complexSYcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

🦋 complexSYcomputational(3) 🦋

okay so this is the manual page for complexSYcomputational — it's giving complex computational energy for SY matrices, and we're about to go through every single function like it's a group chat, no cap. 💅

✨ NAME ✨

complexSYcomputational — complex

🌸 SYNOPSIS 🌸

💅 Functions 💅

subroutine chesv_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, WORK, LWORK, INFO) CHESV_AA_2STAGE computes the solution to system of linear equations A * X = B for HE matrices.
subroutine chetrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO) the description is literally CHETRF_AA_2STAGE 💅
subroutine chetrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO) the description is literally CHETRS_AA_2STAGE 💅
subroutine cla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) CLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
real function cla_syrcond_c (UPLO, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK, RWORK) CLA_SYRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.
real function cla_syrcond_x (UPLO, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK) CLA_SYRCOND_X computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.
subroutine cla_syrfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO) CLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
real function cla_syrpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK) CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
subroutine clahef_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK) the description is literally CLAHEF_AA 💅
subroutine clasyf (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) CLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine clasyf_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK) the description is literally CLASYF_AA 💅
subroutine clasyf_rk (UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO) CLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
subroutine clasyf_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) CLASYF_ROOK computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
subroutine csycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO) the description is literally CSYCON 💅
subroutine csycon_3 (UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, INFO) the description is literally CSYCON_3 💅
subroutine csycon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO) the description is literally CSYCON_ROOK 💅
subroutine csyconv (UPLO, WAY, N, A, LDA, IPIV, E, INFO) the description is literally CSYCONV 💅
subroutine csyconvf (UPLO, WAY, N, A, LDA, E, IPIV, INFO) the description is literally CSYCONVF 💅
subroutine csyconvf_rook (UPLO, WAY, N, A, LDA, E, IPIV, INFO) the description is literally CSYCONVF_ROOK 💅
subroutine csyequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO) the description is literally CSYEQUB 💅
subroutine csyrfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) the description is literally CSYRFS 💅
subroutine csyrfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO) the description is literally CSYRFSX 💅
subroutine csysv_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, WORK, LWORK, INFO) CSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices.
subroutine csytf2 (UPLO, N, A, LDA, IPIV, INFO) CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine csytf2_rk (UPLO, N, A, LDA, E, IPIV, INFO) CSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
subroutine csytf2_rook (UPLO, N, A, LDA, IPIV, INFO) CSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
subroutine csytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) the description is literally CSYTRF 💅
subroutine csytrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) the description is literally CSYTRF_AA 💅
subroutine csytrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO) the description is literally CSYTRF_AA_2STAGE 💅
subroutine csytrf_rk (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO) CSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine csytrf_rook (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) the description is literally CSYTRF_ROOK 💅
subroutine csytri (UPLO, N, A, LDA, IPIV, WORK, INFO) the description is literally CSYTRI 💅
subroutine csytri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) the description is literally CSYTRI2 💅
subroutine csytri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO) the description is literally CSYTRI2X 💅
subroutine csytri_3 (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO) the description is literally CSYTRI_3 💅
subroutine csytri_3x (UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO) the description is literally CSYTRI_3X 💅
subroutine csytri_rook (UPLO, N, A, LDA, IPIV, WORK, INFO) the description is literally CSYTRI_ROOK 💅
subroutine csytrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) the description is literally CSYTRS 💅
subroutine csytrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO) the description is literally CSYTRS2 💅
subroutine csytrs_3 (UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO) the description is literally CSYTRS_3 💅
subroutine csytrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO) the description is literally CSYTRS_AA 💅
subroutine csytrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO) the description is literally CSYTRS_AA_2STAGE 💅
subroutine csytrs_rook (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) the description is literally CSYTRS_ROOK 💅
subroutine ctgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) the description is literally CTGSYL 💅
subroutine ctrsyl (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO) the description is literally CTRSYL 💅
subroutine ctrsyl3 (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, SWORK, LDSWORK, INFO) the description is literally CTRSYL3 💅

🦋 Detailed Description 🦋

This is the group of complex computational functions for SY matrices. like, the whole squad is here, bestie. 💖

💖 Function Documentation 💖

🌸 subroutine chesv_aa_2stage 🌸

subroutine chesv_aa_2stage (character UPLO, integer N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CHESV_AA_2STAGE computes the solution to system of linear equations A * X = B for HE matrices

🎀 Purpose 🎀

okay so CHESV_AA_2STAGE 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.

Aasen's 2-stage algorithm is used to factor A as

   A = U**H * T * U,  if UPLO = 'U', or
   A = L * T * L**H,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is Hermitian and band. The matrix T is then LU-factored with partial pivoting. The factored form of A is then used to solve the system of equations A * X = B.

This is the blocked version of the algorithm, calling Level 3 BLAS.

