LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ cunhr_col()

 subroutine cunhr_col ( integer M, integer N, integer NB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( * ) D, integer INFO )

CUNHR_COL

Purpose:
```  CUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns
as input, stored in A, and performs Householder Reconstruction (HR),
i.e. reconstructs Householder vectors V(i) implicitly representing
another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
where S is an N-by-N diagonal matrix with diagonal entries
equal to +1 or -1. The Householder vectors (columns V(i) of V) are
stored in A on output, and the diagonal entries of S are stored in D.
Block reflectors are also returned in T
(same output format as CGEQRT).```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. M >= N >= 0.``` [in] NB ``` NB is INTEGER The column block size to be used in the reconstruction of Householder column vector blocks in the array A and corresponding block reflectors in the array T. NB >= 1. (Note that if NB > N, then N is used instead of NB as the column block size.)``` [in,out] A ``` A is COMPLEX array, dimension (LDA,N) On entry: The array A contains an M-by-N orthonormal matrix Q_in, i.e the columns of A are orthogonal unit vectors. On exit: The elements below the diagonal of A represent the unit lower-trapezoidal matrix V of Householder column vectors V(i). The unit diagonal entries of V are not stored (same format as the output below the diagonal in A from CGEQRT). The matrix T and the matrix V stored on output in A implicitly define Q_out. The elements above the diagonal contain the factor U of the "modified" LU-decomposition: Q_in - ( S ) = V * U ( 0 ) where 0 is a (M-N)-by-(M-N) zero matrix.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] T ``` T is COMPLEX array, dimension (LDT, N) Let NOCB = Number_of_output_col_blocks = CEIL(N/NB) On exit, T(1:NB, 1:N) contains NOCB upper-triangular block reflectors used to define Q_out stored in compact form as a sequence of upper-triangular NB-by-NB column blocks (same format as the output T in CGEQRT). The matrix T and the matrix V stored on output in A implicitly define Q_out. NOTE: The lower triangles below the upper-triangular blocks will be filled with zeros. See Further Details.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= max(1,min(NB,N)).``` [out] D ``` D is COMPLEX array, dimension min(M,N). The elements can be only plus or minus one. D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where 1 <= i <= min(M,N), and Q_in_i is Q_in after performing i-1 steps of “modified” Gaussian elimination. See Further Details.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
``` The computed M-by-M unitary factor Q_out is defined implicitly as
a product of unitary matrices Q_out(i). Each Q_out(i) is stored in
the compact WY-representation format in the corresponding blocks of
matrices V (stored in A) and T.

The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
matrix A contains the column vectors V(i) in NB-size column
blocks VB(j). For example, VB(1) contains the columns
V(1), V(2), ... V(NB). NOTE: The unit entries on
the diagonal of Y are not stored in A.

The number of column blocks is

NOCB = Number_of_output_col_blocks = CEIL(N/NB)

where each block is of order NB except for the last block, which
is of order LAST_NB = N - (NOCB-1)*NB.

For example, if M=6,  N=5 and NB=2, the matrix V is

V = (    VB(1),   VB(2), VB(3) ) =

= (   1                      )
( v21    1                 )
( v31  v32    1            )
( v41  v42  v43   1        )
( v51  v52  v53  v54    1  )
( v61  v62  v63  v54   v65 )

For each of the column blocks VB(i), an upper-triangular block
reflector TB(i) is computed. These blocks are stored as
a sequence of upper-triangular column blocks in the NB-by-N
matrix T. The size of each TB(i) block is NB-by-NB, except
for the last block, whose size is LAST_NB-by-LAST_NB.

For example, if M=6,  N=5 and NB=2, the matrix T is

T  = (    TB(1),    TB(2), TB(3) ) =

= ( t11  t12  t13  t14   t15  )
(      t22       t24        )

The M-by-M factor Q_out is given as a product of NOCB
unitary M-by-M matrices Q_out(i).

Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),

where each matrix Q_out(i) is given by the WY-representation
using corresponding blocks from the matrices V and T:

Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,

where I is the identity matrix. Here is the formula with matrix
dimensions:

Q(i){M-by-M} = I{M-by-M} -
VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},

where INB = NB, except for the last block NOCB
for which INB=LAST_NB.

=====
NOTE:
=====

If Q_in is the result of doing a QR factorization
B = Q_in * R_in, then:

B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.

So if one wants to interpret Q_out as the result
of the QR factorization of B, then the corresponding R_out
should be equal to R_out = S * R_in, i.e. some rows of R_in
should be multiplied by -1.

For the details of the algorithm, see [1].

[1] "Reconstructing Householder vectors from tall-skinny QR",
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.```
Contributors:
``` November   2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 258 of file cunhr_col.f.

259  IMPLICIT NONE
260 *
261 * -- LAPACK computational routine --
262 * -- LAPACK is a software package provided by Univ. of Tennessee, --
263 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
264 *
265 * .. Scalar Arguments ..
266  INTEGER INFO, LDA, LDT, M, N, NB
267 * ..
268 * .. Array Arguments ..
269  COMPLEX A( LDA, * ), D( * ), T( LDT, * )
270 * ..
271 *
272 * =====================================================================
273 *
274 * .. Parameters ..
275  COMPLEX CONE, CZERO
276  parameter( cone = ( 1.0e+0, 0.0e+0 ),
277  \$ czero = ( 0.0e+0, 0.0e+0 ) )
278 * ..
279 * .. Local Scalars ..
280  INTEGER I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
281  \$ NPLUSONE
282 * ..
283 * .. External Subroutines ..
284  EXTERNAL ccopy, claunhr_col_getrfnp, cscal, ctrsm,
285  \$ xerbla
286 * ..
287 * .. Intrinsic Functions ..
288  INTRINSIC max, min
289 * ..
290 * .. Executable Statements ..
291 *
292 * Test the input parameters
293 *
294  info = 0
295  IF( m.LT.0 ) THEN
296  info = -1
297  ELSE IF( n.LT.0 .OR. n.GT.m ) THEN
298  info = -2
299  ELSE IF( nb.LT.1 ) THEN
300  info = -3
301  ELSE IF( lda.LT.max( 1, m ) ) THEN
302  info = -5
303  ELSE IF( ldt.LT.max( 1, min( nb, n ) ) ) THEN
304  info = -7
305  END IF
306 *
307 * Handle error in the input parameters.
308 *
309  IF( info.NE.0 ) THEN
310  CALL xerbla( 'CUNHR_COL', -info )
311  RETURN
312  END IF
313 *
314 * Quick return if possible
315 *
316  IF( min( m, n ).EQ.0 ) THEN
317  RETURN
318  END IF
319 *
320 * On input, the M-by-N matrix A contains the unitary
321 * M-by-N matrix Q_in.
322 *
323 * (1) Compute the unit lower-trapezoidal V (ones on the diagonal
324 * are not stored) by performing the "modified" LU-decomposition.
325 *
326 * Q_in - ( S ) = V * U = ( V1 ) * U,
327 * ( 0 ) ( V2 )
328 *
329 * where 0 is an (M-N)-by-N zero matrix.
330 *
331 * (1-1) Factor V1 and U.
332
333  CALL claunhr_col_getrfnp( n, n, a, lda, d, iinfo )
334 *
335 * (1-2) Solve for V2.
336 *
337  IF( m.GT.n ) THEN
338  CALL ctrsm( 'R', 'U', 'N', 'N', m-n, n, cone, a, lda,
339  \$ a( n+1, 1 ), lda )
340  END IF
341 *
342 * (2) Reconstruct the block reflector T stored in T(1:NB, 1:N)
343 * as a sequence of upper-triangular blocks with NB-size column
344 * blocking.
345 *
346 * Loop over the column blocks of size NB of the array A(1:M,1:N)
347 * and the array T(1:NB,1:N), JB is the column index of a column
348 * block, JNB is the column block size at each step JB.
349 *
350  nplusone = n + 1
351  DO jb = 1, n, nb
352 *
353 * (2-0) Determine the column block size JNB.
354 *
355  jnb = min( nplusone-jb, nb )
