LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ dlabrd()

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

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

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

Purpose:
 DLABRD 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 DGEBRD
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB)
          The off-diagonal elements of the first NB rows and columns of
          the reduced matrix.
[out]TAUQ
          TAUQ is DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q. See Further Details.
[out]TAUP
          TAUP is DOUBLE PRECISION array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix P. See Further Details.
[out]X
          X is DOUBLE PRECISION 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 DOUBLE PRECISION 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
June 2017
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 212 of file dlabrd.f.

212 *
213 * -- LAPACK auxiliary routine (version 3.7.1) --
214 * -- LAPACK is a software package provided by Univ. of Tennessee, --
215 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216 * June 2017
217 *
218 * .. Scalar Arguments ..
219  INTEGER lda, ldx, ldy, m, n, nb
220 * ..
221 * .. Array Arguments ..
222  DOUBLE PRECISION a( lda, * ), d( * ), e( * ), taup( * ),
223  $ tauq( * ), x( ldx, * ), y( ldy, * )
224 * ..
225 *
226 * =====================================================================
227 *
228 * .. Parameters ..
229  DOUBLE PRECISION zero, one
230  parameter( zero = 0.0d0, one = 1.0d0 )
231 * ..
232 * .. Local Scalars ..
233  INTEGER i
234 * ..
235 * .. External Subroutines ..
236  EXTERNAL dgemv, dlarfg, dscal
237 * ..
238 * .. Intrinsic Functions ..
239  INTRINSIC min
240 * ..
241 * .. Executable Statements ..
242 *
243 * Quick return if possible
244 *
245  IF( m.LE.0 .OR. n.LE.0 )
246  $ RETURN
247 *
248  IF( m.GE.n ) THEN
249 *
250 * Reduce to upper bidiagonal form
251 *
252  DO 10 i = 1, nb
253 *
254 * Update A(i:m,i)
255 *
256  CALL dgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
257  $ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
258  CALL dgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
259  $ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
260 *
261 * Generate reflection Q(i) to annihilate A(i+1:m,i)
262 *
263  CALL dlarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
264  $ tauq( i ) )
265  d( i ) = a( i, i )
266  IF( i.LT.n ) THEN
267  a( i, i ) = one
268 *
269 * Compute Y(i+1:n,i)
270 *
271  CALL dgemv( 'Transpose', m-i+1, n-i, one, a( i, i+1 ),
272  $ lda, a( i, i ), 1, zero, y( i+1, i ), 1 )
273  CALL dgemv( 'Transpose', m-i+1, i-1, one, a( i, 1 ), lda,
274  $ a( i, i ), 1, zero, y( 1, i ), 1 )
275  CALL dgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
276  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
277  CALL dgemv( 'Transpose', m-i+1, i-1, one, x( i, 1 ), ldx,
278  $ a( i, i ), 1, zero, y( 1, i ), 1 )
279  CALL dgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
280  $ lda, y( 1, i ), 1, one, y( i+1, i ), 1 )
281  CALL dscal( n-i, tauq( i ), y( i+1, i ), 1 )
282 *
283 * Update A(i,i+1:n)
284 *
285  CALL dgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
286  $ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
287  CALL dgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
288  $ lda, x( i, 1 ), ldx, one, a( i, i+1 ), lda )
289 *
290 * Generate reflection P(i) to annihilate A(i,i+2:n)
291 *
292  CALL dlarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),
293  $ lda, taup( i ) )
294  e( i ) = a( i, i+1 )
295  a( i, i+1 ) = one
296 *
297 * Compute X(i+1:m,i)
298 *
299  CALL dgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
300  $ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
301  CALL dgemv( 'Transpose', n-i, i, one, y( i+1, 1 ), ldy,
302  $ a( i, i+1 ), lda, zero, x( 1, i ), 1 )
303  CALL dgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
304  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
305  CALL dgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
306  $ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
307  CALL dgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
308  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
309  CALL dscal( m-i, taup( i ), x( i+1, i ), 1 )
310  END IF
311  10 CONTINUE
312  ELSE
313 *
314 * Reduce to lower bidiagonal form
315 *
316  DO 20 i = 1, nb
317 *
318 * Update A(i,i:n)
319 *
320  CALL dgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
321  $ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
322  CALL dgemv( 'Transpose', i-1, n-i+1, -one, a( 1, i ), lda,
323  $ x( i, 1 ), ldx, one, a( i, i ), lda )
324 *
325 * Generate reflection P(i) to annihilate A(i,i+1:n)
326 *
327  CALL dlarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
328  $ taup( i ) )
329  d( i ) = a( i, i )
330  IF( i.LT.m ) THEN
331  a( i, i ) = one
332 *
333 * Compute X(i+1:m,i)
334 *
335  CALL dgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
336  $ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
337  CALL dgemv( 'Transpose', n-i+1, i-1, one, y( i, 1 ), ldy,
338  $ a( i, i ), lda, zero, x( 1, i ), 1 )
339  CALL dgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
340  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
341  CALL dgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
342  $ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
343  CALL dgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
344  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
345  CALL dscal( m-i, taup( i ), x( i+1, i ), 1 )
346 *
347 * Update A(i+1:m,i)
348 *
349  CALL dgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
350  $ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
351  CALL dgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
352  $ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
353 *
354 * Generate reflection Q(i) to annihilate A(i+2:m,i)
355 *
356  CALL dlarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1,
357  $ tauq( i ) )
358  e( i ) = a( i+1, i )
359  a( i+1, i ) = one
360 *
361 * Compute Y(i+1:n,i)
362 *
363  CALL dgemv( 'Transpose', m-i, n-i, one, a( i+1, i+1 ),
364  $ lda, a( i+1, i ), 1, zero, y( i+1, i ), 1 )
365  CALL dgemv( 'Transpose', m-i, i-1, one, a( i+1, 1 ), lda,
366  $ a( i+1, i ), 1, zero, y( 1, i ), 1 )
367  CALL dgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
368  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
369  CALL dgemv( 'Transpose', m-i, i, one, x( i+1, 1 ), ldx,
370  $ a( i+1, i ), 1, zero, y( 1, i ), 1 )
371  CALL dgemv( 'Transpose', i, n-i, -one, a( 1, i+1 ), lda,
372  $ y( 1, i ), 1, one, y( i+1, i ), 1 )
373  CALL dscal( n-i, tauq( i ), y( i+1, i ), 1 )
374  END IF
375  20 CONTINUE
376  END IF
377  RETURN
378 *
379 * End of DLABRD
380 *
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:81
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:158
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:108
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