SRC/zunhr_col.f(3)

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✨ SRC/zunhr_col.f(3) ✨

okay so this page is straight out of the Library Functions Manual, and it’s all about SRC/zunhr_col.f. Bestie, it’s a LAPACK routine page — very main character linear algebra energy. 🫶

🎀 NAME 🎀

SRC/zunhr_col.f

🫶 SYNOPSIS 🫶

Functions/Subroutines

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

ZUNHR_COL

💅 Function/Subroutine Documentation 💅

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: 🌸

okay so ZUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns as input, stored in A, and performs Householder Reconstruction (HR). Translation: 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 ZGEQRT). 💖

✨ 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*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 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*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.

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

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: fortran 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 fortran 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 zunhr_col.f.

💜 Author 💜

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