LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ zlabrd()

subroutine zlabrd ( integer  m,
integer  n,
integer  nb,
complex*16, dimension( lda, * )  a,
integer  lda,
double precision, dimension( * )  d,
double precision, dimension( * )  e,
complex*16, dimension( * )  tauq,
complex*16, dimension( * )  taup,
complex*16, dimension( ldx, * )  x,
integer  ldx,
complex*16, dimension( ldy, * )  y,
integer  ldy 
)

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

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

Purpose:
 ZLABRD 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 ZGEBRD
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*16 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 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 COMPLEX*16 array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Q. See Further Details.
[out]TAUP
          TAUP is COMPLEX*16 array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix P. See Further Details.
[out]X
          X is COMPLEX*16 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*16 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 zlabrd.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 DOUBLE PRECISION D( * ), E( * )
222 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
223 $ Y( LDY, * )
224* ..
225*
226* =====================================================================
227*
228* .. Parameters ..
229 COMPLEX*16 ZERO, ONE
230 parameter( zero = ( 0.0d+0, 0.0d+0 ),
231 $ one = ( 1.0d+0, 0.0d+0 ) )
232* ..
233* .. Local Scalars ..
234 INTEGER I
235 COMPLEX*16 ALPHA
236* ..
237* .. External Subroutines ..
238 EXTERNAL zgemv, zlacgv, zlarfg, zscal
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 zlacgv( i-1, y( i, 1 ), ldy )
259 CALL zgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
260 $ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
261 CALL zlacgv( i-1, y( i, 1 ), ldy )
262 CALL zgemv( '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 zlarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
269 $ tauq( i ) )
270 d( i ) = dble( alpha )
271 IF( i.LT.n ) THEN
272 a( i, i ) = one
273*
274* Compute Y(i+1:n,i)
275*
276 CALL zgemv( '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 zgemv( '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 zgemv( '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 zgemv( '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 zgemv( '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 zscal( n-i, tauq( i ), y( i+1, i ), 1 )
291*
292* Update A(i,i+1:n)
293*
294 CALL zlacgv( n-i, a( i, i+1 ), lda )
295 CALL zlacgv( i, a( i, 1 ), lda )
296 CALL zgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
297 $ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
298 CALL zlacgv( i, a( i, 1 ), lda )
299 CALL zlacgv( i-1, x( i, 1 ), ldx )
300 CALL zgemv( '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 zlacgv( 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 zlarfg( n-i, alpha, a( i, min( i+2, n ) ), lda,
309 $ taup( i ) )
310 e( i ) = dble( alpha )
311 a( i, i+1 ) = one
312*
313* Compute X(i+1:m,i)
314*
315 CALL zgemv( '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 zgemv( '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 zgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
321 $ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
322 CALL zgemv( '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 zgemv( '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 zscal( m-i, taup( i ), x( i+1, i ), 1 )
327 CALL zlacgv( 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 zlacgv( n-i+1, a( i, i ), lda )
339 CALL zlacgv( i-1, a( i, 1 ), lda )
340 CALL zgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
341 $ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
342 CALL zlacgv( i-1, a( i, 1 ), lda )
343 CALL zlacgv( i-1, x( i, 1 ), ldx )
344 CALL zgemv( '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 zlacgv( 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 zlarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
353 $ taup( i ) )
354 d( i ) = dble( alpha )
355 IF( i.LT.m ) THEN
356 a( i, i ) = one
357*
358* Compute X(i+1:m,i)
359*
360 CALL zgemv( '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 zgemv( '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 zgemv( '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 zgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
368 $ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
369 CALL zgemv( '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 zscal( m-i, taup( i ), x( i+1, i ), 1 )
372 CALL zlacgv( n-i+1, a( i, i ), lda )
373*
374* Update A(i+1:m,i)
375*
376 CALL zlacgv( i-1, y( i, 1 ), ldy )
377 CALL zgemv( '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 zlacgv( i-1, y( i, 1 ), ldy )
380 CALL zgemv( '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 zlarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
387 $ tauq( i ) )
388 e( i ) = dble( alpha )
389 a( i+1, i ) = one
390*
391* Compute Y(i+1:n,i)
392*
393 CALL zgemv( '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 zgemv( '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 zgemv( '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 zgemv( '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 zgemv( '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 zscal( n-i, tauq( i ), y( i+1, i ), 1 )
408 ELSE
409 CALL zlacgv( n-i+1, a( i, i ), lda )
410 END IF
411 20 CONTINUE
412 END IF
413 RETURN
414*
415* End of ZLABRD
416*
zgemv
subroutine zgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
ZGEMV
Definition zgemv.f:160
zlacgv
subroutine zlacgv(n, x, incx)
ZLACGV conjugates a complex vector.
Definition zlacgv.f:74
zlarfg
subroutine zlarfg(n, alpha, x, incx, tau)
ZLARFG generates an elementary reflector (Householder matrix).
Definition zlarfg.f:106
zscal
subroutine zscal(n, za, zx, incx)
ZSCAL
Definition zscal.f:78
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