!>
!>    SCHKHS  checks the nonsymmetric eigenvalue problem routines.
!>
!>            SGEHRD factors A as  U H U' , where ' means transpose,
!>            H is hessenberg, and U is an orthogonal matrix.
!>
!>            SORGHR generates the orthogonal matrix U.
!>
!>            SORMHR multiplies a matrix by the orthogonal matrix U.
!>
!>            SHSEQR factors H as  Z T Z' , where Z is orthogonal and
!>            T is , and the eigenvalue vector W.
!>
!>            STREVC computes the left and right eigenvector matrices
!>            L and R for T.
!>
!>            SHSEIN computes the left and right eigenvector matrices
!>            Y and X for H, using inverse iteration.
!>
!>            STREVC3 computes left and right eigenvector matrices
!>            from a Schur matrix T and backtransforms them with Z
!>            to eigenvector matrices L and R for A. L and R are
!>            GE matrices.
!>
!>    When SCHKHS is called, a number of matrix  () and a
!>    number of matrix  are specified.  For each size ()
!>    and each type of matrix, one matrix will be generated and used
!>    to test the nonsymmetric eigenroutines.  For each matrix, 16
!>    tests will be performed:
!>
!>    (1)     | A - U H U**T | / ( |A| n ulp )
!>
!>    (2)     | I - UU**T | / ( n ulp )
!>
!>    (3)     | H - Z T Z**T | / ( |H| n ulp )
!>
!>    (4)     | I - ZZ**T | / ( n ulp )
!>
!>    (5)     | A - UZ H (UZ)**T | / ( |A| n ulp )
!>
!>    (6)     | I - UZ (UZ)**T | / ( n ulp )
!>
!>    (7)     | T(Z computed) - T(Z not computed) | / ( |T| ulp )
!>
!>    (8)     | W(Z computed) - W(Z not computed) | / ( |W| ulp )
!>
!>    (9)     | TR - RW | / ( |T| |R| ulp )
!>
!>    (10)    | L**H T - W**H L | / ( |T| |L| ulp )
!>
!>    (11)    | HX - XW | / ( |H| |X| ulp )
!>
!>    (12)    | Y**H H - W**H Y | / ( |H| |Y| ulp )
!>
!>    (13)    | AX - XW | / ( |A| |X| ulp )
!>
!>    (14)    | Y**H A - W**H Y | / ( |A| |Y| ulp )
!>
!>    (15)    | AR - RW | / ( |A| |R| ulp )
!>
!>    (16)    | LA - WL | / ( |A| |L| ulp )
!>
!>    The  are specified by an array NN(1:NSIZES); the value of
!>    each element NN(j) specifies one size.
!>    The  are specified by a logical array DOTYPE( 1:NTYPES );
!>    if DOTYPE(j) is .TRUE., then matrix type  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  entries 1, ULP, ..., ULP
!>         and random signs.
!>
!>    (7)  Same as (4), but multiplied by SQRT( overflow threshold )
!>    (8)  Same as (4), but multiplied by SQRT( 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  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  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 SQRT( overflow threshold )
!>    (18) Same as (16), but multiplied by SQRT( underflow threshold )
!>
!>    (19) Nonsymmetric matrix with random entries chosen from (-1,1).
!>    (20) Same as (19), but multiplied by SQRT( overflow threshold )
!>    (21) Same as (19), but multiplied by SQRT( underflow threshold )
!> 

!>  NSIZES - INTEGER
!>           The number of sizes of matrices to use.  If it is zero,
!>           SCHKHS does nothing.  It must be at least zero.
!>           Not modified.
!>
!>  NN     - 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.
!>           Not modified.
!>
!>  NTYPES - INTEGER
!>           The number of elements in DOTYPE.   If it is zero, SCHKHS
!>           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. .
!>           Not modified.
!>
!>  DOTYPE - 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.
!>           Not modified.
!>
!>  ISEED  - 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 SCHKHS to continue the same random number
!>           sequence.
!>           Modified.
!>
!>  THRESH - REAL
!>           A test will count as  if the , 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.
!>           Not modified.
!>
!>  NOUNIT - INTEGER
!>           The FORTRAN unit number for printing out error messages
!>           (e.g., if a routine returns IINFO not equal to 0.)
!>           Not modified.
!>
!>  A      - 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.
!>           Modified.
!>
!>  LDA    - INTEGER
!>           The leading dimension of A, H, T1 and T2.  It must be at
!>           least 1 and at least max( NN ).
!>           Not modified.
!>
!>  H      - REAL array, dimension (LDA,max(NN))
!>           The upper hessenberg matrix computed by SGEHRD.  On exit,
!>           H contains the Hessenberg form of the matrix in A.
!>           Modified.
!>
!>  T1     - REAL array, dimension (LDA,max(NN))
!>           The Schur (=) matrix computed by SHSEQR
!>           if Z is computed.  On exit, T1 contains the Schur form of
!>           the matrix in A.
!>           Modified.
!>
!>  T2     - REAL array, dimension (LDA,max(NN))
!>           The Schur matrix computed by SHSEQR when Z is not computed.
!>           This should be identical to T1.
!>           Modified.
!>
!>  LDU    - INTEGER
!>           The leading dimension of U, Z, UZ and UU.  It must be at
!>           least 1 and at least max( NN ).
!>           Not modified.
!>
!>  U      - REAL array, dimension (LDU,max(NN))
!>           The orthogonal matrix computed by SGEHRD.
!>           Modified.
!>
!>  Z      - REAL array, dimension (LDU,max(NN))
!>           The orthogonal matrix computed by SHSEQR.
!>           Modified.
!>
!>  UZ     - REAL array, dimension (LDU,max(NN))
!>           The product of U times Z.
!>           Modified.
!>
!>  WR1    - REAL array, dimension (max(NN))
!>  WI1    - REAL array, dimension (max(NN))
!>           The real and imaginary parts of the eigenvalues of A,
!>           as computed when Z is computed.
!>           On exit, WR1 + WI1*i are the eigenvalues of the matrix in A.
!>           Modified.
!>
!>  WR2    - REAL array, dimension (max(NN))
!>  WI2    - REAL array, dimension (max(NN))
!>           The real and imaginary parts of the eigenvalues of A,
!>           as computed when T is computed but not Z.
!>           On exit, WR2 + WI2*i are the eigenvalues of the matrix in A.
!>           Modified.
!>
!>  WR3    - REAL array, dimension (max(NN))
!>  WI3    - REAL array, dimension (max(NN))
!>           Like WR1, WI1, these arrays contain the eigenvalues of A,
!>           but those computed when SHSEQR only computes the
!>           eigenvalues, i.e., not the Schur vectors and no more of the
!>           Schur form than is necessary for computing the
!>           eigenvalues.
!>           Modified.
!>
!>  EVECTL - REAL array, dimension (LDU,max(NN))
!>           The (upper triangular) left eigenvector matrix for the
!>           matrix in T1.  For complex conjugate pairs, the real part
!>           is stored in one row and the imaginary part in the next.
!>           Modified.
!>
!>  EVECTR - REAL array, dimension (LDU,max(NN))
!>           The (upper triangular) right eigenvector matrix for the
!>           matrix in T1.  For complex conjugate pairs, the real part
!>           is stored in one column and the imaginary part in the next.
!>           Modified.
!>
!>  EVECTY - REAL array, dimension (LDU,max(NN))
!>           The left eigenvector matrix for the
!>           matrix in H.  For complex conjugate pairs, the real part
!>           is stored in one row and the imaginary part in the next.
!>           Modified.
!>
!>  EVECTX - REAL array, dimension (LDU,max(NN))
!>           The right eigenvector matrix for the
!>           matrix in H.  For complex conjugate pairs, the real part
!>           is stored in one column and the imaginary part in the next.
!>           Modified.
!>
!>  UU     - REAL array, dimension (LDU,max(NN))
!>           Details of the orthogonal matrix computed by SGEHRD.
!>           Modified.
!>
!>  TAU    - REAL array, dimension(max(NN))
!>           Further details of the orthogonal matrix computed by SGEHRD.
!>           Modified.
!>
!>  WORK   - REAL array, dimension (NWORK)
!>           Workspace.
!>           Modified.
!>
!>  NWORK  - INTEGER
!>           The number of entries in WORK.  NWORK >= 4*NN(j)*NN(j) + 2.
!>
!>  IWORK  - INTEGER array, dimension (max(NN))
!>           Workspace.
!>           Modified.
!>
!>  SELECT - LOGICAL array, dimension (max(NN))
!>           Workspace.
!>           Modified.
!>
!>  RESULT - REAL array, dimension (16)
!>           The values computed by the fourteen tests described above.
!>           The values are currently limited to 1/ulp, to avoid
!>           overflow.
!>           Modified.
!>
!>  INFO   - 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) ).
!>           -14: LDU < 1 or LDU < NMAX.
!>           -28: NWORK too small.
!>           If  SLATMR, SLATMS, or SLATME returns an error code, the
!>               absolute value of it is returned.
!>           If 1, then SHSEQR could not find all the shifts.
!>           If 2, then the EISPACK code (for small blocks) failed.
!>           If >2, then 30*N iterations were not enough to find an
!>               eigenvalue or to decompose the problem.
!>           Modified.
!>
!>-----------------------------------------------------------------------
!>
!>     Some Local Variables and Parameters:
!>     ---- ----- --------- --- ----------
!>
!>     ZERO, ONE       Real 0 and 1.
!>     MAXTYP          The number of types defined.
!>     MTEST           The number of tests defined: care must be taken
!>                     that (1) the size of RESULT, (2) the number of
!>                     tests actually performed, and (3) MTEST agree.
!>     NTEST           The number of tests performed on this matrix
!>                     so far.  This should be less than MTEST, and
!>                     equal to it by the last test.  It will be less
!>                     if any of the routines being tested indicates
!>                     that it could not compute the matrices that
!>                     would be tested.
!>     NMAX            Largest value in NN.
!>     NMATS           The number of matrices generated so far.
!>     NERRS           The number of tests which have exceeded THRESH
!>                     so far (computed by SLAFTS).
!>     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.
!>     RTOVFL, RTUNFL,
!>     RTULP, RTULPI   Square roots of the previous 4 values.
!>
!>             The following four arrays decode JTYPE:
!>     KTYPE(j)        The general type (1-10) for type .
!>     KMODE(j)        The MODE value to be passed to the matrix
!>                     generator for type .
!>     KMAGN(j)        The order of magnitude ( O(1),
!>                     O(overflow^(1/2) ), O(underflow^(1/2) )
!>     KCONDS(j)       Selects 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.