356 *
357 * (2-1) Copy the upper-triangular part of the current JNB-by-JNB
358 * diagonal block U(JB) (of the N-by-N matrix U) stored
359 * in A(JB:JB+JNB-1,JB:JB+JNB-1) into the upper-triangular part
360 * of the current JNB-by-JNB block T(1:JNB,JB:JB+JNB-1)
361 * column-by-column, total JNB*(JNB+1)/2 elements.
362 *
363  jbtemp1 = jb - 1
364  DO j = jb, jb+jnb-1
365  CALL ccopy( j-jbtemp1, a( jb, j ), 1, t( 1, j ), 1 )
366  END DO
367 *
368 * (2-2) Perform on the upper-triangular part of the current
369 * JNB-by-JNB diagonal block U(JB) (of the N-by-N matrix U) stored
370 * in T(1:JNB,JB:JB+JNB-1) the following operation in place:
371 * (-1)*U(JB)*S(JB), i.e the result will be stored in the upper-
372 * triangular part of T(1:JNB,JB:JB+JNB-1). This multiplication
373 * of the JNB-by-JNB diagonal block U(JB) by the JNB-by-JNB
374 * diagonal block S(JB) of the N-by-N sign matrix S from the
375 * right means changing the sign of each J-th column of the block
376 * U(JB) according to the sign of the diagonal element of the block
377 * S(JB), i.e. S(J,J) that is stored in the array element D(J).
378 *
379  DO j = jb, jb+jnb-1
380  IF( d( j ).EQ.cone ) THEN
381  CALL cscal( j-jbtemp1, -cone, t( 1, j ), 1 )
382  END IF
383  END DO
384 *
385 * (2-3) Perform the triangular solve for the current block
386 * matrix X(JB):
387 *
388 * X(JB) * (A(JB)**T) = B(JB), where:
389 *
390 * A(JB)**T is a JNB-by-JNB unit upper-triangular
391 * coefficient block, and A(JB)=V1(JB), which
392 * is a JNB-by-JNB unit lower-triangular block
393 * stored in A(JB:JB+JNB-1,JB:JB+JNB-1).
394 * The N-by-N matrix V1 is the upper part
395 * of the M-by-N lower-trapezoidal matrix V
396 * stored in A(1:M,1:N);
397 *
398 * B(JB) is a JNB-by-JNB upper-triangular right-hand
399 * side block, B(JB) = (-1)*U(JB)*S(JB), and
400 * B(JB) is stored in T(1:JNB,JB:JB+JNB-1);
401 *
402 * X(JB) is a JNB-by-JNB upper-triangular solution
403 * block, X(JB) is the upper-triangular block
404 * reflector T(JB), and X(JB) is stored
405 * in T(1:JNB,JB:JB+JNB-1).
406 *
407 * In other words, we perform the triangular solve for the
408 * upper-triangular block T(JB):
409 *
410 * T(JB) * (V1(JB)**T) = (-1)*U(JB)*S(JB).
411 *
412 * Even though the blocks X(JB) and B(JB) are upper-
413 * triangular, the routine CTRSM will access all JNB**2
414 * elements of the square T(1:JNB,JB:JB+JNB-1). Therefore,
415 * we need to set to zero the elements of the block
416 * T(1:JNB,JB:JB+JNB-1) below the diagonal before the call
417 * to CTRSM.
418 *
419 * (2-3a) Set the elements to zero.
420 *
421  jbtemp2 = jb - 2
422  DO j = jb, jb+jnb-2
423  DO i = j-jbtemp2, nb
424  t( i, j ) = czero
425  END DO
426  END DO
427 *
428 * (2-3b) Perform the triangular solve.
429 *
430  CALL ctrsm( 'R', 'L', 'C', 'U', jnb, jnb, cone,
431  \$ a( jb, jb ), lda, t( 1, jb ), ldt )
432 *
433  END DO
434 *
435  RETURN
436 *
437 * End of CUNHR_COL
438 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine ctrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRSM
Definition: ctrsm.f:180
subroutine claunhr_col_getrfnp(M, N, A, LDA, D, INFO)
CLAUNHR_COL_GETRFNP
Here is the call graph for this function:
Here is the caller graph for this function: