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*> \brief \b CLAGHE
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE CLAGHE( N, K, D, A, LDA, ISEED, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, N
* ..
* .. Array Arguments ..
* INTEGER ISEED( 4 )
* REAL D( * )
* COMPLEX A( LDA, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CLAGHE generates a complex hermitian matrix A, by pre- and post-
*> multiplying a real diagonal matrix D with a random unitary matrix:
*> A = U*D*U'. The semi-bandwidth may then be reduced to k by additional
*> unitary transformations.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of nonzero subdiagonals within the band of A.
*> 0 <= K <= N-1.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is REAL array, dimension (N)
*> The diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> The generated n by n hermitian matrix A (the full matrix is
*> stored).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= N.
*> \endverbatim
*>
*> \param[in,out] ISEED
*> \verbatim
*> ISEED is INTEGER array, dimension (4)
*> On entry, the seed of the random number generator; the array
*> elements must be between 0 and 4095, and ISEED(4) must be
*> odd.
*> On exit, the seed is updated.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex_matgen
*
* =====================================================================
SUBROUTINE CLAGHE( N, K, D, A, LDA, ISEED, WORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, N
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
REAL D( * )
COMPLEX A( LDA, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX ZERO, ONE, HALF
PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ),
$ ONE = ( 1.0E+0, 0.0E+0 ),
$ HALF = ( 0.5E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, J
REAL WN
COMPLEX ALPHA, TAU, WA, WB
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CGEMV, CGERC, CHEMV, CHER2, CLARNV,
$ CSCAL, XERBLA
* ..
* .. External Functions ..
REAL SCNRM2
COMPLEX CDOTC
EXTERNAL SCNRM2, CDOTC
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX, REAL
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( K.LT.0 .OR. K.GT.N-1 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
END IF
IF( INFO.LT.0 ) THEN
CALL XERBLA( 'CLAGHE', -INFO )
RETURN
END IF
*
* initialize lower triangle of A to diagonal matrix
*
DO 20 J = 1, N
DO 10 I = J + 1, N
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
DO 30 I = 1, N
A( I, I ) = D( I )
30 CONTINUE
*
* Generate lower triangle of hermitian matrix
*
DO 40 I = N - 1, 1, -1
*
* generate random reflection
*
CALL CLARNV( 3, ISEED, N-I+1, WORK )
WN = SCNRM2( N-I+1, WORK, 1 )
WA = ( WN / ABS( WORK( 1 ) ) )*WORK( 1 )
IF( WN.EQ.ZERO ) THEN
TAU = ZERO
ELSE
WB = WORK( 1 ) + WA
CALL CSCAL( N-I, ONE / WB, WORK( 2 ), 1 )
WORK( 1 ) = ONE
TAU = REAL( WB / WA )
END IF
*
* apply random reflection to A(i:n,i:n) from the left
* and the right
*
* compute y := tau * A * u
*
CALL CHEMV( 'Lower', N-I+1, TAU, A( I, I ), LDA, WORK, 1, ZERO,
$ WORK( N+1 ), 1 )
*
* compute v := y - 1/2 * tau * ( y, u ) * u
*
ALPHA = -HALF*TAU*CDOTC( N-I+1, WORK( N+1 ), 1, WORK, 1 )
CALL CAXPY( N-I+1, ALPHA, WORK, 1, WORK( N+1 ), 1 )
*
* apply the transformation as a rank-2 update to A(i:n,i:n)
*
CALL CHER2( 'Lower', N-I+1, -ONE, WORK, 1, WORK( N+1 ), 1,
$ A( I, I ), LDA )
40 CONTINUE
*
* Reduce number of subdiagonals to K
*
DO 60 I = 1, N - 1 - K
*
* generate reflection to annihilate A(k+i+1:n,i)
*
WN = SCNRM2( N-K-I+1, A( K+I, I ), 1 )
WA = ( WN / ABS( A( K+I, I ) ) )*A( K+I, I )
IF( WN.EQ.ZERO ) THEN
TAU = ZERO
ELSE
WB = A( K+I, I ) + WA
CALL CSCAL( N-K-I, ONE / WB, A( K+I+1, I ), 1 )
A( K+I, I ) = ONE
TAU = REAL( WB / WA )
END IF
*
* apply reflection to A(k+i:n,i+1:k+i-1) from the left
*
CALL CGEMV( 'Conjugate transpose', N-K-I+1, K-1, ONE,
$ A( K+I, I+1 ), LDA, A( K+I, I ), 1, ZERO, WORK, 1 )
CALL CGERC( N-K-I+1, K-1, -TAU, A( K+I, I ), 1, WORK, 1,
$ A( K+I, I+1 ), LDA )
*
* apply reflection to A(k+i:n,k+i:n) from the left and the right
*
* compute y := tau * A * u
*
CALL CHEMV( 'Lower', N-K-I+1, TAU, A( K+I, K+I ), LDA,
$ A( K+I, I ), 1, ZERO, WORK, 1 )
*
* compute v := y - 1/2 * tau * ( y, u ) * u
*
ALPHA = -HALF*TAU*CDOTC( N-K-I+1, WORK, 1, A( K+I, I ), 1 )
CALL CAXPY( N-K-I+1, ALPHA, A( K+I, I ), 1, WORK, 1 )
*
* apply hermitian rank-2 update to A(k+i:n,k+i:n)
*
CALL CHER2( 'Lower', N-K-I+1, -ONE, A( K+I, I ), 1, WORK, 1,
$ A( K+I, K+I ), LDA )
*
A( K+I, I ) = -WA
DO 50 J = K + I + 1, N
A( J, I ) = ZERO
50 CONTINUE
60 CONTINUE
*
* Store full hermitian matrix
*
DO 80 J = 1, N
DO 70 I = J + 1, N
A( J, I ) = CONJG( A( I, J ) )
70 CONTINUE
80 CONTINUE
RETURN
*
* End of CLAGHE
*
END
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