ZHETF2(3)

ZHETF2(3) · LAPACK Library Functions Manual · okay bestie, let’s get into the unblocked Bunch-Kaufman tea 💅

💖 NAME 💖

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zhetf2.f

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✨ 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).

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🌸 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). It’s giving “Level 2 BLAS but make it direct” energy. 🦋

🎯 Purpose

Okay so here’s the whole point, 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. Block diagonal? We love a structured queen. 👑

This is the unblocked version of the algorithm, calling Level 2 BLAS. No cap. ✨

🧸 Parameters

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

Pick your triangle, pick your fighter. 💅

N

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

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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).

LDA

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

IPIV

IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D.

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.

The pivot lore is real, and you gotta read it carefully. 🫶

INFO

INFO is INTEGER.

• = 0: successful exit
• < 0: if INFO = -k, the k-th argument had an illegal value
• > 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.

⚠️ That INFO > 0 case is a real warning, bestie: the factorization finished, but D(k,k) is exactly zero, the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. Not a drill. 💀

👑 Author

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

🔎 Further Details

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).

🧚 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

Definition at line 190 of file zhetf2.f.

🤖 Author

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