LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
subroutine clabrd ( integer  M,
integer  N,
integer  NB,
complex, dimension( lda, * )  A,
integer  LDA,
real, dimension( * )  D,
real, dimension( * )  E,
complex, dimension( * )  TAUQ,
complex, dimension( * )  TAUP,
complex, dimension( ldx, * )  X,
integer  LDX,
complex, dimension( ldy, * )  Y,
integer  LDY 
)

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

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

Purpose: 
 CLABRD reduces the first NB rows and columns of a complex general
 m by n matrix A to upper or lower real bidiagonal form by a unitary
 transformation Q**H * 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 CGEBRD
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 COMPLEX 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 unitary
            matrix Q as a product of elementary reflectors; and
            elements above the diagonal in the first NB rows, with the
            array TAUP, represent the unitary 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 unitary
            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 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 (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 COMPLEX array dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Q. See Further Details.
[out]TAUP
          TAUP is COMPLEX array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix P. See Further Details.
[out]X
          X is COMPLEX 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 COMPLEX 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**H  and G(i) = I - taup * u * u**H

  where tauq and taup are complex scalars, and v and u are complex
  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**H 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**H - X*U**H.

  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 214 of file clabrd.f.

214 *
215 * -- LAPACK auxiliary routine (version 3.4.2) --
216 * -- LAPACK is a software package provided by Univ. of Tennessee, --
217 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
218 * September 2012
219 *
220 * .. Scalar Arguments ..
221  INTEGER lda, ldx, ldy, m, n, nb
222 * ..
223 * .. Array Arguments ..
224  REAL d( * ), e( * )
225  COMPLEX a( lda, * ), taup( * ), tauq( * ), x( ldx, * ),
226  $ y( ldy, * )
227 * ..
228 *
229 * =====================================================================
230 *
231 * .. Parameters ..
232  COMPLEX zero, one
233  parameter ( zero = ( 0.0e+0, 0.0e+0 ),
234  $ one = ( 1.0e+0, 0.0e+0 ) )
235 * ..
236 * .. Local Scalars ..
237  INTEGER i
238  COMPLEX alpha
239 * ..
240 * .. External Subroutines ..
241  EXTERNAL cgemv, clacgv, clarfg, cscal
242 * ..
243 * .. Intrinsic Functions ..
244  INTRINSIC min
245 * ..
246 * .. Executable Statements ..
247 *
248 * Quick return if possible
249 *
250  IF( m.LE.0 .OR. n.LE.0 )
251  $ RETURN
252 *
253  IF( m.GE.n ) THEN
254 *
255 * Reduce to upper bidiagonal form
256 *
257  DO 10 i = 1, nb
258 *
259 * Update A(i:m,i)
260 *
261  CALL clacgv( i-1, y( i, 1 ), ldy )
262  CALL cgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
263  $ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
264  CALL clacgv( i-1, y( i, 1 ), ldy )
265  CALL cgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
266  $ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
267 *
268 * Generate reflection Q(i) to annihilate A(i+1:m,i)
269 *
270  alpha = a( i, i )
271  CALL clarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
272  $ tauq( i ) )
273  d( i ) = alpha
274  IF( i.LT.n ) THEN
275  a( i, i ) = one
276 *
277 * Compute Y(i+1:n,i)
278 *
279  CALL cgemv( 'Conjugate transpose', m-i+1, n-i, one,
280  $ a( i, i+1 ), lda, a( i, i ), 1, zero,
281  $ y( i+1, i ), 1 )
282  CALL cgemv( 'Conjugate transpose', m-i+1, i-1, one,
283  $ a( i, 1 ), lda, a( i, i ), 1, zero,
284  $ y( 1, i ), 1 )
285  CALL cgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
286  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
287  CALL cgemv( 'Conjugate transpose', m-i+1, i-1, one,
288  $ x( i, 1 ), ldx, a( i, i ), 1, zero,
289  $ y( 1, i ), 1 )
290  CALL cgemv( 'Conjugate transpose', i-1, n-i, -one,
291  $ a( 1, i+1 ), lda, y( 1, i ), 1, one,
292  $ y( i+1, i ), 1 )
293  CALL cscal( n-i, tauq( i ), y( i+1, i ), 1 )
294 *
295 * Update A(i,i+1:n)
296 *
297  CALL clacgv( n-i, a( i, i+1 ), lda )
298  CALL clacgv( i, a( i, 1 ), lda )
299  CALL cgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
300  $ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
