!>
!> SCHKST  checks the symmetric eigenvalue problem routines.
!>
!>    SSYTRD factors A as  U S U' , where ' means transpose,
!>    S is symmetric tridiagonal, and U is orthogonal.
!>    SSYTRD can use either just the lower or just the upper triangle
!>    of A; SCHKST checks both cases.
!>    U is represented as a product of Householder
!>    transformations, whose vectors are stored in the first
!>    n-1 columns of V, and whose scale factors are in TAU.
!>
!>    SSPTRD does the same as SSYTRD, except that A and V are stored
!>    in  format.
!>
!>    SORGTR constructs the matrix U from the contents of V and TAU.
!>
!>    SOPGTR constructs the matrix U from the contents of VP and TAU.
!>
!>    SSTEQR factors S as  Z D1 Z' , where Z is the orthogonal
!>    matrix of eigenvectors and D1 is a diagonal matrix with
!>    the eigenvalues on the diagonal.  D2 is the matrix of
!>    eigenvalues computed when Z is not computed.
!>
!>    SSTERF computes D3, the matrix of eigenvalues, by the
!>    PWK method, which does not yield eigenvectors.
!>
!>    SPTEQR factors S as  Z4 D4 Z4' , for a
!>    symmetric positive definite tridiagonal matrix.
!>    D5 is the matrix of eigenvalues computed when Z is not
!>    computed.
!>
!>    SSTEBZ computes selected eigenvalues.  WA1, WA2, and
!>    WA3 will denote eigenvalues computed to high
!>    absolute accuracy, with different range options.
!>    WR will denote eigenvalues computed to high relative
!>    accuracy.
!>
!>    SSTEIN computes Y, the eigenvectors of S, given the
!>    eigenvalues.
!>
!>    SSTEDC factors S as Z D1 Z' , where Z is the orthogonal
!>    matrix of eigenvectors and D1 is a diagonal matrix with
!>    the eigenvalues on the diagonal ('I' option). It may also
!>    update an input orthogonal matrix, usually the output
!>    from SSYTRD/SORGTR or SSPTRD/SOPGTR ('V' option). It may
!>    also just compute eigenvalues ('N' option).
!>
!>    SSTEMR factors S as Z D1 Z' , where Z is the orthogonal
!>    matrix of eigenvectors and D1 is a diagonal matrix with
!>    the eigenvalues on the diagonal ('I' option).  SSTEMR
!>    uses the Relatively Robust Representation whenever possible.
!>
!> When SCHKST 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 symmetric eigenroutines.  For each matrix, a number
!> of tests will be performed:
!>
!> (1)     | A - V S V' | / ( |A| n ulp ) SSYTRD( UPLO='U', ... )
!>
!> (2)     | I - UV' | / ( n ulp )        SORGTR( UPLO='U', ... )
!>
!> (3)     | A - V S V' | / ( |A| n ulp ) SSYTRD( UPLO='L', ... )
!>
!> (4)     | I - UV' | / ( n ulp )        SORGTR( UPLO='L', ... )
!>
!> (5-8)   Same as 1-4, but for SSPTRD and SOPGTR.
!>
!> (9)     | S - Z D Z' | / ( |S| n ulp ) SSTEQR('V',...)
!>
!> (10)    | I - ZZ' | / ( n ulp )        SSTEQR('V',...)
!>
!> (11)    | D1 - D2 | / ( |D1| ulp )        SSTEQR('N',...)
!>
!> (12)    | D1 - D3 | / ( |D1| ulp )        SSTERF
!>
!> (13)    0 if the true eigenvalues (computed by sturm count)
!>         of S are within THRESH of
!>         those in D1.  2*THRESH if they are not.  (Tested using
!>         SSTECH)
!>
!> For S positive definite,
!>
!> (14)    | S - Z4 D4 Z4' | / ( |S| n ulp ) SPTEQR('V',...)
!>
!> (15)    | I - Z4 Z4' | / ( n ulp )        SPTEQR('V',...)
!>
!> (16)    | D4 - D5 | / ( 100 |D4| ulp )       SPTEQR('N',...)
!>
!> When S is also diagonally dominant by the factor gamma < 1,
!>
!> (17)    max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
!>          i
!>         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
!>                                              SSTEBZ( 'A', 'E', ...)
!>
!> (18)    | WA1 - D3 | / ( |D3| ulp )          SSTEBZ( 'A', 'E', ...)
!>
!> (19)    ( max { min | WA2(i)-WA3(j) | } +
!>            i     j
!>           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
!>            i     j
!>                                              SSTEBZ( 'I', 'E', ...)
!>
!> (20)    | S - Y WA1 Y' | / ( |S| n ulp )  SSTEBZ, SSTEIN
!>
!> (21)    | I - Y Y' | / ( n ulp )          SSTEBZ, SSTEIN
!>
!> (22)    | S - Z D Z' | / ( |S| n ulp )    SSTEDC('I')
!>
!> (23)    | I - ZZ' | / ( n ulp )           SSTEDC('I')
!>
!> (24)    | S - Z D Z' | / ( |S| n ulp )    SSTEDC('V')
!>
!> (25)    | I - ZZ' | / ( n ulp )           SSTEDC('V')
!>
!> (26)    | D1 - D2 | / ( |D1| ulp )           SSTEDC('V') and
!>                                              SSTEDC('N')
!>
!> Test 27 is disabled at the moment because SSTEMR does not
!> guarantee high relatvie accuracy.
!>
!> (27)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
!>          i
!>         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
!>                                              SSTEMR('V', 'A')
!>
!> (28)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
!>          i
!>         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
!>                                              SSTEMR('V', 'I')
!>
!> Tests 29 through 34 are disable at present because SSTEMR
!> does not handle partial spectrum requests.
!>
!> (29)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'I')
!>
!> (30)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'I')
!>
!> (31)    ( max { min | WA2(i)-WA3(j) | } +
!>            i     j
!>           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
!>            i     j
!>         SSTEMR('N', 'I') vs. SSTEMR('V', 'I')
!>
!> (32)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'V')
!>
!> (33)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'V')
!>
!> (34)    ( max { min | WA2(i)-WA3(j) | } +
!>            i     j
!>           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
!>            i     j
!>         SSTEMR('N', 'V') vs. SSTEMR('V', 'V')
!>
!> (35)    | S - Z D Z' | / ( |S| n ulp )    SSTEMR('V', 'A')
!>
!> (36)    | I - ZZ' | / ( n ulp )           SSTEMR('V', 'A')
!>
!> (37)    ( max { min | WA2(i)-WA3(j) | } +
!>            i     j
!>           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
!>            i     j
!>         SSTEMR('N', 'A') vs. SSTEMR('V', 'A')
!>
!> 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 diagonal matrix with evenly spaced entries
!>      1, ..., ULP  and random signs.
!>      (ULP = (first number larger than 1) - 1 )
!> (4)  A diagonal matrix with geometrically spaced entries
!>      1, ..., ULP  and random signs.
!> (5)  A diagonal matrix with  entries 1, ULP, ..., ULP
!>      and random signs.
!>
!> (6)  Same as (4), but multiplied by SQRT( overflow threshold )
!> (7)  Same as (4), but multiplied by SQRT( underflow threshold )
!>
!> (8)  A matrix of the form  U' D U, where U is orthogonal and
!>      D has evenly spaced entries 1, ..., ULP with random signs
!>      on the diagonal.
!>
!> (9)  A matrix of the form  U' D U, where U is orthogonal and
!>      D has geometrically spaced entries 1, ..., ULP with random
!>      signs on the diagonal.
!>
!> (10) A matrix of the form  U' D U, where U is orthogonal and
!>      D has  entries 1, ULP,..., ULP with random
!>      signs on the diagonal.
!>
!> (11) Same as (8), but multiplied by SQRT( overflow threshold )
!> (12) Same as (8), but multiplied by SQRT( underflow threshold )
!>
!> (13) Symmetric matrix with random entries chosen from (-1,1).
!> (14) Same as (13), but multiplied by SQRT( overflow threshold )
!> (15) Same as (13), but multiplied by SQRT( underflow threshold )
!> (16) Same as (8), but diagonal elements are all positive.
!> (17) Same as (9), but diagonal elements are all positive.
!> (18) Same as (10), but diagonal elements are all positive.
!> (19) Same as (16), but multiplied by SQRT( overflow threshold )
!> (20) Same as (16), but multiplied by SQRT( underflow threshold )
!> (21) A diagonally dominant tridiagonal matrix with geometrically
!>      spaced diagonal entries 1, ..., ULP.
!> 

