*> \brief \b ZLAHEF_AASEN
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLAHEF_AASEN + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLAHEF_AASEN( UPLO, J1, M, NB, A, LDA, IPIV,
* H, LDH, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER J1, M, NB, LDA, LDH, INFO
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX*16 A( LDA, * ), H( LDH, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATRF_AASEN factorizes a panel of a real hermitian matrix A using
*> the Aasen's algorithm. The panel consists of a set of NB rows of A
*> when UPLO is U, or a set of NB columns when UPLO is L.
*>
*> In order to factorize the panel, the Aasen's algorithm requires the
*> last row, or column, of the previous panel. The first row, or column,
*> of A is set to be the first row, or column, of an identity matrix,
*> which is used to factorize the first panel.
*>
*> The resulting J-th row of U, or J-th column of L, is stored in the
*> (J-1)-th row, or column, of A (without the unit diatonals), while
*> the diagonal and subdiagonal of A are overwritten by those of T.
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] J1
*> \verbatim
*> J1 is INTEGER
*> The location of the first row, or column, of the panel
*> within the submatrix of A, passed to this routine, e.g.,
*> when called by ZHETRF_AASEN, for the first panel, J1 is 1,
*> while for the remaining panels, J1 is 2.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The dimension of the submatrix. M >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The dimension of the panel to be facotorized.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,M) for
*> the first panel, while dimension (LDA,M+1) for the
*> remaining panels.
*>
*> On entry, A contains the last row, or column, of
*> the previous panel, and the trailing submatrix of A
*> to be factorized, except for the first panel, only
*> the panel is passed.
*>
*> On exit, the leading panel is factorized.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the row and column interchanges,
*> the row and column k were interchanged with the row and
*> column IPIV(k).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is COMPLEX*16 workspace, dimension (LDH,NB).
*>
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the workspace H. LDH >= max(1,M).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 workspace, dimension (M).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular, and division by zero will occur if it
*> is used to solve a system of equations.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2016
*
*> \ingroup complex16SYcomputational
*
* @precisions fortran z -> c
*
* =====================================================================
SUBROUTINE ZLAHEF_AASEN( UPLO, J1, M, NB, A, LDA, IPIV,
$ H, LDH, WORK, INFO )
*
* -- LAPACK computational 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 2016
*
IMPLICIT NONE
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER M, NB, J1, LDA, LDH, INFO
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * ), H( LDH, * ), WORK( * )
* ..
*
* =====================================================================
* .. Parameters ..
COMPLEX*16 ZERO, ONE
PARAMETER ( ZERO = (0.0D+0, 0.0D+0), ONE = (1.0D+0, 0.0D+0) )
*
* .. Local Scalars ..
INTEGER J, K, K1, I1, I2
COMPLEX*16 PIV, ALPHA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IZAMAX, ILAENV
EXTERNAL LSAME, ILAENV, IZAMAX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DCONJG, MAX
* ..
* .. Executable Statements ..
*
INFO = 0
J = 1
*
* K1 is the first column of the panel to be factorized
