diff --git a/dtool/src/dtoolbase/nearly_zero.h b/dtool/src/dtoolbase/nearly_zero.h
index 0b36d4652d..a0ffa0b333 100644
--- a/dtool/src/dtoolbase/nearly_zero.h
+++ b/dtool/src/dtoolbase/nearly_zero.h
@@ -25,17 +25,17 @@
 // const identifier, and then returning the value of that identifier,
 // seems to lead to compilation errors (at least in VC7) in which
 // sometimes IS_THRESHOLD_COMPEQ(a, a, get_nearly_zero_value(a)) != 0.
-INLINE double
+CONSTEXPR double
 get_nearly_zero_value(double) {
   return 1.0e-12;
 }
 
-INLINE float
+CONSTEXPR float
 get_nearly_zero_value(float) {
   return 1.0e-6f;
 }
 
-INLINE int
+CONSTEXPR int
 get_nearly_zero_value(int) {
   // This is a bit silly, but we should nevertheless define it in
   // case it is called for an integer type.
diff --git a/panda/src/linmath/lmatrix3_src.I b/panda/src/linmath/lmatrix3_src.I
index b6ede0627c..6b12ee6642 100644
--- a/panda/src/linmath/lmatrix3_src.I
+++ b/panda/src/linmath/lmatrix3_src.I
@@ -998,11 +998,9 @@ INLINE_LINMATH void FLOATNAME(LMatrix3)::
 transpose_in_place() {
   TAU_PROFILE("void LMatrix3::transpose_in_place()", " ", TAU_USER);
 
-  #define SWAP__(x,y) { FLOATTYPE temp = (x);  (x) = (y);  (y) = temp;}
-  SWAP__(_m(0, 1),_m(1, 0));
-  SWAP__(_m(0, 2),_m(2, 0));
-  SWAP__(_m(1, 2),_m(2, 1));
-  #undef SWAP__
+  std::swap(_m(0, 1), _m(1, 0));
+  std::swap(_m(0, 2), _m(2, 0));
+  std::swap(_m(1, 2), _m(2, 1));
 }
 
 // Matrix inversion code from Numerical Recipes in C.
@@ -1050,12 +1048,16 @@ INLINE_LINMATH bool FLOATNAME(LMatrix3)::
 invert_from(const FLOATNAME(LMatrix3) &other) {
   TAU_PROFILE("bool LMatrix3::invert_from(const LMatrix3 &)", " ", TAU_USER);
 
+  // We throw the value out only if it's smaller than our "small"
+  // threshold squared.  This helps reduce overly-sensitive
+  // rejections.
 #ifdef HAVE_EIGEN
   bool invertible;
-  other._m.computeInverseWithCheck(_m, invertible);
+  other._m.computeInverseWithCheck(_m, invertible,
+    NEARLY_ZERO(FLOATTYPE) * NEARLY_ZERO(FLOATTYPE));
 
   if (!invertible) {
-#ifdef NDEBUG
+#ifdef NOTIFY_DEBUG
     linmath_cat.warning() << "Tried to invert singular LMatrix3.\n";
 #endif
     (*this) = ident_mat();
@@ -1068,11 +1070,7 @@ invert_from(const FLOATNAME(LMatrix3) &other) {
 
   FLOATTYPE other_det = MATRIX3_DETERMINANT(other._m);
 
-  // We throw the value out only if it's smaller than our "small"
-  // threshold squared.  This helps reduce overly-sensitive
-  // rejections.
   if (IS_THRESHOLD_ZERO(other_det, (NEARLY_ZERO(FLOATTYPE) * NEARLY_ZERO(FLOATTYPE)))) {
-    //    if (IS_NEARLY_ZERO(other_det)) {
 #ifdef NOTIFY_DEBUG
     linmath_cat.warning() << "Tried to invert singular LMatrix3.\n";
 #endif
@@ -1125,10 +1123,11 @@ invert_transpose_from(const FLOATNAME(LMatrix3) &other) {
 #ifdef HAVE_EIGEN
   bool invertible;
   EMatrix3 temp;
-  other._m.computeInverseWithCheck(temp, invertible);
+  other._m.computeInverseWithCheck(temp, invertible,
+    NEARLY_ZERO(FLOATTYPE) * NEARLY_ZERO(FLOATTYPE));
 
   if (!invertible) {
-#ifdef NDEBUG
+#ifdef NOTIFY_DEBUG
     linmath_cat.warning() << "Tried to invert singular LMatrix3.\n";
 #endif
     (*this) = ident_mat();
@@ -1139,6 +1138,7 @@ invert_transpose_from(const FLOATNAME(LMatrix3) &other) {
   return true;
 
