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RedHat 9 (Linux i386) - man page for cgebd2 (redhat section l)

CGEBD2(l)					)					CGEBD2(l)

NAME
       CGEBD2 - reduce a complex general m by n matrix A to upper or lower real bidiagonal form B
       by a unitary transformation

SYNOPSIS
       SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

	   INTEGER	  INFO, LDA, M, N

	   REAL 	  D( * ), E( * )

	   COMPLEX	  A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )

PURPOSE
       CGEBD2 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 array, dimension (LDA,N)
	       On entry, the m by n general matrix to be reduced.  On exit, if m >= n, the diago-
	       nal  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 over-
	       written with the lower bidiagonal matrix B; the elements below the first subdiago-
	       nal,  with  the array TAUQ, represent the unitary matrix Q as a product of elemen-
	       tary reflectors, and the elements above the diagonal, with the array TAUP,  repre-
	       sent  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) REAL array, dimension (min(M,N))
	       The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).

       E       (output) 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.

       TAUQ    (output) COMPLEX array dimension (min(M,N))
	       The scalar factors of the elementary reflectors which represent the unitary matrix
	       Q.  See Further Details.  TAUP	 (output) COMPLEX array, dimension (min(M,N)) The
	       scalar factors of the elementary reflectors which represent the unitary matrix  P.
	       See Further Details.  WORK    (workspace) COMPLEX 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).

LAPACK version 3.0			   15 June 2000 				CGEBD2(l)


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