hetrf_rk(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

💖 NAME 💖

hetrf_rk - {he,sy}trf_rk: triangular factor

okay bestie, welcome to the Library Functions Manual page for hetrf_rk(3) — this is the {he,sy}trf_rk family, and they’re all about that triangular factor life. 💅

✨ SYNOPSIS ✨

Functions

okay so here’s the whole routine squad, no cap — every signature is exactly what you type, and every description keeps the real facts:

subroutine chetrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info) — CHETRF_RK is the complex-Hermitian-indefinite one: it computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). 🦋
subroutine csytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info) — CSYTRF_RK handles the complex-symmetric-indefinite case: it computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). ✨
subroutine dsytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info) — DSYTRF_RK is the real-symmetric-indefinite girlie: it computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). 🎀
subroutine ssytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info) — SSYTRF_RK also does the real-symmetric-indefinite thing: it computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). 🌸
subroutine zhetrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info) — ZHETRF_RK is the complex-Hermitian-indefinite icon: it computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). 💜
subroutine zsytrf_rk (uplo, n, a, lda, e, ipiv, work, lwork, info) — ZSYTRF_RK rounds out the set for complex-symmetric-indefinite matrices: it computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). 🔥

🌸 Detailed Description 🌸

okay so this heading exists and then we immediately roll into the function docs — that’s the tea. ☕

🦋 Function Documentation 🦋

🎀 subroutine chetrf_rk 🎀

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

CHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm). 💅

💜 Purpose: 💜

bestie, CHETRF_RK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T)

where U (or L) is unit upper (or lower) triangular matrix, U**H (or L**H) is the conjugate of U (or L), P is a permutation matrix, P**T is the transpose of P, 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. For more information see Further Details section. ✨

🍓 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.
A — A is COMPLEX 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, contains:
  - a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
  - b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
E — E is COMPLEX array, dimension (N). On exit, contains the superdiagonal (or subdiagonal) elements of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:
- If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
- If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
- NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.
IPIV — IPIV is INTEGER array, dimension (N). IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see Further Details section.

If UPLO = 'U', ( in factorization order, k decreases from N to 1 ):

a) A single positive entry IPIV(k) > 0 means:
- D(k,k) is a 1-by-1 diagonal block.
- If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N);
- If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means:
- D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
- (NOTE: negative entries in IPIV appear ONLY in pairs).
- 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
- 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L', ( in factorization order, k increases from 1 to N ):

a) A single positive entry IPIV(k) > 0 means:
- D(k,k) is a 1-by-1 diagonal block.
- If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N).
- If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means:
- D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
- (NOTE: negative entries in IPIV appear ONLY in pairs).
- 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
- 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

WORK — WORK is COMPLEX 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.
- 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.
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: If INFO = -k, the k-th argument had an illegal value
- > 0: If INFO = k, the matrix A is singular, because:
  - If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
  - If UPLO = 'L': column k in the lower triangular part of A contains all zeros.
  - Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. 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.
  - NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.

👑 Author 👑

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

🧚 Further Details: 🧚

TODO: put correct description

🫶 Contributors: 🫶

and the contributor list for chetrf_rk keeps the same energy:

December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

Definition at line 257 of file chetrf_rk.f.

💖 `csytrf_rk` — complex symmetric, rook pivoting, blocked era

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

CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method — the BLAS3 blocked algorithm version, bestie. It’s giving serious linear algebra main character energy. 💅

🎯 Purpose

okay so here’s the whole vibe: CSYTRF_RK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, 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. For more information see Further Details section. iykyk 👑

💅 Parameters

UPLO — is CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
N — is INTEGER. The order of the matrix A. N >= 0.
A — is COMPLEX 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, contains:

a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

LDA — is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
E — is COMPLEX array, dimension (N). On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where
- If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
- If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases. don’t forget it 💛

IPIV — is INTEGER array, dimension (N). IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see Further Details section.
- If UPLO = 'U', ( in factorization order, k decreases from N to 1 ):
  - a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.
  - b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
    1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
    2. If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
  - c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
  - d) NOTE: Any entry IPIV(k) is always NONZERO on output.
- If UPLO = 'L', ( in factorization order, k increases from 1 to N ):
  - a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
  - b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
    1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
    2. If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
  - c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
  - d) NOTE: Any entry IPIV(k) is always NONZERO on output.
WORK — is COMPLEX array, dimension ( MAX(1,LWORK) ). On exit, if INFO = 0, WORK(1) returns the optimal 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. <butterfly>🦋</butterfly>

INFO — is INTEGER.
- = 0: successful exit
- < 0: If INFO = -k, the k-th argument had an illegal value
- > 0: If INFO = k, the matrix A is singular, because:
  - If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
  - If UPLO = 'L': column k in the lower triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. 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.

NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes. this part is important, bestie — division by zero is not a vibe 😭

💖 Author

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

🔍 Further Details

!> TODO: put correct description

yes, the details section is literally a TODO. iconic behavior, honestly. 🎀

✨ Contributors

December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

Definition at line 257 of file csytrf_rk.f.

🦋 `dsytrf_rk` — real symmetric, rook pivoting, blocked era

subroutine dsytrf_rk (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, integer, dimension( * ) ipiv, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method — the BLAS3 blocked algorithm version. same energy, real arithmetic, still slaying ✨

🎯 Purpose

🎀 Parameters 🎀

bestie. the numbers are the moment. here's what DSYTRF_RK actually means by all of it: it computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method, and the whole thing is:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix, UT (or LT) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, 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. For more information see Further Details section. iykyk 🔥

UPLO — CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
N — INTEGER. The order of the matrix A. N >= 0.
A — DOUBLE PRECISION 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, contains:

a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
E — DOUBLE PRECISION array, dimension (N). On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where
- If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
- If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.

IPIV — INTEGER array, dimension (N). IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see Further Details section. 👑

If UPLO = 'U', ( in factorization order, k decreases from N to 1 ):

a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
2. If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L', ( in factorization order, k increases from 1 to N ):

a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).
1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
2. If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

WORK — DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — 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. no cap, that's the shortcut 💅

INFO — INTEGER.
- = 0: successful exit
- < 0: If INFO = -k, the k-th argument had an illegal value
- > 0: If INFO = k, the matrix A is singular, because: If UPLO = 'U': column k in the upper triangular part of A contains all zeros. If UPLO = 'L': column k in the lower triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. 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. 😭 literally do not do that

NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes. it's giving "we only log the first one" energy

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌸 Further Details: 🌸

!> TODO: put correct description
!>

not the TODO in the official docs 💀 iconic honestly. anyway, the page says exactly that, so that's what you get.

🦋 Contributors: 🦋

December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

Definition at line 257 of file dsytrf_rk.f.

subroutine ssytrf_rk (character uplo, integer n, real, dimension( lda, * )
    a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv,
    real, dimension( * ) work, integer lwork, integer info)

SSYTRF_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

⭐ Purpose: ⭐

so SSYTRF_RK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

this is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section. same vibe, single precision edition ✨

💖 Parameters 💖

UPLO — CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
N — INTEGER. The order of the matrix A. N >= 0.
A —

okay so A is the REAL 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, contains:

a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

`LDA` 📏

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

`E` 🦋

E is REAL array, dimension (N).

On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where

If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.

`IPIV` 👑

IPIV is INTEGER array, dimension (N).

IPIV describes the permutation matrix P in the factorization of matrix A as follows. The absolute value of IPIV(k) represents the index of row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see Further Details section.

If UPLO = 'U', ( in factorization order, k decreases from N to 1 ):

a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block.
- If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N);
- If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
- (NOTE: negative entries in IPIV appear ONLY in pairs).
- 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
- 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L', ( in factorization order, k increases from 1 to N ):

a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block.
- If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N).
- If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
- (NOTE: negative entries in IPIV appear ONLY in pairs).
- 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
- 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

`WORK` 🧚

WORK is REAL 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 = -k, the k-th argument had an illegal value
> 0: If INFO = k, the matrix A is singular, because:
- If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
- If UPLO = 'L': column k in the lower triangular part of A contains all zeros.

NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.

Author ✨

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

Further Details: 🔍

TODO: put correct description

Contributors: 🫶

December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

Definition at line 257 of file ssytrf_rk.f.

`zhetrf_rk` 💜

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

ZHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

Purpose 💫

ZHETRF_RK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),

This is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section.

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
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, contains:

a) ONLY diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

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

E — okay so E is a COMPLEX*16 array, dimension (N). On exit it’s carrying the superdiagonal (or subdiagonal) elements of the Hermitian block diagonal matrix D, and D has 1-by-1 or 2-by-2 diagonal blocks, where:

If UPLO = 'U': E(i) = D(i-1,i), i=2:N, and E(1) is set to 0.
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, and E(N) is set to 0.

📝 NOTE: For a 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both the UPLO = 'U' and UPLO = 'L' cases.

IPIV — IPIV is an INTEGER array, dimension (N). IPIV describes the permutation matrix P in the factorization of matrix A like this: the absolute value of IPIV(k) represents the index of the row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, which correspond to 1 or 2 interchanges at each factorization step. For more info see the Further Details section.

