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version 1.1, 2010/01/26 15:22:45 version 1.20, 2023/08/07 08:39:16
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   *> \brief \b ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download ZGEBD2 + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgebd2.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgebd2.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgebd2.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, LDA, M, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   D( * ), E( * )
   *       COMPLEX*16         A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGEBD2 reduces a complex general m by n matrix A to upper or lower
   *> real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
   *>
   *> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of rows in the matrix A.  M >= 0.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The number of columns in the matrix A.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA,N)
   *>          On entry, the m by n general matrix to be reduced.
   *>          On exit,
   *>          if m >= n, the diagonal and the first superdiagonal are
   *>            overwritten with the upper bidiagonal matrix B; the
   *>            elements below the diagonal, with the array TAUQ, represent
   *>            the unitary matrix Q as a product of elementary
   *>            reflectors, and the elements above the first superdiagonal,
   *>            with the array TAUP, represent the unitary matrix P as
   *>            a product of elementary reflectors;
   *>          if m < n, the diagonal and the first subdiagonal are
   *>            overwritten with the lower bidiagonal matrix B; the
   *>            elements below the first subdiagonal, with the array TAUQ,
   *>            represent the unitary matrix Q as a product of
   *>            elementary reflectors, and the elements above the diagonal,
   *>            with the array TAUP, represent the unitary matrix P 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,M).
   *> \endverbatim
   *>
   *> \param[out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (min(M,N))
   *>          The diagonal elements of the bidiagonal matrix B:
   *>          D(i) = A(i,i).
   *> \endverbatim
   *>
   *> \param[out] E
   *> \verbatim
   *>          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
   *>          The off-diagonal elements of the bidiagonal matrix B:
   *>          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
   *>          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
   *> \endverbatim
   *>
   *> \param[out] TAUQ
   *> \verbatim
   *>          TAUQ is COMPLEX*16 array, dimension (min(M,N))
   *>          The scalar factors of the elementary reflectors which
   *>          represent the unitary matrix Q. See Further Details.
   *> \endverbatim
   *>
   *> \param[out] TAUP
   *> \verbatim
   *>          TAUP is COMPLEX*16 array, dimension (min(M,N))
   *>          The scalar factors of the elementary reflectors which
   *>          represent the unitary matrix P. See Further Details.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (max(M,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.
   *
   *> \ingroup complex16GEcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The matrices Q and P are represented as products of elementary
   *>  reflectors:
   *>
   *>  If m >= n,
   *>
   *>     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
   *>
   *>  Each H(i) and G(i) has the form:
   *>
   *>     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
   *>
   *>  where tauq and taup are complex scalars, and v and u are complex
   *>  vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
   *>  A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
   *>  A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
   *>
   *>  If m < n,
   *>
   *>     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
   *>
   *>  Each H(i) and G(i) has the form:
   *>
   *>     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H
   *>
   *>  where tauq and taup are complex scalars, v and u are complex vectors;
   *>  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
   *>  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
   *>  tauq is stored in TAUQ(i) and taup in TAUP(i).
   *>
   *>  The contents of A on exit are illustrated by the following examples:
   *>
   *>  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
   *>
   *>    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
   *>    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
   *>    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
   *>    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
   *>    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
   *>    (  v1  v2  v3  v4  v5 )
   *>
   *>  where d and e denote diagonal and off-diagonal elements of B, vi
   *>  denotes an element of the vector defining H(i), and ui an element of
   *>  the vector defining G(i).
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )        SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, M, N        INTEGER            INFO, LDA, M, N
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       COMPLEX*16         A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )        COMPLEX*16         A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGEBD2 reduces a complex general m by n matrix A to upper or lower  
 *  real bidiagonal form B by a unitary transformation: Q' * A * P = B.  
 *  
 *  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  M       (input) INTEGER  
 *          The number of rows in the matrix A.  M >= 0.  
 *  
 *  N       (input) INTEGER  
 *          The number of columns in the matrix A.  N >= 0.  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)  
 *          On entry, the m by n general matrix to be reduced.  
 *          On exit,  
 *          if m >= n, the diagonal and the first superdiagonal are  
 *            overwritten with the upper bidiagonal matrix B; the  
 *            elements below the diagonal, with the array TAUQ, represent  
 *            the unitary matrix Q as a product of elementary  
 *            reflectors, and the elements above the first superdiagonal,  
 *            with the array TAUP, represent the unitary matrix P as  
 *            a product of elementary reflectors;  
 *          if m < n, the diagonal and the first subdiagonal are  
 *            overwritten with the lower bidiagonal matrix B; the  
 *            elements below the first subdiagonal, with the array TAUQ,  
 *            represent the unitary matrix Q as a product of  
 *            elementary reflectors, and the elements above the diagonal,  
 *            with the array TAUP, represent the unitary matrix P as  
 *            a product of elementary reflectors.  
