 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zunhr_col()

 subroutine zunhr_col ( integer M, integer N, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( * ) D, integer INFO )

ZUNHR_COL

Purpose:
```  ZUNHR_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 ZGEQRT).```
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*16 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 ZGEQRT). 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*16 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 ZGEQRT). 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*16 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 .

 "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 zunhr_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*16 A( LDA, * ), D( * ), T( LDT, * )
270* ..
271*
272* =====================================================================
273*
274* .. Parameters ..
275 COMPLEX*16 CONE, CZERO
276 parameter( cone = ( 1.0d+0, 0.0d+0 ),
277 \$ czero = ( 0.0d+0, 0.0d+0 ) )
278* ..
279* .. Local Scalars ..
280 INTEGER I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
281 \$ NPLUSONE
282* ..
283* .. External Subroutines ..
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( 'ZUNHR_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 zlaunhr_col_getrfnp( n, n, a, lda, d, iinfo )
334*
335* (1-2) Solve for V2.
336*
337 IF( m.GT.n ) THEN
338 CALL ztrsm( '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 zcopy( 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 zscal( 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 ZTRSM 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 ZTRSM.
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 ztrsm( '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 ZUNHR_COL
438*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine ztrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
ZTRSM
Definition: ztrsm.f:180
subroutine zlaunhr_col_getrfnp(M, N, A, LDA, D, INFO)
ZLAUNHR_COL_GETRFNP
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