 #else  // HAVE_EIGEN
+
   FLOATTYPE other_det = MATRIX3_DETERMINANT(other._m);
 
   if (IS_THRESHOLD_ZERO(other_det, (NEARLY_ZERO(FLOATTYPE) * NEARLY_ZERO(FLOATTYPE)))) {
@@ -1180,10 +1180,11 @@ invert_transpose_from(const FLOATNAME(LMatrix4) &other) {
 #ifdef HAVE_EIGEN
   bool invertible;
   EMatrix3 temp;
-  other._m.block<3, 3>(0, 0).computeInverseWithCheck(temp, invertible);
+  other._m.block<3, 3>(0, 0).computeInverseWithCheck(temp, invertible,
+    NEARLY_ZERO(FLOATTYPE) * NEARLY_ZERO(FLOATTYPE));
 
   if (!invertible) {
-#ifdef NDEBUG
+#ifdef NOTIFY_DEBUG
     linmath_cat.warning() << "Tried to invert singular LMatrix3.\n";
 #endif
     (*this) = ident_mat();
diff --git a/panda/src/linmath/lmatrix4_src.I b/panda/src/linmath/lmatrix4_src.I
index 389b472aea..38fe84b0ad 100644
--- a/panda/src/linmath/lmatrix4_src.I
+++ b/panda/src/linmath/lmatrix4_src.I
@@ -831,7 +831,7 @@ xform(const FLOATNAME(LVecBase4) &v) const {
 
 #ifdef HAVE_EIGEN
   v_res._v.noalias() = v._v * _m;
-#else 
+#else
   VECTOR4_MATRIX4_PRODUCT(v_res, v,(*this));
 #endif  // HAVE_EIGEN
   return v_res;
@@ -853,12 +853,12 @@ xform_point(const FLOATNAME(LVecBase3) &v) const {
 
 #ifdef HAVE_EIGEN
   v_res._v.noalias() = v._v * _m.block<3, 3>(0, 0) + _m.block<1, 3>(3, 0);
-#else 
+#else
   v_res._v(0) = v._v(0)*_m(0, 0) + v._v(1)*_m(1, 0) + v._v(2)*_m(2, 0) + _m(3, 0);
   v_res._v(1) = v._v(0)*_m(0, 1) + v._v(1)*_m(1, 1) + v._v(2)*_m(2, 1) + _m(3, 1);
   v_res._v(2) = v._v(0)*_m(0, 2) + v._v(1)*_m(1, 2) + v._v(2)*_m(2, 2) + _m(3, 2);
 #endif  // HAVE_EIGEN
- 
+
   return v_res;
 }
 
@@ -888,17 +888,17 @@ INLINE_LINMATH FLOATNAME(LVecBase3) FLOATNAME(LMatrix4)::
 xform_vec(const FLOATNAME(LVecBase3) &v) const {
   TAU_PROFILE("LVecBase3 LMatrix4::xform_vec(const LVecBase3 &)", " ", TAU_USER);
   FLOATNAME(LVecBase3) v_res;
- 
+
   // v._v(3) == 0.0f for this case
- 
+
 #ifdef HAVE_EIGEN
   v_res._v.noalias() = v._v * _m.block<3, 3>(0, 0);
-#else 
+#else
   v_res._v(0) = v._v(0)*_m(0, 0) + v._v(1)*_m(1, 0) + v._v(2)*_m(2, 0);
   v_res._v(1) = v._v(0)*_m(0, 1) + v._v(1)*_m(1, 1) + v._v(2)*_m(2, 1);
   v_res._v(2) = v._v(0)*_m(0, 2) + v._v(1)*_m(1, 2) + v._v(2)*_m(2, 2);
 #endif  // HAVE_EIGEN
- 
+
   return v_res;
 }
 
@@ -934,7 +934,7 @@ xform_in_place(FLOATNAME(LVecBase4) &v) const {
 
 #ifdef HAVE_EIGEN
   v._v = v._v * _m;
-#else 
+#else
   v = xform(v);
 #endif  // HAVE_EIGEN
 }
@@ -953,7 +953,7 @@ xform_point_in_place(FLOATNAME(LVecBase3) &v) const {
 
 #ifdef HAVE_EIGEN
   v._v = v._v * _m.block<3, 3>(0, 0) + _m.block<1, 3>(3, 0);
-#else 
+#else
   v = xform_point(v);
 #endif  // HAVE_EIGEN
 }
@@ -981,10 +981,10 @@ INLINE_LINMATH void FLOATNAME(LMatrix4)::
 xform_vec_in_place(FLOATNAME(LVecBase3) &v) const {
   TAU_PROFILE("void LMatrix4::xform_vec_in_place(LVecBase3 &)", " ", TAU_USER);
   // v._v(3) == 0.0f for this case
- 
+
 #ifdef HAVE_EIGEN
   v._v = v._v * _m.block<3, 3>(0, 0);
-#else 
+#else
   v = xform_vec(v);
 #endif  // HAVE_EIGEN
 }
@@ -1325,14 +1325,12 @@ transpose_in_place() {
   _m.transposeInPlace();
 
