LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ zgebrd()

subroutine zgebrd ( integer  M,
integer  N,
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( * )  WORK,
integer  LWORK,
integer  INFO 
)

ZGEBRD

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

Purpose:
 ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
 bidiagonal form B by a unitary transformation: Q**H * A * P = B.

 If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
Parameters
[in]M
          M is INTEGER
          The number of rows in the matrix A.  M >= 0.
[in]N
          N is INTEGER
          The number of columns in the matrix A.  N >= 0.
[in,out]A
          A is COMPLEX*16 array, dimension (LDA,N)
          On entry, the M-by-N general matrix to be reduced.
          On exit,
          if m >= n, the diagonal and the first superdiagonal are
            overwritten with the upper bidiagonal matrix B; the
            elements below the diagonal, with the array TAUQ, represent
            the unitary matrix Q as a product of elementary
            reflectors, and the elements above the first superdiagonal,
            with the array TAUP, represent the unitary matrix P as
            a product of elementary reflectors;
          if m < n, the diagonal and the first subdiagonal are
            overwritten with the lower bidiagonal matrix B; the
            elements below the first subdiagonal, with the array TAUQ,
            represent the unitary matrix Q as a product of
            elementary reflectors, and the elements above the diagonal,
            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 (min(M,N))
          The diagonal elements of the bidiagonal matrix B:
          D(i) = A(i,i).
[out]E
          E is DOUBLE PRECISION array, dimension (min(M,N)-1)
          The off-diagonal elements of the bidiagonal matrix B:
          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
[out]TAUQ
          TAUQ is COMPLEX*16 array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Q. See Further Details.
[out]TAUP
          TAUP is COMPLEX*16 array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the unitary matrix P. See Further Details.
[out]WORK
          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The length of the array WORK.  LWORK >= max(1,M,N).
          For optimum performance LWORK >= (M+N)*NB, where NB
          is the optimal blocksize.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
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:

  If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

  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; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
  A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
  A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

  If m < n,

     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

  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; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
  A(i+2:m,i); u(1:i-1) = 0, u(i) = 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).

  The contents of A on exit are illustrated by the following examples:

  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
    (  v1  v2  v3  v4  v5 )

  where d and e denote diagonal and off-diagonal elements of B, vi
  denotes an element of the vector defining H(i), and ui an element of
  the vector defining G(i).

Definition at line 203 of file zgebrd.f.

205*
206* -- LAPACK computational routine --
207* -- LAPACK is a software package provided by Univ. of Tennessee, --
208* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
209*
210* .. Scalar Arguments ..
211 INTEGER INFO, LDA, LWORK, M, N
212* ..
213* .. Array Arguments ..
214 DOUBLE PRECISION D( * ), E( * )
215 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
216* ..
217*
218* =====================================================================
219*
220* .. Parameters ..
221 COMPLEX*16 ONE
222 parameter( one = ( 1.0d+0, 0.0d+0 ) )
223* ..
224* .. Local Scalars ..
225 LOGICAL LQUERY
226 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
227 $ NBMIN, NX, WS
228* ..
229* .. External Subroutines ..
230 EXTERNAL xerbla, zgebd2, zgemm, zlabrd
231* ..
232* .. Intrinsic Functions ..
233 INTRINSIC dble, max, min
234* ..
235* .. External Functions ..
236 INTEGER ILAENV
237 EXTERNAL ilaenv
238* ..
239* .. Executable Statements ..
240*
241* Test the input parameters
242*
243 info = 0
244 nb = max( 1, ilaenv( 1, 'ZGEBRD', ' ', m, n, -1, -1 ) )
245 lwkopt = ( m+n )*nb
246 work( 1 ) = dble( lwkopt )
247 lquery = ( lwork.EQ.-1 )
248 IF( m.LT.0 ) THEN
249 info = -1
250 ELSE IF( n.LT.0 ) THEN
251 info = -2
252 ELSE IF( lda.LT.max( 1, m ) ) THEN
253 info = -4
254 ELSE IF( lwork.LT.max( 1, m, n ) .AND. .NOT.lquery ) THEN
255 info = -10
256 END IF
257 IF( info.LT.0 ) THEN
258 CALL xerbla( 'ZGEBRD', -info )
259 RETURN
260 ELSE IF( lquery ) THEN
261 RETURN
262 END IF
263*
264* Quick return if possible
265*
266 minmn = min( m, n )
267 IF( minmn.EQ.0 ) THEN
268 work( 1 ) = 1
269 RETURN
270 END IF
271*
272 ws = max( m, n )
273 ldwrkx = m
274 ldwrky = n
275*
276 IF( nb.GT.1 .AND. nb.LT.minmn ) THEN
277*
278* Set the crossover point NX.
279*
280 nx = max( nb, ilaenv( 3, 'ZGEBRD', ' ', m, n, -1, -1 ) )
281*
282* Determine when to switch from blocked to unblocked code.
283*
284 IF( nx.LT.minmn ) THEN
285 ws = ( m+n )*nb
286 IF( lwork.LT.ws ) THEN
287*
288* Not enough work space for the optimal NB, consider using
289* a smaller block size.
290*
291 nbmin = ilaenv( 2, 'ZGEBRD', ' ', m, n, -1, -1 )
292 IF( lwork.GE.( m+n )*nbmin ) THEN
293 nb = lwork / ( m+n )
294 ELSE
295 nb = 1
296 nx = minmn
297 END IF
298 END IF
299 END IF
300 ELSE
301 nx = minmn
302 END IF
303*
304 DO 30 i = 1, minmn - nx, nb
305*
306* Reduce rows and columns i:i+ib-1 to bidiagonal form and return
307* the matrices X and Y which are needed to update the unreduced
308* part of the matrix
309*
310 CALL zlabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),
311 $ tauq( i ), taup( i ), work, ldwrkx,
312 $ work( ldwrkx*nb+1 ), ldwrky )
313*
314* Update the trailing submatrix A(i+ib:m,i+ib:n), using
315* an update of the form A := A - V*Y**H - X*U**H
316*
317 CALL zgemm( 'No transpose', 'Conjugate transpose', m-i-nb+1,
318 $ n-i-nb+1, nb, -one, a( i+nb, i ), lda,
319 $ work( ldwrkx*nb+nb+1 ), ldwrky, one,
320 $ a( i+nb, i+nb ), lda )
321 CALL zgemm( 'No transpose', 'No transpose', m-i-nb+1, n-i-nb+1,
322 $ nb, -one, work( nb+1 ), ldwrkx, a( i, i+nb ), lda,
323 $ one, a( i+nb, i+nb ), lda )
324*
325* Copy diagonal and off-diagonal elements of B back into A
326*
327 IF( m.GE.n ) THEN
328 DO 10 j = i, i + nb - 1
329 a( j, j ) = d( j )
330 a( j, j+1 ) = e( j )
331 10 CONTINUE
332 ELSE
333 DO 20 j = i, i + nb - 1
334 a( j, j ) = d( j )
335 a( j+1, j ) = e( j )
336 20 CONTINUE
337 END IF
338 30 CONTINUE
339*
340* Use unblocked code to reduce the remainder of the matrix
341*
342 CALL zgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),
343 $ tauq( i ), taup( i ), work, iinfo )
344 work( 1 ) = ws
345 RETURN
346*
347* End of ZGEBRD
348*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zgebd2(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
ZGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
Definition: zgebd2.f:189
subroutine zlabrd(M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Definition: zlabrd.f:212
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