LAPACK  3.9.0 LAPACK: Linear Algebra PACKage

## ◆ clabrd()

 subroutine clabrd ( integer M, integer N, integer NB, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( ldx, * ) X, integer LDX, complex, dimension( ldy, * ) Y, integer LDY )

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

Purpose:
``` CLABRD 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 CGEBRD```
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 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 REAL 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 REAL array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix.``` [out] TAUQ ``` TAUQ is COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.``` [out] TAUP ``` TAUP is COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.``` [out] X ``` X is COMPLEX 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 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).```
Date
June 2017
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 214 of file clabrd.f.

214 *
215 * -- LAPACK auxiliary routine (version 3.7.1) --
216 * -- LAPACK is a software package provided by Univ. of Tennessee, --
217 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
218 * June 2017
219 *
220 * .. Scalar Arguments ..
221  INTEGER LDA, LDX, LDY, M, N, NB
222 * ..
223 * .. Array Arguments ..
224  REAL D( * ), E( * )
225  COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
226  \$ Y( LDY, * )
227 * ..
228 *
229 * =====================================================================
230 *
231 * .. Parameters ..
232  COMPLEX ZERO, ONE
233  parameter( zero = ( 0.0e+0, 0.0e+0 ),
234  \$ one = ( 1.0e+0, 0.0e+0 ) )
235 * ..
236 * .. Local Scalars ..
237  INTEGER I
238  COMPLEX ALPHA
239 * ..
240 * .. External Subroutines ..
241  EXTERNAL cgemv, clacgv, clarfg, cscal
242 * ..
243 * .. Intrinsic Functions ..
244  INTRINSIC min
245 * ..
246 * .. Executable Statements ..
247 *
248 * Quick return if possible
249 *
250  IF( m.LE.0 .OR. n.LE.0 )
251  \$ RETURN
252 *
253  IF( m.GE.n ) THEN
254 *
255 * Reduce to upper bidiagonal form
256 *
257  DO 10 i = 1, nb
258 *
259 * Update A(i:m,i)
260 *
261  CALL clacgv( i-1, y( i, 1 ), ldy )
262  CALL cgemv( 'No transpose', m-i+1, i-1, -one, a( i, 1 ),
263  \$ lda, y( i, 1 ), ldy, one, a( i, i ), 1 )
264  CALL clacgv( i-1, y( i, 1 ), ldy )
265  CALL cgemv( 'No transpose', m-i+1, i-1, -one, x( i, 1 ),
266  \$ ldx, a( 1, i ), 1, one, a( i, i ), 1 )
267 *
268 * Generate reflection Q(i) to annihilate A(i+1:m,i)
269 *
270  alpha = a( i, i )
271  CALL clarfg( m-i+1, alpha, a( min( i+1, m ), i ), 1,
272  \$ tauq( i ) )
273  d( i ) = alpha
274  IF( i.LT.n ) THEN
275  a( i, i ) = one
276 *
277 * Compute Y(i+1:n,i)
278 *
279  CALL cgemv( 'Conjugate transpose', m-i+1, n-i, one,
280  \$ a( i, i+1 ), lda, a( i, i ), 1, zero,
281  \$ y( i+1, i ), 1 )
282  CALL cgemv( 'Conjugate transpose', m-i+1, i-1, one,
283  \$ a( i, 1 ), lda, a( i, i ), 1, zero,
284  \$ y( 1, i ), 1 )
285  CALL cgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
286  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
287  CALL cgemv( 'Conjugate transpose', m-i+1, i-1, one,
288  \$ x( i, 1 ), ldx, a( i, i ), 1, zero,
289  \$ y( 1, i ), 1 )
290  CALL cgemv( 'Conjugate transpose', i-1, n-i, -one,
291  \$ a( 1, i+1 ), lda, y( 1, i ), 1, one,
292  \$ y( i+1, i ), 1 )
293  CALL cscal( n-i, tauq( i ), y( i+1, i ), 1 )
294 *
295 * Update A(i,i+1:n)
296 *
297  CALL clacgv( n-i, a( i, i+1 ), lda )
298  CALL clacgv( i, a( i, 1 ), lda )
299  CALL cgemv( 'No transpose', n-i, i, -one, y( i+1, 1 ),
300  \$ ldy, a( i, 1 ), lda, one, a( i, i+1 ), lda )
301  CALL clacgv( i, a( i, 1 ), lda )
302  CALL clacgv( i-1, x( i, 1 ), ldx )
303  CALL cgemv( 'Conjugate transpose', i-1, n-i, -one,
304  \$ a( 1, i+1 ), lda, x( i, 1 ), ldx, one,
305  \$ a( i, i+1 ), lda )
306  CALL clacgv( i-1, x( i, 1 ), ldx )
307 *
308 * Generate reflection P(i) to annihilate A(i,i+2:n)
309 *
310  alpha = a( i, i+1 )
311  CALL clarfg( n-i, alpha, a( i, min( i+2, n ) ),
