Applied polar decomposition patch. Fixes Issue 621. Thanks to Christian for the report, Joshua for the fix, Dongsoo for checking the fix.

Applied picking cloth patch. Fixes Issue 646. Thanks to Dongsoo.
Applied patch Softbody updateConstraints. Fixes Issue 503. Thanks to Dave Bruce Phillips and Dongsoo.
Fix various warnigns under Mac OSX.

UnitTests/BulletUnitTests/TestPolarDecomposition.cpp

@@ -0,0 +1,103 @@
+#include "TestPolarDecomposition.h"
+#include "BulletSoftBody/btSoftBodyInternals.h"
+#include "LinearMath/btPolarDecomposition.h"
+#include "btCholeskyDecomposition.h"
+
+namespace
+{
+  const int MAX_ITERATIONS = btPolarDecomposition::DEFAULT_MAX_ITERATIONS;
+  const btScalar TOLERANCE = btPolarDecomposition::DEFAULT_TOLERANCE;
+}
+
+void TestPolarDecomposition::testZeroMatrix()
+{
+  const btMatrix3x3 A(0,0,0,0,0,0,0,0,0);
+  const int iterations = PolarDecompose(A, U, H);
+
+  // The decomposition should fail because the zero matrix is singular
+  CPPUNIT_ASSERT(iterations == MAX_ITERATIONS);
+}
+
+void TestPolarDecomposition::testSingularMatrix()
+{
+  const btMatrix3x3 A(1,0,0,1,0,0,0,0,1);
+
+  const int iterations = PolarDecompose(A, U, H);
+
+  // The cdecomposition should fail because the matrix is singular.
+  CPPUNIT_ASSERT(iterations == MAX_ITERATIONS);
+}
+
+void TestPolarDecomposition::testPoorlyConditionedMatrix()
+{
+  const btScalar e = btScalar(TOLERANCE);
+  const btMatrix3x3 A(1,0,0,1,e,0,0,0,1);
+
+  const int iterations = PolarDecompose(A, U, H);
+
+  // The decomposition should succeed, however, the matrix is poorly
+  // conditioned, i.e. as 'e' approaches zero, the matrix approaches a singular
+  // matrix (no inverse).
+  CPPUNIT_ASSERT(iterations < MAX_ITERATIONS);
+  CPPUNIT_ASSERT(equal(A, U * H));
+  CPPUNIT_ASSERT(isOrthogonal(U));
+  CPPUNIT_ASSERT(isSymmetric(H));
+  CPPUNIT_ASSERT(isPositiveDefinite(H));
+}
+
+void TestPolarDecomposition::testIdentityMatrix()
+{
+  const btMatrix3x3 A = I;
+  const int iterations = PolarDecompose(A, U, H);
+
+  // The identity is a special case. The decomposition should succeed and both
+  // the U and H matrices should be equal to the identity.
+  CPPUNIT_ASSERT(iterations < MAX_ITERATIONS);
+  CPPUNIT_ASSERT(equal(A, U * H));
+  CPPUNIT_ASSERT(equal(U, I));
+  CPPUNIT_ASSERT(equal(H, I));
+}
+
+void TestPolarDecomposition::testNonSingularMatrix()
+{
+  const btMatrix3x3 A(4, 6, 6, 9, 2, 0, 1, 6, 0);
+  const int iterations = PolarDecompose(A, U, H);
+
+  // The decomposition should succeed so that A = U*H, where U is orthogonal and
+  // H is symmetric and positive definite.
+  CPPUNIT_ASSERT(iterations < MAX_ITERATIONS);
+  CPPUNIT_ASSERT(equal(A, U * H));
+  CPPUNIT_ASSERT(isOrthogonal(U));
+  CPPUNIT_ASSERT(isSymmetric(H));
+  CPPUNIT_ASSERT(isPositiveDefinite(H));
+}
+
+bool TestPolarDecomposition::equal(const btMatrix3x3& A, const btMatrix3x3& B) const
+{
+  for (unsigned int i = 0; i < 3; ++i)
+    for (unsigned int j = 0; j < 3; ++j)
+      if (btFabs(A[i][j] - B[i][j]) > TOLERANCE)
+        return false;
+
+  return true;
+}
+
+bool TestPolarDecomposition::isOrthogonal(const btMatrix3x3& A) const
+{
+  return equal(A.transpose() * A, I);
+}
+
+bool TestPolarDecomposition::isPositiveDefinite(const btMatrix3x3& A) const
+{
+  static btMatrix3x3 storage;
+  return (0 == choleskyDecompose(A, storage));
+}
+
+bool TestPolarDecomposition::isSymmetric(const btMatrix3x3& A) const
+{
+  return 
+    (btFabs(A[0][1] - A[1][0]) < TOLERANCE) &&
+    (btFabs(A[0][2] - A[2][0]) < TOLERANCE) &&
+    (btFabs(A[1][2] - A[2][1]) < TOLERANCE);
+}
+