SRC/zhetf2.f(3)

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version 3.12.1 · 4 versions
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💅 SRC/zhetf2.f — Library Functions Manual 💅

okay bestie, so you're looking at the SRC/zhetf2.f page from the Library Functions Manual, and honestly? it's giving main character energy. this is the unblocked girlie of the ZHETF2 family, and she is literally about to factorize a complex Hermitian matrix for you. no cap. let's get into it. ✨

✨ SYNOPSIS ✨

Functions/Subroutines

  • subroutine zhetf2 (uplo, n, a, lda, ipiv, info) — ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS).

🦋 Function/Subroutine Documentation 🦋

subroutine zhetf2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS). she's iconic, she's unblocked, she's calling Level 2 BLAS, and she's about to make your matrix make sense. 💖

🎀 Purpose 🎀

okay so here's the tea, bestie:

!>
!> ZHETF2 computes the factorization of a complex Hermitian matrix A
!> using the Bunch-Kaufman diagonal pivoting method:
!>
!>    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, U**H is the conjugate transpose of U, and D is
!> Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
!>
!> This is the unblocked version of the algorithm, calling Level 2 BLAS.
!>

the way that she splits A into U*D*U**H or L*D*L**H… obsessed. U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. and yes, this is the unblocked version, calling Level 2 BLAS. periodt. 👑

💜 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. fr fr, don't come at her with a negative N.
  • 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 (she said don't look at it 💅). 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 (see below for further details). she leaves you with everything you need. 🫶
  • LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,N). lowkey important, don't skip this.
  • IPIV — IPIV is INTEGER array, dimension (N). details of the interchanges and the block structure of D. this is the lore:
    • if UPLO = 'U':
      • 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 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':
      • 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 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.
  • INFO — INFO is INTEGER. the vibe check on your exit status:
    • = 0: successful exit. slay. ✨
    • < 0: if INFO = -k, the k-th argument had an illegal value. not me making a typo in the arguments 😭
    • > 0: if INFO = k, D(k,k) 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. bestie this is the danger zone, heed the warning. 💀

🌸 Author 🌸

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

legendary institutions honestly, iykyk. 🎓

🔥 Further Details: 🔥

okay so 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).

and 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).

literally so much math but she ate and left no crumbs. 🧚

🧚 Contributors: 🧚

  • 09-29-06 — patch from Bobby Cheng, MathWorks
    • Replace l.210 and l.393:
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN

by:

IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
  • 01-01-96 — Based on modifications by
    • J. Lewis, Boeing Computer Services Company
    • A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

support your local contributors bestie, they're the real MVPs. 🫡

📌 Definition at line 190 of file zhetf2.f.

Author: Generated automatically by Doxygen for LAPACK from the source code. told you she's automated and iconic.

and that's the whole vibe, bestie. LAPACK Version 3.12.0 is the version, and this whole page is from source file SRC/zhetf2.f(3) — catch you in the next factorization. mwah! 💋🌈✨

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