SDRVVX  checks the nonsymmetric eigenvalue problem expert driver


    SGEEVX.


    SDRVVX uses both test matrices generated randomly depending on


    data supplied in the calling sequence, as well as on data


    read from an input file and including precomputed condition


    numbers to which it compares the ones it computes.


    When SDRVVX is called, a number of matrix 'sizes' ('n's') and a


    number of matrix 'types' are specified in the calling sequence.


    For each size ('n') and each type of matrix, one matrix will be


    generated and used to test the nonsymmetric eigenroutines.  For


    each matrix, 9 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)     W(full) = W(partial)


      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)     VR(full) = VR(partial)


      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)     VL(full) = VL(partial)


      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)     RCONDV(full) = RCONDV(partial)


      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.


    The 'sizes' are specified by an array NN(1:NSIZES); the value of


    each element NN(j) specifies one size.


    The 'types' are specified by a logical array DOTYPE( 1:NTYPES );


    if DOTYPE(j) is .TRUE., then matrix type 'j' will be generated.


    Currently, the list of possible types is:


    (1)  The zero matrix.


    (2)  The identity matrix.


    (3)  A (transposed) Jordan block, with 1's on the diagonal.


    (4)  A diagonal matrix with evenly spaced entries


         1, ..., ULP  and random signs.


         (ULP = (first number larger than 1) - 1 )


    (5)  A diagonal matrix with geometrically spaced entries


         1, ..., ULP  and random signs.


    (6)  A diagonal matrix with 'clustered' entries 1, ULP, ..., ULP


         and random signs.


    (7)  Same as (4), but multiplied by a constant near


         the overflow threshold


    (8)  Same as (4), but multiplied by a constant near


         the underflow threshold


    (9)  A matrix of the form  U' T U, where U is orthogonal and


         T has evenly spaced entries 1, ..., ULP with random signs


         on the diagonal and random O(1) entries in the upper


         triangle.


    (10) A matrix of the form  U' T U, where U is orthogonal and


         T has geometrically spaced entries 1, ..., ULP with random


         signs on the diagonal and random O(1) entries in the upper


         triangle.


    (11) A matrix of the form  U' T U, where U is orthogonal and


         T has 'clustered' entries 1, ULP,..., ULP with random


         signs on the diagonal and random O(1) entries in the upper


         triangle.


    (12) A matrix of the form  U' T U, where U is orthogonal and


         T has real or complex conjugate paired eigenvalues randomly


         chosen from ( ULP, 1 ) and random O(1) entries in the upper


         triangle.


    (13) A matrix of the form  X' T X, where X has condition


         SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP


         with random signs on the diagonal and random O(1) entries


         in the upper triangle.


    (14) A matrix of the form  X' T X, where X has condition


         SQRT( ULP ) and T has geometrically spaced entries


         1, ..., ULP with random signs on the diagonal and random


         O(1) entries in the upper triangle.


    (15) A matrix of the form  X' T X, where X has condition


         SQRT( ULP ) and T has 'clustered' entries 1, ULP,..., ULP


         with random signs on the diagonal and random O(1) entries


         in the upper triangle.


    (16) A matrix of the form  X' T X, where X has condition


         SQRT( ULP ) and T has real or complex conjugate paired


         eigenvalues randomly chosen from ( ULP, 1 ) and random


         O(1) entries in the upper triangle.


    (17) Same as (16), but multiplied by a constant


         near the overflow threshold


    (18) Same as (16), but multiplied by a constant


         near the underflow threshold


    (19) Nonsymmetric matrix with random entries chosen from (-1,1).


         If N is at least 4, all entries in first two rows and last


         row, and first column and last two columns are zero.


    (20) Same as (19), but multiplied by a constant


         near the overflow threshold


    (21) Same as (19), but multiplied by a constant


         near the underflow threshold


    In addition, an input file will be read from logical unit number


    NIUNIT. The file contains matrices along with precomputed


    eigenvalues and reciprocal condition numbers for the eigenvalues


    and right eigenvectors. For these matrices, in addition to tests


    (1) to (9) we will compute the following two tests:


   (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.

NSIZES

          NSIZES is INTEGER


          The number of sizes of matrices to use.  NSIZES must be at


          least zero. If it is zero, no randomly generated matrices


          are tested, but any test matrices read from NIUNIT will be


          tested.

NN

          NN is INTEGER array, dimension (NSIZES)


          An array containing the sizes to be used for the matrices.


          Zero values will be skipped.  The values must be at least


          zero.

