ZSYTRF(3)

💅 ZSYTRF — zsytrf.f(3) 🎀

okay bestie so this page lives in the Library Functions Manual, and the file it's documenting is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zsytrf.f ✨ yes that's a whole entire path, stay with me, it's just where the source lives 🫶

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zsytrf.f — that's literally the whole name section, icon, no notes 💅

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🌸 SYNOPSIS 🌸

Functions/Subroutines

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

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🦋 Function/Subroutine Documentation 🦋

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

💖 Purpose 💖

!> !> ZSYTRF computes the factorization of a complex symmetric matrix A !> using the Bunch-Kaufman diagonal pivoting method. The form of the !> factorization is !> !> A = U*D*UT or A = L*D*LT !> !> where U (or L) is a product of permutation and unit upper (lower) !> triangular matrices, and D is symmetric 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 real talk: this one factorizes a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method, and the factorization comes out as A = U*D*U**T or A = L*D*L**T — where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks 🔥

and heads up: this is the blocked version of the algorithm, calling Level 3 BLAS. main character energy, it's the fast one ✨

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🧚 Parameters 🧚

• UPLO — UPLO is CHARACTER*1. it tells you which triangle of A is stored:
  • = '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. (fr fr, valid input only bestie 🙈)
• A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the symmetric 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). the details of the interchanges and the block structure of D 😤 this part is lowkey the spicy one:
  • 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. cute 🎀
• 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.

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. (so it won't yell at you, bestie, it's being polite)

• INFO — INFO is INTEGER:
  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value (oops 🙈)
  • > 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. 💀 like it literally tells you division by zero is coming, do not ignore this

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👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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⭐ Further Details ⭐

!> !> If UPLO = 'U', then A = U*D*UT, 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 = LD*LT, 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). !

so if UPLO = 'U', then A = U*D*U**T, 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 — and when the diagonal block D(k) is of order s (s = 1 or 2), you get the shape up there ⬆️

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).

flip it to UPLO = 'L' and then A = L*D*L**T, 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, with the shape shown above ⬆️

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). 🌈

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Definition at line 181 of file zsytrf.f. 📍

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🍓 Author 🍓

Generated automatically by Doxygen for LAPACK from the source code. (not me writing it by hand, the robots did that one 🤖✨)

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okay that's the whole page bestie, you ate that 💅✨ this was LAPACK, Version 3.12.0, footer and all — go factorize something symmetric, ily, bye 🫶🎀