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diff --git a/src/armadillo/include/armadillo_bits/fn_approx_equal.hpp b/src/armadillo/include/armadillo_bits/fn_approx_equal.hpp
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+++ b/src/armadillo/include/armadillo_bits/fn_approx_equal.hpp
@@ -0,0 +1,471 @@
+// SPDX-License-Identifier: Apache-2.0
+// 
+// Copyright 2008-2016 Conrad Sanderson (http://conradsanderson.id.au)
+// Copyright 2008-2016 National ICT Australia (NICTA)
+// 
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+// http://www.apache.org/licenses/LICENSE-2.0
+// 
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+// ------------------------------------------------------------------------
+
+
+//! \addtogroup fn_approx_equal
+//! @{
+
+
+
+template<typename eT>
+arma_inline
+bool
+internal_approx_equal_abs_diff(const eT& x, const eT& y, const typename get_pod_type<eT>::result tol)
+  {
+  typedef typename get_pod_type<eT>::result T;
+  
+  if(x != y)
+    {
+    if(is_real<T>::value)  // also true for eT = std::complex<float> or eT = std::complex<double>
+      {
+      if( arma_isnan(x) || arma_isnan(y) || (eop_aux::arma_abs(x - y) > tol) )  { return false; }
+      }
+    else
+      {
+      if( eop_aux::arma_abs( ( cond_rel< is_cx<eT>::no >::gt(x, y) ) ? (x-y) : (y-x) ) > tol )  { return false; }
+      }
+    }
+  
+  return true;
+  }
+
+
+
+template<typename eT>
+arma_inline
+bool
+internal_approx_equal_rel_diff(const eT& a, const eT& b, const typename get_pod_type<eT>::result tol)
+  {
+  typedef typename get_pod_type<eT>::result T;
+  
+  if(a != b)
+    {
+    if(is_real<T>::value)  // also true for eT = std::complex<float> or eT = std::complex<double>
+      {
+      if( arma_isnan(a) || arma_isnan(b) )  { return false; }
+      
+      const T abs_a = eop_aux::arma_abs(a);
+      const T abs_b = eop_aux::arma_abs(b);
+      
+      const T max_c = (std::max)(abs_a,abs_b);
+      
+      const T abs_d = eop_aux::arma_abs(a - b);
+      
+      if(max_c >= T(1))
+        {
+        if( abs_d > (tol * max_c) )  { return false; }
+        }
+      else
+        {
+        if( (abs_d / max_c) > tol )  { return false; }
+        }
+      }
+    else
+      {
+      const T abs_a = eop_aux::arma_abs(a);
+      const T abs_b = eop_aux::arma_abs(b);
+      
+      const T max_c = (std::max)(abs_a,abs_b);
+      
+      const T abs_d = eop_aux::arma_abs( ( cond_rel< is_cx<eT>::no >::gt(a, b) ) ? (a-b) : (b-a) );
+      
+      if( abs_d > (tol * max_c) )  { return false; }
+      }
+    }
+  
+  return true;
+  }
+
+
+
+template<bool use_abs_diff, bool use_rel_diff, typename T1, typename T2>
+inline
+bool
+internal_approx_equal_worker
+  (
+  const Base<typename T1::elem_type,T1>& A,
+  const Base<typename T1::elem_type,T2>& B,
+  const typename T1::pod_type abs_tol,
+  const typename T1::pod_type rel_tol
+  )
+  {
+  arma_extra_debug_sigprint();
+  
+  typedef typename T1::elem_type eT;
+  typedef typename T1::pod_type   T;
+  
+  arma_debug_check( ((use_abs_diff == false) && (use_rel_diff == false)), "internal_approx_equal_worker(): both 'use_abs_diff' and 'use_rel_diff' are false" );
+  
+  if(use_abs_diff)  { arma_debug_check( cond_rel< is_signed<T>::value >::lt(abs_tol, T(0)), "approx_equal(): argument 'abs_tol' must be >= 0" ); }
+  if(use_rel_diff)  { arma_debug_check( cond_rel< is_signed<T>::value >::lt(rel_tol, T(0)), "approx_equal(): argument 'rel_tol' must be >= 0" ); }
+  
+  const Proxy<T1> PA(A.get_ref());
+  const Proxy<T2> PB(B.get_ref());
+  
+  if( (PA.get_n_rows() != PB.get_n_rows()) || (PA.get_n_cols() != PB.get_n_cols()) )  { return false; }
