sspt21.f

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*> \brief \b SSPT21
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE SSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
*                          TAU, WORK, RESULT )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            ITYPE, KBAND, LDU, N
*       ..
*       .. Array Arguments ..
*       REAL               AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ),
*      $                   U( LDU, * ), VP( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SSPT21  generally checks a decomposition of the form
*>
*>         A = U S U**T
*>
*> where **T means transpose, A is symmetric (stored in packed format), 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 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) = | A - U S U**T | / ( |A| n ulp ) and
*>         RESULT(2) = | I - U U**T | / ( n ulp )
*>
*> If ITYPE=2, then:
*>
*>         RESULT(1) = | A - V S V**T | / ( |A| n ulp )
*>
*> If ITYPE=3, then:
*>
*>         RESULT(1) = | I - V U**T | / ( n ulp )
*>
*> Packed storage means that, for example, if UPLO='U', then the columns
*> of the upper triangle of A are stored one after another, so that
*> A(1,j+1) immediately follows A(j,j) in the array AP.  Similarly, if
*> UPLO='L', then the columns of the lower triangle of A are stored one
*> after another in AP, so that A(j+1,j+1) immediately follows A(n,j)
*> in the array AP.  This means that A(i,j) is stored in:
*>
*>    AP( i + j*(j-1)/2 )                 if UPLO='U'
*>
*>    AP( i + (2*n-j)*(j-1)/2 )           if UPLO='L'
*>
*> The array VP bears the same relation to the matrix V that A does to
*> AP.
*>
*> For ITYPE > 1, the transformation U is expressed as a product
*> of Householder transformations:
*>
*>    If UPLO='U', then  V = H(n-1)...H(1),  where
*>
*>        H(j) = I  -  tau(j) v(j) v(j)**T
*>
*>    and the first j-1 elements of v(j) are stored in V(1:j-1,j+1),
*>    (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ),
*>    the j-th element is 1, and the last n-j elements are 0.
*>
*>    If UPLO='L', then  V = H(1)...H(n-1),  where
*>
*>        H(j) = I  -  tau(j) v(j) v(j)**T
*>
*>    and the first j elements of v(j) are 0, the (j+1)-st is 1, and the
*>    (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e.,
*>    in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ITYPE
*> \verbatim
*>          ITYPE is 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 )
*>
*>          2: U expressed as a product V of Housholder transformations:
*>             RESULT(1) = | A - V S V**T | / ( |A| n ulp )
*>
*>          3: U expressed both as a dense orthogonal matrix and
*>             as a product of Housholder transformations:
*>             RESULT(1) = | I - V U**T | / ( n ulp )
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER
*>          If UPLO='U', AP and VP are considered to contain the upper
*>          triangle of A and V.
*>          If UPLO='L', AP and VP are considered to contain the lower
*>          triangle of A and V.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The size of the matrix.  If it is zero, SSPT21 does nothing.
*>          It must be at least zero.
*> \endverbatim
*>
*> \param[in] KBAND
*> \verbatim
*>          KBAND is 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.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*>          AP is REAL array, dimension (N*(N+1)/2)
*>          The original (unfactored) matrix.  It is assumed to be
*>          symmetric, and contains the columns of just the upper
*>          triangle (UPLO='U') or only the lower triangle (UPLO='L'),
*>          packed one after another.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is REAL array, dimension (N)
*>          The diagonal of the (symmetric tri-) diagonal matrix.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is REAL 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.
*> \endverbatim
*>
*> \param[in] U
*> \verbatim
*>          U is 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.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of U.  LDU must be at least N and
*>          at least 1.
*> \endverbatim
*>
*> \param[in] VP
*> \verbatim
*>          VP is REAL array, dimension (N*(N+1)/2)
*>          If ITYPE=2 or 3, the columns of this array contain the
*>          Householder vectors used to describe the orthogonal matrix
*>          in the decomposition, as described in purpose.
*>          *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.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is 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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (N**2+N)
*>          Workspace.
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*>          RESULT is 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 ITYPE=1.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup single_eig
*
*  =====================================================================
      SUBROUTINE sspt21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
     $                   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, LDU, N
*     ..
*     .. Array Arguments ..
      REAL               AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ),
     $                   u( ldu, * ), vp( * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE, TEN
      parameter( zero = 0.0e0, one = 1.0e0, ten = 10.0e0 )
      REAL               HALF
      parameter( half = 1.0e+0 / 2.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER
      CHARACTER          CUPLO
      INTEGER            IINFO, J, JP, JP1, JR, LAP
      REAL               ANORM, TEMP, ULP, UNFL, VSAVE, WNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SDOT, SLAMCH, SLANGE, SLANSP
      EXTERNAL           lsame, sdot, slamch, slange, slansp
*     ..
*     .. External Subroutines ..
      EXTERNAL           saxpy, scopy, sgemm, slacpy, slaset, sopmtr,
     $                   sspmv, sspr, sspr2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real
*     ..
*     .. Executable Statements ..
*
*     1)      Constants
*
      result( 1 ) = zero
      IF( itype.EQ.1 )
     $   result( 2 ) = zero
      IF( n.LE.0 )
     $   RETURN
*
      lap = ( n*( n+1 ) ) / 2
*
      IF( lsame( uplo, 'U' ) ) THEN
         lower = .false.
         cuplo = 'U'
      ELSE
         lower = .true.
         cuplo = 'L'
      END IF
*
      unfl = slamch( 'Safe minimum' )
      ulp = slamch( 'Epsilon' )*slamch( '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( slansp( '1', cuplo, n, ap, work ), unfl )
      END IF
*
*     Compute error matrix:
*
      IF( itype.EQ.1 ) THEN
*
*        ITYPE=1: error = A - U S U**T
*
         CALL slaset( 'Full', n, n, zero, zero, work, n )
         CALL scopy( lap, ap, 1, work, 1 )
*
         DO 10 j = 1, n
            CALL sspr( cuplo, n, -d( j ), u( 1, j ), 1, work )
   10    CONTINUE
*
         IF( n.GT.1 .AND. kband.EQ.1 ) THEN
            DO 20 j = 1, n - 1
               CALL sspr2( cuplo, n, -e( j ), u( 1, j ), 1, u( 1, j+1 ),
     $                     1, work )
   20       CONTINUE
         END IF
         wnorm = slansp( '1', cuplo, n, work, work( n**2+1 ) )
*
      ELSE IF( itype.EQ.2 ) THEN
*
*        ITYPE=2: error = V S V**T - A
*
         CALL slaset( 'Full', n, n, zero, zero, work, n )
*
         IF( lower ) THEN
            work( lap ) = d( n )
            DO 40 j = n - 1, 1, -1
               jp = ( ( 2*n-j )*( j-1 ) ) / 2
               jp1 = jp + n - j
               IF( kband.EQ.1 ) THEN
                  work( jp+j+1 ) = ( one-tau( j ) )*e( j )
                  DO 30 jr = j + 2, n
                     work( jp+jr ) = -tau( j )*e( j )*vp( jp+jr )
   30             CONTINUE
               END IF
*
               IF( tau( j ).NE.zero ) THEN
                  vsave = vp( jp+j+1 )
                  vp( jp+j+1 ) = one
                  CALL sspmv( 'L', n-j, one, work( jp1+j+1 ),
     $                        vp( jp+j+1 ), 1, zero, work( lap+1 ), 1 )
                  temp = -half*tau( j )*sdot( n-j, work( lap+1 ), 1,
     $                   vp( jp+j+1 ), 1 )
                  CALL saxpy( n-j, temp, vp( jp+j+1 ), 1, work( lap+1 ),
     $                        1 )
                  CALL sspr2( 'L', n-j, -tau( j ), vp( jp+j+1 ), 1,
     $                        work( lap+1 ), 1, work( jp1+j+1 ) )
                  vp( jp+j+1 ) = vsave
               END IF
               work( jp+j ) = d( j )
   40       CONTINUE
         ELSE
            work( 1 ) = d( 1 )
            DO 60 j = 1, n - 1
               jp = ( j*( j-1 ) ) / 2
               jp1 = jp + j
               IF( kband.EQ.1 ) THEN
                  work( jp1+j ) = ( one-tau( j ) )*e( j )
                  DO 50 jr = 1, j - 1
                     work( jp1+jr ) = -tau( j )*e( j )*vp( jp1+jr )
   50             CONTINUE
               END IF
*
               IF( tau( j ).NE.zero ) THEN
                  vsave = vp( jp1+j )
                  vp( jp1+j ) = one
                  CALL sspmv( 'U', j, one, work, vp( jp1+1 ), 1, zero,
     $                        work( lap+1 ), 1 )
                  temp = -half*tau( j )*sdot( j, work( lap+1 ), 1,
     $                   vp( jp1+1 ), 1 )
                  CALL saxpy( j, temp, vp( jp1+1 ), 1, work( lap+1 ),
     $                        1 )
                  CALL sspr2( 'U', j, -tau( j ), vp( jp1+1 ), 1,
     $                        work( lap+1 ), 1, work )
                  vp( jp1+j ) = vsave
               END IF
               work( jp1+j+1 ) = d( j+1 )
   60       CONTINUE
         END IF
*
         DO 70 j = 1, lap
            work( j ) = work( j ) - ap( j )
   70    CONTINUE
         wnorm = slansp( '1', cuplo, n, work, work( lap+1 ) )
*
      ELSE IF( itype.EQ.3 ) THEN
*
*        ITYPE=3: error = U V**T - I
*
         IF( n.LT.2 )
     $      RETURN
         CALL slacpy( ' ', n, n, u, ldu, work, n )
         CALL sopmtr( 'R', cuplo, 'T', n, n, vp, tau, work, n,
     $                work( n**2+1 ), iinfo )
         IF( iinfo.NE.0 ) THEN
            result( 1 ) = ten / ulp
            RETURN
         END IF
*
         DO 80 j = 1, n
            work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - one
   80    CONTINUE
*
         wnorm = slange( '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, real( n ) ) / ( n*ulp )
         END IF
      END IF
*
*     Do Test 2
*
*     Compute  U U**T - I
*
      IF( itype.EQ.1 ) THEN
         CALL sgemm( 'N', 'C', n, n, n, one, u, ldu, u, ldu, zero, work,
     $               n )
*
         DO 90 j = 1, n
            work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - one
   90    CONTINUE
*
         result( 2 ) = min( slange( '1', n, n, work, n,
     $                 work( n**2+1 ) ), real( n ) ) / ( n*ulp )
      END IF
*
      RETURN
*
*     End of SSPT21
*
      END