!>
!> DCHKSB2STG tests the reduction of a symmetric band matrix to tridiagonal
!> form, used with the symmetric eigenvalue problem.
!>
!> DSBTRD factors a symmetric band matrix A as  U S U' , where ' means
!> transpose, S is symmetric tridiagonal, and U is orthogonal.
!> DSBTRD can use either just the lower or just the upper triangle
!> of A; DCHKSB2STG checks both cases.
!>
!> DSYTRD_SB2ST factors a symmetric band matrix A as  U S U' ,
!> where ' means transpose, S is symmetric tridiagonal, and U is
!> orthogonal. DSYTRD_SB2ST can use either just the lower or just
!> the upper triangle of A; DCHKSB2STG checks both cases.
!>
!> DSTEQR factors S as  Z D1 Z'.
!> D1 is the matrix of eigenvalues computed when Z is not computed
!> and from the S resulting of DSBTRD  (used as reference for DSYTRD_SB2ST)
!> D2 is the matrix of eigenvalues computed when Z is not computed
!> and from the S resulting of DSYTRD_SB2ST .
!> D3 is the matrix of eigenvalues computed when Z is not computed
!> and from the S resulting of DSYTRD_SB2ST .
!>
!> When DCHKSB2STG is called, a number of matrix  (), a number
!> of bandwidths (), and a number of matrix  are
!> specified.  For each size (), each bandwidth () less than or
!> equal to , and each type of matrix, one matrix will be generated
!> and used to test the symmetric banded reduction routine.  For each
!> matrix, a number of tests will be performed:
!>
!> (1)     | A - V S V' | / ( |A| n ulp )  computed by DSBTRD with
!>                                         UPLO='U'
!>
!> (2)     | I - UU' | / ( n ulp )
!>
!> (3)     | A - V S V' | / ( |A| n ulp )  computed by DSBTRD with
!>                                         UPLO='L'
!>
!> (4)     | I - UU' | / ( n ulp )
!>
!> (5)     | D1 - D2 | / ( |D1| ulp )      where D1 is computed by
!>                                         DSBTRD with UPLO='U' and
!>                                         D2 is computed by
!>                                         DSYTRD_SB2ST with UPLO='U'
!>
!> (6)     | D1 - D3 | / ( |D1| ulp )      where D1 is computed by
!>                                         DSBTRD with UPLO='U' and
!>                                         D3 is computed by
!>                                         DSYTRD_SB2ST with UPLO='L'
!>
!> 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 )
!> 

NSIZES

!>          NSIZES is INTEGER
!>          The number of sizes of matrices to use.  If it is zero,
!>          DCHKSB2STG 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.
!> 

NWDTHS

!>          NWDTHS is INTEGER
!>          The number of bandwidths to use.  If it is zero,
!>          DCHKSB2STG does nothing.  It must be at least zero.
!> 

KK

!>          KK is INTEGER array, dimension (NWDTHS)
!>          An array containing the bandwidths to be used for the band
!>          matrices.  The values must be at least zero.
!> 

NTYPES

!>          NTYPES is INTEGER
!>          The number of elements in DOTYPE.   If it is zero, DCHKSB2STG
!>          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 DCHKSB2STG to continue the same random number
!>          sequence.
!> 

THRESH

!>          THRESH is DOUBLE PRECISION
!>          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 DOUBLE PRECISION array, dimension
!>                            (LDA, max(NN))
!>          Used to hold the matrix whose eigenvalues are to be
!>          computed.
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of A.  It must be at least 2 (not 1!)
!>          and at least max( KK )+1.
!> 

SD

!>          SD is DOUBLE PRECISION array, dimension (max(NN))
!>          Used to hold the diagonal of the tridiagonal matrix computed
!>          by DSBTRD.
!> 

SE

!>          SE is DOUBLE PRECISION array, dimension (max(NN))
!>          Used to hold the off-diagonal of the tridiagonal matrix
!>          computed by DSBTRD.
!> 

D1

!>          D1 is DOUBLE PRECISION array, dimension (max(NN))
!> 

D2

!>          D2 is DOUBLE PRECISION array, dimension (max(NN))
!> 

D3

!>          D3 is DOUBLE PRECISION array, dimension (max(NN))
!> 

U

!>          U is DOUBLE PRECISION array, dimension (LDU, max(NN))
!>          Used to hold the orthogonal matrix computed by DSBTRD.
!> 

LDU

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

WORK

!>          WORK is DOUBLE PRECISION array, dimension (LWORK)
!> 

LWORK

!>          LWORK is INTEGER
!>          The number of entries in WORK.  This must be at least
!>          max( LDA+1, max(NN)+1 )*max(NN).
!> 

RESULT

!>          RESULT is DOUBLE PRECISION array, dimension (4)
!>          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.
!>
!>-----------------------------------------------------------------------
!>
!>       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.
!>       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) )
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.