NSIZES

!>          NSIZES is INTEGER
!>          The number of sizes of matrices to use.  If it is zero,
!>          SCHKST 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, SCHKST
!>          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 SCHKST to continue the same random number
!>          sequence.
!> 

THRESH

!>          THRESH is 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.
!> 

NOUNIT

!>          NOUNIT is INTEGER
!>          The FORTRAN unit number for printing out error messages
!>          (e.g., if a routine returns IINFO not equal to 0.)
!> 

A

!>          A is REAL array of
!>                                  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.  It must be at
!>          least 1 and at least max( NN ).
!> 

AP

!>          AP is REAL array of
!>                      dimension( max(NN)*max(NN+1)/2 )
!>          The matrix A stored in packed format.
!> 

SD

!>          SD is REAL array of
!>                             dimension( max(NN) )
!>          The diagonal of the tridiagonal matrix computed by SSYTRD.
!>          On exit, SD and SE contain the tridiagonal form of the
!>          matrix in A.
!> 

SE

!>          SE is REAL array of
!>                             dimension( max(NN) )
!>          The off-diagonal of the tridiagonal matrix computed by
!>          SSYTRD.  On exit, SD and SE contain the tridiagonal form of
!>          the matrix in A.
!> 

D1

!>          D1 is REAL array of
!>                             dimension( max(NN) )
!>          The eigenvalues of A, as computed by SSTEQR simultaneously
!>          with Z.  On exit, the eigenvalues in D1 correspond with the
!>          matrix in A.
!> 

D2

!>          D2 is REAL array of
!>                             dimension( max(NN) )
!>          The eigenvalues of A, as computed by SSTEQR if Z is not
!>          computed.  On exit, the eigenvalues in D2 correspond with
!>          the matrix in A.
!> 

D3

!>          D3 is REAL array of
!>                             dimension( max(NN) )
!>          The eigenvalues of A, as computed by SSTERF.  On exit, the
!>          eigenvalues in D3 correspond with the matrix in A.
!> 

D4

!>          D4 is REAL array of
!>                             dimension( max(NN) )
!>          The eigenvalues of A, as computed by SPTEQR(V).
!>          ZPTEQR factors S as  Z4 D4 Z4*
!>          On exit, the eigenvalues in D4 correspond with the matrix in A.
!> 

D5

!>          D5 is REAL array of
!>                             dimension( max(NN) )
!>          The eigenvalues of A, as computed by SPTEQR(N)
!>          when Z is not computed. On exit, the
!>          eigenvalues in D4 correspond with the matrix in A.
!> 

WA1

!>          WA1 is REAL array of
!>                             dimension( max(NN) )
!>          All eigenvalues of A, computed to high
!>          absolute accuracy, with different range options.
!>          as computed by SSTEBZ.
!> 

WA2

!>          WA2 is REAL array of
!>                             dimension( max(NN) )
!>          Selected eigenvalues of A, computed to high
!>          absolute accuracy, with different range options.
!>          as computed by SSTEBZ.
!>          Choose random values for IL and IU, and ask for the
!>          IL-th through IU-th eigenvalues.
!> 

