zhet21()

subroutine zhet21	(	integer	ITYPE,
		character	UPLO,
		integer	N,
		integer	KBAND,
		complex*16, dimension( lda, * )	A,
		integer	LDA,
		double precision, dimension( * )	D,
		double precision, dimension( * )	E,
		complex*16, dimension( ldu, * )	U,
		integer	LDU,
		complex*16, dimension( ldv, * )	V,
		integer	LDV,
		complex*16, dimension( * )	TAU,
		complex*16, dimension( * )	WORK,
		double precision, dimension( * )	RWORK,
		double precision, dimension( 2 )	RESULT
	)

ZHET21

Purpose:

 ZHET21 generally checks a decomposition of the form

    A = U S U**H

 where **H means conjugate transpose, A is hermitian, U is unitary, and
 S is diagonal (if KBAND=0) or (real) 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**H | / ( |A| n ulp ) and
    RESULT(2) = | I - U U**H | / ( n ulp )

 If ITYPE=2, then:

    RESULT(1) = | A - V S V**H | / ( |A| n ulp )

 If ITYPE=3, then:

    RESULT(1) = | I - U V**H | / ( 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)**H and each
 vector v(j) has its first j elements 0 and the remaining n-j elements
 stored in V(j+1:n,j).

Parameters

[in]	ITYPE	ITYPE is INTEGER Specifies the type of tests to be performed. 1: U expressed as a dense unitary matrix: RESULT(1) = | A - U S U**H | / ( |A| n ulp ) and RESULT(2) = | I - U U**H | / ( n ulp ) 2: U expressed as a product V of Housholder transformations: RESULT(1) = | A - V S V**H | / ( |A| n ulp ) 3: U expressed both as a dense unitary matrix and as a product of Housholder transformations: RESULT(1) = | I - U V**H | / ( n ulp )
[in]	UPLO	UPLO is 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.
[in]	N	N is INTEGER The size of the matrix. If it is zero, ZHET21 does nothing. It must be at least zero.
[in]	KBAND	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.
[in]	A	A is COMPLEX*16 array, dimension (LDA, N) The original (unfactored) matrix. It is assumed to be hermitian, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced.
[in]	LDA	LDA is INTEGER The leading dimension of A. It must be at least 1 and at least N.
[in]	D	D is DOUBLE PRECISION array, dimension (N) The diagonal of the (symmetric tri-) diagonal matrix.
[in]	E	E is 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.
[in]	U	U is COMPLEX*16 array, dimension (LDU, N) If ITYPE=1 or 3, this contains the unitary matrix in the decomposition, expressed as a dense matrix. If ITYPE=2, then it is not referenced.
[in]	LDU	LDU is INTEGER The leading dimension of U. LDU must be at least N and at least 1.
[in]	V	V is COMPLEX*16 array, dimension (LDV, N) If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the unitary 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.
[in]	LDV	LDV is INTEGER The leading dimension of V. LDV must be at least N and at least 1.
[in]	TAU	TAU is COMPLEX*16 array, dimension (N) If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)**H in the Householder transformation H(j) of the product U = H(1)...H(n-2) If ITYPE < 2, then TAU is not referenced.
[out]	WORK	WORK is COMPLEX*16 array, dimension (2*N**2)
[out]	RWORK	RWORK is DOUBLE PRECISION array, dimension (N)
[out]	RESULT	RESULT is 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

Definition at line 216 of file zhet21.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, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), E( * ), RESULT( 2 ), RWORK( * )
      COMPLEX*16         A( LDA, * ), TAU( * ), U( LDU, * ),
     $                   V( LDV, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TEN
      parameter( zero = 0.0d+0, one = 1.0d+0, ten = 10.0d+0 )
      COMPLEX*16         CZERO, CONE
      parameter( czero = ( 0.0d+0, 0.0d+0 ),
     $                   cone = ( 1.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER
      CHARACTER          CUPLO
      INTEGER            IINFO, J, JCOL, JR, JROW
      DOUBLE PRECISION   ANORM, ULP, UNFL, WNORM
      COMPLEX*16         VSAVE
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH, ZLANGE, ZLANHE
      EXTERNAL           lsame, dlamch, zlange, zlanhe
*     ..
*     .. External Subroutines ..
      EXTERNAL           zgemm, zher, zher2, zlacpy, zlarfy, zlaset,
     $                   zunm2l, zunm2r
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          dble, dcmplx, max, min
*     ..
*     .. Executable Statements ..
*
      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( zlanhe( '1', cuplo, n, a, lda, rwork ), unfl )
      END IF
*
*     Compute error matrix:
*
      IF( itype.EQ.1 ) THEN
*
*        ITYPE=1: error = A - U S U**H
*
         CALL zlaset( 'Full', n, n, czero, czero, work, n )
         CALL zlacpy( cuplo, n, n, a, lda, work, n )
*
         DO 10 j = 1, n
            CALL zher( 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 zher2( cuplo, n, -dcmplx( e( j ) ), u( 1, j ), 1,
     $                     u( 1, j-1 ), 1, work, n )
   20       CONTINUE
         END IF
         wnorm = zlanhe( '1', cuplo, n, work, n, rwork )
*
      ELSE IF( itype.EQ.2 ) THEN
*
*        ITYPE=2: error = V S V**H - A
*
         CALL zlaset( 'Full', n, n, czero, czero, 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 ) = ( cone-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 zlarfy( '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 ) = ( cone-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 zlarfy( '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 = zlanhe( '1', cuplo, n, work, n, rwork )
*
      ELSE IF( itype.EQ.3 ) THEN
*
*        ITYPE=3: error = U V**H - I
*
         IF( n.LT.2 )
     $      RETURN
         CALL zlacpy( ' ', n, n, u, ldu, work, n )
         IF( lower ) THEN
            CALL zunm2r( 'R', 'C', n, n-1, n-1, v( 2, 1 ), ldv, tau,
     $                   work( n+1 ), n, work( n**2+1 ), iinfo )
         ELSE
            CALL zunm2l( 'R', 'C', 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 ) - cone
  100    CONTINUE
*
         wnorm = zlange( '1', n, n, work, n, rwork )
      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  U U**H - I
*
      IF( itype.EQ.1 ) THEN
         CALL zgemm( 'N', 'C', n, n, n, cone, u, ldu, u, ldu, czero,
     $               work, n )
*
         DO 110 j = 1, n
            work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - cone
  110    CONTINUE
*
         result( 2 ) = min( zlange( '1', n, n, work, n, rwork ),
     $                 dble( n ) ) / ( n*ulp )
      END IF
*
      RETURN
*
*     End of ZHET21
*

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