NTYPES

          NTYPES is INTEGER


          The number of elements in DOTYPE. NTYPES must be at least


          zero. If it is zero, no randomly generated test matrices


          are tested, but and test matrices read from NIUNIT will be


          tested. If it is MAXTYP+1 and NSIZES is 1, then an


          additional type, MAXTYP+1 is defined, which is to use


          whatever matrix is in A.  This is only useful if


          DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .

DOTYPE

          DOTYPE is LOGICAL array, dimension (NTYPES)


          If DOTYPE(j) is .TRUE., then for each size in NN a


          matrix of that size and of type j will be generated.


          If NTYPES is smaller than the maximum number of types


          defined (PARAMETER MAXTYP), then types NTYPES+1 through


          MAXTYP will not be generated.  If NTYPES is larger


          than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)


          will be ignored.

ISEED

          ISEED is INTEGER array, dimension (4)


          On entry ISEED specifies the seed of the random number


          generator. The array elements should be between 0 and 4095;


          if not they will be reduced mod 4096.  Also, ISEED(4) must


          be odd.  The random number generator uses a linear


          congruential sequence limited to small integers, and so


          should produce machine independent random numbers. The


          values of ISEED are changed on exit, and can be used in the


          next call to SDRVVX to continue the same random number


          sequence.

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.

NIUNIT

          NIUNIT is INTEGER


          The FORTRAN unit number for reading in the data file of


          problems to solve.

NOUNIT

          NOUNIT is INTEGER


          The FORTRAN unit number for printing out error messages


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

A

          A is REAL array, dimension


                      (LDA, max(NN,12))


          Used to hold the matrix whose eigenvalues are to be


          computed.  On exit, A contains the last matrix actually used.

LDA

          LDA is INTEGER


          The leading dimension of the arrays A and H.


          LDA >= max(NN,12), since 12 is the dimension of the largest


          matrix in the precomputed input file.

H

          H is REAL array, dimension


                      (LDA, max(NN,12))


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

WR

          WR is REAL array, dimension (max(NN))

WI

          WI is REAL array, dimension (max(NN))


          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 (max(NN,12))

WI1

          WI1 is REAL array, dimension (max(NN,12))


          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, max(NN,12))


          VL holds the computed left eigenvectors.

LDVL

          LDVL is INTEGER


          Leading dimension of VL. Must be at least max(1,max(NN,12)).

VR

          VR is REAL array, dimension


                      (LDVR, max(NN,12))


          VR holds the computed right eigenvectors.

LDVR

          LDVR is INTEGER


          Leading dimension of VR. Must be at least max(1,max(NN,12)).

LRE

          LRE is REAL array, dimension


                      (LDLRE, max(NN,12))


          LRE holds the computed right or left eigenvectors.

LDLRE

          LDLRE is INTEGER


          Leading dimension of LRE. Must be at least max(1,max(NN,12))

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 seven tests described above.


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

WORK

          WORK is REAL array, dimension (NWORK)

NWORK

          NWORK is INTEGER


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


          max(6*12+2*12**2,6*NN(j)+2*NN(j)**2) =


          max(    360     ,6*NN(j)+2*NN(j)**2)    for all j.

IWORK

          IWORK is INTEGER array, dimension (2*max(NN,12))

INFO

          INFO is INTEGER


          If 0,  then successful exit.


          If <0, then input parameter -INFO is incorrect.


          If >0, SLATMR, SLATMS, SLATME or SGET23 returned an error


                 code, and INFO is its absolute value.
-----------------------------------------------------------------------


     Some Local Variables and Parameters:


     ---- ----- --------- --- ----------


     ZERO, ONE       Real 0 and 1.


     MAXTYP          The number of types defined.


     NMAX            Largest value in NN or 12.


     NERRS           The number of tests which have exceeded THRESH


     COND, CONDS,


     IMODE           Values to be passed to the matrix generators.


     ANORM           Norm of A; passed to matrix generators.


     OVFL, UNFL      Overflow and underflow thresholds.


     ULP, ULPINV     Finest relative precision and its inverse.


     RTULP, RTULPI   Square roots of the previous 4 values.


             The following four arrays decode JTYPE:


     KTYPE(j)        The general type (1-10) for type 'j'.


     KMODE(j)        The MODE value to be passed to the matrix


                     generator for type 'j'.


     KMAGN(j)        The order of magnitude ( O(1),


                     O(overflow^(1/2) ), O(underflow^(1/2) )


     KCONDS(j)       Selectw whether CONDS is to be 1 or


                     1/sqrt(ulp).  (0 means irrelevant.)

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