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*> \brief \b SLAGSY
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SLAGSY( N, K, D, A, LDA, ISEED, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDA, N
* ..
* .. Array Arguments ..
* INTEGER ISEED( 4 )
* REAL A( LDA, * ), D( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAGSY generates a real symmetric matrix A, by pre- and post-
*> multiplying a real diagonal matrix D with a random orthogonal matrix:
*> A = U*D*U'. The semi-bandwidth may then be reduced to k by additional
*> orthogonal 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 REAL array, dimension (LDA,N)
*> The generated n by n symmetric 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 REAL 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 real_matgen
*
* =====================================================================
SUBROUTINE SLAGSY( N, K, D, A, LDA, ISEED, WORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* -- 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 A( LDA, * ), D( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, HALF
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, HALF = 0.5E+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
REAL ALPHA, TAU, WA, WB, WN
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SGEMV, SGER, SLARNV, SSCAL, SSYMV,
$ SSYR2, XERBLA
* ..
* .. External Functions ..
REAL SDOT, SNRM2
EXTERNAL SDOT, SNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN
* ..
* .. 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( 'SLAGSY', -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 symmetric matrix
*
DO 40 I = N - 1, 1, -1
*
* generate random reflection
*
CALL SLARNV( 3, ISEED, N-I+1, WORK )
WN = SNRM2( N-I+1, WORK, 1 )
WA = SIGN( WN, WORK( 1 ) )
IF( WN.EQ.ZERO ) THEN
TAU = ZERO
ELSE
WB = WORK( 1 ) + WA
CALL SSCAL( N-I, ONE / WB, WORK( 2 ), 1 )
WORK( 1 ) = ONE
TAU = 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 SSYMV( '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*SDOT( N-I+1, WORK( N+1 ), 1, WORK, 1 )
CALL SAXPY( 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 SSYR2( '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 = SNRM2( N-K-I+1, A( K+I, I ), 1 )
WA = SIGN( WN, A( K+I, I ) )
IF( WN.EQ.ZERO ) THEN
TAU = ZERO
ELSE
WB = A( K+I, I ) + WA
CALL SSCAL( N-K-I, ONE / WB, A( K+I+1, I ), 1 )
A( K+I, I ) = ONE
TAU = WB / WA
END IF
*
* apply reflection to A(k+i:n,i+1:k+i-1) from the left
*
CALL SGEMV( 'Transpose', N-K-I+1, K-1, ONE, A( K+I, I+1 ), LDA,
$ A( K+I, I ), 1, ZERO, WORK, 1 )
CALL SGER( 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 SSYMV( '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*SDOT( N-K-I+1, WORK, 1, A( K+I, I ), 1 )
CALL SAXPY( N-K-I+1, ALPHA, A( K+I, I ), 1, WORK, 1 )
*
* apply symmetric rank-2 update to A(k+i:n,k+i:n)
*
CALL SSYR2( '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 symmetric matrix
*
DO 80 J = 1, N
DO 70 I = J + 1, N
A( J, I ) = A( I, J )
70 CONTINUE
80 CONTINUE
RETURN
*
* End of SLAGSY
*
END
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