SDRVES checks the nonsymmetric eigenvalue (Schur form) problem


    driver SGEES.


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


    number of matrix 'types' are specified.  For each size ('n')


    and each type of matrix, one matrix will be generated and used


    to test the nonsymmetric eigenroutines.  For each matrix, 13


    tests will be performed:


    (1)     0 if T is in Schur form, 1/ulp otherwise


           (no sorting of eigenvalues)


    (2)     | A - VS T VS' | / ( n |A| ulp )


      Here VS is the matrix of Schur eigenvectors, and T is in Schur


      form  (no sorting of eigenvalues).


    (3)     | I - VS VS' | / ( n ulp ) (no sorting of eigenvalues).


    (4)     0     if WR+sqrt(-1)*WI are eigenvalues of T


            1/ulp otherwise


            (no sorting of eigenvalues)


    (5)     0     if T(with VS) = T(without VS),


            1/ulp otherwise


            (no sorting of eigenvalues)


    (6)     0     if eigenvalues(with VS) = eigenvalues(without VS),


            1/ulp otherwise


            (no sorting of eigenvalues)


    (7)     0 if T is in Schur form, 1/ulp otherwise


            (with sorting of eigenvalues)


    (8)     | A - VS T VS' | / ( n |A| ulp )


      Here VS is the matrix of Schur eigenvectors, and T is in Schur


      form  (with sorting of eigenvalues).


    (9)     | I - VS VS' | / ( n ulp ) (with sorting of eigenvalues).


    (10)    0     if WR+sqrt(-1)*WI are eigenvalues of T


            1/ulp otherwise


            (with sorting of eigenvalues)


    (11)    0     if T(with VS) = T(without VS),


            1/ulp otherwise


            (with sorting of eigenvalues)


    (12)    0     if eigenvalues(with VS) = eigenvalues(without VS),


            1/ulp otherwise


            (with sorting of eigenvalues)


    (13)    if sorting worked and SDIM is the number of


            eigenvalues which were SELECTed


    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

NSIZES

          NSIZES is INTEGER


          The number of sizes of matrices to use.  If it is zero,


          SDRVES does nothing.  It must be at least zero.

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.   If it is zero, SDRVES


          does nothing.  It must be at least zero.  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 SDRVES 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.

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))


          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 A, and H. LDA must be at


          least 1 and at least max(NN).

H

          H is REAL array, dimension (LDA, max(NN))


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

HT

          HT is REAL array, dimension (LDA, max(NN))


          Yet another copy of the test matrix A, modified by SGEES.

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.

WRT

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

WIT

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


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


          but those computed when SGEES only computes a partial


          eigendecomposition, i.e. not Schur vectors

VS

          VS is REAL array, dimension (LDVS, max(NN))


          VS holds the computed Schur vectors.

LDVS

          LDVS is INTEGER


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

RESULT

          RESULT is REAL array, dimension (13)


          The values computed by the 13 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


          5*NN(j)+2*NN(j)**2 for all j.

IWORK

          IWORK is INTEGER array, dimension (max(NN))

BWORK

          BWORK is LOGICAL array, dimension (max(NN))

INFO

          INFO is INTEGER


          If 0, then everything ran OK.


           -1: NSIZES < 0


           -2: Some NN(j) < 0


           -3: NTYPES < 0


           -6: THRESH < 0


           -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).


          -17: LDVS < 1 or LDVS < NMAX, where NMAX is max( NN(j) ).


          -20: NWORK too small.


          If  SLATMR, SLATMS, SLATME or SGEES returns an error code,


              the absolute value of it is returned.
-----------------------------------------------------------------------


     Some Local Variables and Parameters:


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


     ZERO, ONE       Real 0 and 1.


     MAXTYP          The number of types defined.


     NMAX            Largest value in NN.


     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.)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.