+  
+  if( (Proxy<T1>::use_at == false) && (Proxy<T2>::use_at == false) )
+    {
+    const uword N = PA.get_n_elem();
+    
+    const typename Proxy<T1>::ea_type PA_ea = PA.get_ea(); 
+    const typename Proxy<T2>::ea_type PB_ea = PB.get_ea(); 
+    
+    for(uword i=0; i<N; ++i)
+      {
+      const eT x = PA_ea[i];
+      const eT y = PB_ea[i];
+      
+      const bool state_abs_diff = (use_abs_diff) ? internal_approx_equal_abs_diff(x, y, abs_tol) : true;
+      const bool state_rel_diff = (use_rel_diff) ? internal_approx_equal_rel_diff(x, y, rel_tol) : true;
+      
+      if(use_abs_diff && use_rel_diff)
+        {
+        if((state_abs_diff == false) && (state_rel_diff == false))  { return false; }
+        }
+      else
+      if(use_abs_diff)
+        {
+        if(state_abs_diff == false)  { return false; }
+        }
+      else
+      if(use_rel_diff)
+        {
+        if(state_rel_diff == false)  { return false; }
+        }
+      }
+    }
+  else
+    {
+    const uword n_rows = PA.get_n_rows();
+    const uword n_cols = PA.get_n_cols();
+    
+    for(uword col=0; col < n_cols; ++col)
+    for(uword row=0; row < n_rows; ++row)
+      {
+      const eT x = PA.at(row,col);
+      const eT y = PB.at(row,col);
+      
+      const bool state_abs_diff = (use_abs_diff) ? internal_approx_equal_abs_diff(x, y, abs_tol) : true;
+      const bool state_rel_diff = (use_rel_diff) ? internal_approx_equal_rel_diff(x, y, rel_tol) : true;
+      
+      if(use_abs_diff && use_rel_diff)
+        {
+        if((state_abs_diff == false) && (state_rel_diff == false))  { return false; }
+        }
+      else
+      if(use_abs_diff)
+        {
+        if(state_abs_diff == false)  { return false; }
+        }
+      else
+      if(use_rel_diff)
+        {
+        if(state_rel_diff == false)  { return false; }
+        }
+      }
+    }
+  
+  return true;
+  }
+
+
+
+template<bool use_abs_diff, bool use_rel_diff, typename T1, typename T2>
+inline
+bool
+internal_approx_equal_worker
+  (
+  const BaseCube<typename T1::elem_type,T1>& A,
+  const BaseCube<typename T1::elem_type,T2>& B,
+  const typename T1::pod_type abs_tol,
+  const typename T1::pod_type rel_tol
+  )
+  {
+  arma_extra_debug_sigprint();
+  
+  typedef typename T1::elem_type eT;
+  typedef typename T1::pod_type   T;
+  
+  arma_debug_check( ((use_abs_diff == false) && (use_rel_diff == false)), "internal_approx_equal_worker(): both 'use_abs_diff' and 'use_rel_diff' are false" );
+  
+  if(use_abs_diff)  { arma_debug_check( cond_rel< is_signed<T>::value >::lt(abs_tol, T(0)), "approx_equal(): argument 'abs_tol' must be >= 0" ); }
+  if(use_rel_diff)  { arma_debug_check( cond_rel< is_signed<T>::value >::lt(rel_tol, T(0)), "approx_equal(): argument 'rel_tol' must be >= 0" ); }
+  
+  const ProxyCube<T1> PA(A.get_ref());
+  const ProxyCube<T2> PB(B.get_ref());
+  
+  if( (PA.get_n_rows() != PB.get_n_rows()) || (PA.get_n_cols() != PB.get_n_cols()) || (PA.get_n_slices() != PB.get_n_slices()) )  { return false; }
+  
+  if( (ProxyCube<T1>::use_at == false) && (ProxyCube<T2>::use_at == false) )
+    {
+    const uword N = PA.get_n_elem();
+    
+    const typename ProxyCube<T1>::ea_type PA_ea = PA.get_ea(); 
+    const typename ProxyCube<T2>::ea_type PB_ea = PB.get_ea(); 
+    
+    for(uword i=0; i<N; ++i)
+      {
+      const eT x = PA_ea[i];
+      const eT y = PB_ea[i];
+      
+      const bool state_abs_diff = (use_abs_diff) ? internal_approx_equal_abs_diff(x, y, abs_tol) : true;
+      const bool state_rel_diff = (use_rel_diff) ? internal_approx_equal_rel_diff(x, y, rel_tol) : true;
+      
+      if(use_abs_diff && use_rel_diff)