💅 Parameters 💅

UPLO UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
N N is INTEGER The order of the matrix A. N >= 0.
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 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, L is stored below (or above) the subdiaonal blocks, when UPLO is 'L' (or 'U').

LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
TB TB is COMPLEX array, dimension (LTB) On exit, details of the LU factorization of the band matrix.
LTB LTB is INTEGER The size of the array TB. LTB >= 4*N, internally used to select NB such that LTB >= (3*NB+1)*N.

If LTB = -1, then a workspace query is assumed; the routine only calculates the optimal size of LTB, returns this value as the first entry of TB, and no error message related to LTB is issued by XERBLA.

IPIV IPIV is INTEGER array, dimension (N) On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k).
IPIV2 IPIV2 is INTEGER array, dimension (N) On exit, it contains the details of the interchanges, i.e., the row and column k of T were interchanged with the row and column IPIV(k).
B B is COMPLEX array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
WORK WORK is COMPLEX workspace of size LWORK
LWORK LWORK is INTEGER The size of WORK. LWORK >= N, internally used to select NB such that LWORK >= N*NB.

If LWORK = -1? okay bestie, that’s the secret workspace-query handshake: 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 giving “I’m just measuring, not solving” energy. 💅✨

INFO

INFO is INTEGER

= 0: successful exit. Slay.
< 0: if INFO = -i, the i-th argument had an illegal value. Not iconic.
> 0: if INFO = i, band LU factorization failed on i-th column. Big yikes.

Author

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

👑 `CHETRF_AA_2STAGE` 👑

   subroutine chetrf_aa_2stage (character UPLO, integer N, complex, dimension(
       lda, * ) A, integer LDA, complex, dimension( * ) TB, integer LTB,
       integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex,
       dimension( * ) WORK, integer LWORK, integer INFO)
       CHETRF_AA_2STAGE

Purpose:

CHETRF_AA_2STAGE computes the factorization of a real hermitian matrix A using the Aasen's algorithm. The form of the factorization is

   A = U**T*T*U  or  A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a hermitian band matrix with the bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is LU factorized with partial pivoting).

This is the blocked version of the algorithm, calling Level 3 BLAS. It’s the main character of factorizations, fr fr. 🚀

Parameters

UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX 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, L is stored below (or above) the subdiaonal blocks, when UPLO is 'L' (or 'U'). (Yes, the original says “subdiaonal” — we’re not fixing it, we’re respecting the lore. 💀)
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
TB — TB is COMPLEX array, dimension (LTB). On exit, details of the LU factorization of the band matrix.
LTB — LTB is INTEGER. The size of the array TB. LTB >= 4*N, internally used to select NB such that LTB >= (3*NB+1)*N. If LTB = -1, then a workspace query is assumed; the routine only calculates the optimal size of LTB, returns this value as the first entry of TB, and no error message related to LTB is issued by XERBLA.
IPIV — IPIV is INTEGER array, dimension (N). On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k).
IPIV2 — IPIV2 is INTEGER array, dimension (N). On exit, it contains the details of the interchanges, i.e., the row and column k of T were interchanged with the row and column IPIV(k). (Yes, it says IPIV(k) here, not IPIV2(k) — we keep it exactly as the original, no editing. 🫶)
WORK — WORK is COMPLEX workspace of size LWORK.
LWORK — LWORK is INTEGER. The size of WORK. LWORK >= N, internally used to select NB such that LWORK >= N*NB. 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 = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, band LU factorization failed on i-th column.

Author

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

🌸 `CHETRS_AA_2STAGE` 🌸

   subroutine chetrs_aa_2stage (character UPLO, integer N, integer NRHS,
       complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB,
       integer LTB, integer, dimension( * ) IPIV, integer, dimension( * )
       IPIV2, complex, dimension( ldb, * ) B, integer LDB, integer INFO)
       CHETRS_AA_2STAGE

Purpose:

CHETRS_AA_2STAGE solves a system of linear equations A*X = B with a real hermitian matrix A using the factorization A = U**T*T*U or A = L*T*L**T computed by CHETRF_AA_2STAGE. 🧚

Parameters

UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
- = 'U': Upper triangular, form is A = U**T*T*U;
- = 'L': Lower triangular, form is A = L*T*L**T.
N — N is INTEGER. 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 array, dimension (LDA,N). Details of factors computed by CHETRF_AA_2STAGE.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
TB — TB is COMPLEX array, dimension (LTB). Details of factors computed by CHETRF_AA_2STAGE.
LTB — LTB is INTEGER. The size of the array TB. LTB >= 4*N.
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges as computed by CHETRF_AA_2STAGE.
IPIV2 — IPIV2 is INTEGER array, dimension (N). Details of the interchanges as computed by CHETRF_AA_2STAGE.
B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
INFO — INFO is INTEGER
- = 0: successful exit.
- < 0: if INFO = -i, the i-th argument had an illegal value.

