SRC/zhetrf.f(3)

🎀 NAME 🎀

SRC/zhetrf.f — bestie, you're on the SRC/zhetrf.f(3) page from the Library Functions Manual, and it's all about the ZHETRF subroutine. Let's get into it 💅

🦋 SYNOPSIS 🦋

Functions/Subroutines

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

The main-character routine here is ZHETRF. Periodt.

💖 Function/Subroutine Documentation 💖

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

Okay so 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. Slay. ✨

🌸 Parameters

• UPLO — UPLO is CHARACTER*1; = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
• 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 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.

• WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — LWORK is INTEGER; the length of WORK. LWORK >= 1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV.
• INFO — INFO is INTEGER; = 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.

🫶 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).

📍 Definition

Definition at line 176 of file zhetrf.f.

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