SRC/zhetf2_rk.f(3)

💅 SRC/zhetf2_rk.f(3) 💅

okay so bestie, you've stumbled into the LAPACK library functions manual and this page is all about one gloriously specific little Fortran subroutine. it's the <pink>zhetf2_rk</pink> page, from LAPACK Version 3.12.0, and we're gonna go through every single parameter together because that's what friends do 💖

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✨ NAME ✨

SRC/zhetf2_rk.f

like, that's literally it. that's the name. she's a file, she's in SRC, she's got a .f extension, she's iconic 🎀

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🌸 SYNOPSIS 🌸

Functions/Subroutines

• subroutine zhetf2_rk (uplo, n, a, lda, e, ipiv, info) — ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

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🦋 Function/Subroutine Documentation 🦋

subroutine zhetf2_rk (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, integer info)

ZHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

Purpose:

lowkey the whole point of this routine is factored right here — ZHETF2_RK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

literally this is the unblocked version of the algorithm, calling Level 2 BLAS. for more info see the Further Details section (spoiler: it's a TODO, we'll get there 😭).

Parameters

UPLO

UPLO is CHARACTER*1 and it 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. yes bestie, it can be zero, no drama.

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:

• a) 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
• b) 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). don't skimp on it 🫶

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. For more info see Further Details section.

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

• a) 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 matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.
• b) 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,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 matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
• c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
• d) NOTE: Any entry IPIV(k) is always NONZERO on output.

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

• a) 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 matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
• b) 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 matrix A(1:N,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 matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
• c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
• d) NOTE: Any entry IPIV(k) is always NONZERO on output.

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. 💀 like, genuinely, watch out.

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.

Further Details:

TODO: put further details

😭 bestie they left it as a TODO. no cap. that's what's on the page.

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
• 01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company; A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

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Definition at line 240 of file zhetf2_rk.f.

Author

Generated automatically by Doxygen for LAPACK from the source code. yeah, a robot wrote the boring parts, we just gave it the vibe check ✨

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and that's the whole page, bestie — every flag, every index, every IPIV sign convention, all of it. slay 🫶

🌈 sign-off: LAPACK, Version 3.12.0 — page SRC/zhetf2_rk.f(3). periodt 💅