cgebd2()

subroutine cgebd2	(	integer	m,
		integer	n,
		complex, dimension( lda, * )	a,
		integer	lda,
		real, dimension( * )	d,
		real, dimension( * )	e,
		complex, dimension( * )	tauq,
		complex, dimension( * )	taup,
		complex, dimension( * )	work,
		integer	info )

CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.

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Purpose:

!>
!> CGEBD2 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.
!> 

Parameters

[in]	M	!> M is INTEGER !> The number of rows in the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns in the matrix A. N >= 0. !>
[in,out]	A	!> A is COMPLEX 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. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
[out]	D	!> D is REAL array, dimension (min(M,N)) !> The diagonal elements of the bidiagonal matrix B: !> D(i) = A(i,i). !>
[out]	E	!> E is REAL 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. !>
[out]	TAUQ	!> TAUQ is COMPLEX array, dimension (min(M,N)) !> The scalar factors of the elementary reflectors which !> represent the unitary matrix Q. See Further Details. !>
[out]	TAUP	!> TAUP is COMPLEX array, dimension (min(M,N)) !> The scalar factors of the elementary reflectors which !> represent the unitary matrix P. See Further Details. !>
[out]	WORK	!> WORK is COMPLEX array, dimension (max(M,N)) !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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**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).
!> 

Definition at line 187 of file cgebd2.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      REAL               D( * ), E( * )
      COMPLEX            A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX            ZERO
      parameter( zero = ( 0.0e+0, 0.0e+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      COMPLEX            ALPHA
*     ..
*     .. External Subroutines ..
      EXTERNAL           clacgv, clarf1f, clarfg, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          conjg, max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      info = 0
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -4
      END IF
      IF( info.LT.0 ) THEN
         CALL xerbla( 'CGEBD2', -info )
         RETURN
      END IF
*
      IF( m.GE.n ) THEN
*
*        Reduce to upper bidiagonal form
*
         DO 10 i = 1, n
*
*           Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
            alpha = a( i, i )
            CALL clarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
     $                   tauq( i ) )
            d( i ) = real( alpha )
*
*           Apply H(i)**H to A(i:m,i+1:n) from the left
*
            IF( i.LT.n )
     $         CALL clarf1f( 'Left', m-i+1, n-i, a( i, i ), 1,
     $                       conjg( tauq( i ) ), a( i, i+1 ), lda,
     $                       work )
            a( i, i ) = d( i )
*
            IF( i.LT.n ) THEN
*
*              Generate elementary reflector G(i) to annihilate
*              A(i,i+2:n)
*
               CALL clacgv( n-i, a( i, i+1 ), lda )
               alpha = a( i, i+1 )
               CALL clarfg( n-i, alpha, a( i, min( i+2, n ) ),
     $                      lda, taup( i ) )
               e( i ) = real( alpha )
*
*              Apply G(i) to A(i+1:m,i+1:n) from the right
*
               CALL clarf1f( 'Right', m-i, n-i, a( i, i+1 ), lda,
     $                       taup( i ), a( i+1, i+1 ), lda, work )
               CALL clacgv( n-i, a( i, i+1 ), lda )
               a( i, i+1 ) = e( i )
            ELSE
               taup( i ) = zero
            END IF
   10    CONTINUE
      ELSE
*
*        Reduce to lower bidiagonal form
*
         DO 20 i = 1, m
*
*           Generate elementary reflector G(i) to annihilate A(i,i+1:n)
*
            CALL clacgv( n-i+1, a( i, i ), lda )
            alpha = a( i, i )
            CALL clarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
     $                   taup( i ) )
            d( i ) = real( alpha )
*
*           Apply G(i) to A(i+1:m,i:n) from the right
*
            IF( i.LT.m )
     $         CALL clarf1f( 'Right', m-i, n-i+1, a( i, i ), lda,
     $                       taup( i ), a( i+1, i ), lda, work )
            CALL clacgv( n-i+1, a( i, i ), lda )
            a( i, i ) = d( i )
*
            IF( i.LT.m ) THEN
*
*              Generate elementary reflector H(i) to annihilate
*              A(i+2:m,i)
*
               alpha = a( i+1, i )
               CALL clarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
     $                      tauq( i ) )
               e( i ) = real( alpha )
*
*              Apply H(i)**H to A(i+1:m,i+1:n) from the left
*
               CALL clarf1f( 'Left', m-i, n-i, a( i+1, i ), 1,
     $                       conjg( tauq( i ) ), a( i+1, i+1 ), lda,
     $                       work )
               a( i+1, i ) = e( i )
            ELSE
               tauq( i ) = zero
            END IF
   20    CONTINUE
      END IF
      RETURN
*
*     End of CGEBD2
*

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