Author

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

🦋 `CLA_SYAMV` — computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. 🦋

   subroutine cla_syamv (integer UPLO, integer N, real ALPHA, complex,
       dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, integer
       INCX, real BETA, real, dimension( * ) Y, integer INCY)
       CLA_SYAMV computes a matrix-vector product using a symmetric indefinite
       matrix to calculate error bounds.

Purpose:

CLA_SYAMV performs the matrix-vector operation

        y := alpha*abs(A)*abs(x) + beta*abs(y),

where alpha and beta are scalars, x and y are vectors and A is an n by n symmetric matrix.

This function is primarily used in calculating error bounds. To protect against underflow during evaluation, components in the resulting vector are perturbed away from zero by (N+1) times the underflow threshold. To prevent unnecessarily large errors for block-structure embedded in general matrices, 'symbolically' zero components are not perturbed. A zero entry is considered 'symbolic' if all multiplications involved in computing that entry have at least one zero multiplicand. 🔥

Parameters

UPLO — UPLO is INTEGER. On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows:
- UPLO = BLAS_UPPER — Only the upper triangular part of A is to be referenced.
- UPLO = BLAS_LOWER — Only the lower triangular part of A is to be referenced. Unchanged on exit.
N — N is INTEGER. On entry, N specifies the number of columns of the matrix A. N must be at least zero. Unchanged on exit.
ALPHA — ALPHA is REAL. On entry, ALPHA specifies the scalar alpha. Unchanged on exit.
A — A is COMPLEX array, dimension ( LDA, n ). Before entry, the leading m by n part of the array A must contain the matrix of coefficients. Unchanged on exit.
LDA — LDA is INTEGER. On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). Unchanged on exit.
X — X is COMPLEX array, dimension ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the vector x. Unchanged on exit.
INCX — INCX is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.
BETA — BETA is REAL. On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit.
Y — Y is REAL array, dimension ( 1 + ( n - 1 )*abs( INCY ) ). Before entry with BETA non-zero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y.
INCY — INCY is INTEGER. On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit.

Author

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

Further Details:

Level 2 Blas routine. ✨

-- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs. -- Modified for the absolute-value product, April 2006 Jason Riedy, UC Berkeley

cla_syrcond_c 💖

real function cla_syrcond_c (character UPLO, integer N, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,
    integer, dimension( * ) IPIV, real, dimension( * ) C, logical CAPPLY,
    integer INFO, complex, dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_SYRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices. bestie it's literally just vibing with the condition number, no drama 💅

✨ Purpose ✨

CLA_SYRCOND_C computes the infinity norm condition number of op(A) * inv(diag(C)) where C is a REAL vector. that's the whole assignment, periodt 🎀

🌸 Parameters 🌸

UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF.
C — C is REAL array, dimension (N). The vector C in the formula op(A) * inv(diag(C)).
CAPPLY — CAPPLY is LOGICAL. If .TRUE. then access the vector C in the formula above.
INFO — INFO is INTEGER
- = 0: Successful exit.
- i > 0: The ith argument is invalid.
WORK — WORK is COMPLEX array, dimension (2*N). Workspace.
RWORK — RWORK is REAL array, dimension (N). Workspace.

💜 Author

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

cla_syrcond_x 🦋

real function cla_syrcond_x (character UPLO, integer N, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF, integer LDAF,
    integer, dimension( * ) IPIV, complex, dimension( * ) X, integer INFO,
    complex, dimension( * ) WORK, real, dimension( * ) RWORK)

CLA_SYRCOND_X computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices. same energy as cla_syrcond_c but the diag friend is a whole COMPLEX vector this time, iconic ✨

✨ Purpose ✨

CLA_SYRCOND_X computes the infinity norm condition number of op(A) * diag(X) where X is a COMPLEX vector. no notes, that's the tea 🍓

🌸 Parameters 🌸

UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF.
X — X is COMPLEX array, dimension (N). The vector X in the formula op(A) * diag(X).
INFO — INFO is INTEGER
- = 0: Successful exit.
- i > 0: The ith argument is invalid.
WORK — WORK is COMPLEX array, dimension (2*N). Workspace.
RWORK — RWORK is REAL array, dimension (N). Workspace.

💜 Author

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

cla_syrfsx_extended 👑

subroutine cla_syrfsx_extended (integer PREC_TYPE, character UPLO, integer
    N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex,
    dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV,
    logical COLEQU, real, dimension( * ) C, complex, dimension( ldb, * ) B,
    integer LDB, complex, dimension( ldy, * ) Y, integer LDY, real,
    dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * )
    ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, complex,
    dimension( * ) RES, real, dimension( * ) AYB, complex, dimension( * )
    DY, complex, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real
    RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)

CLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. it's giving ✨main character energy✨ — this one does the extra work so your answer is actually trustworthy 🫶

✨ Purpose ✨

CLA_SYRCOND_C... okay no wait, it's CLA_SYRFSX_EXTENDED that improves the computed solution to a system of linear equations by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. This subroutine is called by CSYRFSX to perform iterative refinement. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. Note that this subroutine is only responsible for setting the second fields of ERR_BNDS_NORM and ERR_BNDS_COMP.