+        {
+        if((state_abs_diff == false) && (state_rel_diff == false))  { return false; }
+        }
+      else
+      if(use_abs_diff)
+        {
+        if(state_abs_diff == false)  { return false; }
+        }
+      else
+      if(use_rel_diff)
+        {
+        if(state_rel_diff == false)  { return false; }
+        }
+      }
+    }
+  else
+    {
+    const uword n_rows   = PA.get_n_rows();
+    const uword n_cols   = PA.get_n_cols();
+    const uword n_slices = PA.get_n_slices();
+    
+    for(uword slice=0; slice < n_slices; ++slice)
+    for(uword   col=0;   col < n_cols;   ++col  )
+    for(uword   row=0;   row < n_rows;   ++row  )
+      {
+      const eT x = PA.at(row,col,slice);
+      const eT y = PB.at(row,col,slice);
+      
+      const bool state_abs_diff = (use_abs_diff) ? internal_approx_equal_abs_diff(x, y, abs_tol) : true;
+      const bool state_rel_diff = (use_rel_diff) ? internal_approx_equal_rel_diff(x, y, rel_tol) : true;
+      
+      if(use_abs_diff && use_rel_diff)
+        {
+        if((state_abs_diff == false) && (state_rel_diff == false))  { return false; }
+        }
+      else
+      if(use_abs_diff)
+        {
+        if(state_abs_diff == false)  { return false; }
+        }
+      else
+      if(use_rel_diff)
+        {
+        if(state_rel_diff == false)  { return false; }
+        }
+      }
+    }
+  
+  return true;
+  }
+
+
+
+template<typename T1, typename T2>
+inline
+bool
+internal_approx_equal_handler(const T1& A, const T2& B, const char* method, const typename T1::pod_type abs_tol, const typename T1::pod_type rel_tol)
+  {
+  arma_extra_debug_sigprint();
+  
+  typedef typename T1::pod_type T;
+  
+  const char sig = (method != nullptr) ? method[0] : char(0);
+  
+  arma_debug_check( ((sig != 'a') && (sig != 'r') && (sig != 'b')), "approx_equal(): argument 'method' must be \"absdiff\" or \"reldiff\" or \"both\"" );
+  
+  bool status = false;
+  
+  if(sig == 'a')
+    {
+    status = internal_approx_equal_worker<true,false>(A, B, abs_tol, T(0));
+    }
+  else
+  if(sig == 'r')
+    {
+    status = internal_approx_equal_worker<false,true>(A, B, T(0), rel_tol);
+    }
+  else
+  if(sig == 'b')
+    {
+    status = internal_approx_equal_worker<true,true>(A, B, abs_tol, rel_tol);
+    }
+  
+  return status;
+  }
+
+
+
+template<typename T1, typename T2>
+inline
+bool
+internal_approx_equal_handler(const T1& A, const T2& B, const char* method, const typename T1::pod_type tol)
+  {
+  arma_extra_debug_sigprint();
+  
+  typedef typename T1::pod_type T;
+  
+  const char sig = (method != nullptr) ? method[0] : char(0);
+  
+  arma_debug_check( ((sig != 'a') && (sig != 'r') && (sig != 'b')), "approx_equal(): argument 'method' must be \"absdiff\" or \"reldiff\" or \"both\"" );
+  
+  arma_debug_check( (sig == 'b'), "approx_equal(): argument 'method' is \"both\", but only one 'tol' argument has been given" );
+  
+  bool status = false;
+  
+  if(sig == 'a')
+    {
+    status = internal_approx_equal_worker<true,false>(A, B, tol, T(0));
+    }
+  else
+  if(sig == 'r')
+    {
+    status = internal_approx_equal_worker<false,true>(A, B, T(0), tol);
+    }
+  
+  return status;
+  }
+
+
+
+template<typename T1, typename T2>
+arma_warn_unused
+inline
+bool
+approx_equal(const Base<typename T1::elem_type,T1>& A, const Base<typename T1::elem_type,T2>& B, const char* method, const typename T1::pod_type tol)
+  {
+  arma_extra_debug_sigprint();
+  
+  return internal_approx_equal_handler(A.get_ref(), B.get_ref(), method, tol);
+  }
+
+
+
+template<typename T1, typename T2>
+arma_warn_unused
+inline
+bool
