALL_SUBTEST_19((transposeInPlace_strided<float, ColMajor, 2>()));
  CALL_SUBTEST_19((transposeInPlace_strided<float, RowMajor, 2>()));
  CALL_SUBTEST_19((transposeInPlace_strided<double, ColMajor, 2>()));
  CALL_SUBTEST_19((transposeInPlace_strided<double, RowMajor, 2>()));
}