 
 *********
 EXAMPLE 1
 *********
 
 SYSTEM MATRIX A:
 
    0.100000E+01   0.500000E+00   0.333333E+00   0.250000E+00   0.200000E+00
    0.200000E+01   0.100000E+01   0.666667E+00   0.500000E+00   0.400000E+00
    0.000000E+00   0.150000E+01   0.100000E+01   0.750000E+00   0.600000E+00
    0.000000E+00   0.000000E+00   0.133333E+01   0.100000E+01   0.800000E+00
    0.000000E+00   0.000000E+00   0.000000E+00   0.125000E+01   0.100000E+01
 
 RESULT VECTOR F:
 
   -0.100000E+02
    0.195000E+02
   -0.256667E+02
    0.202500E+02
   -0.780000E+01
 
 EIGENVALUES OF (A - B*F): 
 
    0.500000E+01   0.000000E+00
    0.400000E+01   0.000000E+00
    0.300000E+01   0.000000E+00
    0.100000E+01   0.000000E+00
    0.200000E+01   0.000000E+00
 
   (THE MATRIX (A - B*F) SHOULD HAVE EIGENVALUES
   1.0, 2.0, 3.0, 4.0 and 5.0, BUT NOT NECESSARILY
   IN THIS ORDER. THIS EXAMPLE USES A SYSTEM IN
   CONTROLLABILITY FORM.)
 
 
 *********
 EXAMPLE 2
 *********
 
 SYSTEM MATRIX A:
 
    0.100000E+01   0.500000E+00   0.333333E+00   0.250000E+00   0.200000E+00
    0.200000E+01   0.100000E+01   0.666667E+00   0.500000E+00   0.400000E+00
    0.000000E+00   0.000000E+00   0.100000E+01   0.750000E+00   0.600000E+00
    0.000000E+00   0.000000E+00   0.133333E+01   0.100000E+01   0.800000E+00
    0.000000E+00   0.000000E+00   0.000000E+00   0.125000E+01   0.100000E+01
 SYSTEM IS CLOSE TO UNCONTROLLABLE
 USING TOLERANCE   0.166533453694E-14
    0 EIGENVALUES ALLOCATED.
 
 RESULT VECTOR F:
 
    0.000000E+00
    0.000000E+00
    0.000000E+00
    0.000000E+00
    0.000000E+00
 
 EIGENVALUES OF (A - B*F):
 
    0.319742E+01   0.000000E+00
    0.219512E+00   0.000000E+00
    0.158307E+01   0.000000E+00
   -0.130451E-14   0.271205E-07
   -0.130451E-14  -0.271205E-07
 
   (SINCE THE SYSTEM (b,A) IS UNCONTROLLABLE,
   NO EIGENVALUES ARE ALLOCATED.  THE VECTOR F
   SHOULD BE ZERO,  AND THE EIGENVALUES OF
   (A - B*F) SHOULD BE THOSE OF THE MATRIX A)
 
 
 *********
 EXAMPLE 3 
 *********
 
 SYSTEM MATRIX A:
 
   -0.200000E+01  -0.100000E+01  -0.666667E+00  -0.500000E+00  -0.400000E+00
   -0.200000E+02  -0.500000E+00  -0.333333E+00  -0.250000E+00  -0.200000E+00
    0.000000E+00   0.000000E+00   0.000000E+00   0.000000E+00   0.000000E+00
    0.100000E+01   0.500000E+00   0.333333E+00   0.250000E+00   0.200000E+00
    0.200000E+01   0.100000E+01   0.666667E+00   0.500000E+00   0.400000E+00
 SYSTEM IS CLOSE TO UNCONTROLLABLE
 USING TOLERANCE   0.104348655600E-13
    2 EIGENVALUES ALLOCATED.
 
 RESULT VECTOR F:
 
    0.614761E-01
   -0.549937E+00
   -0.150751E+00
   -0.369929E+00
   -0.604181E+00
 
 EIGENVALUES OF (A - B*F):
 
   -0.437294E+01   0.000000E+00
    0.200000E+01   0.000000E+00
    0.100000E+01   0.000000E+00
   -0.444089E-15   0.000000E+00
    0.693889E-16   0.000000E+00
 
 MACHINE UNIT:    4.44089209850063D-016
 
  (THE SUBROUTINE SHOULD ALLOCATE TWO
  EIGENVALUES AND THEN IF THE CORRECT MACHINE
  UNIT IS COMPUTED THE SUBROUTINE SHOULD
  DETECT AN UNCONTROLLABLE SYSTEM.
  NOTE THAT SOME MACHINES/COMPILERS MAY NOT COMPUTE
  THE CORRECT MACHINE UNIT. INSTEAD THEY MAY
  COMPUTE A MACHINE UNIT AFFECTED BY THE LENGTH OF
  THE REGISTER. IN SUCH A CASE THE SUBROUTINE MAY
  NOT DETECT UNCONTROLLABILITY AND IT MAY CONTINUE
  THE ALLOCATIONS WITH AN ILL-CONDITIONED
  EIGENVALUE ALLOCATION PROBLEM. THIS WILL RESULT
  TO AN INACCURATE  F  AND THUS TO INACCURATE
  EIGENVALUES FOR  (A - B*F).
  FOR MORE DETAILS SEE REFERENCES IN THE PAPER.
  IF UNCONTROLLABILITY IS DETECTED THE MATRIX
  (A - B*F)  SHOULD HAVE EIGENVALUES 1.0, 2.0,
  AND THREE OTHER UNSPECIFIED EIGENVALUES.)
 
 
 *********
 EXAMPLE 4 
 *********
 
 SYSTEM MATRIX A:
 
    0.200000E+01   0.225000E+02   0.133333E+01   0.125000E+01   0.120000E+01   0.116667E+01
    0.300000E+01   0.250000E+01   0.350000E+02   0.225000E+01   0.220000E+01   0.216667E+01
    0.400000E+01   0.350000E+01   0.333333E+01   0.487500E+02   0.320000E+01   0.316667E+01
    0.500000E+01   0.450000E+01   0.433333E+01   0.425000E+01   0.630000E+02   0.416667E+01
    0.600000E+01   0.550000E+01   0.533333E+01   0.525000E+01   0.520000E+01   0.775000E+02
    0.700000E+01   0.650000E+01   0.633333E+01   0.625000E+01   0.620000E+01   0.616667E+01
 
 RESULT VECTOR F:
 
   -0.752710E+00
    0.136653E+01
    0.155286E+01
    0.981425E+00
   -0.225093E+01
    0.123992E+01
 
 EIGENVALUES OF (A - B*F): 
 
    0.630000E+02   0.000000E+00
    0.700000E+01   0.280000E+02
    0.700000E+01  -0.280000E+02
   -0.240000E+02   0.000000E+00
   -0.150000E+02   0.200000E+02
   -0.150000E+02  -0.200000E+02
 
   (THE MATRIX (A - B*F) SHOULD HAVE
   EIGENVALUES 63.0, -24.0, 7.0 + 28.0i,
   7.0 - 28.0i, -15.0 + 20.0i, and -15.0 - 20.0i,
   IN ANY ORDER.)  THIS EXAMPLE ALLOCATES
   EIGENVALUES TO A GENERAL SYSTEM.
 
 
 *********
 EXAMPLE 5 
 *********
 
 SYSTEM MATRIX A:
 
    0.200000E+01   0.225000E+02   0.133333E+01   0.125000E+01   0.120000E+01   0.116667E+01   0.114286E+01   0.112500E+01
    0.300000E+01   0.250000E+01   0.350000E+02   0.225000E+01   0.220000E+01   0.216667E+01   0.214286E+01   0.212500E+01
    0.400000E+01   0.350000E+01   0.333333E+01   0.487500E+02   0.320000E+01   0.316667E+01   0.314286E+01   0.312500E+01
    0.500000E+01   0.450000E+01   0.433333E+01   0.425000E+01   0.630000E+02   0.416667E+01   0.414286E+01   0.412500E+01
    0.600000E+01   0.550000E+01   0.533333E+01   0.525000E+01   0.520000E+01   0.775000E+02   0.514286E+01   0.512500E+01
    0.700000E+01   0.650000E+01   0.633333E+01   0.625000E+01   0.620000E+01   0.616667E+01   0.921429E+02   0.612500E+01
    0.800000E+01   0.750000E+01   0.733333E+01   0.725000E+01   0.720000E+01   0.716667E+01   0.714286E+01   0.106875E+03
    0.900000E+01   0.850000E+01   0.833333E+01   0.825000E+01   0.820000E+01   0.816667E+01   0.814286E+01   0.812500E+01
 
 RESULT VECTOR F:
 
   -0.306557E+02
   -0.139444E+02
   -0.939827E+01
   -0.317777E+01
    0.147384E+01
    0.919658E+01
   -0.179245E+02
    0.192049E+03
 
 EIGENVALUES OF (A - B*F): 
 
    0.100000E+01   0.000000E+00
    0.830762E+01   0.148595E+00
    0.830762E+01  -0.148595E+00
    0.807486E+01   0.332714E+00
    0.807486E+01  -0.332714E+00
    0.778723E+01   0.265196E+00
    0.778723E+01  -0.265196E+00
    0.766058E+01   0.000000E+00
 
  (THIS EXAMPLE ALLOCATES THE EIGENVALUE 8.0
  SEVEN TIMES. HOWEVER, THE COMPUTED MULTIPLE
  EIGENVALUE IS NOT ACCURATE TO MACHINE PRECISION.
  THIS IS DUE TO THE FACT THAT UPPER HESSENBERG
  MATRICES WITH NON-ZERO SUBDIAGONAL ELEMENTS ARE
  DEFECTIVE IF THEY HAVE MULTIPLE EIGENVALUES. THIS
  RESULTS IN A JORDAN CANONICAL FORM WITH
  JORDAN BLOCKS AS LARGE AS THE MULTIPLICITY
  OF THE EIGENVALUES, INDICATING A MATRIX THAT
  IS ILL-CONDITIONED WITH RESPECT TO THE
  EIGENVALUE PROBLEM. THE VECTOR  F  IS ACCURATE,
  BUT THE COMPUTED EIGENVALUES FROM  DEIGVL  ARE
  NOT. THIS IS A RESULT OF THE EIGENPROBLEM
  OF  (A - B*F), AND NOT THE POLE PLACEMENT
  ALGORITHM ITSELF.  A COMPARISON OF THE SINGLE 
  AND DOUBLE PRECISION VERSIONS OF THIS PROGRAM 
  WILL DEMONSTRATE THIS DISTINCTION.)
 
