*> \brief \b DORHR_COL
*
* =========== DOCUMENTATION ===========
*
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* http://www.netlib.org/lapack/explore-html/
*
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*
* Definition:
* ===========
*
* SUBROUTINE DORHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, M, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), D( * ), T( LDT, * )
* ..
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DORHR_COL takes an M-by-N real 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 DGEQRT).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. M >= N >= 0.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> 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.)
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION 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
*> DGEQRT). 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION 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 DGEQRT).
*> 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.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T.
*> LDT >= max(1,min(NB,N)).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The computed M-by-M orthogonal factor Q_out is defined implicitly as
*> a product of orthogonal 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
*> orthogonal 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.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> November 2019, Igor Kozachenko,
*> Computer Science Division,
*> University of California, Berkeley
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE DORHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
IMPLICIT NONE
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDT, M, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), D( * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
$ NPLUSONE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAORHR_COL_GETRFNP, DSCAL, DTRSM,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
INFO = -2
ELSE IF( NB.LT.1 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
ELSE IF( LDT.LT.MAX( 1, MIN( NB, N ) ) ) THEN
INFO = -7
END IF
*
* Handle error in the input parameters.
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORHR_COL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N ).EQ.0 ) THEN
RETURN
END IF
*
* On input, the M-by-N matrix A contains the orthogonal
* M-by-N matrix Q_in.
*
* (1) Compute the unit lower-trapezoidal V (ones on the diagonal
* are not stored) by performing the "modified" LU-decomposition.
*
* Q_in - ( S ) = V * U = ( V1 ) * U,
* ( 0 ) ( V2 )
*
* where 0 is an (M-N)-by-N zero matrix.
*
* (1-1) Factor V1 and U.
CALL DLAORHR_COL_GETRFNP( N, N, A, LDA, D, IINFO )
*
* (1-2) Solve for V2.
*
IF( M.GT.N ) THEN
CALL DTRSM( 'R', 'U', 'N', 'N', M-N, N, ONE, A, LDA,
$ A( N+1, 1 ), LDA )
END IF
*
* (2) Reconstruct the block reflector T stored in T(1:NB, 1:N)
* as a sequence of upper-triangular blocks with NB-size column
* blocking.
*
* Loop over the column blocks of size NB of the array A(1:M,1:N)
* and the array T(1:NB,1:N), JB is the column index of a column
* block, JNB is the column block size at each step JB.
*
NPLUSONE = N + 1
DO JB = 1, N, NB
*
* (2-0) Determine the column block size JNB.
*
JNB = MIN( NPLUSONE-JB, NB )
*
* (2-1) Copy the upper-triangular part of the current JNB-by-JNB
* diagonal block U(JB) (of the N-by-N matrix U) stored
* in A(JB:JB+JNB-1,JB:JB+JNB-1) into the upper-triangular part
* of the current JNB-by-JNB block T(1:JNB,JB:JB+JNB-1)
* column-by-column, total JNB*(JNB+1)/2 elements.
*
JBTEMP1 = JB - 1
DO J = JB, JB+JNB-1
CALL DCOPY( J-JBTEMP1, A( JB, J ), 1, T( 1, J ), 1 )
END DO
*
* (2-2) Perform on the upper-triangular part of the current
* JNB-by-JNB diagonal block U(JB) (of the N-by-N matrix U) stored
* in T(1:JNB,JB:JB+JNB-1) the following operation in place:
* (-1)*U(JB)*S(JB), i.e the result will be stored in the upper-
* triangular part of T(1:JNB,JB:JB+JNB-1). This multiplication
* of the JNB-by-JNB diagonal block U(JB) by the JNB-by-JNB
* diagonal block S(JB) of the N-by-N sign matrix S from the
* right means changing the sign of each J-th column of the block
* U(JB) according to the sign of the diagonal element of the block
* S(JB), i.e. S(J,J) that is stored in the array element D(J).
*
DO J = JB, JB+JNB-1
IF( D( J ).EQ.ONE ) THEN
CALL DSCAL( J-JBTEMP1, -ONE, T( 1, J ), 1 )
END IF
END DO
*
* (2-3) Perform the triangular solve for the current block
* matrix X(JB):
*
* X(JB) * (A(JB)**T) = B(JB), where:
*
* A(JB)**T is a JNB-by-JNB unit upper-triangular
* coefficient block, and A(JB)=V1(JB), which
* is a JNB-by-JNB unit lower-triangular block
* stored in A(JB:JB+JNB-1,JB:JB+JNB-1).
* The N-by-N matrix V1 is the upper part
* of the M-by-N lower-trapezoidal matrix V
* stored in A(1:M,1:N);
*
* B(JB) is a JNB-by-JNB upper-triangular right-hand
* side block, B(JB) = (-1)*U(JB)*S(JB), and
* B(JB) is stored in T(1:JNB,JB:JB+JNB-1);
*
* X(JB) is a JNB-by-JNB upper-triangular solution
* block, X(JB) is the upper-triangular block
* reflector T(JB), and X(JB) is stored
* in T(1:JNB,JB:JB+JNB-1).
*
* In other words, we perform the triangular solve for the
* upper-triangular block T(JB):
*
* T(JB) * (V1(JB)**T) = (-1)*U(JB)*S(JB).
*
* Even though the blocks X(JB) and B(JB) are upper-
* triangular, the routine DTRSM will access all JNB**2
* elements of the square T(1:JNB,JB:JB+JNB-1). Therefore,
* we need to set to zero the elements of the block
* T(1:JNB,JB:JB+JNB-1) below the diagonal before the call
* to DTRSM.
*
* (2-3a) Set the elements to zero.
*
JBTEMP2 = JB - 2
DO J = JB, JB+JNB-2
DO I = J-JBTEMP2, NB
T( I, J ) = ZERO
END DO
END DO
*
* (2-3b) Perform the triangular solve.
*
CALL DTRSM( 'R', 'L', 'T', 'U', JNB, JNB, ONE,
$ A( JB, JB ), LDA, T( 1, JB ), LDT )
*
END DO
*
RETURN
*
* End of DORHR_COL
*
END