      SUBROUTINE DSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V,
     $                   LDV, TAU, WORK, RESULT )
*
*  -- LAPACK test routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            ITYPE, KBAND, LDA, LDU, LDV, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
     $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  DSYT21 generally checks a decomposition of the form
*
*     A = U S U'
*
*  where ' means transpose, A is symmetric, U is orthogonal, 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 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) = | A - U S U' | / ( |A| n ulp ) *and*
*     RESULT(2) = | I - UU' | / ( n ulp )
*
*  If ITYPE=2, then:
*
*     RESULT(1) = | A - V S V' | / ( |A| n ulp )
*
*  If ITYPE=3, then:
*
*     RESULT(1) = | I - VU' | / ( n ulp )
*
*  For ITYPE > 1, the transformation U is expressed as a product
*  V = H(1)...H(n-2),  where H(j) = I  -  tau(j) v(j) v(j)' and each
*  vector v(j) has its first j elements 0 and the remaining n-j elements
*  stored in V(j+1:n,j).
*
*  Arguments
*  =========
*
*  ITYPE   (input) 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 )   *and*
*             RESULT(2) = | I - UU' | / ( n ulp )
*
*          2: U expressed as a product V of Housholder transformations:
*             RESULT(1) = | A - V S V' | / ( |A| n ulp )
*
*          3: U expressed both as a dense orthogonal matrix and
*             as a product of Housholder transformations:
*             RESULT(1) = | I - VU' | / ( n ulp )
*
*  UPLO    (input) CHARACTER
*          If UPLO='U', the upper triangle of A and V will be used and
*          the (strictly) lower triangle will not be referenced.
*          If UPLO='L', the lower triangle of A and V will be used and
*          the (strictly) upper triangle will not be referenced.
*
*  N       (input) INTEGER
*          The size of the matrix.  If it is zero, DSYT21 does nothing.
*          It must be at least zero.
*
*  KBAND   (input) 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.
*
*  A       (input) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of A.  It must be at least 1
*          and at least N.
*
*  D       (input) DOUBLE PRECISION array, dimension (N)
*          The diagonal of the (symmetric tri-) diagonal matrix.
*
*  E       (input) DOUBLE PRECISION array, dimension (N-1)
*          The off-diagonal of the (symmetric tri-) diagonal matrix.
*          E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
*          (3,2) element, etc.
*          Not referenced if KBAND=0.
*
*  U       (input) 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.
*
*  LDU     (input) INTEGER
*          The leading dimension of U.  LDU must be at least N and
*          at least 1.
*
*  V       (input) DOUBLE PRECISION array, dimension (LDV, N)
*          If ITYPE=2 or 3, the columns of this array contain the
*          Householder vectors used to describe the orthogonal matrix
*          in the decomposition.  If UPLO='L', then the vectors are in
*          the lower triangle, if UPLO='U', then in the upper
*          triangle.
*          *NOTE* If ITYPE=2 or 3, V is modified and restored.  The
*          subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
*          is set to one, and later reset to its original value, during
*          the course of the calculation.
*          If ITYPE=1, then it is neither referenced nor modified.
*
*  LDV     (input) INTEGER
*          The leading dimension of V.  LDV must be at least N and
*          at least 1.
*
*  TAU     (input) 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.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N**2)
*
*  RESULT  (output) 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 ITYPE=1.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TEN
      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TEN = 10.0D0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER
      CHARACTER          CUPLO
      INTEGER            IINFO, J, JCOL, JR, JROW
      DOUBLE PRECISION   ANORM, ULP, UNFL, VSAVE, WNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, DLANGE, DLANSY
      EXTERNAL           LSAME, DLAMCH, DLANGE, DLANSY
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DLACPY, DLARFY, DLASET, DORM2L, DORM2R,
     $                   DSYR, DSYR2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          DBLE, MAX, MIN
*     ..
*     .. Executable Statements ..
*
      WNORM = ZERO
      RESULT( 1 ) = ZERO
      IF( ITYPE.EQ.1 )
     $   RESULT( 2 ) = ZERO
      IF( N.LE.0 )
     $   RETURN
*
      IF( LSAME( UPLO, 'U' ) ) THEN
         LOWER = .FALSE.
         CUPLO = 'U'
      ELSE
         LOWER = .TRUE.
