LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches

## ◆ cgebrd()

 subroutine cgebrd ( integer m, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension( * ) work, integer lwork, integer info )

CGEBRD

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

Purpose:
``` CGEBRD 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 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 REAL array, dimension (min(M,N)) The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).``` [out] E ``` E is REAL 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 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 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 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**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 204 of file cgebrd.f.

206*
207* -- LAPACK computational routine --
208* -- LAPACK is a software package provided by Univ. of Tennessee, --
209* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
210*
211* .. Scalar Arguments ..
212 INTEGER INFO, LDA, LWORK, M, N
213* ..
214* .. Array Arguments ..
215 REAL D( * ), E( * )
216 COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ),
217 \$ WORK( * )
218* ..
219*
220* =====================================================================
221*
222* .. Parameters ..
223 COMPLEX ONE
224 parameter( one = ( 1.0e+0, 0.0e+0 ) )
225* ..
226* .. Local Scalars ..
227 LOGICAL LQUERY
228 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
229 \$ NBMIN, NX, WS
230* ..
231* .. External Subroutines ..
232 EXTERNAL cgebd2, cgemm, clabrd, xerbla
233* ..
234* .. Intrinsic Functions ..
235 INTRINSIC max, min, real
236* ..
237* .. External Functions ..
238 INTEGER ILAENV
239 EXTERNAL ilaenv
240* ..
241* .. Executable Statements ..
242*
243* Test the input parameters
244*
245 info = 0
246 nb = max( 1, ilaenv( 1, 'CGEBRD', ' ', m, n, -1, -1 ) )
247 lwkopt = ( m+n )*nb
248 work( 1 ) = real( lwkopt )
249 lquery = ( lwork.EQ.-1 )
250 IF( m.LT.0 ) THEN
251 info = -1
252 ELSE IF( n.LT.0 ) THEN
253 info = -2
254 ELSE IF( lda.LT.max( 1, m ) ) THEN
255 info = -4
256 ELSE IF( lwork.LT.max( 1, m, n ) .AND. .NOT.lquery ) THEN
257 info = -10
258 END IF
259 IF( info.LT.0 ) THEN
260 CALL xerbla( 'CGEBRD', -info )
261 RETURN
262 ELSE IF( lquery ) THEN
263 RETURN
264 END IF
265*
266* Quick return if possible
267*
268 minmn = min( m, n )
269 IF( minmn.EQ.0 ) THEN
270 work( 1 ) = 1
271 RETURN
272 END IF
273*
274 ws = max( m, n )
275 ldwrkx = m
276 ldwrky = n
277*
278 IF( nb.GT.1 .AND. nb.LT.minmn ) THEN
279*
280* Set the crossover point NX.
281*
282 nx = max( nb, ilaenv( 3, 'CGEBRD', ' ', m, n, -1, -1 ) )
283*
284* Determine when to switch from blocked to unblocked code.
285*
286 IF( nx.LT.minmn ) THEN
287 ws = ( m+n )*nb
288 IF( lwork.LT.ws ) THEN
289*
290* Not enough work space for the optimal NB, consider using
291* a smaller block size.
292*
293 nbmin = ilaenv( 2, 'CGEBRD', ' ', m, n, -1, -1 )
294 IF( lwork.GE.( m+n )*nbmin ) THEN
295 nb = lwork / ( m+n )
296 ELSE
297 nb = 1
298 nx = minmn
299 END IF
300 END IF
301 END IF
302 ELSE
303 nx = minmn
304 END IF
305*
306 DO 30 i = 1, minmn - nx, nb
307*
308* Reduce rows and columns i:i+ib-1 to bidiagonal form and return
309* the matrices X and Y which are needed to update the unreduced
310* part of the matrix
311*
312 CALL clabrd( m-i+1, n-i+1, nb, a( i, i ), lda, d( i ), e( i ),
313 \$ tauq( i ), taup( i ), work, ldwrkx,
314 \$ work( ldwrkx*nb+1 ), ldwrky )
315*
316* Update the trailing submatrix A(i+ib:m,i+ib:n), using
317* an update of the form A := A - V*Y**H - X*U**H
318*
319 CALL cgemm( 'No transpose', 'Conjugate transpose', m-i-nb+1,
320 \$ n-i-nb+1, nb, -one, a( i+nb, i ), lda,
321 \$ work( ldwrkx*nb+nb+1 ), ldwrky, one,
322 \$ a( i+nb, i+nb ), lda )
323 CALL cgemm( 'No transpose', 'No transpose', m-i-nb+1, n-i-nb+1,
324 \$ nb, -one, work( nb+1 ), ldwrkx, a( i, i+nb ), lda,
325 \$ one, a( i+nb, i+nb ), lda )
326*
327* Copy diagonal and off-diagonal elements of B back into A
328*
329 IF( m.GE.n ) THEN
330 DO 10 j = i, i + nb - 1
331 a( j, j ) = d( j )
332 a( j, j+1 ) = e( j )
333 10 CONTINUE
334 ELSE
335 DO 20 j = i, i + nb - 1
336 a( j, j ) = d( j )
337 a( j+1, j ) = e( j )
338 20 CONTINUE
339 END IF
340 30 CONTINUE
341*
342* Use unblocked code to reduce the remainder of the matrix
343*
344 CALL cgebd2( m-i+1, n-i+1, a( i, i ), lda, d( i ), e( i ),
345 \$ tauq( i ), taup( i ), work, iinfo )
346 work( 1 ) = ws
347 RETURN
348*
349* End of CGEBRD
350*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine cgebd2(m, n, a, lda, d, e, tauq, taup, work, info)
CGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm.
Definition cgebd2.f:190
subroutine cgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
CGEMM
Definition cgemm.f:188
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
subroutine clabrd(m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Definition clabrd.f:212
Here is the call graph for this function:
Here is the caller graph for this function: