# 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 [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.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

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).

Mon Jun 28 2021 Version 3.10.0 LAPACK