hesv_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

, in arch rolling

🌸 NAME 🌸

okay so this is the hesv_rk(3) page from the Library Functions Manual, and the name is literally:

hesv_rk — {he,sy}sv_rk: rook (v3) 💅

✨ SYNOPSIS ✨

🧚 Functions 🧚

chesv_rk — CHESV_RK computes the solution to system of linear equations A * X = B for SY matrices

subroutine chesv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)

csysv_rk — CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

subroutine csysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)

dsysv_rk — DSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

subroutine dsysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)

ssysv_rk — SSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

subroutine ssysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)

zhesv_rk — ZHESV_RK computes the solution to system of linear equations A * X = B for SY matrices

subroutine zhesv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)

zsysv_rk — ZSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

subroutine zsysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work, lwork, info)

💫 Detailed Description 💫

bestie, this section is literally just Detailed Description energy, and then the function docs come in right after. No side quests here. ✨

🔥 Function Documentation 🔥

🌟 subroutine `chesv_rk` ...

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

CHESV_RK computes the solution to system of linear equations A * X = B for SY matrices

💖 Purpose 💖

okay so CHESV_RK 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. slay. 💅

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as

A = P*U*D*(U**H)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**H)*(P**T),  if UPLO = 'L',

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. 🦋

CHETRF_RK is called to compute the factorization of a complex Hermitian matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine CHETRS_3. 🔥

🎀 Parameters 🎀

UPLO — CHARACTER*1. Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A — 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, if INFO = 0, diagonal of the block diagonal matrix D and factors U or L as computed by CHETRF_RK:
  - 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. For more info see the description of CHETRF_RK routine.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
E — COMPLEX array, dimension (N). On exit, contains the output computed by the factorization routine CHETRF_RK, i.e. 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.

For more info see the description of CHETRF_RK routine.

IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D, as determined by CHETRF_RK. For more info see the description of CHETRF_RK routine.
B — COMPLEX 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.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
WORK — COMPLEX array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — INTEGER. The length of WORK. LWORK >= 1. For best performance of factorization stage LWORK >= max(1,N*NB), where NB is the optimal blocksize for CHETRF_RK. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array for factorization stage, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
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.

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.

🫶 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 226 of file chesv_rk.f.

🌈 subroutine `csysv_rk` ...

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

CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

💖 Purpose 💖

CSYSV_RK computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. ✨

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as

A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',

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. 🦋

CSYTRF_RK is called to compute the factorization of a complex symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine CSYTRS_3. 🔥

🎀 Parameters 🎀

UPLO — CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A — this is where part 1 cuts off, bestie — the A description continues in the next part 💅
A — 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, if INFO = 0, diagonal of the block diagonal matrix D and factors U or L as computed by CSYTRF_RK: 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.

For more info see the description of CSYTRF_RK routine.

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 output computed by the factorization routine CSYTRF_RK, i.e. 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.

For more info see the description of CSYTRF_RK routine.

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

For more info see the description of CSYTRF_RK routine.

B — B is COMPLEX 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.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
WORK — WORK is COMPLEX array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER. The length of WORK. LWORK >= 1. For best performance of factorization stage LWORK >= max(1,N*NB), where NB is the optimal blocksize for CSYTRF_RK.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array for factorization stage, 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.

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 226 of file csysv_rk.f.

🦋 `subroutine dsysv_rk`

   subroutine dsysv_rk (character uplo, integer n, integer nrhs, double
       precision, dimension( lda, * ) a, integer lda, double precision,
       dimension( * ) e, integer, dimension( * ) ipiv, double precision,
       dimension( ldb, * ) b, integer ldb, double precision, dimension( * )
       work, integer lwork, integer info)

DSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices

Purpose 🎀

okay so DSYSV_RK computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as

A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',

DSYTRF_RK is called to compute the factorization of a real symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine DSYTRS_3.

