SRC/zlahef_rk.f(3)

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version 3.12.1 · 7 versions
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💅 SRC/zlahef_rk.f — straight out of the LAPACK Library Functions Manual, bestie

okay so picture this: page SRC/zlahef_rk.f(3), a whole manual page for one little Fortran subroutine that is lowkey doing the most. it's giving linear algebra excellence, and yes, we're gonna read the whole thing together. grab a snack 🍓

✨ NAME

SRC/zlahef_rk.f — that's the file, that's the name, that's the whole vibe. 💫

🦋 SYNOPSIS

Functions/Subroutines

  • subroutine zlahef_rk (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info) — ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method.

👑 Function/Subroutine Documentation

Here's the full signature, copy-pasted by me with love (and zero typos, because accuracy is my whole personality):

subroutine zlahef_rk (character uplo, integer n, integer nb, integer kb,
    complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( *
    ) e, integer, dimension( * ) ipiv, complex*16, dimension( ldw, * ) w,
    integer ldw, integer info)

ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method.

🌸 Purpose

So what does it actually do? ZLAHEF_RK computes a partial factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method. The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**H U22**H )

A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  if UPLO = 'L',
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.

ZLAHEF_RK is an auxiliary routine called by ZHETRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). 💪

🎀 Parameters

bestie there are a LOT of them, so buckle up — every single one matters, no skipping, iykyk.

UPLO

UPLO is CHARACTER*1. Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:

  • = 'U': Upper triangular
  • = 'L': Lower triangular

N

N is INTEGER. The order of the matrix A. N >= 0.

NB

NB is INTEGER. The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks. (so like, 2 is the bare minimum, don't be shy with it though)

KB

KB is INTEGER. The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.

A

A is COMPLEX*16 array, dimension (LDA,N).

On entry, the Hermitian matrix A.

  • If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
  • If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, contains:

  1. ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
  2. If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

LDA

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

E

E is COMPLEX*16 array, dimension (N).

On exit, contains the superdiagonal (or subdiagonal) elements of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:

  • If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
  • If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases. 🫶

IPIV

IPIV is INTEGER array, dimension (N).

IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step.

okay this next bit is a LOT but it's important so I'm laying it out exactly, no cap:

If UPLO = 'U' (in factorization order, k decreases from N to 1):

  1. A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,N-KB+1:N); if IPIV(k) = k, no interchange occurred.
  2. A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
    1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N). If -IPIV(k) = k, no interchange occurred.
    2. If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N). If -IPIV(k-1) = k-1, no interchange occurred.
  3. In both cases a) and b) is always ABS( IPIV(k) ) <= k.
  4. NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L' (in factorization order, k increases from 1 to N):

  1. A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If IPIV(k) = k, no interchange occurred.
  2. A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
    1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k) = k, no interchange occurred.
    2. If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k+1) = k+1, no interchange occurred.
  3. In both cases a) and b) is always ABS( IPIV(k) ) >= k.
  4. NOTE: Any entry IPIV(k) is always NONZERO on output.

W

W is COMPLEX*16 array, dimension (LDW,NB).

LDW

LDW is INTEGER. The leading dimension of the array W. LDW >= max(1,N).

INFO

INFO is INTEGER.

  • = 0: successful exit 🎉
  • < 0: If INFO = -k, the k-th argument had an illegal value
  • > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. 💀

NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.

🌈 Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

🧚 Contributors

  • December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
  • September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

Definition at line 260 of file zlahef_rk.f.

⭐ Author (of this page, not the code)

Generated automatically by Doxygen for LAPACK from the source code. (not me writing it by hand, bestie — the robots did that part 🤖✨)

and that's the whole page!! mwah 💋 this little icon of a subroutine ate and left no crumbs. page SRC/zlahef_rk.f(3), LAPACK Version 3.12.0 — go forth and factorize responsibly, bestie 🎀🦋💖

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