SRC/clahef_rk.f(3)

✨ NAME ✨

Okay bestie, welcome to the Library Functions Manual page for SRC/clahef_rk.f(3). The file is literally SRC/clahef_rk.f, and the main-character subroutine inside is clahef_rk — she’s an auxiliary routine and she knows it 💅.

🌸 SYNOPSIS 🌸

Functions/Subroutines

• subroutine clahef_rk (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info) — CLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. No cap, that’s the whole vibe.

💖 Function/Subroutine Documentation 💖

Here’s the full signature, copied exactly so nobody comes for me:

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

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

🎯 Purpose

So, like, CLAHEF_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    )

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.

CLAHEF_RK is an auxiliary routine called by CHETRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). She’s helping, not doing the whole main quest 🌟.

🧸 Parameters

🔤 UPLO

UPLO is CHARACTER*1. 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.

🔢 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.

🔢 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 array, dimension (LDA,N). On entry, it’s 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, A 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); the 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).

🎀 E

E is COMPLEX array, dimension (N). On exit, it 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 a 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. Literally both. Respect the consistency 💅.

🔁 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.

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 submatrix A(1:N,N-KB+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,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.
• c) In both cases a) and b) it is 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 submatrix A(1:N,1:KB).
  • 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 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.
• c) In both cases a) and b) it is always ABS( IPIV(k) ) >= k.
• d) NOTE: Any entry IPIV(k) is always NONZERO on output.

🗂️ W

W is COMPLEX 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. That’s the “do not pass go, do not divide by zero” moment 😭.

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. She remembers the first red flag only 💀.

👑 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

Definition at line 260 of file clahef_rk.f.

📝 Author 📝

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