301  CALL clacgv( i, a( i, 1 ), lda )
302  CALL clacgv( i-1, x( i, 1 ), ldx )
303  CALL cgemv( 'Conjugate transpose', i-1, n-i, -one,
304  $ a( 1, i+1 ), lda, x( i, 1 ), ldx, one,
305  $ a( i, i+1 ), lda )
306  CALL clacgv( i-1, x( i, 1 ), ldx )
307 *
308 * Generate reflection P(i) to annihilate A(i,i+2:n)
309 *
310  alpha = a( i, i+1 )
311  CALL clarfg( n-i, alpha, a( i, min( i+2, n ) ),
312  $ lda, taup( i ) )
313  e( i ) = alpha
314  a( i, i+1 ) = one
315 *
316 * Compute X(i+1:m,i)
317 *
318  CALL cgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
319  $ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
320  CALL cgemv( 'Conjugate transpose', n-i, i, one,
321  $ y( i+1, 1 ), ldy, a( i, i+1 ), lda, zero,
322  $ x( 1, i ), 1 )
323  CALL cgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
324  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
325  CALL cgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
326  $ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
327  CALL cgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
328  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
329  CALL cscal( m-i, taup( i ), x( i+1, i ), 1 )
330  CALL clacgv( n-i, a( i, i+1 ), lda )
331  END IF
332  10 CONTINUE
333  ELSE
334 *
335 * Reduce to lower bidiagonal form
336 *
337  DO 20 i = 1, nb
338 *
339 * Update A(i,i:n)
340 *
341  CALL clacgv( n-i+1, a( i, i ), lda )
342  CALL clacgv( i-1, a( i, 1 ), lda )
343  CALL cgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
344  $ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
345  CALL clacgv( i-1, a( i, 1 ), lda )
346  CALL clacgv( i-1, x( i, 1 ), ldx )
347  CALL cgemv( 'Conjugate transpose', i-1, n-i+1, -one,
348  $ a( 1, i ), lda, x( i, 1 ), ldx, one, a( i, i ),
349  $ lda )
350  CALL clacgv( i-1, x( i, 1 ), ldx )
351 *
352 * Generate reflection P(i) to annihilate A(i,i+1:n)
353 *
354  alpha = a( i, i )
355  CALL clarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
356  $ taup( i ) )
357  d( i ) = alpha
358  IF( i.LT.m ) THEN
359  a( i, i ) = one
360 *
361 * Compute X(i+1:m,i)
362 *
363  CALL cgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
364  $ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
365  CALL cgemv( 'Conjugate transpose', n-i+1, i-1, one,
366  $ y( i, 1 ), ldy, a( i, i ), lda, zero,
367  $ x( 1, i ), 1 )
368  CALL cgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
369  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
370  CALL cgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
371  $ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
372  CALL cgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
373  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
374  CALL cscal( m-i, taup( i ), x( i+1, i ), 1 )
375  CALL clacgv( n-i+1, a( i, i ), lda )
376 *
377 * Update A(i+1:m,i)
378 *
379  CALL clacgv( i-1, y( i, 1 ), ldy )
380  CALL cgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
381  $ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
382  CALL clacgv( i-1, y( i, 1 ), ldy )
383  CALL cgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
384  $ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
385 *
386 * Generate reflection Q(i) to annihilate A(i+2:m,i)
387 *
388  alpha = a( i+1, i )
389  CALL clarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
390  $ tauq( i ) )
391  e( i ) = alpha
392  a( i+1, i ) = one
393 *
394 * Compute Y(i+1:n,i)
395 *
396  CALL cgemv( 'Conjugate transpose', m-i, n-i, one,
397  $ a( i+1, i+1 ), lda, a( i+1, i ), 1, zero,
398  $ y( i+1, i ), 1 )
399  CALL cgemv( 'Conjugate transpose', m-i, i-1, one,
400  $ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
401  $ y( 1, i ), 1 )
402  CALL cgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
403  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
404  CALL cgemv( 'Conjugate transpose', m-i, i, one,
405  $ x( i+1, 1 ), ldx, a( i+1, i ), 1, zero,
406  $ y( 1, i ), 1 )
407  CALL cgemv( 'Conjugate transpose', i, n-i, -one,
408  $ a( 1, i+1 ), lda, y( 1, i ), 1, one,
409  $ y( i+1, i ), 1 )
410  CALL cscal( n-i, tauq( i ), y( i+1, i ), 1 )
411  ELSE
412  CALL clacgv( n-i+1, a( i, i ), lda )
413  END IF
414  20 CONTINUE
415  END IF
416  RETURN
417 *
418 * End of CLABRD
419 *
cscal
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:54
cgemv
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:160
clacgv
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:76
clarfg
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:108

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