Functions/Subroutines
subroutine	slabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
	SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Function/Subroutine Documentation

subroutine slabrd	(	integer	M,
		integer	N,
		integer	NB,
		real, dimension( lda, * )	A,
		integer	LDA,
		real, dimension( * )	D,
		real, dimension( * )	E,
		real, dimension( * )	TAUQ,
		real, dimension( * )	TAUP,
		real, dimension( ldx, * )	X,
		integer	LDX,
		real, dimension( ldy, * )	Y,
		integer	LDY
	)

SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Download SLABRD + dependencies [TGZ] [ZIP] [TXT]

Purpose:

 SLABRD reduces the first NB rows and columns of a real general
 m by n matrix A to upper or lower bidiagonal form by an orthogonal
 transformation Q**T * A * P, and returns the matrices X and Y which
 are needed to apply the transformation to the unreduced part of A.

 If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
 bidiagonal form.

 This is an auxiliary routine called by SGEBRD

Parameters:

[in]	M	M is INTEGER The number of rows in the matrix A.
[in]	N	N is INTEGER The number of columns in the matrix A.
[in]	NB	NB is INTEGER The number of leading rows and columns of A to be reduced.
[in,out]	A	A is REAL array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal 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 (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i).
[out]	E	E is REAL array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.
[out]	TAUQ	TAUQ is REAL array dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.
[out]	TAUP	TAUP is REAL array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.
[out]	X	X is REAL array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A.
[in]	LDX	LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).
[out]	Y	Y is REAL array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A.
[in]	LDY	LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N).

Author:: Univ. of Tennessee; Univ. of California Berkeley; Univ. of Colorado Denver; NAG Ltd.

Date:: September 2012

Further Details:

  The matrices Q and P are represented as products of elementary
  reflectors:

     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v**T  and G(i) = I - taup * u * u**T

  where tauq and taup are real scalars, and v and u are real vectors.

  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
  A(i:m,i); u(1:i) = 0, u(i+1) = 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).

  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

  The elements of the vectors v and u together form the m-by-nb matrix
  V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
  the transformation to the unreduced part of the matrix, using a block
  update of the form:  A := A - V*Y**T - X*U**T.

  The contents of A on exit are illustrated by the following examples
  with nb = 2:

  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )

  where a denotes an element of the original matrix which is unchanged,
  vi denotes an element of the vector defining H(i), and ui an element
  of the vector defining G(i).

Definition at line 210 of file slabrd.f.

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