Warn about power of a matrix with non-semisimple 0 eigenvalue.

unsupported/Eigen/MatrixFunctions

@@ -262,6 +262,10 @@ where \f$ T_1 \f$ is invertible. Then \f$ T^p \f$ is given by
       0     & 0
     \end{array}. \right] \f]
 
+\warning Fractional power of a matrix with a non-semisimple zero
+eigenvalue is not well-defined. We introduce an assertion failure
+against inaccurate result, e.g. \include MatrixPower_failure.cpp
+
 Details of the algorithm can be found in: Nicholas J. Higham and
 Lijing Lin, "A Schur-Padé algorithm for fractional powers of a
 matrix," <em>SIAM J. %Matrix Anal. Applic.</em>,

unsupported/doc/examples/MatrixPower_failure.cpp

@@ -0,0 +1,20 @@
+#include <unsupported/Eigen/MatrixFunctions>
+#include <iostream>
+
+int main()
+{
+  Eigen::Matrix4d A;
+  A << 0, 0, 2, 3,
+       0, 0, 4, 5,
+       0, 0, 6, 7,
+       0, 0, 8, 9;
+  std::cout << A.pow(0.37) << std::endl;
+  
+  // The 1 makes eigenvalue 0 non-semisimple.
+  A.coeffRef(0, 1) = 1;
+
+  // This fails if EIGEN_NO_DEBUG is undefined.
+  std::cout << A.pow(0.37) << std::endl;
+
+  return 0;
+}