🌸 Parameters 🌸

PREC_TYPE — PREC_TYPE is INTEGER. Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P
- = 'S': Single
- = 'D': Double
- = 'I': Indigenous
- = 'X' or 'E': Extra
UPLO — UPLO is CHARACTER*1
- = '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.
A — A is COMPLEX array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF.
COLEQU — COLEQU is LOGICAL. If .TRUE. then column equilibration was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.
C — C is REAL array, dimension (N). The column scale factors for A. If COLEQU = .FALSE., C is not accessed. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.

okay bestie listen — that radix-powers thing is not a suggestion, it's a whole thing. if you scale by anything else you might get error estimates that are straight up lying to you 💀

B — B is COMPLEX array, dimension (LDB,NRHS). The right-hand-side matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Y — Y is COMPLEX array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by CSYTRS. On exit, the improved solution matrix Y.
LDY — LDY is INTEGER. The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT — BERR_OUT is REAL array, dimension (NRHS). On exit, BERR_OUT(j) contains the componentwise relative backward error for right-hand-side j from the formula

max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) is the componentwise absolute value of the matrix or vector Z. This is computed by CLA_LIN_BERR.

N_NORMS — N_NORMS is INTEGER. Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP). If N_NORMS >= 1 return normwise error bounds. If N_NORMS >= 2 return componentwise error bounds.
ERR_BNDS_NORM — ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

max_j (abs(XTRUE(j,i) - X(j,i)))
------------------------------
     max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').

err = 2 — 'Guaranteed' error bound: the estimated forward error 💖 almost certainly within a factor of 10 of the true error, so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. Periodt.
err = 3 — Reciprocal condition number: the estimated normwise reciprocal condition number. It gets compared against the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z.

Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1. ✨

This subroutine is only responsible for setting the second field above. That's it, that's the whole job 👑

See Lapack Working Note 165 for further details and extra cautions.

ERR_BNDS_COMP — ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))

The array is indexed by the right-hand side i (on which the componentwise relative error depends), and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 — 'Guaranteed' error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. 👀
err = 3 — Reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z.

Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1. 🦋

This subroutine is only responsible for setting the second field above.

See Lapack Working Note 165 for further details and extra cautions.

RES — RES is COMPLEX array, dimension (N). Workspace to hold the intermediate residual. 🫶
AYB — AYB is REAL array, dimension (N). Workspace. That's all it does, it just holds space, we respect it.
DY — DY is COMPLEX array, dimension (N). Workspace to hold the intermediate solution. 💅
Y_TAIL — Y_TAIL is COMPLEX array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. The little leftover crumbs bestie 🍓
RCOND — RCOND is REAL

Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. 😭

ITHRESH — ITHRESH is INTEGER

The maximum number of residual computations allowed for refinement. The default is 10. For 'aggressive' set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. ⚠️

RTHRESH — RTHRESH is REAL

Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.

DZ_UB — DZ_UB is REAL

Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we define as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details.

IGNORE_CWISE — IGNORE_CWISE is LOGICAL

If .TRUE. then ignore componentwise convergence. Default value is .FALSE..

INFO — INFO is INTEGER

= 0:  Successful exit.
< 0:  if INFO = -i, the ith argument to CLA_SYRFSX_EXTENDED had an illegal
      value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`real function cla_syrpvgrw`

real function cla_syrpvgrw (character*1 UPLO, integer N, integer INFO,
    complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * )
    ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * )
    WORK)

Hold up — that's the signature, and the real one as printed is:

real function cla_syrpvgrw (character*1 UPLO, integer N, integer INFO,
    complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * )
    AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) WORK)

CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix. 👑

Purpose

CLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. 😬

Parameters

UPLO — UPLO is CHARACTER*1

= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

N — N is INTEGER

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

INFO — INFO is INTEGER

The value of INFO returned from CSYTRF, .i.e., the pivot in column INFO is exactly 0.

A — A is COMPLEX array, dimension (LDA,N)

On entry, the N-by-N matrix A.

LDA — LDA is INTEGER

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

AF — AF is COMPLEX array, dimension (LDAF,N)

The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.

LDAF — LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

IPIV — IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D as determined by CSYTRF.

WORK — WORK is REAL array, dimension (2*N)

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`subroutine clahef_aa`

subroutine clahef_aa (character UPLO, integer J1, integer M, integer NB,
    complex, dimension( lda, * ) A, integer LDA, integer, dimension( * )
    IPIV, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( *
    ) WORK)

CLAHEF_AA ✨ (yes the whole subroutine is just its own name up there, iconic behaviour honestly)

Purpose

CLAHEF_AA factorizes a panel of a complex hermitian matrix A using the Aasen's algorithm. The panel consists of a set of NB rows of A when UPLO is U, or a set of NB columns when UPLO is L.