* i.e., K1 is 2 for the first block column, and 1 for the rest of the blocks
*
K1 = (2-J1)+1
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* .....................................................
* Factorize A as U**T*D*U using the upper triangle of A
* .....................................................
*
10 CONTINUE
IF ( J.GT.MIN(M, NB) )
$ GO TO 20
*
* K is the column to be factorized
* when being called from ZHETRF_AASEN,
* > for the first block column, J1 is 1, hence J1+J-1 is J,
* > for the rest of the columns, J1 is 2, and J1+J-1 is J+1,
*
K = J1+J-1
*
* H(J:N, J) := A(J, J:N) - H(J:N, 1:(J-1)) * L(J1:(J-1), J),
* where H(J:N, J) has been initialized to be A(J, J:N)
*
IF( K.GT.2 ) THEN
*
* K is the column to be factorized
* > for the first block column, K is J, skipping the first two
* columns
* > for the rest of the columns, K is J+1, skipping only the
* first column
*
CALL ZLACGV( J-K1, A( 1, J ), 1 )
CALL ZGEMV( 'No transpose', M-J+1, J-K1,
$ -ONE, H( J, K1 ), LDH,
$ A( 1, J ), 1,
$ ONE, H( J, J ), 1 )
CALL ZLACGV( J-K1, A( 1, J ), 1 )
END IF
*
* Copy H(i:n, i) into WORK
*
CALL ZCOPY( M-J+1, H( J, J ), 1, WORK( 1 ), 1 )
*
IF( J.GT.K1 ) THEN
*
* Compute WORK := WORK - L(J-1, J:N) * T(J-1,J),
* where A(J-1, J) stores T(J-1, J) and A(J-2, J:N) stores U(J-1, J:N)
*
ALPHA = -DCONJG( A( K-1, J ) )
CALL ZAXPY( M-J+1, ALPHA, A( K-2, J ), LDA, WORK( 1 ), 1 )
END IF
*
* Set A(J, J) = T(J, J)
*
A( K, J ) = DBLE( WORK( 1 ) )
*
IF( J.LT.M ) THEN
*
* Compute WORK(2:N) = T(J, J) L(J, (J+1):N)
* where A(J, J) stores T(J, J) and A(J-1, (J+1):N) stores U(J, (J+1):N)
*
IF( (J1+J-1).GT.1 ) THEN
ALPHA = -A( K, J )
CALL ZAXPY( M-J, ALPHA, A( K-1, J+1 ), LDA,
$ WORK( 2 ), 1 )
ENDIF
*
* Find max(|WORK(2:n)|)
*
I2 = IZAMAX( M-J, WORK( 2 ), 1 ) + 1
PIV = WORK( I2 )
*
* Apply hermitian pivot
*
IF( (I2.NE.2) .AND. (PIV.NE.0) ) THEN
*
* Swap WORK(I1) and WORK(I2)
*
I1 = 2
WORK( I2 ) = WORK( I1 )
WORK( I1 ) = PIV
*
* Swap A(I1, I1+1:N) with A(I1+1:N, I2)
*
I1 = I1+J-1
I2 = I2+J-1
CALL ZSWAP( I2-I1-1, A( J1+I1-1, I1+1 ), LDA,
$ A( J1+I1, I2 ), 1 )
CALL ZLACGV( I2-I1, A( J1+I1-1, I1+1 ), LDA )
CALL ZLACGV( I2-I1-1, A( J1+I1, I2 ), 1 )
*
* Swap A(I1, I2+1:N) with A(I2, I2+1:N)
*
CALL ZSWAP( M-I2, A( J1+I1-1, I2+1 ), LDA,
$ A( J1+I2-1, I2+1 ), LDA )
*
* Swap A(I1, I1) with A(I2,I2)
*
PIV = A( I1+J1-1, I1 )
A( J1+I1-1, I1 ) = A( J1+I2-1, I2 )
A( J1+I2-1, I2 ) = PIV
*
* Swap H(I1, 1:J1) with H(I2, 1:J1)
*
CALL ZSWAP( I1-1, H( I1, 1 ), LDH, H( I2, 1 ), LDH )
IPIV( I1 ) = I2
*
IF( I1.GT.(K1-1) ) THEN
*
* Swap L(1:I1-1, I1) with L(1:I1-1, I2),
* skipping the first column
*
CALL ZSWAP( I1-K1+1, A( 1, I1 ), 1,
$ A( 1, I2 ), 1 )
END IF
ELSE
IPIV( J+1 ) = J+1
ENDIF
*
* Set A(J, J+1) = T(J, J+1)
*
A( K, J+1 ) = WORK( 2 )
IF( (A( K, J ).EQ.ZERO ) .AND.
$ ( (J.EQ.M) .OR. (A( K, J+1 ).EQ.ZERO))) THEN
IF(INFO .EQ. 0) THEN
INFO = J
END IF
END IF
*
IF( J.LT.NB ) THEN
*
* Copy A(J+1:N, J+1) into H(J:N, J),
*
CALL ZCOPY( M-J, A( K+1, J+1 ), LDA,
$ H( J+1, J+1 ), 1 )
END IF
*
* Compute L(J+2, J+1) = WORK( 3:N ) / T(J, J+1),
* where A(J, J+1) = T(J, J+1) and A(J+2:N, J) = L(J+2:N, J+1)
*
IF( A( K, J+1 ).NE.ZERO ) THEN
ALPHA = ONE / A( K, J+1 )
CALL ZCOPY( M-J-1, WORK( 3 ), 1, A( K, J+2 ), LDA )
CALL ZSCAL( M-J-1, ALPHA, A( K, J+2 ), LDA )
ELSE
CALL ZLASET( 'Full', 1, M-J-1, ZERO, ZERO,
$ A( K, J+2 ), LDA)
END IF
ELSE
IF( (A( K, J ).EQ.ZERO) .AND. (INFO.EQ.0) ) THEN
INFO = J
END IF
END IF
J = J + 1
GO TO 10
20 CONTINUE
*
ELSE
*
* .....................................................
* Factorize A as L*D*L**T using the lower triangle of A
* .....................................................
