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*> \brief \b ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.
*
* =========== DOCUMENTATION ===========
*
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* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
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*
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*
* Definition:
* ===========
*
* SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
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*
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* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION E( * )
* COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
* ..
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*
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*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
*> Hermitian tridiagonal form by a unitary similarity
*> transformation Q**H * A * Q, and returns the matrices V and W which are
*> needed to apply the transformation to the unreduced part of A.
*>
*> If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
*> matrix, of which the upper triangle is supplied;
*> if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
*> matrix, of which the lower triangle is supplied.
*>
*> This is an auxiliary routine called by ZHETRD.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> Hermitian matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of rows and columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
*> n-by-n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n-by-n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit:
*> if UPLO = 'U', the last NB columns have been reduced to
*> tridiagonal form, with the diagonal elements overwriting
*> the diagonal elements of A; the elements above the diagonal
*> with the array TAU, represent the unitary matrix Q as a
*> product of elementary reflectors;
*> if UPLO = 'L', the first NB columns have been reduced to
*> tridiagonal form, with the diagonal elements overwriting
*> the diagonal elements of A; the elements below the diagonal
*> with the array TAU, represent the unitary matrix Q as a
*> product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
*> elements of the last NB columns of the reduced matrix;
*> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
*> the first NB columns of the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is COMPLEX*16 array, dimension (N-1)
*> The scalar factors of the elementary reflectors, stored in
*> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
*> See Further Details.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is COMPLEX*16 array, dimension (LDW,NB)
*> The n-by-nb matrix W required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*> LDW is INTEGER
*> The leading dimension of the array W. LDW >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
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*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
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*
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*> \date December 2016
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*
*> \ingroup complex16OTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n) H(n-1) . . . H(n-nb+1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**H
*>
*> where tau is a complex scalar, and v is a complex vector with
*> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
*> and tau in TAU(i-1).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(nb).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**H
*>
*> where tau is a complex scalar, and v is a complex vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*> and tau in TAU(i).
*>
*> The elements of the vectors v together form the n-by-nb matrix V
*> which is needed, with W, to apply the transformation to the unreduced
*> part of the matrix, using a Hermitian rank-2k update of the form:
*> A := A - V*W**H - W*V**H.
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5 and nb = 2:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( a a a v4 v5 ) ( d )
*> ( a a v4 v5 ) ( 1 d )
*> ( a 1 v5 ) ( v1 1 a )
*> ( d 1 ) ( v1 v2 a a )
*> ( d ) ( v1 v2 a a a )
*>
*> where d denotes a diagonal element of the reduced matrix, a denotes
*> an element of the original matrix that is unchanged, and vi denotes
*> an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZLATRD ( UPLO , N , NB , A , LDA , E , TAU , W , LDW )
*
