CLAUNHR_COL_GETRFNP2(3)

💅 CLAUNHR_COL_GETRFNP2(3) — like, the whole entire manual page, bestie

okay so this is a Library Functions Manual page, and the file it lives in is literally:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/claunhr_col_getrfnp2.f

yes that's a real path, no I'm not shortening it, it's iconic in its own chaotic way 🫶

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines

recursive subroutine CLAUNHR_COL_GETRFNP2 (m, n, a, lda, d, info)
    CLAUNHR_COL_GETRFNP2

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🌸 Function/Subroutine Documentation 🌸

recursive subroutine CLAUNHR_COL_GETRFNP2 (integer m, integer n, complex,
    dimension( lda, * ) a, integer lda, complex, dimension( * ) d, integer
    info)

CLAUNHR_COL_GETRFNP2 — that's her name, that's her whole thing, she's recursive and she's not sorry about it 💜

🌈 Purpose:

!>
!> CLAUNHR_COL_GETRFNP2 computes the modified LU factorization without
!> pivoting of a complex general M-by-N matrix A. The factorization has
!> the form:
!>
!>     A - S = L * U,
!>
!> where:
!>    S is a m-by-n diagonal sign matrix with the diagonal D, so that
!>    D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
!>    as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
!>    i-1 steps of Gaussian elimination. This means that the diagonal
!>    element at each step of  Gaussian elimination is at
!>    least one in absolute value (so that division-by-zero not
!>    possible during the division by the diagonal element);
!>
!>    L is a M-by-N lower triangular matrix with unit diagonal elements
!>    (lower trapezoidal if M > N);
!>
!>    and U is a M-by-N upper triangular matrix
!>    (upper trapezoidal if M < N).
!>
!> This routine is an auxiliary routine used in the Householder
!> reconstruction routine CUNHR_COL. In CUNHR_COL, this routine is
!> applied to an M-by-N matrix A with orthonormal columns, where each
!> element is bounded by one in absolute value. With the choice of
!> the matrix S above, one can show that the diagonal element at each
!> step of Gaussian elimination is the largest (in absolute value) in
!> the column on or below the diagonal, so that no pivoting is required
!> for numerical stability [1].
!>
!> For more details on the Householder reconstruction algorithm,
!> including the modified LU factorization, see [1].
!>
!> This is the recursive version of the LU factorization algorithm.
!> Denote A - S by B. The algorithm divides the matrix B into four
!> submatrices:
!>
!>        [  B11 | B12  ]  where B11 is n1 by n1,
!>    B = [ -----|----- ]        B21 is (m-n1) by n1,
!>        [  B21 | B22  ]        B12 is n1 by n2,
!>                               B22 is (m-n1) by n2,
!>                               with n1 = min(m,n)/2, n2 = n-n1.
!>
!>
!> The subroutine calls itself to factor B11, solves for B21,
!> solves for B12, updates B22, then calls itself to factor B22.
!>
!> For more details on the recursive LU algorithm, see [2].
!>
!> CLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked
!> routine CLAUNHR_COL_GETRFNP, which uses blocked code calling
!> Level 3 BLAS to update the submatrix. However, CLAUNHR_COL_GETRFNP2
!> is self-sufficient and can be used without CLAUNHR_COL_GETRFNP.
!>
!> [1] ,
!>     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
!>     E. Solomonik, J. Parallel Distrib. Comput.,
!>     vol. 85, pp. 3-31, 2015.
!>
!> [2] , F. Gustavson, IBM J. of Res. and Dev.,
!>     vol. 41, no. 6, pp. 737-755, 1997.
!>

so like, in girl terms: she does a modified LU factorization of a complex general M-by-N matrix A without pivoting, and the whole A - S = L * U thing is the setup 💖 And honestly the way she picks that diagonal D(i) = -SIGN(A(i,i)) so the diagonal element is always at least one in absolute value?? so division-by-zero literally can't happen?? that's the self-care we deserve ✨

Also she's an auxiliary routine for CUNHR_COL (the Householder reconstruction routine), where A has orthonormal columns, each element bounded by one in abs value — and with that S choice, the diagonal element at each Gaussian elimination step is the largest in abs value in the column on or below the diagonal, so no pivoting is required for numerical stability [1] 👑

And she's recursive — she splits B = A - S into four submatrices (B11 n1×n1, B21 (m-n1)×n1, B12 n1×n2, B22 (m-n1)×n2, with n1 = min(m,n)/2, n2 = n-n1), calls herself to factor B11, solves for B21, solves for B12, updates B22, then calls herself again to factor B22. literal main character energy 🔥

And yes bestie, CLAUNHR_COL_GETRFNP (the blocked one) calls her to factorize a block, but she's self-sufficient and can be used all on her own without it 💅

The refs she cites (iykyk):

• [1] G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen, E. Solomonik, J. Parallel Distrib. Comput., vol. 85, pp. 3-31, 2015.
• [2] F. Gustavson, IBM J. of Res. and Dev., vol. 41, no. 6, pp. 737-755, 1997.

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🦋 Parameters

• M

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

the row count, she's INTEGER, and M >= 0 fr fr.

• N

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

the column count, also INTEGER, also N >= 0.

• A

!>          A is COMPLEX array, dimension (LDA,N)
!>          On entry, the M-by-N matrix to be factored.
!>          On exit, the factors L and U from the factorization
!>          A-S=L*U; the unit diagonal elements of L are not stored.
!>

COMPLEX array, dimension (LDA,N). On entry: the M-by-N matrix to be factored. On exit: the factors L and U from A-S=L*U — and heads up, the unit diagonal elements of L are not stored 🙈

• LDA

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

leading dimension of A, INTEGER, and LDA >= max(1,M) — no cap.

• D

!>          D is COMPLEX array, dimension min(M,N)
!>          The diagonal elements of the diagonal M-by-N sign matrix S,
!>          D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can be
!>          only ( +1.0, 0.0 ) or (-1.0, 0.0 ).
!>

COMPLEX array, dimension min(M,N) — the diagonal elements of the diagonal M-by-N sign matrix S, with D(i) = S(i,i) for 1 <= i <= min(M,N). And the elements can be only ( +1.0, 0.0 ) or (-1.0, 0.0 ) ⭐ like, pick a lane, and it's one of those two.

• INFO

!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>

INTEGER, and it's giving:

• = 0: successful exit 🎉
• < 0: if INFO = -i, the i-th argument had an illegal value 💀

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👑 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
!>
!>

shoutout Igor, November 2019, Computer Science Division, University of California, Berkeley 🫶

Definition at line 166 of file claunhr_col_getrfnp2.f — that's where she lives, in case you need to find her.

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🌟 Author (the meta one)

Generated automatically by Doxygen for LAPACK from the source code. (yes, a robot wrote this bit, not me, don't @ me 🤖)

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and that's the whole page bestie!! LAPACK, Version 3.12.0, dated /home/abui..._getrfnp2.f(3) — mwah, slay, go factor something 💖✨🎀