 *          See Further Details.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,M).  
 *  
 *  D       (output) DOUBLE PRECISION array, dimension (min(M,N))  
 *          The diagonal elements of the bidiagonal matrix B:  
 *          D(i) = A(i,i).  
 *  
 *  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)  
 *          The off-diagonal elements of the bidiagonal matrix B:  
 *          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;  
 *          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.  
 *  
 *  TAUQ    (output) COMPLEX*16 array dimension (min(M,N))  
 *          The scalar factors of the elementary reflectors which  
 *          represent the unitary matrix Q. See Further Details.  
 *  
 *  TAUP    (output) COMPLEX*16 array, dimension (min(M,N))  
 *          The scalar factors of the elementary reflectors which  
 *          represent the unitary matrix P. See Further Details.  
 *  
 *  WORK    (workspace) COMPLEX*16 array, dimension (max(M,N))  
 *  
 *  INFO    (output) INTEGER  
 *          = 0: successful exit  
 *          < 0: if INFO = -i, the i-th argument had an illegal value.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The matrices Q and P are represented as products of elementary  
 *  reflectors:  
 *  
 *  If m >= n,  
 *  
 *     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)  
 *  
 *  Each H(i) and G(i) has the form:  
 *  
 *     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'  
 *  
 *  where tauq and taup are complex scalars, and v and u are complex  
 *  vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in  
 *  A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in  
 *  A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).  
 *  
 *  If m < n,  
 *  
 *     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)  
 *  
 *  Each H(i) and G(i) has the form:  
 *  
 *     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'  
 *  
 *  where tauq and taup are complex scalars, v and u are complex vectors;  
 *  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);  
 *  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);  
 *  tauq is stored in TAUQ(i) and taup in TAUP(i).  
 *  
 *  The contents of A on exit are illustrated by the following examples:  
 *  
 *  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):  
 *  
 *    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )  
 *    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )  
 *    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )  
 *    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )  
 *    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )  
 *    (  v1  v2  v3  v4  v5 )  
 *  
 *  where d and e denote diagonal and off-diagonal elements of B, vi  
 *  denotes an element of the vector defining H(i), and ui an element of  
 *  the vector defining G(i).  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Line 167 Line 244
             ALPHA = A( I, I )              ALPHA = A( I, I )
             CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,              CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,
      $                   TAUQ( I ) )       $                   TAUQ( I ) )
             D( I ) = ALPHA              D( I ) = DBLE( ALPHA )
             A( I, I ) = ONE              A( I, I ) = ONE
 *  *
 *           Apply H(i)' to A(i:m,i+1:n) from the left  *           Apply H(i)**H to A(i:m,i+1:n) from the left
 *  *
             IF( I.LT.N )              IF( I.LT.N )
      $         CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1,       $         CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1,
Line 186 Line 263
                ALPHA = A( I, I+1 )                 ALPHA = A( I, I+1 )
                CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA,                 CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA,
      $                      TAUP( I ) )       $                      TAUP( I ) )
                E( I ) = ALPHA                 E( I ) = DBLE( ALPHA )
                A( I, I+1 ) = ONE                 A( I, I+1 ) = ONE
 *  *
 *              Apply G(i) to A(i+1:m,i+1:n) from the right  *              Apply G(i) to A(i+1:m,i+1:n) from the right
Line 211 Line 288
             ALPHA = A( I, I )              ALPHA = A( I, I )
             CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,              CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,
      $                   TAUP( I ) )       $                   TAUP( I ) )
             D( I ) = ALPHA              D( I ) = DBLE( ALPHA )
             A( I, I ) = ONE              A( I, I ) = ONE
 *  *
 *           Apply G(i) to A(i+1:m,i:n) from the right  *           Apply G(i) to A(i+1:m,i:n) from the right
Line 230 Line 307
                ALPHA = A( I+1, I )                 ALPHA = A( I+1, I )
                CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,                 CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,
      $                      TAUQ( I ) )       $                      TAUQ( I ) )
                E( I ) = ALPHA                 E( I ) = DBLE( ALPHA )
                A( I+1, I ) = ONE                 A( I+1, I ) = ONE
 *  *
 *              Apply H(i)' to A(i+1:m,i+1:n) from the left  *              Apply H(i)**H to A(i+1:m,i+1:n) from the left
 *  *
                CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1,                 CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1,
      $                     DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA,       $                     DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA,
Removed from v.1.1  
changed lines
  Added in v.1.20

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