In order to factorize the panel, the Aasen's algorithm requires the last row, or column, of the previous panel. The first row, or column, of A is set to be the first row, or column, of an identity matrix, which is used to factorize the first panel.

The resulting J-th row of U, or J-th column of L, is stored in the (J-1)-th row, or column, of A (without the unit diagonals), while the diagonal and subdiagonal of A are overwritten by those of T. 💅

Parameters

UPLO — UPLO is CHARACTER*1

= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

J1 — J1 is INTEGER

The location of the first row, or column, of the panel within the submatrix of A, passed to this routine, e.g., when called by CHETRF_AA, for the first panel, J1 is 1, while for the remaining panels, J1 is 2.

M — M is INTEGER

The dimension of the submatrix. M >= 0.

NB — NB is INTEGER

The dimension of the panel to be facotorized. (their spelling, not mine, and honestly? we stan a legend with a typo 🎀)

A — A is COMPLEX array, dimension (LDA,M) for the first panel, while dimension (LDA,M+1) for the remaining panels.

On entry, A contains the last row, or column, of the previous panel, and the trailing submatrix of A to be factorized, except for the first panel, only the panel is passed.

On exit, the leading panel is factorized. 🌸

LDA —
LDA — INTEGER, bestie. It's the leading dimension of the array A. Rule of the land: LDA >= max(1,N). No exceptions, she's strict 💅
IPIV — INTEGER array, dimension (N). This is where the ✨drama✨ of the interchanges lives — the deets on the row and column swaps. Basically: row and column k got swapped with row and column IPIV(k). Tea spilled.
H — COMPLEX workspace, dimension (LDH,NB). Just a lil scratch pad, she's doing her job 🫶
LDH — INTEGER, the leading dimension of the workspace H. And you guessed it: LDH >= max(1,M).
WORK — COMPLEX workspace, dimension (M). More scratch space, more vibes.

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💜 subroutine `clasyf` 💜

subroutine clasyf (character UPLO, integer N, integer NB, integer KB,
    complex, dimension( lda, * ) A, integer LDA, integer, dimension( * )
    IPIV, complex, dimension( ldw, * ) W, integer LDW, integer INFO)

CLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method. Okay so that's the whole assignment. Let's get into it 💖

✨ Purpose ✨

CLASYF computes a partial factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) ( D    0  ) ( L11**T L21**T )  if UPLO = 'L'
      ( L21  I ) ( 0   A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. Note that U**T denotes the transpose of U.

CLASYF is an auxiliary routine called by CSYTRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

✨ Parameters ✨

UPLO — CHARACTER*1. Tells us 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.
NB — INTEGER. The max number of columns of the matrix A that should be factored. And listen — NB should be at least 2 to allow for 2-by-2 pivot blocks. Don't skimp on this one.
KB — INTEGER. The number of columns of A that were actually factored (the receipts, if you will). KB is either NB-1 or NB, or N if N <= NB.
A — 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, A contains details of the partial factorization. She transforms, periodt 💅
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D. Okay so this part has layers, so let's break it down:

If UPLO = 'U':

Only the last KB elements of IPIV are set.
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 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':

Only the first KB elements of IPIV are set.
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 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.

W — COMPLEX array, dimension (LDW,NB). Over here being the workspace icon that she is ✨
LDW — INTEGER. The leading dimension of the array W. LDW >= max(1,N).
INFO — INTEGER. The vibe check at the end:
- = 0: successful exit 🎉
- > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular. Not a crash, but definitely not slay.

🌈 Author 🌈

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💖 subroutine `clasyf_aa` 💖

subroutine clasyf_aa (character UPLO, integer J1, integer M, integer NB,
    complex, dimension( lda, * ) A, integer LDA, integer, dimension( * )
    IPIV, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( *
    ) WORK)

CLASYF_AA — she's giving very much "here for the Aasen's algorithm" energy, and that's literally it.

✨ Purpose ✨

DLATRF_AA factorizes a panel of a complex symmetric matrix A using the Aasen's algorithm. The panel consists of a set of NB rows of A when UPLO is U, or a set of NB columns when UPLO is L.

