ZHETRF(3)

💖 ZHETRF(3) — the LAPACK glow-up your Hermitian matrix deserves

okay so first things first, bestie: this page is zhetrf.f(3) from the Library Functions Manual, and the source file it's describing lives at

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

and the page itself is at /home/abui...RC/zhetrf.f(3). iconic path, lowkey unhinged, but we move. ✨

---

🌸 NAME

ZHETRF — it factors a complex Hermitian matrix. that's the whole vibe. no cap, that's it.

---

🎀 SYNOPSIS

Functions/Subroutines

subroutine ZHETRF (uplo, n, a, lda, ipiv, work, lwork, info)

ZHETRF (yes that's the whole subtitle, she's lowkey mysterious like that 💅)

---

🦋 Function/Subroutine Documentation

subroutine ZHETRF

subroutine ZHETRF (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRF

✨ Purpose

fr fr, here's what she does:

ZHETRF computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is A = U*D*U**H or A = L*D*L**H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. This is the blocked version of the algorithm, calling Level 3 BLAS.

so like, she does the thing, but the blocked way, and she's calling Level 3 BLAS while she does it. main character energy honestly. 👑

🫶 Parameters

• UPLO — UPLO is CHARACTER*1, and it's literally just "which triangle are we working with":

= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

• N — N is INTEGER. the order of the matrix A. and the rule is N >= 0. (zero is allowed, don't be weird about it.)
• A — A is COMPLEX*16 array, dimension (LDA,N). on entry, she'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 (so like, don't worry about what's down there, she's not looking 🙈). 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L are hanging out in there (peep the Further Details below for the full tea).

• LDA — LDA is INTEGER. it's the leading dimension of the array A, and she needs LDA >= max(1,N). non-negotiable bestie.
• IPIV — IPIV is INTEGER array, dimension (N). this is where the drama lives: details of the interchanges and the block structure of D.
  • if IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • if UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
  • if UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

see what I mean? she's got lore. 📖

• WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal LWORK. she tells you what she wishes she had. queen behavior. 🎀
• LWORK — LWORK is INTEGER. the length of WORK. she needs LWORK >= 1. for best performance though? LWORK >= N*NB, where NB is the block size returned by ILAENV. give her room to breathe and she'll serve. 💅
• INFO — INFO is INTEGER, and it's the vibe check at the end:

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, D(i,i) is exactly zero.  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.

okay real talk on that last one: INFO > 0 means D is exactly singular. the factorization finished, but if you go solve a system with it, you're dividing by zero. that's not a vibe, that's a crash. be warned bestie. 💀

👑 Author

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

(four whole institutions, she's got a whole squad behind her, we love a collaborative queen 🌈)

🌟 Further Details

okay here's the deep lore section, buckle up:

If UPLO = 'U', then A = U*D*U**H, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k). If UPLO = 'L', then A = L*D*L**H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

like… she really said "here's the whole blueprint, no gatekeeping." obsessed. 💜

---

📍 Where she lives

Definition at line 176 of file zhetrf.f.

🌸 Author (the meta one)

Generated automatically by Doxygen for LAPACK from the source code. (so like, a robot wrote the page, but the math is very much human. credit where it's due 🫶)

---

and that's the whole page bestie!! ✨💖

LAPACK — Version 3.12.0 — and the file is /home/abui...RC/zhetrf.f(3). go forth and factor your Hermitian matrices, slay, periodt. 🦋🎀👑