SRC/zhesv.f(3)

💖 SRC/zhesv.f 💖

okay bestie, so this is a page from the LAPACK Library Functions Manual (you know, the one that lives in the SRC/zhesv.f(3) corner of the universe) and it's literally all about one glorious little subroutine. buckle up 🎀

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✨ SYNOPSIS ✨

Functions/Subroutines

• subroutine zhesv (uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info) — the main character fr: ZHESV computes the solution to system of linear equations A * X = B for HE matrices 👑

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🌸 Function/Subroutine Documentation 🌸

🦋 subroutine zhesv (character uplo, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integer lwork, integer info)

ZHESV computes the solution to system of linear equations A * X = B for HE matrices — no cap, that's the whole assignment 💅

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💜 Purpose:

!>
!> ZHESV computes the solution to a complex system of linear equations
!>    A * X = B,
!> where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
!> matrices.
!>
!> The diagonal pivoting method is used to factor A as
!>    A = U * D * U**H,  if UPLO = 'U', or
!>    A = L * D * L**H,  if UPLO = 'L',
!> 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.  The factored form of A is then
!> used to solve the system of equations A * X = B.
!>

so basically it's solving a complex system A * X = B where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS matrices 🫶 the diagonal pivoting method factors A into A = U * D * U**H when UPLO = 'U', or A = L * D * L**H when UPLO = 'L' — 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. then that factored form is used to solve A * X = B. iconic behavior honestly ✨

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🎀 Parameters 🎀

UPLO

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

UPLO is a CHARACTER*1 and it's literally telling you which triangle of A gets stored: 'U' = upper triangle of A is stored, 'L' = lower triangle of A is stored 🦋

N

!>          N is INTEGER
!>          The number of linear equations, i.e., the order of the
!>          matrix A.  N >= 0.
!>

N is an INTEGER: the number of linear equations, i.e., the order of the matrix A. rule: N >= 0 ⭐

NRHS

!>          NRHS is INTEGER
!>          The number of right hand sides, i.e., the number of columns
!>          of the matrix B.  NRHS >= 0.
!>

NRHS is an INTEGER: the number of right hand sides, i.e., the number of columns of the matrix B. rule: NRHS >= 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, if INFO = 0, the block diagonal matrix D and the
!>          multipliers used to obtain the factor U or L from the
!>          factorization A = U*D*U**H or A = L*D*L**H as computed by
!>          ZHETRF.
!>

A is a COMPLEX*16 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 (rude, but okay);
• 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, if INFO = 0, you get the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by ZHETRF 👑

LDA

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

LDA is an INTEGER: the leading dimension of the array A. rule: LDA >= max(1,N) 🌸

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D, as
!>          determined by ZHETRF.  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.
!>

IPIV is an INTEGER array, dimension (N) — the receipts on the interchanges and the block structure of D, as determined by ZHETRF 🧚

• 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.

B

!>          B is COMPLEX*16 array, dimension (LDB,NRHS)
!>          On entry, the N-by-NRHS right hand side matrix B.
!>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
!>

B is a COMPLEX*16 array, dimension (LDB,NRHS): on entry, the N-by-NRHS right hand side matrix B; on exit, if INFO = 0, the N-by-NRHS solution matrix X 💖 glow-up moment fr

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,N).
!>

LDB is an INTEGER: the leading dimension of the array B. rule: LDB >= max(1,N) ✅

WORK

!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>

WORK is a COMPLEX*16 array, dimension (MAX(1,LWORK)). on exit, if INFO = 0, WORK(1) returns the optimal LWORK ✨ (it's giving "look how much room you could have used")

LWORK

!>          LWORK is INTEGER
!>          The length of WORK.  LWORK >= 1, and for best performance
!>          LWORK >= max(1,N*NB), where NB is the optimal blocksize for
!>          ZHETRF.
!>          for LWORK < N, TRS will be done with Level BLAS 2
!>          for LWORK >= N, TRS will be done with Level BLAS 3
!>
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, returns
!>          this value as the first entry of the WORK array, and no error
!>          message related to LWORK is issued by XERBLA.
!>

LWORK is an INTEGER: the length of WORK. rules and vibes:

• LWORK >= 1, and for best performance LWORK >= max(1,N*NB), where NB is the optimal blocksize for ZHETRF;
• for LWORK < N, TRS will be done with Level BLAS 2;
• for LWORK >= N, TRS will be done with Level BLAS 3.

and if LWORK = -1, then a workspace query is assumed: the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA 🙈 it's just peeking, not committing

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, so the solution could not be computed.
!>

INFO is an 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, so the solution could not be computed 😭

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👑 Author 👑

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

Definition at line 169 of file zhesv.f. 📍

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🌈 Author (the meta one) 🌈

Generated automatically by Doxygen for LAPACK from the source code 🤖

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and that's the whole page, bestie — no details left behind, periodt 💅✨

LAPACK · Version 3.12.0 · SRC/zhesv.f(3) 🫶🎀