SGET23  checks the nonsymmetric eigenvalue problem driver SGEEVX.


    If COMP = .FALSE., the first 8 of the following tests will be


    performed on the input matrix A, and also test 9 if LWORK is


    sufficiently large.


    if COMP is .TRUE. all 11 tests will be performed.


    (1)     | A * VR - VR * W | / ( n |A| ulp )


      Here VR is the matrix of unit right eigenvectors.


      W is a block diagonal matrix, with a 1x1 block for each


      real eigenvalue and a 2x2 block for each complex conjugate


      pair.  If eigenvalues j and j+1 are a complex conjugate pair,


      so WR(j) = WR(j+1) = wr and WI(j) = - WI(j+1) = wi, then the


      2 x 2 block corresponding to the pair will be:


              (  wr  wi  )


              ( -wi  wr  )


      Such a block multiplying an n x 2 matrix  ( ur ui ) on the


      right will be the same as multiplying  ur + i*ui  by  wr + i*wi.


    (2)     | A**H * VL - VL * W**H | / ( n |A| ulp )


      Here VL is the matrix of unit left eigenvectors, A**H is the


      conjugate transpose of A, and W is as above.


    (3)     | |VR(i)| - 1 | / ulp and largest component real


      VR(i) denotes the i-th column of VR.


    (4)     | |VL(i)| - 1 | / ulp and largest component real


      VL(i) denotes the i-th column of VL.


    (5)     0 if W(full) = W(partial), 1/ulp otherwise


      W(full) denotes the eigenvalues computed when VR, VL, RCONDV


      and RCONDE are also computed, and W(partial) denotes the


      eigenvalues computed when only some of VR, VL, RCONDV, and


      RCONDE are computed.


    (6)     0 if VR(full) = VR(partial), 1/ulp otherwise


      VR(full) denotes the right eigenvectors computed when VL, RCONDV


      and RCONDE are computed, and VR(partial) denotes the result


      when only some of VL and RCONDV are computed.


    (7)     0 if VL(full) = VL(partial), 1/ulp otherwise


      VL(full) denotes the left eigenvectors computed when VR, RCONDV


      and RCONDE are computed, and VL(partial) denotes the result


      when only some of VR and RCONDV are computed.


    (8)     0 if SCALE, ILO, IHI, ABNRM (full) =


                 SCALE, ILO, IHI, ABNRM (partial)


            1/ulp otherwise


      SCALE, ILO, IHI and ABNRM describe how the matrix is balanced.


      (full) is when VR, VL, RCONDE and RCONDV are also computed, and


      (partial) is when some are not computed.


    (9)     0 if RCONDV(full) = RCONDV(partial), 1/ulp otherwise


      RCONDV(full) denotes the reciprocal condition numbers of the


      right eigenvectors computed when VR, VL and RCONDE are also


      computed. RCONDV(partial) denotes the reciprocal condition


      numbers when only some of VR, VL and RCONDE are computed.


   (10)     |RCONDV - RCDVIN| / cond(RCONDV)


      RCONDV is the reciprocal right eigenvector condition number


      computed by SGEEVX and RCDVIN (the precomputed true value)


      is supplied as input. cond(RCONDV) is the condition number of


      RCONDV, and takes errors in computing RCONDV into account, so


      that the resulting quantity should be O(ULP). cond(RCONDV) is


      essentially given by norm(A)/RCONDE.


   (11)     |RCONDE - RCDEIN| / cond(RCONDE)


      RCONDE is the reciprocal eigenvalue condition number


      computed by SGEEVX and RCDEIN (the precomputed true value)


      is supplied as input.  cond(RCONDE) is the condition number


      of RCONDE, and takes errors in computing RCONDE into account,


      so that the resulting quantity should be O(ULP). cond(RCONDE)


      is essentially given by norm(A)/RCONDV.

COMP

          COMP is LOGICAL


          COMP describes which input tests to perform:


            = .FALSE. if the computed condition numbers are not to


                      be tested against RCDVIN and RCDEIN


            = .TRUE.  if they are to be compared

BALANC

          BALANC is CHARACTER


          Describes the balancing option to be tested.


