LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ slabrd()

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.
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 208 of file slabrd.f.

210 *
211 * -- LAPACK auxiliary routine --
212 * -- LAPACK is a software package provided by Univ. of Tennessee, --
213 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
214 *
215 * .. Scalar Arguments ..
216  INTEGER LDA, LDX, LDY, M, N, NB
217 * ..
218 * .. Array Arguments ..
219  REAL A( LDA, * ), D( * ), E( * ), TAUP( * ),
220  $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
221 * ..
222 *
223 * =====================================================================
224 *
225 * .. Parameters ..
226  REAL ZERO, ONE
227  parameter( zero = 0.0e0, one = 1.0e0 )
228 * ..
229 * .. Local Scalars ..
230  INTEGER I
231 * ..
232 * .. External Subroutines ..
233  EXTERNAL sgemv, slarfg, sscal
234 * ..
235 * .. Intrinsic Functions ..
236  INTRINSIC min
237 * ..
238 * .. Executable Statements ..
239 *
240 * Quick return if possible
241 *
242  IF( m.LE.0 .OR. n.LE.0 )
243  $ RETURN
244 *
245  IF( m.GE.n ) THEN
246 *
247 * Reduce to upper bidiagonal form
248 *
249  DO 10 i = 1, nb
250 *
251 * Update A(i:m,i)
252 *
253  CALL sgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
254  $ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
255  CALL sgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
256  $ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
257 *
258 * Generate reflection Q(i) to annihilate A(i+1:m,i)
259 *
260  CALL slarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
261  $ tauq( i ) )
262  d( i ) = a( i, i )
263  IF( i.LT.n ) THEN
264  a( i, i ) = one
265 *
266 * Compute Y(i+1:n,i)
267 *
268  CALL sgemv( 'Transpose', m-i+1, n-i, one, a( i, i+1 ),
269  $ lda, a( i, i ), 1, zero, y( i+1, i ), 1 )
270  CALL sgemv( 'Transpose', m-i+1, i-1, one, a( i, 1 ), lda,
271  $ a( i, i ), 1, zero, y( 1, i ), 1 )
272  CALL sgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
273  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
274  CALL sgemv( 'Transpose', m-i+1, i-1, one, x( i, 1 ), ldx,
275  $ a( i, i ), 1, zero, y( 1, i ), 1 )
276  CALL sgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
277  $ lda, y( 1, i ), 1, one, y( i+1, i ), 1 )
278  CALL sscal( n-i, tauq( i ), y( i+1, i ), 1 )
279 *
280 * Update A(i,i+1:n)
281 *
282  CALL sgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
283  $ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
284  CALL sgemv( 'Transpose', i-1, n-i, -one, a( 1, i+1 ),
285  $ lda, x( i, 1 ), ldx, one, a( i, i+1 ), lda )
286 *
287 * Generate reflection P(i) to annihilate A(i,i+2:n)
288 *
289  CALL slarfg( n-i, a( i, i+1 ), a( i, min( i+2, n ) ),
290  $ lda, taup( i ) )
291  e( i ) = a( i, i+1 )
292  a( i, i+1 ) = one
293 *
294 * Compute X(i+1:m,i)
295 *
296  CALL sgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
297  $ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
298  CALL sgemv( 'Transpose', n-i, i, one, y( i+1, 1 ), ldy,
299  $ a( i, i+1 ), lda, zero, x( 1, i ), 1 )
300  CALL sgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
301  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
302  CALL sgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
303  $ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
304  CALL sgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
305  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
306  CALL sscal( m-i, taup( i ), x( i+1, i ), 1 )
307  END IF
308  10 CONTINUE
309  ELSE
310 *
311 * Reduce to lower bidiagonal form
312 *
313  DO 20 i = 1, nb
314 *
315 * Update A(i,i:n)
316 *
317  CALL sgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
318  $ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
319  CALL sgemv( 'Transpose', i-1, n-i+1, -one, a( 1, i ), lda,
320  $ x( i, 1 ), ldx, one, a( i, i ), lda )
321 *
322 * Generate reflection P(i) to annihilate A(i,i+1:n)
323 *
324  CALL slarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
325  $ taup( i ) )
326  d( i ) = a( i, i )
327  IF( i.LT.m ) THEN
328  a( i, i ) = one
329 *
330 * Compute X(i+1:m,i)
331 *
332  CALL sgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
333  $ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
334  CALL sgemv( 'Transpose', n-i+1, i-1, one, y( i, 1 ), ldy,
335  $ a( i, i ), lda, zero, x( 1, i ), 1 )
336  CALL sgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
337  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
338  CALL sgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
339  $ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
340  CALL sgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
341  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
342  CALL sscal( m-i, taup( i ), x( i+1, i ), 1 )
343 *
344 * Update A(i+1:m,i)
345 *
346  CALL sgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
347  $ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
348  CALL sgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
349  $ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
350 *
351 * Generate reflection Q(i) to annihilate A(i+2:m,i)
352 *
353  CALL slarfg( m-i, a( i+1, i ), a( min( i+2, m ), i ), 1,
354  $ tauq( i ) )
355  e( i ) = a( i+1, i )
356  a( i+1, i ) = one
357 *
358 * Compute Y(i+1:n,i)
359 *
360  CALL sgemv( 'Transpose', m-i, n-i, one, a( i+1, i+1 ),
361  $ lda, a( i+1, i ), 1, zero, y( i+1, i ), 1 )
362  CALL sgemv( 'Transpose', m-i, i-1, one, a( i+1, 1 ), lda,
363  $ a( i+1, i ), 1, zero, y( 1, i ), 1 )
364  CALL sgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
365  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
366  CALL sgemv( 'Transpose', m-i, i, one, x( i+1, 1 ), ldx,
367  $ a( i+1, i ), 1, zero, y( 1, i ), 1 )
368  CALL sgemv( 'Transpose', i, n-i, -one, a( 1, i+1 ), lda,
369  $ y( 1, i ), 1, one, y( i+1, i ), 1 )
370  CALL sscal( n-i, tauq( i ), y( i+1, i ), 1 )
371  END IF
372  20 CONTINUE
373  END IF
374  RETURN
375 *
376 * End of SLABRD
377 *
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
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