 #else
-  #define SWAP__(x,y) { FLOATTYPE temp = (x);  (x) = (y);  (y) = temp;}
-  SWAP__(_m(0, 1),_m(1, 0));
-  SWAP__(_m(0, 2),_m(2, 0));
-  SWAP__(_m(0, 3),_m(3, 0));
-  SWAP__(_m(1, 2),_m(2, 1));
-  SWAP__(_m(1, 3),_m(3, 1));
-  SWAP__(_m(2, 3),_m(3, 2));
-  #undef SWAP__
+  std::swap(_m(0, 1), _m(1, 0));
+  std::swap(_m(0, 2), _m(2, 0));
+  std::swap(_m(0, 3), _m(3, 0));
+  std::swap(_m(1, 2), _m(2, 1));
+  std::swap(_m(1, 3), _m(3, 1));
+  std::swap(_m(2, 3), _m(3, 2));
 #endif  // HAVE_EIGEN
 }
 
@@ -1356,13 +1354,19 @@ INLINE_LINMATH bool FLOATNAME(LMatrix4)::
 invert_from(const FLOATNAME(LMatrix4) &other) {
   TAU_PROFILE("bool LMatrix4::invert_from(const LMatrix4 &)", " ", TAU_USER);
 #ifdef HAVE_EIGEN
+  // We use the squared nearly_zero value as determinant threshold
+  // for checking whether a matrix is singular, since that's the
+  // same constant we use in the non-Eigen case (see lmatrix3_src.I)
+  // and also because we otherwise run into issues very quickly.
   bool invertible;
-  other._m.computeInverseWithCheck(_m, invertible);
+  other._m.computeInverseWithCheck(_m, invertible,
+    NEARLY_ZERO(FLOATTYPE) * NEARLY_ZERO(FLOATTYPE));
 
   if (!invertible) {
-#ifdef NDEBUG
+#ifdef NOTIFY_DEBUG
     linmath_cat.warning() << "Tried to invert singular LMatrix4.\n";
 #endif
+
     (*this) = ident_mat();
     nassertr(!no_singular_invert, false);
   }
@@ -1382,9 +1386,10 @@ invert_from(const FLOATNAME(LMatrix4) &other) {
   int index[4];
 
   if (!decompose_mat(index)) {
-#ifdef NDEBUG
+#ifdef NOTIFY_DEBUG
     linmath_cat.warning() << "Tried to invert singular LMatrix4.\n";
 #endif
+
     (*this) = ident_mat();
     nassertr(!no_singular_invert, false);
     return false;
@@ -1441,11 +1446,11 @@ invert_affine_from(const FLOATNAME(LMatrix4) &other) {
   _m(3, 0) = -(other._m(3, 0) * _m(0, 0) +
                other._m(3, 1) * _m(1, 0) +
                other._m(3, 2) * _m(2, 0));
- 
+
   _m(3, 1) = -(other._m(3, 0) * _m(0, 1) +
                other._m(3, 1) * _m(1, 1) +
                other._m(3, 2) * _m(2, 1));
- 
+
   _m(3, 2) = -(other._m(3, 0) * _m(0, 2) +
                other._m(3, 1) * _m(1, 2) +
                other._m(3, 2) * _m(2, 2));
@@ -1481,17 +1486,17 @@ accumulate(const FLOATNAME(LMatrix4) &other, FLOATTYPE weight) {
   _m(0, 1) += other._m(0, 1) * weight;
   _m(0, 2) += other._m(0, 2) * weight;
   _m(0, 3) += other._m(0, 3) * weight;
- 
+
   _m(1, 0) += other._m(1, 0) * weight;
   _m(1, 1) += other._m(1, 1) * weight;
   _m(1, 2) += other._m(1, 2) * weight;
   _m(1, 3) += other._m(1, 3) * weight;
- 
+
   _m(2, 0) += other._m(2, 0) * weight;
   _m(2, 1) += other._m(2, 1) * weight;
   _m(2, 2) += other._m(2, 2) * weight;
   _m(2, 3) += other._m(2, 3) * weight;
- 
+
   _m(3, 0) += other._m(3, 0) * weight;
   _m(3, 1) += other._m(3, 1) * weight;
   _m(3, 2) += other._m(3, 2) * weight;