*> \brief \b DSYT22
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE DSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
*                          V, LDV, TAU, WORK, RESULT )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            ITYPE, KBAND, LDA, LDU, LDV, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
*      $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>      DSYT22  generally checks a decomposition of the form
*>
*>              A U = U S
*>
*>      where A is symmetric, the columns of U are orthonormal, and S
*>      is diagonal (if KBAND=0) or symmetric tridiagonal (if
*>      KBAND=1).  If ITYPE=1, then U is represented as a dense matrix,
*>      otherwise the U is expressed as a product of Householder
*>      transformations, whose vectors are stored in the array "V" and
*>      whose scaling constants are in "TAU"; we shall use the letter
*>      "V" to refer to the product of Householder transformations
*>      (which should be equal to U).
*>
*>      Specifically, if ITYPE=1, then:
*>
*>              RESULT(1) = | U' A U - S | / ( |A| m ulp ) *andC>              RESULT(2) = | I - U'U | / ( m ulp )
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \verbatim
*>  ITYPE   INTEGER
*>          Specifies the type of tests to be performed.
*>          1: U expressed as a dense orthogonal matrix:
*>             RESULT(1) = | A - U S U' | / ( |A| n ulp )   *andC>             RESULT(2) = | I - UU' | / ( n ulp )
*>
*>  UPLO    CHARACTER
*>          If UPLO='U', the upper triangle of A will be used and the
*>          (strictly) lower triangle will not be referenced.  If
*>          UPLO='L', the lower triangle of A will be used and the
*>          (strictly) upper triangle will not be referenced.
*>          Not modified.
*>
*>  N       INTEGER
*>          The size of the matrix.  If it is zero, DSYT22 does nothing.
*>          It must be at least zero.
*>          Not modified.
*>
*>  M       INTEGER
*>          The number of columns of U.  If it is zero, DSYT22 does
*>          nothing.  It must be at least zero.
*>          Not modified.
*>
*>  KBAND   INTEGER
*>          The bandwidth of the matrix.  It may only be zero or one.
*>          If zero, then S is diagonal, and E is not referenced.  If
*>          one, then S is symmetric tri-diagonal.
*>          Not modified.
*>
*>  A       DOUBLE PRECISION array, dimension (LDA , N)
*>          The original (unfactored) matrix.  It is assumed to be
*>          symmetric, and only the upper (UPLO='U') or only the lower
*>          (UPLO='L') will be referenced.
*>          Not modified.
*>
*>  LDA     INTEGER
*>          The leading dimension of A.  It must be at least 1
*>          and at least N.
*>          Not modified.
*>
*>  D       DOUBLE PRECISION array, dimension (N)
*>          The diagonal of the (symmetric tri-) diagonal matrix.
*>          Not modified.
*>
*>  E       DOUBLE PRECISION array, dimension (N)
*>          The off-diagonal of the (symmetric tri-) diagonal matrix.
*>          E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
*>          Not referenced if KBAND=0.
*>          Not modified.
*>
*>  U       DOUBLE PRECISION array, dimension (LDU, N)
*>          If ITYPE=1 or 3, this contains the orthogonal matrix in
*>          the decomposition, expressed as a dense matrix.  If ITYPE=2,
*>          then it is not referenced.
*>          Not modified.
*>
*>  LDU     INTEGER
*>          The leading dimension of U.  LDU must be at least N and
*>          at least 1.
*>          Not modified.
*>
*>  V       DOUBLE PRECISION array, dimension (LDV, N)
*>          If ITYPE=2 or 3, the lower triangle of this array contains
*>          the Householder vectors used to describe the orthogonal
*>          matrix in the decomposition.  If ITYPE=1, then it is not
*>          referenced.
*>          Not modified.
*>
*>  LDV     INTEGER
*>          The leading dimension of V.  LDV must be at least N and
*>          at least 1.
*>          Not modified.
*>
*>  TAU     DOUBLE PRECISION array, dimension (N)
*>          If ITYPE >= 2, then TAU(j) is the scalar factor of
*>          v(j) v(j)' in the Householder transformation H(j) of
*>          the product  U = H(1)...H(n-2)
*>          If ITYPE < 2, then TAU is not referenced.
*>          Not modified.
*>
*>  WORK    DOUBLE PRECISION array, dimension (2*N**2)
*>          Workspace.
*>          Modified.
*>
*>  RESULT  DOUBLE PRECISION array, dimension (2)
*>          The values computed by the two tests described above.  The
*>          values are currently limited to 1/ulp, to avoid overflow.
*>          RESULT(1) is always modified.  RESULT(2) is modified only
*>          if LDU is at least N.
*>          Modified.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup double_eig
*
*  =====================================================================
      SUBROUTINE DSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
     $                   V, LDV, TAU, WORK, RESULT )
*
*  -- LAPACK test routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            ITYPE, KBAND, LDA, LDU, LDV, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
     $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J, JJ, JJ1, JJ2, NN, NNP1
      DOUBLE PRECISION   ANORM, ULP, UNFL, WNORM
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH, DLANSY
      EXTERNAL           DLAMCH, DLANSY
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DORT01, DSYMM
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      RESULT( 1 ) = ZERO
      RESULT( 2 ) = ZERO
      IF( N.LE.0 .OR. M.LE.0 )
     $   RETURN
*
      UNFL = DLAMCH( 'Safe minimum' )
      ULP = DLAMCH( 'Precision' )
*
*     Do Test 1
*
*     Norm of A:
*
      ANORM = MAX( DLANSY( '1', UPLO, N, A, LDA, WORK ), UNFL )
*
*     Compute error matrix:
*
*     ITYPE=1: error = U' A U - S
*
      CALL DSYMM( 'L', UPLO, N, M, ONE, A, LDA, U, LDU, ZERO, WORK, N )
      NN = N*N
      NNP1 = NN + 1
      CALL DGEMM( 'T', 'N', M, M, N, ONE, U, LDU, WORK, N, ZERO,
     $            WORK( NNP1 ), N )
      DO 10 J = 1, M
         JJ = NN + ( J-1 )*N + J
         WORK( JJ ) = WORK( JJ ) - D( J )
   10 CONTINUE
      IF( KBAND.EQ.1 .AND. N.GT.1 ) THEN
         DO 20 J = 2, M
            JJ1 = NN + ( J-1 )*N + J - 1
            JJ2 = NN + ( J-2 )*N + J
            WORK( JJ1 ) = WORK( JJ1 ) - E( J-1 )
            WORK( JJ2 ) = WORK( JJ2 ) - E( J-1 )
   20    CONTINUE
      END IF
      WNORM = DLANSY( '1', UPLO, M, WORK( NNP1 ), N, WORK( 1 ) )
*
      IF( ANORM.GT.WNORM ) THEN
         RESULT( 1 ) = ( WNORM / ANORM ) / ( M*ULP )
      ELSE
         IF( ANORM.LT.ONE ) THEN
            RESULT( 1 ) = ( MIN( WNORM, M*ANORM ) / ANORM ) / ( M*ULP )
         ELSE
            RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( M ) ) / ( M*ULP )
         END IF
      END IF
*
*     Do Test 2
*
*     Compute  U'U - I
*
      IF( ITYPE.EQ.1 )
     $   CALL DORT01( 'Columns', N, M, U, LDU, WORK, 2*N*N,
     $                RESULT( 2 ) )
*
      RETURN
*
*     End of DSYT22
*
      END
