SRC/cunhr_col.f(3)

📦

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🌸 NAME 🌸

SRC/cunhr_col.f — okay bestie, this is the Library Functions Manual page for SRC/cunhr_col.f(3). It's the source-file glow-up for the cunhr_col routine. 💅

✨ SYNOPSIS ✨

🧩 Functions/Subroutines 🧩

subroutine cunhr_col (m, n, nb, a, lda, t, ldt, d, info)
    CUNHR_COL

🎀 Function/Subroutine Documentation 🎀

💖 `subroutine 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 💜

okay so CUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns as input, stored in A, and performs Householder Reconstruction (HR). That means it reconstructs Householder vectors V(i) that implicitly represent 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 🌈

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 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 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 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. 💀
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= max(1,min(NB,N)).
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.
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. (yes, the source literally says Y there — I'm not editing it, bestie 🙈)

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] ,
    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 cunhr_col.f.

🪄 Author 🪄

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okay bestie, that's the whole page. Footer sign-off: LAPACK Version 3.12.0, source SRC/cunhr_col.f(3). The footer doesn't print a separate date, so I'm not inventing one — no cap. 💅✨

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