# sorhr_col.f - Man Page

## Synopsis

### Functions/Subroutines

subroutine sorhr_col (M, N, NB, A, LDA, T, LDT, D, INFO)
SORHR_COL

## Function/Subroutine Documentation

### subroutine sorhr_col (integer M, integer N, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) D, integer INFO)

SORHR_COL

Purpose:

```  SORHR_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 SGEQRT).```
Parameters

M

```          M is INTEGER
The number of rows of the matrix A. M >= 0.```

N

```          N is INTEGER
The number of columns of the matrix A. M >= N >= 0.```

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.)```

A

```          A is REAL 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
SGEQRT). 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.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,M).```

T

```          T is REAL 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 SGEQRT).
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.```

LDT

```          LDT is INTEGER
The leading dimension of the array T.
LDT >= max(1,min(NB,N)).```

D

```          D is REAL 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.```

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 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 .

 "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.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

``` November   2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 258 of file sorhr_col.f.

## Author

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## Referenced By

The man page sorhr_col(3) is an alias of sorhr_col.f(3).

Thu Apr 1 2021 Version 3.9.1 LAPACK