The resulting J-th row of U, or J-th column of L, is stored in the (J-1)-th row, or column, of A (without the unit diagonals), while the diagonal and subdiagonal of A are overwritten by those of T. She's reorganizing, not destroying, iykyk 🦋

✨ Parameters ✨

UPLO — CHARACTER*1. The storage situation:
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
J1 — INTEGER. The location of the first row, or column, of the panel within the submatrix of A, passed to this routine. Like, e.g., when called by CSYTRF_AA, for the first panel, J1 is 1, while for the remaining panels, J1 is 2. Cute lil head start.
M — INTEGER. The dimension of the submatrix. M >= 0.
NB — INTEGER. The dimension of the panel to be facotorized. (Yes, we're keeping the typo exactly as she wrote it, no cap 😭)
A — COMPLEX array, dimension (LDA,M) for the first panel, while dimension (LDA,M+1) for the remaining panels. On entry, A contains the last row, or column, of the previous panel, and the trailing submatrix of A to be factorized, except for the first panel, only the panel is passed. On exit, the leading panel is factorized. She ate and left no crumbs 🔥
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
IPIV — INTEGER array, dimension (M). Details of the row and column interchanges — the row and column k were interchanged with the row and column IPIV(k).
H — COMPLEX workspace, dimension (LDH,NB).
LDH — INTEGER. The leading dimension of the workspace H. LDH >= max(1,M).
WORK — COMPLEX workspace, dimension (M).

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💫 Contributors: 💫

November 2013,  Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

Shoutout to Igor, absolute legend 🫶

👑 subroutine `clasyf_rk` 👑

subroutine clasyf_rk (character UPLO, integer N, integer NB, integer KB,
    complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) E,
    integer, dimension( * ) IPIV, complex, dimension( ldw, * ) W, integer
    LDW, integer INFO)

CLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method. The rook method, bestie — she's the careful one 👑

✨ Purpose ✨

CLASYF_RK computes a partial factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method. The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
      ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L',
      ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.

CLASYF_RK is an auxiliary routine called by CSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

✨ Parameters ✨

UPLO — CHARACTER*1. 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.
NB — INTEGER. The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks. Same rule as always, don't fight it.
KB — INTEGER. The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.
A — 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, it's holding:

a) ONLY the diagonal elements of the symmetric block diagonal matrix D sit right on the diagonal of A, i.e. D(k,k) = A(k,k); (the 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). No cap, just give A the room it deserves. 💅

`E`

E is COMPLEX array, dimension (N). On exit, it 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. Both cases, bestie. It's giving equality. ✨

`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.

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 submatrix A(1:N,N-KB+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. ONLY. In. Pairs. 🎀

If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,N-KB+1:N). If -IPIV(k) = k, no interchange occurred.
If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,N-KB+1:N). If -IPIV(k-1) = k-1, no interchange occurred.

c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output. No zeros allowed, periodt. 💖

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 submatrix A(1:N,1:KB).
- 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.

If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k) = k, no interchange occurred.
If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the submatrix A(1:N,1:KB). If -IPIV(k+1) = k+1, no interchange occurred.

c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
d) NOTE: Any entry IPIV(k) is always NONZERO on output. Same energy, flipped inequality. 🦋

`W`

W is COMPLEX array, dimension (LDW,NB). That's the whole vibe.

`LDW`

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

`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. First one only, bestie. The rest are lost to the void. 💀

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

✨ `subroutine clasyf_rook` ✨

   subroutine clasyf_rook (character UPLO, integer N, integer NB, integer KB,
       complex, dimension( lda, * ) A, integer LDA, integer, dimension( * )
       IPIV, complex, dimension( ldw, * ) W, integer LDW, integer INFO)

CLASYF_ROOK computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. Rook energy only. ♟️

Purpose

CLASYF_ROOK computes a partial factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. The partial factorization has the form:

            A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
                  ( 0  U22 ) (  0   D  ) ( U12**T U22**T )

            A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  if UPLO = 'L'
                  ( L21  I ) (  0  A22 ) (  0       I    )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB.

CLASYF_ROOK is an auxiliary routine called by CSYTRF_ROOK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). Blocked code, Level 3 BLAS, main character energy. 🔥

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.

`NB`

NB is INTEGER. The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks. At least 2, or the 2-by-2 blocks simply cannot slay.

`KB`

KB is INTEGER. The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.

`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, A contains details of the partial factorization.

`LDA`

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

`IPIV`

IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D.

If UPLO = 'U':

Only the last KB elements of IPIV are set.
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 IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Yes, inerchaged. We keep the typo. It's iconic. 🙈)

If UPLO = 'L':

Only the first KB elements of IPIV are set.
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 IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block. Same typo, same legend. 💅

`W`

W is COMPLEX array, dimension (LDW,NB).

`LDW`

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

`INFO`

INFO is INTEGER

= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors

November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley

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

✨ `csycon` ✨

   subroutine csycon (character UPLO, integer N, complex, dimension( lda, * )
       A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND,
       complex, dimension( * ) WORK, integer INFO)
       CSYCON

Okay so CSYCON is here to estimate the reciprocal of the condition number (in the 1-norm, no cap) of a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T that CSYTRF already did. It gets an estimate for norm(inv(A)), and then the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). Math but make it fashion. 💅

Parameters

UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF.
ANORM — ANORM is REAL. The 1-norm of the original matrix A.
RCOND — RCOND is REAL. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.
WORK — WORK is COMPLEX array, dimension (2*N).
INFO — INFO is INTEGER. = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

Author

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

✨ `csycon_3` ✨

   subroutine csycon_3 (character UPLO, integer N, complex, dimension( lda, * )
       A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,
       real ANORM, real RCOND, complex, dimension( * ) WORK, integer INFO)
       CSYCON_3

Bestie, CSYCON_3 also estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric matrix A, but it uses the factorization computed by CSYTRF_RK or CSYTRF_BK:

   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.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). This routine uses BLAS3 solver CSYTRS_3. Fancy. 🎀

Parameters

UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix: = 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). Diagonal of the block diagonal matrix D and factors U or L as computed by CSYTRF_RK and CSYTRF_BK:
- 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 should be provided on entry 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 entry, 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) not referenced; If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.

IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF_RK or CSYTRF_BK.
ANORM — ANORM is REAL. The 1-norm of the original matrix A.
RCOND — RCOND is REAL. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.
WORK — WORK is COMPLEX array, dimension (2*N).
INFO — INFO is INTEGER. = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

Author

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

Contributors:

June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

✨ `csycon_rook` ✨

   subroutine csycon_rook (character UPLO, integer N, complex, dimension( lda,
       * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real
       RCOND, complex, dimension( * ) WORK, integer INFO)
        CSYCON_ROOK

CSYCON_ROOK estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF_ROOK.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). Rook energy. ♟️

Parameters

UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF_ROOK.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF_ROOK.
ANORM — ANORM is REAL. The 1-norm of the original matrix A.
RCOND — RCOND is REAL. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.
WORK — WORK is COMPLEX array, dimension (2*N).
INFO — INFO is INTEGER. = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

Author

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

Contributors:

April 2012, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

✨ `csyconv` ✨

   subroutine csyconv (character UPLO, character WAY, integer N, complex,
       dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
       complex, dimension( * ) E, integer INFO)
       CSYCONV

CSYCONV converts A given by TRF into L and D and vice-versa. It gets Non-diag elements of D (returned in workspace) and applies or reverses permutation done in TRF. It’s giving transformation. 🦋

Parameters

UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix. = 'U': Upper triangular, form is A = U*D*U**T; = 'L': Lower triangular, form is A = L*D*L**T.
WAY — WAY is CHARACTER*1. = 'C': Convert; = 'R': Revert.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF.
E — E is COMPLEX array, dimension (N). E stores the supdiagonal/subdiagonal of the symmetric 1-by-1 or 2-by-2 block diagonal matrix D in LDLT.
INFO — INFO is INTEGER. = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

Author

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

🦋 `csyconvf` — the format-swapper that keeps the receipts

   subroutine csyconvf (character UPLO, character WAY, integer N, complex,
       dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,
       dimension( * ) IPIV, integer INFO)
       CSYCONVF

Purpose 🎯

Okay bestie, listen. This one is a translator. If parameter WAY = 'C':

CSYCONVF converts the factorization output format used in CSYTRF provided on entry in parameter A into the factorization output format used in CSYTRF_RK (or CSYTRF_BK) that is stored on exit in parameters A and E. It also converts in place details of the interchanges stored in IPIV from the format used in CSYTRF into the format used in CSYTRF_RK (or CSYTRF_BK). 💅

If parameter WAY = 'R':

CSYCONVF performs the conversion in reverse direction, i.e. converts the factorization output format used in CSYTRF_RK (or CSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in CSYTRF that is stored on exit in parameter A. It also converts in place details of the interchanges stored in IPIV from the format used in CSYTRF_RK (or CSYTRF_BK) into the format used in CSYTRF.

And no cap, CSYCONVF can also convert in Hermitian matrix case, i.e. between formats used in CHETRF and CHETRF_RK (or CHETRF_BK). Versatile queen behavior. ✨