WA3

!>          WA3 is REAL array of
!>                             dimension( max(NN) )
!>          Selected eigenvalues of A, computed to high
!>          absolute accuracy, with different range options.
!>          as computed by SSTEBZ.
!>          Determine the values VL and VU of the IL-th and IU-th
!>          eigenvalues and ask for all eigenvalues in this range.
!> 

WR

!>          WR is REAL array of
!>                             dimension( max(NN) )
!>          All eigenvalues of A, computed to high
!>          absolute accuracy, with different options.
!>          as computed by SSTEBZ.
!> 

U

!>          U is REAL array of
!>                             dimension( LDU, max(NN) ).
!>          The orthogonal matrix computed by SSYTRD + SORGTR.
!> 

LDU

!>          LDU is INTEGER
!>          The leading dimension of U, Z, and V.  It must be at least 1
!>          and at least max( NN ).
!> 

V

!>          V is REAL array of
!>                             dimension( LDU, max(NN) ).
!>          The Housholder vectors computed by SSYTRD in reducing A to
!>          tridiagonal form.  The vectors computed with UPLO='U' are
!>          in the upper triangle, and the vectors computed with UPLO='L'
!>          are in the lower triangle.  (As described in SSYTRD, the
!>          sub- and superdiagonal are not set to 1, although the
!>          true Householder vector has a 1 in that position.  The
!>          routines that use V, such as SORGTR, set those entries to
!>          1 before using them, and then restore them later.)
!> 

VP

!>          VP is REAL array of
!>                      dimension( max(NN)*max(NN+1)/2 )
!>          The matrix V stored in packed format.
!> 

TAU

!>          TAU is REAL array of
!>                             dimension( max(NN) )
!>          The Householder factors computed by SSYTRD in reducing A
!>          to tridiagonal form.
!> 

Z

!>          Z is REAL array of
!>                             dimension( LDU, max(NN) ).
!>          The orthogonal matrix of eigenvectors computed by SSTEQR,
!>          SPTEQR, and SSTEIN.
!> 

WORK

!>          WORK is REAL array of
!>                      dimension( LWORK )
!> 

LWORK

!>          LWORK is INTEGER
!>          The number of entries in WORK.  This must be at least
!>          1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2
!>          where Nmax = max( NN(j), 2 ) and lg = log base 2.
!> 

IWORK

!>          IWORK is INTEGER array,
!>          Workspace.
!> 

LIWORK

!>          LIWORK is INTEGER
!>          The number of entries in IWORK.  This must be at least
!>                  6 + 6*Nmax + 5 * Nmax * lg Nmax
!>          where Nmax = max( NN(j), 2 ) and lg = log base 2.
!> 

RESULT

!>          RESULT is REAL array, dimension (26)
!>          The values computed by the tests described above.
!>          The values are currently limited to 1/ulp, to avoid
!>          overflow.
!> 

INFO

!>          INFO is INTEGER
!>          If 0, then everything ran OK.
!>           -1: NSIZES < 0
!>           -2: Some NN(j) < 0
!>           -3: NTYPES < 0
!>           -5: THRESH < 0
!>           -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
!>          -23: LDU < 1 or LDU < NMAX.
!>          -29: LWORK too small.
!>          If  SLATMR, SLATMS, SSYTRD, SORGTR, SSTEQR, SSTERF,
!>              or SORMC2 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.
!>       NTEST           The number of tests performed, or which can
!>                       be performed so far, for the current matrix.
!>       NTESTT          The total number of tests performed so far.
!>       NBLOCK          Blocksize as returned by ENVIR.
!>       NMAX            Largest value in NN.
!>       NMATS           The number of matrices generated so far.
!>       NERRS           The number of tests which have exceeded THRESH
!>                       so far.
!>       COND, 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  Square roots of the previous 2 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) )
!> 

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