LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ clabrd()

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.
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 210 of file clabrd.f.

212 *
213 * -- LAPACK auxiliary routine --
214 * -- LAPACK is a software package provided by Univ. of Tennessee, --
215 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216 *
217 * .. Scalar Arguments ..
218  INTEGER LDA, LDX, LDY, M, N, NB
219 * ..
220 * .. Array Arguments ..
221  REAL D( * ), E( * )
222  COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
223  $ Y( LDY, * )
224 * ..
225 *
226 * =====================================================================
227 *
228 * .. Parameters ..
229  COMPLEX ZERO, ONE
230  parameter( zero = ( 0.0e+0, 0.0e+0 ),
231  $ one = ( 1.0e+0, 0.0e+0 ) )
232 * ..
233 * .. Local Scalars ..
234  INTEGER I
235  COMPLEX ALPHA
236 * ..
237 * .. External Subroutines ..
238  EXTERNAL cgemv, clacgv, clarfg, cscal
239 * ..
240 * .. Intrinsic Functions ..
241  INTRINSIC min
242 * ..
243 * .. Executable Statements ..
244 *
245 * Quick return if possible
246 *
247  IF( m.LE.0 .OR. n.LE.0 )
248  $ RETURN
249 *
250  IF( m.GE.n ) THEN
251 *
252 * Reduce to upper bidiagonal form
253 *
254  DO 10 i = 1, nb
255 *
256 * Update A(i:m,i)
257 *
258  CALL clacgv( i-1, y( i, 1 ), ldy )
259  CALL cgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
260  $ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
261  CALL clacgv( i-1, y( i, 1 ), ldy )
262  CALL cgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
263  $ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
264 *
265 * Generate reflection Q(i) to annihilate A(i+1:m,i)
266 *
267  alpha = a( i, i )
268  CALL clarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
269  $ tauq( i ) )
270  d( i ) = real( alpha )
271  IF( i.LT.n ) THEN
272  a( i, i ) = one
273 *
274 * Compute Y(i+1:n,i)
275 *
276  CALL cgemv( 'Conjugate transpose', m-i+1, n-i, one,
277  $ a( i, i+1 ), lda, a( i, i ), 1, zero,
278  $ y( i+1, i ), 1 )
279  CALL cgemv( 'Conjugate transpose', m-i+1, i-1, one,
280  $ a( i, 1 ), lda, a( i, i ), 1, zero,
281  $ y( 1, i ), 1 )
282  CALL cgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
283  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
284  CALL cgemv( 'Conjugate transpose', m-i+1, i-1, one,
285  $ x( i, 1 ), ldx, a( i, i ), 1, zero,
286  $ y( 1, i ), 1 )
287  CALL cgemv( 'Conjugate transpose', i-1, n-i, -one,
288  $ a( 1, i+1 ), lda, y( 1, i ), 1, one,
289  $ y( i+1, i ), 1 )
290  CALL cscal( n-i, tauq( i ), y( i+1, i ), 1 )
291 *
292 * Update A(i,i+1:n)
293 *
294  CALL clacgv( n-i, a( i, i+1 ), lda )
295  CALL clacgv( i, a( i, 1 ), lda )
296  CALL cgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
297  $ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
298  CALL clacgv( i, a( i, 1 ), lda )
299  CALL clacgv( i-1, x( i, 1 ), ldx )
300  CALL cgemv( 'Conjugate transpose', i-1, n-i, -one,
301  $ a( 1, i+1 ), lda, x( i, 1 ), ldx, one,
302  $ a( i, i+1 ), lda )
303  CALL clacgv( i-1, x( i, 1 ), ldx )
304 *
305 * Generate reflection P(i) to annihilate A(i,i+2:n)
306 *
307  alpha = a( i, i+1 )
308  CALL clarfg( n-i, alpha, a( i, min( i+2, n ) ),
309  $ lda, taup( i ) )
310  e( i ) = real( alpha )
311  a( i, i+1 ) = one
312 *
313 * Compute X(i+1:m,i)
314 *
315  CALL cgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
316  $ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
317  CALL cgemv( 'Conjugate transpose', n-i, i, one,
318  $ y( i+1, 1 ), ldy, a( i, i+1 ), lda, zero,
319  $ x( 1, i ), 1 )
320  CALL cgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
