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chpt21.f
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*> \brief \b CHPT21
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE CHPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
* TAU, WORK, RWORK, RESULT )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER ITYPE, KBAND, LDU, N
* ..
* .. Array Arguments ..
* REAL D( * ), E( * ), RESULT( 2 ), RWORK( * )
* COMPLEX AP( * ), TAU( * ), U( LDU, * ), VP( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CHPT21 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 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**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 )
*>
*> 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)**H
*>
*> 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)**H
*>
*> 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 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 )
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The size of the matrix. If it is zero, CHPT21 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 COMPLEX array, dimension (N*(N+1)/2)
*> The original (unfactored) matrix. It is assumed to be
*> hermitian, 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)
*> 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 COMPLEX 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.
*> \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 unitary 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 COMPLEX 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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (N**2)
*> Workspace.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (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.
*
*> \ingroup complex_eig
*
* =====================================================================
SUBROUTINE CHPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
$ TAU, WORK, RWORK, RESULT )
*
* -- LAPACK test routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER ITYPE, KBAND, LDU, N
* ..
* .. Array Arguments ..
REAL D( * ), E( * ), RESULT( 2 ), RWORK( * )
COMPLEX AP( * ), TAU( * ), U( LDU, * ), VP( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TEN
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TEN = 10.0E+0 )
REAL HALF
PARAMETER ( HALF = 1.0E+0 / 2.0E+0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LOWER
CHARACTER CUPLO
INTEGER IINFO, J, JP, JP1, JR, LAP
REAL ANORM, ULP, UNFL, WNORM
COMPLEX TEMP, VSAVE
* ..
* .. External Functions ..
LOGICAL LSAME
REAL CLANGE, CLANHP, SLAMCH
COMPLEX CDOTC
EXTERNAL LSAME, CLANGE, CLANHP, SLAMCH, CDOTC
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CCOPY, CGEMM, CHPMV, CHPR, CHPR2,
$ CLACPY, CLASET, CUPMTR
* ..
* .. Intrinsic Functions ..
INTRINSIC CMPLX, MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
* 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( CLANHP( '1', CUPLO, N, AP, RWORK ), UNFL )
END IF
*
* Compute error matrix:
*
IF( ITYPE.EQ.1 ) THEN
*
* ITYPE=1: error = A - U S U**H
*
CALL CLASET( 'Full', N, N, CZERO, CZERO, WORK, N )
CALL CCOPY( LAP, AP, 1, WORK, 1 )
*
DO 10 J = 1, N
CALL CHPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
10 CONTINUE
*
IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
DO 20 J = 2, N - 1
CALL CHPR2( CUPLO, N, -CMPLX( E( J ) ), U( 1, J ), 1,
$ U( 1, J-1 ), 1, WORK )
20 CONTINUE
END IF
WNORM = CLANHP( '1', CUPLO, N, WORK, RWORK )
*
ELSE IF( ITYPE.EQ.2 ) THEN
*
* ITYPE=2: error = V S V**H - A
*
CALL CLASET( 'Full', N, N, CZERO, CZERO, 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 ) = ( CONE-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.CZERO ) THEN
VSAVE = VP( JP+J+1 )
VP( JP+J+1 ) = CONE
CALL CHPMV( 'L', N-J, CONE, WORK( JP1+J+1 ),
$ VP( JP+J+1 ), 1, CZERO, WORK( LAP+1 ), 1 )
TEMP = -HALF*TAU( J )*CDOTC( N-J, WORK( LAP+1 ), 1,
$ VP( JP+J+1 ), 1 )
CALL CAXPY( N-J, TEMP, VP( JP+J+1 ), 1, WORK( LAP+1 ),
$ 1 )
CALL CHPR2( '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 ) = ( CONE-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.CZERO ) THEN
VSAVE = VP( JP1+J )
VP( JP1+J ) = CONE
CALL CHPMV( 'U', J, CONE, WORK, VP( JP1+1 ), 1, CZERO,
$ WORK( LAP+1 ), 1 )
TEMP = -HALF*TAU( J )*CDOTC( J, WORK( LAP+1 ), 1,
$ VP( JP1+1 ), 1 )
CALL CAXPY( J, TEMP, VP( JP1+1 ), 1, WORK( LAP+1 ),
$ 1 )
CALL CHPR2( '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 = CLANHP( '1', CUPLO, N, WORK, RWORK )
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* ITYPE=3: error = U V**H - I
*
IF( N.LT.2 )
$ RETURN
CALL CLACPY( ' ', N, N, U, LDU, WORK, N )
CALL CUPMTR( 'R', CUPLO, 'C', 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 ) - CONE
80 CONTINUE
*
WNORM = CLANGE( '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, REAL( N ) ) / ( N*ULP )
END IF
END IF
*
* Do Test 2
*
* Compute U U**H - I
*
IF( ITYPE.EQ.1 ) THEN
CALL CGEMM( 'N', 'C', N, N, N, CONE, U, LDU, U, LDU, CZERO,
$ WORK, N )
*
DO 90 J = 1, N
WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - CONE
90 CONTINUE
*
RESULT( 2 ) = MIN( CLANGE( '1', N, N, WORK, N, RWORK ),
$ REAL( N ) ) / ( N*ULP )
END IF
*
RETURN
*
* End of CHPT21
*
END