 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dgebrd()

 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.```
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 203 of file dgebrd.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 A( LDA, * ), D( * ), E( * ), TAUP( * ),
215  \$ TAUQ( * ), WORK( * )
216 * ..
217 *
218 * =====================================================================
219 *
220 * .. Parameters ..
221  DOUBLE PRECISION ONE
222  parameter( one = 1.0d+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 dgebd2, dgemm, dlabrd, xerbla
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, 'DGEBRD', ' ', 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( 'DGEBRD', -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, 'DGEBRD', ' ', 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, 'DGEBRD', ' ', 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+nb-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 dlabrd( 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+nb:m,i+nb:n), using an update
315 * of the form A := A - V*Y**T - X*U**T
316 *
317  CALL dgemm( 'No transpose', 'Transpose', m-i-nb+1, n-i-nb+1,
318  \$ nb, -one, a( i+nb, i ), lda,
319  \$ work( ldwrkx*nb+nb+1 ), ldwrky, one,
320  \$ a( i+nb, i+nb ), lda )
321  CALL dgemm( '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 dgebd2( 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 DGEBRD
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 dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
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:189
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:210
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