*
30 CONTINUE
IF( J.GT.MIN( M, NB ) )
$ GO TO 40
*
* K is the column to be factorized
* when being called from ZHETRF_AASEN,
* > for the first block column, J1 is 1, hence J1+J-1 is J,
* > for the rest of the columns, J1 is 2, and J1+J-1 is J+1,
*
K = J1+J-1
*
* H(J:N, J) := A(J:N, J) - H(J:N, 1:(J-1)) * L(J, J1:(J-1))^T,
* where H(J:N, J) has been initialized to be A(J:N, J)
*
IF( K.GT.2 ) THEN
*
* K is the column to be factorized
* > for the first block column, K is J, skipping the first two
* columns
* > for the rest of the columns, K is J+1, skipping only the
* first column
*
CALL ZLACGV( J-K1, A( J, 1 ), LDA )
CALL ZGEMV( 'No transpose', M-J+1, J-K1,
$ -ONE, H( J, K1 ), LDH,
$ A( J, 1 ), LDA,
$ ONE, H( J, J ), 1 )
CALL ZLACGV( J-K1, A( J, 1 ), LDA )
END IF
*
* Copy H(J:N, J) into WORK
*
CALL ZCOPY( M-J+1, H( J, J ), 1, WORK( 1 ), 1 )
*
IF( J.GT.K1 ) THEN
*
* Compute WORK := WORK - L(J:N, J-1) * T(J-1,J),
* where A(J-1, J) = T(J-1, J) and A(J, J-2) = L(J, J-1)
*
ALPHA = -DCONJG( A( J, K-1 ) )
CALL ZAXPY( M-J+1, ALPHA, A( J, K-2 ), 1, WORK( 1 ), 1 )
END IF
*
* Set A(J, J) = T(J, J)
*
A( J, K ) = DBLE( WORK( 1 ) )
*
IF( J.LT.M ) THEN
*
* Compute WORK(2:N) = T(J, J) L((J+1):N, J)
* where A(J, J) = T(J, J) and A((J+1):N, J-1) = L((J+1):N, J)
*
IF( (J1+J-1).GT.1 ) THEN
ALPHA = -A( J, K )
CALL ZAXPY( M-J, ALPHA, A( J+1, K-1 ), 1,
$ WORK( 2 ), 1 )
ENDIF
*
* Find max(|WORK(2:n)|)
*
I2 = IZAMAX( M-J, WORK( 2 ), 1 ) + 1
PIV = WORK( I2 )
*
* Apply hermitian pivot
*
IF( (I2.NE.2) .AND. (PIV.NE.0) ) THEN
*
* Swap WORK(I1) and WORK(I2)
*
I1 = 2
WORK( I2 ) = WORK( I1 )
WORK( I1 ) = PIV
*
* Swap A(I1+1:N, I1) with A(I2, I1+1:N)
*
I1 = I1+J-1
I2 = I2+J-1
CALL ZSWAP( I2-I1-1, A( I1+1, J1+I1-1 ), 1,
$ A( I2, J1+I1 ), LDA )
CALL ZLACGV( I2-I1, A( I1+1, J1+I1-1 ), 1 )
CALL ZLACGV( I2-I1-1, A( I2, J1+I1 ), LDA )
*
* Swap A(I2+1:N, I1) with A(I2+1:N, I2)
*
CALL ZSWAP( M-I2, A( I2+1, J1+I1-1 ), 1,
$ A( I2+1, J1+I2-1 ), 1 )
*
* Swap A(I1, I1) with A(I2, I2)
*
PIV = A( I1, J1+I1-1 )
A( I1, J1+I1-1 ) = A( I2, J1+I2-1 )
A( I2, J1+I2-1 ) = PIV
*
* Swap H(I1, I1:J1) with H(I2, I2:J1)
*
CALL ZSWAP( I1-1, H( I1, 1 ), LDH, H( I2, 1 ), LDH )
IPIV( I1 ) = I2
*
IF( I1.GT.(K1-1) ) THEN
*
* Swap L(1:I1-1, I1) with L(1:I1-1, I2),
* skipping the first column
*
CALL ZSWAP( I1-K1+1, A( I1, 1 ), LDA,
$ A( I2, 1 ), LDA )
END IF
ELSE
IPIV( J+1 ) = J+1
ENDIF
*
* Set A(J+1, J) = T(J+1, J)
*
A( J+1, K ) = WORK( 2 )
IF( (A( J, K ).EQ.ZERO) .AND.
$ ( (J.EQ.M) .OR. (A( J+1, K ).EQ.ZERO)) ) THEN
IF (INFO .EQ. 0)
$ INFO = J
END IF
*
IF( J.LT.NB ) THEN
*
* Copy A(J+1:N, J+1) into H(J+1:N, J),
*
CALL ZCOPY( M-J, A( J+1, K+1 ), 1,
$ H( J+1, J+1 ), 1 )
END IF
*
* Compute L(J+2, J+1) = WORK( 3:N ) / T(J, J+1),
* where A(J, J+1) = T(J, J+1) and A(J+2:N, J) = L(J+2:N, J+1)
*
IF( A( J+1, K ).NE.ZERO ) THEN
ALPHA = ONE / A( J+1, K )
CALL ZCOPY( M-J-1, WORK( 3 ), 1, A( J+2, K ), 1 )
CALL ZSCAL( M-J-1, ALPHA, A( J+2, K ), 1 )
ELSE
CALL ZLASET( 'Full', M-J-1, 1, ZERO, ZERO,
$ A( J+2, K ), LDA )
END IF
ELSE
IF( (A( J, K ).EQ.ZERO) .AND. (J.EQ.M)
$ .AND. (INFO.EQ.0) ) INFO = J
END IF
J = J + 1
GO TO 30
40 CONTINUE
END IF
RETURN
*
* End of ZLAHEF_AASEN
*
END