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* -- LAPACK auxiliary routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* December 2016
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*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA , LDW , N , NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION E ( * )
COMPLEX * 16 A ( LDA , * ) , TAU ( * ) , W ( LDW , * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX * 16 ZERO , ONE , HALF
PARAMETER ( ZERO = ( 0.0D+0 , 0.0D+0 ) ,
$ ONE = ( 1.0D+0 , 0.0D+0 ) ,
$ HALF = ( 0.5D+0 , 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I , IW
COMPLEX * 16 ALPHA
* ..
* .. External Subroutines ..
EXTERNAL ZAXPY , ZGEMV , ZHEMV , ZLACGV , ZLARFG , ZSCAL
* ..
* .. External Functions ..
LOGICAL LSAME
COMPLEX * 16 ZDOTC
EXTERNAL LSAME , ZDOTC
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE , MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF ( N . LE . 0 )
$ RETURN
*
IF ( LSAME ( UPLO , 'U' ) ) THEN
*
* Reduce last NB columns of upper triangle
*
DO 10 I = N , N - NB + 1 , - 1
IW = I - N + NB
IF ( I . LT . N ) THEN
*
* Update A(1:i,i)
*
A ( I , I ) = DBLE ( A ( I , I ) )
CALL ZLACGV ( N - I , W ( I , IW + 1 ) , LDW )
CALL ZGEMV ( 'No transpose' , I , N - I , - ONE , A ( 1 , I + 1 ) ,
$ LDA , W ( I , IW + 1 ) , LDW , ONE , A ( 1 , I ) , 1 )
CALL ZLACGV ( N - I , W ( I , IW + 1 ) , LDW )
CALL ZLACGV ( N - I , A ( I , I + 1 ) , LDA )
CALL ZGEMV ( 'No transpose' , I , N - I , - ONE , W ( 1 , IW + 1 ) ,
$ LDW , A ( I , I + 1 ) , LDA , ONE , A ( 1 , I ) , 1 )
CALL ZLACGV ( N - I , A ( I , I + 1 ) , LDA )
A ( I , I ) = DBLE ( A ( I , I ) )
END IF
IF ( I . GT . 1 ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(1:i-2,i)
*
ALPHA = A ( I - 1 , I )
CALL ZLARFG ( I - 1 , ALPHA , A ( 1 , I ) , 1 , TAU ( I - 1 ) )
E ( I - 1 ) = ALPHA
A ( I - 1 , I ) = ONE
*
* Compute W(1:i-1,i)
*
CALL ZHEMV ( 'Upper' , I - 1 , ONE , A , LDA , A ( 1 , I ) , 1 ,
$ ZERO , W ( 1 , IW ) , 1 )
IF ( I . LT . N ) THEN
CALL ZGEMV ( 'Conjugate transpose' , I - 1 , N - I , ONE ,
$ W ( 1 , IW + 1 ) , LDW , A ( 1 , I ) , 1 , ZERO ,
$ W ( I + 1 , IW ) , 1 )
CALL ZGEMV ( 'No transpose' , I - 1 , N - I , - ONE ,
$ A ( 1 , I + 1 ) , LDA , W ( I + 1 , IW ) , 1 , ONE ,
$ W ( 1 , IW ) , 1 )
CALL ZGEMV ( 'Conjugate transpose' , I - 1 , N - I , ONE ,
$ A ( 1 , I + 1 ) , LDA , A ( 1 , I ) , 1 , ZERO ,
$ W ( I + 1 , IW ) , 1 )
CALL ZGEMV ( 'No transpose' , I - 1 , N - I , - ONE ,
$ W ( 1 , IW + 1 ) , LDW , W ( I + 1 , IW ) , 1 , ONE ,
$ W ( 1 , IW ) , 1 )
END IF
CALL ZSCAL ( I - 1 , TAU ( I - 1 ) , W ( 1 , IW ) , 1 )
ALPHA = - HALF * TAU ( I - 1 ) * ZDOTC ( I - 1 , W ( 1 , IW ) , 1 ,
$ A ( 1 , I ) , 1 )
CALL ZAXPY ( I - 1 , ALPHA , A ( 1 , I ) , 1 , W ( 1 , IW ) , 1 )
END IF
*
10 CONTINUE
ELSE
*
* Reduce first NB columns of lower triangle
*
DO 20 I = 1 , NB
*
* Update A(i:n,i)
*
A ( I , I ) = DBLE ( A ( I , I ) )
CALL ZLACGV ( I - 1 , W ( I , 1 ) , LDW )
CALL ZGEMV ( 'No transpose' , N - I + 1 , I - 1 , - ONE , A ( I , 1 ) ,
$ LDA , W ( I , 1 ) , LDW , ONE , A ( I , I ) , 1 )
CALL ZLACGV ( I - 1 , W ( I , 1 ) , LDW )
CALL ZLACGV ( I - 1 , A ( I , 1 ) , LDA )
CALL ZGEMV ( 'No transpose' , N - I + 1 , I - 1 , - ONE , W ( I , 1 ) ,
$ LDW , A ( I , 1 ) , LDA , ONE , A ( I , I ) , 1 )
CALL ZLACGV ( I - 1 , A ( I , 1 ) , LDA )
A ( I , I ) = DBLE ( A ( I , I ) )
IF ( I . LT . N ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(i+2:n,i)
*
ALPHA = A ( I + 1 , I )
CALL ZLARFG ( N - I , ALPHA , A ( MIN ( I + 2 , N ) , I ) , 1 ,
$ TAU ( I ) )
E ( I ) = ALPHA
A ( I + 1 , I ) = ONE
*
* Compute W(i+1:n,i)
*
CALL ZHEMV ( 'Lower' , N - I , ONE , A ( I + 1 , I + 1 ) , LDA ,
$ A ( I + 1 , I ) , 1 , ZERO , W ( I + 1 , I ) , 1 )
CALL ZGEMV ( 'Conjugate transpose' , N - I , I - 1 , ONE ,
$ W ( I + 1 , 1 ) , LDW , A ( I + 1 , I ) , 1 , ZERO ,
$ W ( 1 , I ) , 1 )
CALL ZGEMV ( 'No transpose' , N - I , I - 1 , - ONE , A ( I + 1 , 1 ) ,
$ LDA , W ( 1 , I ) , 1 , ONE , W ( I + 1 , I ) , 1 )
CALL ZGEMV ( 'Conjugate transpose' , N - I , I - 1 , ONE ,
$ A ( I + 1 , 1 ) , LDA , A ( I + 1 , I ) , 1 , ZERO ,
$ W ( 1 , I ) , 1 )
CALL ZGEMV ( 'No transpose' , N - I , I - 1 , - ONE , W ( I + 1 , 1 ) ,
$ LDW , W ( 1 , I ) , 1 , ONE , W ( I + 1 , I ) , 1 )
CALL ZSCAL ( N - I , TAU ( I ) , W ( I + 1 , I ) , 1 )
ALPHA = - HALF * TAU ( I ) * ZDOTC ( N - I , W ( I + 1 , I ) , 1 ,
$ A ( I + 1 , I ) , 1 )
CALL ZAXPY ( N - I , ALPHA , A ( I + 1 , I ) , 1 , W ( I + 1 , I ) , 1 )
END IF
*
20 CONTINUE
END IF
*
RETURN
*
* End of ZLATRD
*
END