ssyt22()

subroutine ssyt22	(	integer	ITYPE,
		character	UPLO,
		integer	N,
		integer	M,
		integer	KBAND,
		real, dimension( lda, * )	A,
		integer	LDA,
		real, dimension( * )	D,
		real, dimension( * )	E,
		real, dimension( ldu, * )	U,
		integer	LDU,
		real, dimension( ldv, * )	V,
		integer	LDV,
		real, dimension( * )	TAU,
		real, dimension( * )	WORK,
		real, dimension( 2 )	RESULT
	)

SSYT22

Purpose:

      SSYT22  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**T A U - S | / ( |A| m ulp ) and
              RESULT(2) = | I - U**T U | / ( m ulp )

  ITYPE   INTEGER
          Specifies the type of tests to be performed.
          1: U expressed as a dense orthogonal matrix:
             RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and
             RESULT(2) = | I - U U**T | / ( 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, SSYT22 does nothing.
          It must be at least zero.
          Not modified.

  M       INTEGER
          The number of columns of U.  If it is zero, SSYT22 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       REAL 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       REAL array, dimension (N)
          The diagonal of the (symmetric tri-) diagonal matrix.
          Not modified.

  E       REAL 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       REAL 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       REAL 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     REAL array, dimension (N)
          If ITYPE >= 2, then TAU(j) is the scalar factor of
          v(j) v(j)**T 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    REAL array, dimension (2*N**2)
          Workspace.
          Modified.

  RESULT  REAL 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

Definition at line 159 of file ssyt22.f.

*
*  -- 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 ..
      REAL               A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
     $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. Local Scalars ..
      INTEGER            J, JJ, JJ1, JJ2, NN, NNP1
      REAL               ANORM, ULP, UNFL, WNORM
*     ..
*     .. External Functions ..
      REAL               SLAMCH, SLANSY
      EXTERNAL           slamch, slansy
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgemm, ssymm
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real
*     ..
*     .. Executable Statements ..
*
      result( 1 ) = zero
      result( 2 ) = zero
      IF( n.LE.0 .OR. m.LE.0 )
     $   RETURN
*
      unfl = slamch( 'Safe minimum' )
      ulp = slamch( 'Precision' )
*
*     Do Test 1
*
*     Norm of A:
*
      anorm = max( slansy( '1', uplo, n, a, lda, work ), unfl )
*
*     Compute error matrix:
*
*     ITYPE=1: error = U**T A U - S
*
      CALL ssymm( 'L', uplo, n, m, one, a, lda, u, ldu, zero, work, n )
      nn = n*n
      nnp1 = nn + 1
      CALL sgemm( '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 = slansy( '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, real( m ) ) / ( m*ulp )
         END IF
      END IF
*
*     Do Test 2
*
*     Compute  U**T U - I
*
      IF( itype.EQ.1 )
     $   CALL sort01( 'Columns', n, m, u, ldu, work, 2*n*n,
     $                result( 2 ) )
*
      RETURN
*
*     End of SSYT22
*

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