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SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION E( * )
COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
* ..
*
* Purpose
* =======
*
* ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
* Hermitian tridiagonal form by a unitary similarity
* transformation Q' * 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.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the upper or lower triangular part of the
* Hermitian matrix A is stored:
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (input) INTEGER
* The order of the matrix A.
*
* NB (input) INTEGER
* The number of rows and columns to be reduced.
*
* A (input/output) 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* E (output) 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.
*
* TAU (output) 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.
*
* W (output) COMPLEX*16 array, dimension (LDW,NB)
* The n-by-nb matrix W required to update the unreduced part
* of A.
*
* LDW (input) INTEGER
* The leading dimension of the array W. LDW >= max(1,N).
*
* Further Details
* ===============
*
* 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'
*
* 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'
*
* 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' - W*V'.
*
* 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).
*
* =====================================================================
*
* .. 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
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