         CUPLO = 'L'
      END IF
*
      UNFL = DLAMCH( 'Safe minimum' )
      ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
*
*     Some Error Checks
*
      IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
         RESULT( 1 ) = TEN / ULP
         RETURN
      END IF
*
*     Do Test 1
*
*     Norm of A:
*
      IF( ITYPE.EQ.3 ) THEN
         ANORM = ONE
      ELSE
         ANORM = MAX( DLANSY( '1', CUPLO, N, A, LDA, WORK ), UNFL )
      END IF
*
*     Compute error matrix:
*
      IF( ITYPE.EQ.1 ) THEN
*
*        ITYPE=1: error = A - U S U'
*
         CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
         CALL DLACPY( CUPLO, N, N, A, LDA, WORK, N )
*
         DO 10 J = 1, N
            CALL DSYR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK, N )
   10    CONTINUE
*
         IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
            DO 20 J = 1, N - 1
               CALL DSYR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ),
     $                     1, WORK, N )
   20       CONTINUE
         END IF
         WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) )
*
      ELSE IF( ITYPE.EQ.2 ) THEN
*
*        ITYPE=2: error = V S V' - A
*
         CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
*
         IF( LOWER ) THEN
            WORK( N**2 ) = D( N )
            DO 40 J = N - 1, 1, -1
               IF( KBAND.EQ.1 ) THEN
                  WORK( ( N+1 )*( J-1 )+2 ) = ( ONE-TAU( J ) )*E( J )
                  DO 30 JR = J + 2, N
                     WORK( ( J-1 )*N+JR ) = -TAU( J )*E( J )*V( JR, J )
   30             CONTINUE
               END IF
*
               VSAVE = V( J+1, J )
               V( J+1, J ) = ONE
               CALL DLARFY( 'L', N-J, V( J+1, J ), 1, TAU( J ),
     $                      WORK( ( N+1 )*J+1 ), N, WORK( N**2+1 ) )
               V( J+1, J ) = VSAVE
               WORK( ( N+1 )*( J-1 )+1 ) = D( J )
   40       CONTINUE
         ELSE
            WORK( 1 ) = D( 1 )
            DO 60 J = 1, N - 1
               IF( KBAND.EQ.1 ) THEN
                  WORK( ( N+1 )*J ) = ( ONE-TAU( J ) )*E( J )
                  DO 50 JR = 1, J - 1
                     WORK( J*N+JR ) = -TAU( J )*E( J )*V( JR, J+1 )
   50             CONTINUE
               END IF
*
               VSAVE = V( J, J+1 )
               V( J, J+1 ) = ONE
               CALL DLARFY( 'U', J, V( 1, J+1 ), 1, TAU( J ), WORK, N,
     $                      WORK( N**2+1 ) )
               V( J, J+1 ) = VSAVE
               WORK( ( N+1 )*J+1 ) = D( J+1 )
   60       CONTINUE
         END IF
*
         DO 90 JCOL = 1, N
            IF( LOWER ) THEN
               DO 70 JROW = JCOL, N
                  WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) )
     $                - A( JROW, JCOL )
   70          CONTINUE
            ELSE
               DO 80 JROW = 1, JCOL
                  WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) )
     $                - A( JROW, JCOL )
   80          CONTINUE
            END IF
   90    CONTINUE
         WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) )
*
      ELSE IF( ITYPE.EQ.3 ) THEN
*
*        ITYPE=3: error = U V' - I
*
         IF( N.LT.2 )
     $      RETURN
         CALL DLACPY( ' ', N, N, U, LDU, WORK, N )
         IF( LOWER ) THEN
            CALL DORM2R( 'R', 'T', N, N-1, N-1, V( 2, 1 ), LDV, TAU,
     $                   WORK( N+1 ), N, WORK( N**2+1 ), IINFO )
         ELSE
            CALL DORM2L( 'R', 'T', N, N-1, N-1, V( 1, 2 ), LDV, TAU,
     $                   WORK, N, WORK( N**2+1 ), IINFO )
         END IF
         IF( IINFO.NE.0 ) THEN
            RESULT( 1 ) = TEN / ULP
            RETURN
         END IF
*
         DO 100 J = 1, N
            WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
  100    CONTINUE
*
         WNORM = DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) )
      END IF
*
      IF( ANORM.GT.WNORM ) THEN
         RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
      ELSE
         IF( ANORM.LT.ONE ) THEN
            RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
         ELSE
            RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
         END IF
      END IF
*
*     Do Test 2
*
*     Compute  UU' - I
*
      IF( ITYPE.EQ.1 ) THEN
         CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
     $               N )
*
         DO 110 J = 1, N
            WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
  110    CONTINUE
*
         RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N,
     $                 WORK( N**2+1 ) ), DBLE( N ) ) / ( N*ULP )
      END IF
*
      RETURN
*
*     End of DSYT21
*
      END
