LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine dgebrd ( integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E, double precision, dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double precision, dimension( * ) WORK, integer LWORK, integer INFO )

DGEBRD

Purpose:
``` DGEBRD reduces a general real M-by-N matrix A to upper or lower
bidiagonal form B by an orthogonal transformation: Q**T * 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 DOUBLE PRECISION 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of elementary reflectors, and the elements above the diagonal, 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 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 DOUBLE PRECISION array dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details.``` [out] TAUP ``` TAUP is DOUBLE PRECISION array, dimension (min(M,N)) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details.``` [out] WORK ``` WORK is DOUBLE PRECISION 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.```
Date
November 2011
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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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**T  and G(i) = I - taup * u * u**T

where tauq and taup are real scalars, and v and u are real 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 207 of file dgebrd.f.

207 *
208 * -- LAPACK computational routine (version 3.4.0) --
209 * -- LAPACK is a software package provided by Univ. of Tennessee, --
210 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
211 * November 2011
212 *
213 * .. Scalar Arguments ..
214  INTEGER info, lda, lwork, m, n
215 * ..
216 * .. Array Arguments ..
217  DOUBLE PRECISION a( lda, * ), d( * ), e( * ), taup( * ),
218  \$ tauq( * ), work( * )
219 * ..
220 *
221 * =====================================================================
222 *
223 * .. Parameters ..
224  DOUBLE PRECISION one
225  parameter ( one = 1.0d+0 )
226 * ..
227 * .. Local Scalars ..
228  LOGICAL lquery
229  INTEGER i, iinfo, j, ldwrkx, ldwrky, lwkopt, minmn, nb,
230  \$ nbmin, nx
231  DOUBLE PRECISION ws
232 * ..
233 * .. External Subroutines ..
234  EXTERNAL dgebd2, dgemm, dlabrd, xerbla
235 * ..
236 * .. Intrinsic Functions ..
237  INTRINSIC dble, max, min
238 * ..
239 * .. External Functions ..
240  INTEGER ilaenv
241  EXTERNAL ilaenv
242 * ..
243 * .. Executable Statements ..
244 *
245 * Test the input parameters
246 *
247  info = 0
248  nb = max( 1, ilaenv( 1, 'DGEBRD', ' ', m, n, -1, -1 ) )
249  lwkopt = ( m+n )*nb
250  work( 1 ) = dble( lwkopt )
251  lquery = ( lwork.EQ.-1 )
252  IF( m.LT.0 ) THEN
253  info = -1
254  ELSE IF( n.LT.0 ) THEN
255  info = -2
256  ELSE IF( lda.LT.max( 1, m ) ) THEN
257  info = -4
258  ELSE IF( lwork.LT.max( 1, m, n ) .AND. .NOT.lquery ) THEN
259  info = -10
260  END IF
261  IF( info.LT.0 ) THEN
262  CALL xerbla( 'DGEBRD', -info )
263  RETURN
264  ELSE IF( lquery ) THEN
265  RETURN
266  END IF
267 *
268 * Quick return if possible
269 *
270  minmn = min( m, n )
271  IF( minmn.EQ.0 ) THEN
272  work( 1 ) = 1
273  RETURN
274  END IF
275 *
276  ws = max( m, n )
277  ldwrkx = m
278  ldwrky = n
279 *
280  IF( nb.GT.1 .AND. nb.LT.minmn ) THEN
281 *
282 * Set the crossover point NX.
283 *
284  nx = max( nb, ilaenv( 3, 'DGEBRD', ' ', m, n, -1, -1 ) )
285 *
286 * Determine when to switch from blocked to unblocked code.
287 *
288  IF( nx.LT.minmn ) THEN
289  ws = ( m+n )*nb
290  IF( lwork.LT.ws ) THEN
291 *
292 * Not enough work space for the optimal NB, consider using
293 * a smaller block size.
294 *
295  nbmin = ilaenv( 2, 'DGEBRD', ' ', m, n, -1, -1 )
296  IF( lwork.GE.( m+n )*nbmin ) THEN
297  nb = lwork / ( m+n )
298  ELSE
299  nb = 1
300  nx = minmn
301  END IF
302  END IF
303  END IF
304  ELSE
305  nx = minmn
306  END IF
307 *
308  DO 30 i = 1, minmn - nx, nb
309 *
310 * Reduce rows and columns i:i+nb-1 to bidiagonal form and return
311 * the matrices X and Y which are needed to update the unreduced
312 * part of the matrix
313 *
314  CALL dlabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),
315  \$ tauq( i ), taup( i ), work, ldwrkx,
316  \$ work( ldwrkx*nb+1 ), ldwrky )
317 *
318 * Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
319 * of the form A := A - V*Y**T - X*U**T
320 *
321  CALL dgemm( 'No transpose', 'Transpose', m-i-nb+1, n-i-nb+1,
322  \$ nb, -one, a( i+nb, i ), lda,
323  \$ work( ldwrkx*nb+nb+1 ), ldwrky, one,
324  \$ a( i+nb, i+nb ), lda )
325  CALL dgemm( 'No transpose', 'No transpose', m-i-nb+1, n-i-nb+1,
326  \$ nb, -one, work( nb+1 ), ldwrkx, a( i, i+nb ), lda,
327  \$ one, a( i+nb, i+nb ), lda )
328 *
329 * Copy diagonal and off-diagonal elements of B back into A
330 *
331  IF( m.GE.n ) THEN
332  DO 10 j = i, i + nb - 1
333  a( j, j ) = d( j )
334  a( j, j+1 ) = e( j )
335  10 CONTINUE
336  ELSE
337  DO 20 j = i, i + nb - 1
338  a( j, j ) = d( j )
339  a( j+1, j ) = e( j )
340  20 CONTINUE
341  END IF
342  30 CONTINUE
343 *
344 * Use unblocked code to reduce the remainder of the matrix
345 *
346  CALL dgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),
347  \$ tauq( i ), taup( i ), work, iinfo )
348  work( 1 ) = ws
349  RETURN
350 *
351 * End of DGEBRD
352 *
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:189
subroutine dgebd2(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO)
DGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
Definition: dgebd2.f:191
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dlabrd(M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Definition: dlabrd.f:212
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83

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