321  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
322  CALL cgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
323  $ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
324  CALL cgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
325  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
326  CALL cscal( m-i, taup( i ), x( i+1, i ), 1 )
327  CALL clacgv( n-i, a( i, i+1 ), lda )
328  END IF
329  10 CONTINUE
330  ELSE
331 *
332 * Reduce to lower bidiagonal form
333 *
334  DO 20 i = 1, nb
335 *
336 * Update A(i,i:n)
337 *
338  CALL clacgv( n-i+1, a( i, i ), lda )
339  CALL clacgv( i-1, a( i, 1 ), lda )
340  CALL cgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
341  $ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
342  CALL clacgv( i-1, a( i, 1 ), lda )
343  CALL clacgv( i-1, x( i, 1 ), ldx )
344  CALL cgemv( 'Conjugate transpose', i-1, n-i+1, -one,
345  $ a( 1, i ), lda, x( i, 1 ), ldx, one, a( i, i ),
346  $ lda )
347  CALL clacgv( i-1, x( i, 1 ), ldx )
348 *
349 * Generate reflection P(i) to annihilate A(i,i+1:n)
350 *
351  alpha = a( i, i )
352  CALL clarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
353  $ taup( i ) )
354  d( i ) = real( alpha )
355  IF( i.LT.m ) THEN
356  a( i, i ) = one
357 *
358 * Compute X(i+1:m,i)
359 *
360  CALL cgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
361  $ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
362  CALL cgemv( 'Conjugate transpose', n-i+1, i-1, one,
363  $ y( i, 1 ), ldy, a( i, i ), lda, zero,
364  $ x( 1, i ), 1 )
365  CALL cgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
366  $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
367  CALL cgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
368  $ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
369  CALL cgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
370  $ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
371  CALL cscal( m-i, taup( i ), x( i+1, i ), 1 )
372  CALL clacgv( n-i+1, a( i, i ), lda )
373 *
374 * Update A(i+1:m,i)
375 *
376  CALL clacgv( i-1, y( i, 1 ), ldy )
377  CALL cgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
378  $ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
379  CALL clacgv( i-1, y( i, 1 ), ldy )
380  CALL cgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
381  $ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
382 *
383 * Generate reflection Q(i) to annihilate A(i+2:m,i)
384 *
385  alpha = a( i+1, i )
386  CALL clarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
387  $ tauq( i ) )
388  e( i ) = real( alpha )
389  a( i+1, i ) = one
390 *
391 * Compute Y(i+1:n,i)
392 *
393  CALL cgemv( 'Conjugate transpose', m-i, n-i, one,
394  $ a( i+1, i+1 ), lda, a( i+1, i ), 1, zero,
395  $ y( i+1, i ), 1 )
396  CALL cgemv( 'Conjugate transpose', m-i, i-1, one,
397  $ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
398  $ y( 1, i ), 1 )
399  CALL cgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
400  $ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
401  CALL cgemv( 'Conjugate transpose', m-i, i, one,
402  $ x( i+1, 1 ), ldx, a( i+1, i ), 1, zero,
403  $ y( 1, i ), 1 )
404  CALL cgemv( 'Conjugate transpose', i, n-i, -one,
405  $ a( 1, i+1 ), lda, y( 1, i ), 1, one,
406  $ y( i+1, i ), 1 )
407  CALL cscal( n-i, tauq( i ), y( i+1, i ), 1 )
408  ELSE
409  CALL clacgv( n-i+1, a( i, i ), lda )
410  END IF
411  20 CONTINUE
412  END IF
413  RETURN
414 *
415 * End of CLABRD
416 *
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:158
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:74
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
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