312  \$ lda, taup( i ) )
313  e( i ) = alpha
314  a( i, i+1 ) = one
315 *
316 * Compute X(i+1:m,i)
317 *
318  CALL cgemv( 'No transpose', m-i, n-i, one, a( i+1, i+1 ),
319  \$ lda, a( i, i+1 ), lda, zero, x( i+1, i ), 1 )
320  CALL cgemv( 'Conjugate transpose', n-i, i, one,
321  \$ y( i+1, 1 ), ldy, a( i, i+1 ), lda, zero,
322  \$ x( 1, i ), 1 )
323  CALL cgemv( 'No transpose', m-i, i, -one, a( i+1, 1 ),
324  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
325  CALL cgemv( 'No transpose', i-1, n-i, one, a( 1, i+1 ),
326  \$ lda, a( i, i+1 ), lda, zero, x( 1, i ), 1 )
327  CALL cgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
328  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
329  CALL cscal( m-i, taup( i ), x( i+1, i ), 1 )
330  CALL clacgv( n-i, a( i, i+1 ), lda )
331  END IF
332  10 CONTINUE
333  ELSE
334 *
335 * Reduce to lower bidiagonal form
336 *
337  DO 20 i = 1, nb
338 *
339 * Update A(i,i:n)
340 *
341  CALL clacgv( n-i+1, a( i, i ), lda )
342  CALL clacgv( i-1, a( i, 1 ), lda )
343  CALL cgemv( 'No transpose', n-i+1, i-1, -one, y( i, 1 ),
344  \$ ldy, a( i, 1 ), lda, one, a( i, i ), lda )
345  CALL clacgv( i-1, a( i, 1 ), lda )
346  CALL clacgv( i-1, x( i, 1 ), ldx )
347  CALL cgemv( 'Conjugate transpose', i-1, n-i+1, -one,
348  \$ a( 1, i ), lda, x( i, 1 ), ldx, one, a( i, i ),
349  \$ lda )
350  CALL clacgv( i-1, x( i, 1 ), ldx )
351 *
352 * Generate reflection P(i) to annihilate A(i,i+1:n)
353 *
354  alpha = a( i, i )
355  CALL clarfg( n-i+1, alpha, a( i, min( i+1, n ) ), lda,
356  \$ taup( i ) )
357  d( i ) = alpha
358  IF( i.LT.m ) THEN
359  a( i, i ) = one
360 *
361 * Compute X(i+1:m,i)
362 *
363  CALL cgemv( 'No transpose', m-i, n-i+1, one, a( i+1, i ),
364  \$ lda, a( i, i ), lda, zero, x( i+1, i ), 1 )
365  CALL cgemv( 'Conjugate transpose', n-i+1, i-1, one,
366  \$ y( i, 1 ), ldy, a( i, i ), lda, zero,
367  \$ x( 1, i ), 1 )
368  CALL cgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
369  \$ lda, x( 1, i ), 1, one, x( i+1, i ), 1 )
370  CALL cgemv( 'No transpose', i-1, n-i+1, one, a( 1, i ),
371  \$ lda, a( i, i ), lda, zero, x( 1, i ), 1 )
372  CALL cgemv( 'No transpose', m-i, i-1, -one, x( i+1, 1 ),
373  \$ ldx, x( 1, i ), 1, one, x( i+1, i ), 1 )
374  CALL cscal( m-i, taup( i ), x( i+1, i ), 1 )
375  CALL clacgv( n-i+1, a( i, i ), lda )
376 *
377 * Update A(i+1:m,i)
378 *
379  CALL clacgv( i-1, y( i, 1 ), ldy )
380  CALL cgemv( 'No transpose', m-i, i-1, -one, a( i+1, 1 ),
381  \$ lda, y( i, 1 ), ldy, one, a( i+1, i ), 1 )
382  CALL clacgv( i-1, y( i, 1 ), ldy )
383  CALL cgemv( 'No transpose', m-i, i, -one, x( i+1, 1 ),
384  \$ ldx, a( 1, i ), 1, one, a( i+1, i ), 1 )
385 *
386 * Generate reflection Q(i) to annihilate A(i+2:m,i)
387 *
388  alpha = a( i+1, i )
389  CALL clarfg( m-i, alpha, a( min( i+2, m ), i ), 1,
390  \$ tauq( i ) )
391  e( i ) = alpha
392  a( i+1, i ) = one
393 *
394 * Compute Y(i+1:n,i)
395 *
396  CALL cgemv( 'Conjugate transpose', m-i, n-i, one,
397  \$ a( i+1, i+1 ), lda, a( i+1, i ), 1, zero,
398  \$ y( i+1, i ), 1 )
399  CALL cgemv( 'Conjugate transpose', m-i, i-1, one,
400  \$ a( i+1, 1 ), lda, a( i+1, i ), 1, zero,
401  \$ y( 1, i ), 1 )
402  CALL cgemv( 'No transpose', n-i, i-1, -one, y( i+1, 1 ),
403  \$ ldy, y( 1, i ), 1, one, y( i+1, i ), 1 )
404  CALL cgemv( 'Conjugate transpose', m-i, i, one,
405  \$ x( i+1, 1 ), ldx, a( i+1, i ), 1, zero,
406  \$ y( 1, i ), 1 )
407  CALL cgemv( 'Conjugate transpose', i, n-i, -one,
408  \$ a( 1, i+1 ), lda, y( 1, i ), 1, one,
409  \$ y( i+1, i ), 1 )
410  CALL cscal( n-i, tauq( i ), y( i+1, i ), 1 )
411  ELSE
412  CALL clacgv( n-i+1, a( i, i ), lda )
413  END IF
414  20 CONTINUE
415  END IF
416  RETURN
417 *
418 * End of CLABRD
419 *
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clarfg
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:108
cgemv
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:160
clacgv
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:76
cscal
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:80