            = 'N' for no permuting or diagonal scaling


            = 'P' for permuting but no diagonal scaling


            = 'S' for no permuting but diagonal scaling


            = 'B' for permuting and diagonal scaling

JTYPE

          JTYPE is INTEGER


          Type of input matrix. Used to label output if error occurs.

THRESH

          THRESH is REAL


          A test will count as 'failed' if the 'error', computed as


          described above, exceeds THRESH.  Note that the error


          is scaled to be O(1), so THRESH should be a reasonably


          small multiple of 1, e.g., 10 or 100.  In particular,


          it should not depend on the precision (single vs. double)


          or the size of the matrix.  It must be at least zero.

ISEED

          ISEED is INTEGER array, dimension (4)


          If COMP = .FALSE., the random number generator seed


          used to produce matrix.


          If COMP = .TRUE., ISEED(1) = the number of the example.


          Used to label output if error occurs.

NOUNIT

          NOUNIT is INTEGER


          The FORTRAN unit number for printing out error messages


          (e.g., if a routine returns INFO not equal to 0.)

N

          N is INTEGER


          The dimension of A. N must be at least 0.

A

          A is REAL array, dimension (LDA,N)


          Used to hold the matrix whose eigenvalues are to be


          computed.

LDA

          LDA is INTEGER


          The leading dimension of A, and H. LDA must be at


          least 1 and at least N.

H

          H is REAL array, dimension (LDA,N)


          Another copy of the test matrix A, modified by SGEEVX.

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)


          The real and imaginary parts of the eigenvalues of A.


          On exit, WR + WI*i are the eigenvalues of the matrix in A.

WR1

          WR1 is REAL array, dimension (N)

WI1

          WI1 is REAL array, dimension (N)


          Like WR, WI, these arrays contain the eigenvalues of A,


          but those computed when SGEEVX only computes a partial


          eigendecomposition, i.e. not the eigenvalues and left


          and right eigenvectors.

VL

          VL is REAL array, dimension (LDVL,N)


          VL holds the computed left eigenvectors.

LDVL

          LDVL is INTEGER


          Leading dimension of VL. Must be at least max(1,N).

VR

          VR is REAL array, dimension (LDVR,N)


          VR holds the computed right eigenvectors.

LDVR

          LDVR is INTEGER


          Leading dimension of VR. Must be at least max(1,N).

LRE

          LRE is REAL array, dimension (LDLRE,N)


          LRE holds the computed right or left eigenvectors.

LDLRE

          LDLRE is INTEGER


          Leading dimension of LRE. Must be at least max(1,N).

RCONDV

          RCONDV is REAL array, dimension (N)


          RCONDV holds the computed reciprocal condition numbers


          for eigenvectors.

RCNDV1

          RCNDV1 is REAL array, dimension (N)


          RCNDV1 holds more computed reciprocal condition numbers


          for eigenvectors.

RCDVIN

          RCDVIN is REAL array, dimension (N)


          When COMP = .TRUE. RCDVIN holds the precomputed reciprocal


          condition numbers for eigenvectors to be compared with


          RCONDV.

RCONDE

          RCONDE is REAL array, dimension (N)


          RCONDE holds the computed reciprocal condition numbers


          for eigenvalues.

RCNDE1

          RCNDE1 is REAL array, dimension (N)


          RCNDE1 holds more computed reciprocal condition numbers


          for eigenvalues.

RCDEIN

          RCDEIN is REAL array, dimension (N)


          When COMP = .TRUE. RCDEIN holds the precomputed reciprocal


          condition numbers for eigenvalues to be compared with


          RCONDE.

SCALE

          SCALE is REAL array, dimension (N)


          Holds information describing balancing of matrix.

SCALE1

          SCALE1 is REAL array, dimension (N)


          Holds information describing balancing of matrix.

RESULT

          RESULT is REAL array, dimension (11)


          The values computed by the 11 tests described above.


          The values are currently limited to 1/ulp, to avoid


          overflow.

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER


          The number of entries in WORK.  This must be at least


          3*N, and 6*N+N**2 if tests 9, 10 or 11 are to be performed.

IWORK

          IWORK is INTEGER array, dimension (2*N)

INFO

          INFO is INTEGER


          If 0,  successful exit.


          If <0, input parameter -INFO had an incorrect value.


          If >0, SGEEVX returned an error code, the absolute


                 value of which is returned.

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