+approx_equal(const BaseCube<typename T1::elem_type,T1>& A, const BaseCube<typename T1::elem_type,T2>& B, const char* method, const typename T1::pod_type tol)
+  {
+  arma_extra_debug_sigprint();
+  
+  return internal_approx_equal_handler(A.get_ref(), B.get_ref(), method, tol);
+  }
+
+
+
+template<typename T1, typename T2>
+arma_warn_unused
+inline
+bool
+approx_equal(const Base<typename T1::elem_type,T1>& A, const Base<typename T1::elem_type,T2>& B, const char* method, const typename T1::pod_type abs_tol, const typename T1::pod_type rel_tol)
+  {
+  arma_extra_debug_sigprint();
+  
+  return internal_approx_equal_handler(A.get_ref(), B.get_ref(), method, abs_tol, rel_tol);
+  }
+
+
+
+template<typename T1, typename T2>
+arma_warn_unused
+inline
+bool
+approx_equal(const BaseCube<typename T1::elem_type,T1>& A, const BaseCube<typename T1::elem_type,T2>& B, const char* method, const typename T1::pod_type abs_tol, const typename T1::pod_type rel_tol)
+  {
+  arma_extra_debug_sigprint();
+  
+  return internal_approx_equal_handler(A.get_ref(), B.get_ref(), method, abs_tol, rel_tol);
+  }
+
+
+
+template<typename T1, typename T2>
+arma_warn_unused
+inline
+bool
+approx_equal(const SpBase<typename T1::elem_type,T1>& A, const SpBase<typename T1::elem_type,T2>& B, const char* method, const typename T1::pod_type tol)
+  {
+  arma_extra_debug_sigprint();
+  
+  typedef typename T1::elem_type eT;
+  typedef typename T1::pod_type   T;
+  
+  const char sig = (method != nullptr) ? method[0] : char(0);
+  
+  arma_debug_check( ((sig != 'a') && (sig != 'r') && (sig != 'b')), "approx_equal(): argument 'method' must be \"absdiff\" or \"reldiff\" or \"both\"" );
+  
+  arma_debug_check( (sig == 'b'), "approx_equal(): argument 'method' is \"both\", but only one 'tol' argument has been given" );
+  
+  arma_debug_check( (sig == 'r'), "approx_equal(): only the \"absdiff\" method is currently implemented for sparse matrices" );
+  
+  arma_debug_check( cond_rel< is_signed<T>::value >::lt(tol, T(0)), "approx_equal(): argument 'tol' must be >= 0" );
+  
+  const unwrap_spmat<T1> UA(A.get_ref());
+  const unwrap_spmat<T2> UB(B.get_ref());
+  
+  if( (UA.M.n_rows != UB.M.n_rows) || (UA.M.n_cols != UB.M.n_cols) )  { return false; }
+  
+  const SpMat<eT> C = UA.M - UB.M;
+  
+  typename SpMat<eT>::const_iterator it     = C.begin();
+  typename SpMat<eT>::const_iterator it_end = C.end();
+  
+  while(it != it_end)
+    {
+    const eT val = (*it);
+    
+    if( arma_isnan(val) || (eop_aux::arma_abs(val) > tol) )  { return false; }
+    
+    ++it;
+    }
+  
+  return true;
+  }
+
+
+
+template<typename T1, typename T2>
+arma_warn_unused
+inline
+bool
+approx_equal(const SpBase<typename T1::elem_type,T1>& A, const SpBase<typename T1::elem_type,T2>& B, const char* method, const typename T1::pod_type abs_tol, const typename T1::pod_type rel_tol)
+  {
+  arma_extra_debug_sigprint();
+  
+  typedef typename T1::pod_type T;
+  
+  const char sig = (method != nullptr) ? method[0] : char(0);
+  
+  arma_debug_check( ((sig != 'a') && (sig != 'r') && (sig != 'b')), "approx_equal(): argument 'method' must be \"absdiff\" or \"reldiff\" or \"both\"" );
+  
+  arma_debug_check( ((sig == 'r') || (sig == 'b')), "approx_equal(): only the \"absdiff\" method is currently implemented for sparse matrices" );
+  
+  arma_debug_check( cond_rel< is_signed<T>::value >::lt(abs_tol, T(0)), "approx_equal(): argument 'abs_tol' must be >= 0" );
+  arma_debug_check( cond_rel< is_signed<T>::value >::lt(rel_tol, T(0)), "approx_equal(): argument 'rel_tol' must be >= 0" );
+  
+  return approx_equal(A.get_ref(), B.get_ref(), "abs", abs_tol);
+  }
+
+
+
+//! @}