--- rpl/lapack/lapack/zhetd2.f 2010/08/06 15:32:41 1.4
+++ rpl/lapack/lapack/zhetd2.f 2017/06/17 10:54:15 1.16
@@ -1,9 +1,184 @@
+*> \brief \b ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm).
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZHETD2 + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER UPLO
+* INTEGER INFO, LDA, N
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION D( * ), E( * )
+* COMPLEX*16 A( LDA, * ), TAU( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZHETD2 reduces a complex Hermitian matrix A to real symmetric
+*> tridiagonal form T by a unitary similarity transformation:
+*> Q**H * A * Q = T.
+*> \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. N >= 0.
+*> \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 diagonal and first superdiagonal
+*> of A are overwritten by the corresponding elements of the
+*> tridiagonal matrix T, and the elements above the first
+*> superdiagonal, with the array TAU, represent the unitary
+*> matrix Q as a product of elementary reflectors; if UPLO
+*> = 'L', the diagonal and first subdiagonal of A are over-
+*> written by the corresponding elements of the tridiagonal
+*> matrix T, and the elements below the first subdiagonal, 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] D
+*> \verbatim
+*> D is DOUBLE PRECISION array, dimension (N)
+*> The diagonal elements of the tridiagonal matrix T:
+*> D(i) = A(i,i).
+*> \endverbatim
+*>
+*> \param[out] E
+*> \verbatim
+*> E is DOUBLE PRECISION array, dimension (N-1)
+*> The off-diagonal elements of the tridiagonal matrix T:
+*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
+*> \endverbatim
+*>
+*> \param[out] TAU
+*> \verbatim
+*> TAU is COMPLEX*16 array, dimension (N-1)
+*> The scalar factors of the elementary reflectors (see Further
+*> Details).
+*> \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 December 2016
+*
+*> \ingroup complex16HEcomputational
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*>
+*> If UPLO = 'U', the matrix Q is represented as a product of elementary
+*> reflectors
+*>
+*> Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
+*> A(1:i-1,i+1), and tau in TAU(i).
+*>
+*> If UPLO = 'L', the matrix Q is represented as a product of elementary
+*> reflectors
+*>
+*> Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
+*> and tau in TAU(i).
+*>
+*> The contents of A on exit are illustrated by the following examples
+*> with n = 5:
+*>
+*> if UPLO = 'U': if UPLO = 'L':
+*>
+*> ( d e v2 v3 v4 ) ( d )
+*> ( d e v3 v4 ) ( e d )
+*> ( d e v4 ) ( v1 e d )
+*> ( d e ) ( v1 v2 e d )
+*> ( d ) ( v1 v2 v3 e d )
+*>
+*> where d and e denote diagonal and off-diagonal elements of T, and vi
+*> denotes an element of the vector defining H(i).
+*> \endverbatim
+*>
+* =====================================================================
SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
*
-* -- LAPACK routine (version 3.2) --
+* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
+* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
@@ -14,106 +189,6 @@
COMPLEX*16 A( LDA, * ), TAU( * )
* ..
*
-* Purpose
-* =======
-*
-* ZHETD2 reduces a complex Hermitian matrix A to real symmetric
-* tridiagonal form T by a unitary similarity transformation:
-* Q' * A * Q = T.
-*
-* 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. N >= 0.
-*
-* 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 diagonal and first superdiagonal
-* of A are overwritten by the corresponding elements of the
-* tridiagonal matrix T, and the elements above the first
-* superdiagonal, with the array TAU, represent the unitary
-* matrix Q as a product of elementary reflectors; if UPLO
-* = 'L', the diagonal and first subdiagonal of A are over-
-* written by the corresponding elements of the tridiagonal
-* matrix T, and the elements below the first subdiagonal, 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).
-*
-* D (output) DOUBLE PRECISION array, dimension (N)
-* The diagonal elements of the tridiagonal matrix T:
-* D(i) = A(i,i).
-*
-* E (output) DOUBLE PRECISION array, dimension (N-1)
-* The off-diagonal elements of the tridiagonal matrix T:
-* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
-*
-* TAU (output) COMPLEX*16 array, dimension (N-1)
-* The scalar factors of the elementary reflectors (see Further
-* Details).
-*
-* INFO (output) INTEGER
-* = 0: successful exit
-* < 0: if INFO = -i, the i-th argument had an illegal value.
-*
-* Further Details
-* ===============
-*
-* If UPLO = 'U', the matrix Q is represented as a product of elementary
-* reflectors
-*
-* Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
-* A(1:i-1,i+1), and tau in TAU(i).
-*
-* If UPLO = 'L', the matrix Q is represented as a product of elementary
-* reflectors
-*
-* Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
-* and tau in TAU(i).
-*
-* The contents of A on exit are illustrated by the following examples
-* with n = 5:
-*
-* if UPLO = 'U': if UPLO = 'L':
-*
-* ( d e v2 v3 v4 ) ( d )
-* ( d e v3 v4 ) ( e d )
-* ( d e v4 ) ( v1 e d )
-* ( d e ) ( v1 v2 e d )
-* ( d ) ( v1 v2 v3 e d )
-*
-* where d and e denote diagonal and off-diagonal elements of T, and vi
-* denotes an element of the vector defining H(i).
-*
* =====================================================================
*
* .. Parameters ..
@@ -143,7 +218,7 @@
* Test the input parameters
*
INFO = 0
- UPPER = LSAME( UPLO, 'U' )
+ UPPER = LSAME( UPLO, 'U')
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
@@ -168,7 +243,7 @@
A( N, N ) = DBLE( A( N, N ) )
DO 10 I = N - 1, 1, -1
*
-* Generate elementary reflector H(i) = I - tau * v * v'
+* Generate elementary reflector H(i) = I - tau * v * v**H
* to annihilate A(1:i-1,i+1)
*
ALPHA = A( I, I+1 )
@@ -186,13 +261,13 @@
CALL ZHEMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
$ TAU, 1 )
*
-* Compute w := x - 1/2 * tau * (x'*v) * v
+* Compute w := x - 1/2 * tau * (x**H * v) * v
*
ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, A( 1, I+1 ), 1 )
CALL ZAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
*
* Apply the transformation as a rank-2 update:
-* A := A - v * w' - w * v'
+* A := A - v * w**H - w * v**H
*
CALL ZHER2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
$ LDA )
@@ -212,7 +287,7 @@
A( 1, 1 ) = DBLE( A( 1, 1 ) )
DO 20 I = 1, N - 1
*
-* Generate elementary reflector H(i) = I - tau * v * v'
+* Generate elementary reflector H(i) = I - tau * v * v**H
* to annihilate A(i+2:n,i)
*
ALPHA = A( I+1, I )
@@ -230,14 +305,14 @@
CALL ZHEMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
$ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
*
-* Compute w := x - 1/2 * tau * (x'*v) * v
+* Compute w := x - 1/2 * tau * (x**H * v) * v
*
ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, A( I+1, I ),
$ 1 )
CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
*
* Apply the transformation as a rank-2 update:
-* A := A - v * w' - w * v'
+* A := A - v * w**H - w * v**H
*
CALL ZHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
$ A( I+1, I+1 ), LDA )