Parameters 🌸

UPLO — UPLO is CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is 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.
NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A — A is 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, if INFO = 0, diagonal of the block diagonal matrix D and factors U or L as computed by DSYTRF_RK: 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.

For more info see the description of DSYTRF_RK routine.

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
E — E is DOUBLE PRECISION array, dimension (N). On exit, contains the output computed by the factorization routine DSYTRF_RK, i.e. 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.

For more info see the description of DSYTRF_RK routine.

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

For more info see the description of DSYTRF_RK routine.

B — B is DOUBLE PRECISION 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.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
WORK — WORK is DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK —
LWORK — INTEGER. It's the length of WORK, and LWORK >= 1. Now, for the factorization stage to actually slay (peak performance, no lag), you want LWORK >= max(1,N*NB), where NB is the optimal blocksize for DSYTRF_RK. 💅

And the iconic shortcut: if LWORK = -1, the routine assumes a workspace query — it only calculates the optimal size of the WORK array for the factorization stage, returns that value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. So you get the tea without committing to the whole array. ✨

INFO — INTEGER. Exit report card, and it's giving three moods:
- = 0: successful exit. we love a clean run 🎀
- < 0: if INFO = -k, the k-th argument had an illegal value. bestie, you passed something cursed 😭
- > 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. 🫠 like it finished the job but the vibes are catastrophic.

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. So it's not a full receipt, it's just the first red flag. 🙈

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🫶 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 226 of file dsysv_rk.f. 📍

🦋 `ssysv_rk` — the single-precision twin, no cap

subroutine ssysv_rk (character uplo, integer n, integer nrhs, real,
    dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer,
    dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, real,
    dimension( * ) work, integer lwork, integer info)

SSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices. ✨

💖 Purpose:

So here's the whole vibe, bestie: SSYSV_RK computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. 🫶

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as

A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',

SSYTRF_RK is called to compute the factorization of a real symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine SSYTRS_3. two-step process, both steps ate. 🔥

🌸 Parameters

UPLO — CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': upper triangle of A is stored
- = 'L': lower triangle of A is stored
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. 💜
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A — 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. (half the matrix is just ✨not invited✨ to the function.)

On exit, if INFO = 0, diagonal of the block diagonal matrix D and factors U or L as computed by SSYTRF_RK:

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
If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.

For more info see the description of DSYTRF_RK routine. (yes, it really says DSYTRF_RK in the SSYSV_RK docs — that's what the source says, iykyk, don't come for me 😌)

LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🎀
E — REAL array, dimension (N). On exit, contains the output computed by the factorization routine DSYTRF_RK, i.e. 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. zeroed out, respectfully 💀

For more info see the description of DSYTRF_RK routine.

IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D, as determined by SSYTRF_RK. 🦋

For more info see the description of DSYTRF_RK routine.

B — REAL 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 complete ✨
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
WORK — REAL array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🌈
LWORK — INTEGER. The length of WORK. LWORK >= 1. For best performance of factorization stage LWORK >= max(1,N*NB), where NB is the optimal blocksize for DSYTRF_RK. 💅

INFO — INTEGER. The whole report card, again:
- = 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.

🫶 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 226 of file ssysv_rk.f. 📍

🧚 `zhesv_rk` — the complex*16 one, coming in hot

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

ZHESV_RK computes the solution to system of linear equations A * X = B for SY matrices

⭐ Purpose

…and the rest of this one is literally in the next chunk, stay with me bestie 💜

okay so bestie, ZHESV_RK is literally here to solve 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 bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as

A = P*U*D*(U**H)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**H)*(P**T),  if UPLO = 'L',

ZHETRF_RK is called to compute the factorization of a complex Hermitian matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine ZHETRS_3. no cap. 🫶