If UPLO = 'U' — in factorization order, k decreases from N to 1:
- a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block.
  - If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N).
  - If IPIV(k) = k, no interchange occurred.
- b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
  - (NOTE: negative entries in IPIV appear ONLY in pairs).
  - 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
  - 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
- c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
- d) NOTE: Any entry IPIV(k) is always NONZERO on output.
If UPLO = 'L' — in factorization order, k increases from 1 to N:
- a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block.
  - If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N).
  - If IPIV(k) = k, no interchange occurred.
- b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means: D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  - (NOTE: negative entries in IPIV appear ONLY in pairs).
  - 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
  - 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
- c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
- d) NOTE: Any entry IPIV(k) is always NONZERO on output.

WORK — WORK is a 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 = -k, the k-th argument had an illegal value.
> 0: If INFO = k, the matrix A is singular, because:
- If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
- If UPLO = 'L': column k in the lower triangular part of A contains all zeros.

🚨 NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.

🫶 Author

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

🌸 Further Details

!> TODO: put correct description
!>

✨ Contributors

December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

Definition at line 257 of file zhetrf_rk.f.

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

ZSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).

🎯 Purpose

ZSYTRF_RK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T)

This is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section.

🌈 Parameters

UPLO — UPLO is CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:

= 'U': Upper triangular
= 'L': Lower triangular

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

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, contains:

a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

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

E — E is COMPLEX*16 array, dimension (N). On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:

If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

📝 NOTE: For a 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both the UPLO = 'U' or UPLO = 'L' cases.

IPIV

okay so IPIV is an INTEGER array, dimension (N), and it's literally the receipts for the permutation matrix P in the factorization of matrix A. Here's the tea:

the absolute value of IPIV(k) is the index of the row and column that were interchanged with the k-th row and column.
the value of UPLO describes the order in which those interchanges were applied.
the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks which correspond to 1 or 2 interchanges at each factorization step. For more info see the Further Details section, bestie.

If UPLO = 'U' (in factorization order, k decreases from N to 1):

a) A single positive entry IPIV(k) > 0 means:
- D(k,k) is a 1-by-1 diagonal block.
- If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N).
- If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means:
- D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
- (NOTE: negative entries in IPIV appear ONLY in pairs).
1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
2. If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L' (in factorization order, k increases from 1 to N):

a) A single positive entry IPIV(k) > 0 means:
- D(k,k) is a 1-by-1 diagonal block.
- If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N).
- If IPIV(k) = k, no interchange occurred.
b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means:
- D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
- (NOTE: negative entries in IPIV appear ONLY in pairs).
1. If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
2. If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred. (yes, the doc literally says k-1 and -IPIV(k-1) in this sub-bullet — keep it exact, no cap.)
c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output.

🧰 WORK

WORK is a COMPLEX*16 array, dimension ( MAX(1,LWORK) ).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

📏 LWORK

LWORK is INTEGER.

It's 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. slay.
< 0: If INFO = -k, the k-th argument had an illegal value.
> 0: If INFO = k, the matrix A is singular, because:
- If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
- If UPLO = 'L': column k in the lower triangular part of A contains all zeros.
- Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. 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.
- NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes.

👑 Author

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

🔍 Further Details:

!> TODO: put correct description
!>

🌈 Contributors:

December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

📍 Definition at line 257 of file zsytrf_rk.f.

🤖 Author

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hetrf_rk(3)

💖 NAME 💖

✨ SYNOPSIS ✨

Functions

🌸 Detailed Description 🌸

🦋 Function Documentation 🦋

🎀 subroutine chetrf_rk 🎀

💜 Purpose: 💜

🍓 Parameters 🍓

👑 Author 👑

🧚 Further Details: 🧚

🫶 Contributors: 🫶

💖 csytrf_rk — complex symmetric, rook pivoting, blocked era

🎯 Purpose

💅 Parameters

💖 Author

🔍 Further Details

✨ Contributors

🦋 dsytrf_rk — real symmetric, rook pivoting, blocked era

🎯 Purpose

🎀 Parameters 🎀

👑 Author 👑

🌸 Further Details: 🌸

🦋 Contributors: 🦋

⭐ Purpose: ⭐

💖 Parameters 💖

LDA 📏

E 🦋

IPIV 👑

WORK 🧚

LWORK 🌈

INFO ⭐

Author ✨

Further Details: 🔍

Contributors: 🫶

zhetrf_rk 💜

Purpose 💫

Parameters 💅

🫶 Author

🌸 Further Details

✨ Contributors

🎯 Purpose

🌈 Parameters

🧰 WORK

📏 LWORK

🚨 INFO

👑 Author

🔍 Further Details:

🌈 Contributors:

🤖 Author

💖 `csytrf_rk` — complex symmetric, rook pivoting, blocked era

🦋 `dsytrf_rk` — real symmetric, rook pivoting, blocked era

`LDA` 📏

`E` 🦋

`IPIV` 👑

`WORK` 🧚

`LWORK` 🌈

`INFO` ⭐

`zhetrf_rk` 💜