💖 Parameters 💖

UPLO — UPLO is CHARACTER*1 and says whether the upper or lower triangular part of the Hermitian matrix A is 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.
NRHS — NRHS is INTEGER; the number of right hand sides, i.e. the number of columns of the matrix B. 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, diagonal of the block diagonal matrix D and factors U or L as computed by ZHETRF_RK:
  - 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.
- For more info see the description of ZHETRF_RK routine.
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 output computed by the factorization routine ZHETRF_RK, i.e. 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.
- For more info see the description of ZHETRF_RK routine.
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D, as determined by ZHETRF_RK.
- For more info see the description of ZHETRF_RK routine.
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.
LDB — LDB is INTEGER; the leading dimension of the array B. LDB >= max(1,N).
WORK — WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER; the length of WORK. LWORK >= 1. For best performance of factorization stage LWORK >= max(1,N*NB), where NB is the optimal blocksize for ZHETRF_RK.
- If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array for factorization stage, 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.

🌟 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 226 of file zhesv_rk.f.

🦋 `zsysv_rk` 🦋

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

ZSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices. 🎀

✨ Purpose ✨

ZSYSV_RK computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as

A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',

ZSYTRF_RK is called to compute the factorization of a complex symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine ZSYTRS_3. 🦋

🎀 Parameters 🎀

UPLO — UPLO is CHARACTER*1 and says whether the upper or lower triangular part of the symmetric matrix A is 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.
NRHS — NRHS is INTEGER; the number of right hand sides, i.e. the number of columns of the matrix B. NRHS >= 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, if INFO = 0, diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK:
  - 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.
- For more info see the description of ZSYTRF_RK routine.
LDA — LDA is INTEGER; the leading dimension of the array A. LDA >= max(1,N).
E —

Still in the parameter list, bestie — the rest of the cast is right here, no skips: ✨

E — COMPLEX*16 array, dimension (N). On exit, it contains the output computed by the factorization routine ZSYTRF_RK, i.e. 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, 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 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. For more info see the description of the ZSYTRF_RK routine. 🦋
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D, as determined by ZSYTRF_RK. For more info see the description of the ZSYTRF_RK routine. 💅
B — 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. 💖
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N). ✨
WORK — COMPLEX*16 array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🎀
LWORK — INTEGER. The length of WORK. LWORK >= 1. For best performance of factorization stage LWORK >= max(1,N*NB), where NB is the optimal blocksize for ZSYTRF_RK. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array for factorization stage, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 🌸
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. 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 👑

The author squad is literally:

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

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 226 of file zsysv_rk.f.

Author 💅

Generated automatically by Doxygen for LAPACK from the source code. 🧚

And that's the whole page, bestie — the footer energy is LAPACK Version 3.12.0, and the page tag is hesv_rk(3). Go forth and factorize responsibly 👑✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

hesv_rk(3)

🌸 NAME 🌸

✨ SYNOPSIS ✨

🧚 Functions 🧚

💫 Detailed Description 💫

🔥 Function Documentation 🔥

🌟 subroutine chesv_rk ...

💖 Purpose 💖

🎀 Parameters 🎀

👑 Author 👑

🫶 Contributors 🫶

🌈 subroutine csysv_rk ...

💖 Purpose 💖

🎀 Parameters 🎀

Author 👑

Contributors 🫶

🦋 subroutine dsysv_rk

Purpose 🎀

Parameters 🌸

👑 Author

🫶 Contributors:

🦋 ssysv_rk — the single-precision twin, no cap

💖 Purpose:

🌸 Parameters

👑 Author

🫶 Contributors:

🧚 zhesv_rk — the complex*16 one, coming in hot

⭐ Purpose

💖 Parameters 💖

👑 Author 👑

🌟 Contributors 🌟

🦋 zsysv_rk 🦋

✨ Purpose ✨

🎀 Parameters 🎀

Author 👑

Contributors 🫶

Author 💅

🌟 subroutine `chesv_rk` ...

🌈 subroutine `csysv_rk` ...

🦋 `subroutine dsysv_rk`

🦋 `ssysv_rk` — the single-precision twin, no cap

🧚 `zhesv_rk` — the complex*16 one, coming in hot

🦋 `zsysv_rk` 🦋