Parameters 📋

UPLO — UPLO is CHARACTER*1. Tells us whether the details of the factorization are stored as an upper or lower triangular matrix A. 💅
- = 'U': Upper triangular
- = 'L': Lower triangular
WAY — WAY is CHARACTER*1.
- = 'C': Convert
- = 'R': Revert
N — N is INTEGER. The order of the matrix A. N >= 0. (No negativity allowed, she's not about that life.)
A — A is COMPLEX array, dimension (LDA,N). Two whole moods depending on WAY:
- 1) If WAY ='C':
  - On entry, contains factorization details in format used in CSYTRF:
    - a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and
    - b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A.
  - On exit, contains factorization details in format used in CSYTRF_RK or CSYTRF_BK:
    - 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.
- 2) If WAY = 'R':
  - On entry, contains factorization details in format used in CSYTRF_RK or CSYTRF_BK:
    - 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.
  - On exit, contains factorization details in format used in CSYTRF:
    - a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and
    - b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal 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).
- 1) If WAY ='C':
  - On entry, just a workspace. (She's showing up with nothing and leaving with everything. Growth. 🌱)
  - 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.
- 2) If WAY = 'R':
  - On entry, 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) not referenced;
    - If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
  - On exit, is not changed. (Iconic: untouched, unbothered, thriving. 💅)
IPIV — IPIV is INTEGER array, dimension (N).
- 1) If WAY ='C': On entry, details of the interchanges and the block structure of D in the format used in CSYTRF. On exit, details of the interchanges and the block structure of D in the format used in CSYTRF_RK ( or CSYTRF_BK).
- 1) If WAY ='R': On entry, details of the interchanges and the block structure of D in the format used in CSYTRF_RK ( or CSYTRF_BK). On exit, details of the interchanges and the block structure of D in the format used in CSYTRF.
INFO — INFO is INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

Author ✍️

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

Contributors: 👑

November 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley

🌸 `csyconvf_rook` — the other format-swapper, ROOK edition

   subroutine csyconvf_rook (character UPLO, character WAY, integer N, complex,
       dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,
       dimension( * ) IPIV, integer INFO)
       CSYCONVF_ROOK

Purpose 🎯

If parameter WAY = 'C':

CSYCONVF_ROOK converts the factorization output format used in CSYTRF_ROOK provided on entry in parameter A into the factorization output format used in CSYTRF_RK (or CSYTRF_BK) that is stored on exit in parameters A and E. IPIV format for CSYTRF_ROOK and CSYTRF_RK (or CSYTRF_BK) is the same and is not converted. (Not her job, not her problem. 😌)

If parameter WAY = 'R':

CSYCONVF_ROOK performs the conversion in reverse direction, i.e. converts the factorization output format used in CSYTRF_RK (or CSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in CSYTRF_ROOK that is stored on exit in parameter A. IPIV format for CSYTRF_ROOK and CSYTRF_RK (or CSYTRF_BK) is the same and is not converted.

And yeah, CSYCONVF_ROOK can also convert in Hermitian matrix case, i.e. between formats used in CHETRF_ROOK and CHETRF_RK (or CHETRF_BK). She contains multitudes. 🦋

Parameters 📋

UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix A.
- = 'U': Upper triangular
- = 'L': Lower triangular
WAY — WAY is CHARACTER*1.
- = 'C': Convert
- = 'R': Revert
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N).
- 1) If WAY ='C':
  - On entry, contains factorization details in format used in CSYTRF_ROOK:
    - a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and
    - b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal part of A.
  - On exit, contains factorization details in format used in CSYTRF_RK or CSYTRF_BK:
    - 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.
- 2) If WAY = 'R':
  - On entry, contains factorization details in format used in CSYTRF_RK or CSYTRF_BK:
    - 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.
  - On exit, contains factorization details in format used in CSYTRF_ROOK:
    - a) all elements of the symmetric block diagonal matrix D on the diagonal of A and on superdiagonal (or subdiagonal) of A, and
    - b) If UPLO = 'U': multipliers used to obtain factor U in the superdiagonal part of A. If UPLO = 'L': multipliers used to obtain factor L in the superdiagonal 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).
- 1) If WAY ='C':
  - On entry, just a workspace.
  - 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.
- 2) If WAY = 'R':
E — bestie, this is an array holding the superdiagonal (or subdiagonal, depending on the vibe) elements of the symmetric block diagonal matrix D, which is built out of 1-by-1 or 2-by-2 diagonal blocks. Here's the tea:
- If UPLO = 'U': E(i) = D(i-1,i), i=2:N, and E(1) is not referenced.
- If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, and E(N) is not referenced.

On exit, is not changed. 💅 (she's just here to look pretty, she doesn't get touched)

IPIV — IPIV is INTEGER array, dimension (N).

On entry, it's got all the details of the interchanges and the block structure of D, as determined by:

CSYTRF_ROOK, if WAY ='C';
CSYTRF_RK (or CSYTRF_BK), if WAY ='R'.

The IPIV format is the same for all these routines. Fr fr, all of them. ✨

On exit, is not changed.

INFO — INFO is INTEGER
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 😭

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🫶 Contributors:

November 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley

🦋 CSYEQUB

subroutine csyequb (character UPLO, integer N, complex, dimension( lda, * )
    A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, complex,
    dimension( * ) WORK, integer INFO)

✨ Purpose:

CSYEQUB computes row and column scalings that are meant to equilibrate a symmetric matrix A (with respect to the Euclidean norm, no cap) and reduce its condition number. The scale factors S are computed by the BIN algorithm (see references, bestie) so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of the smallest possible condition number over all possible diagonal scalings. Like, she's that good. 💅

🌸 Parameters

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

The order of the matrix A. N >= 0.

A — A is COMPLEX array, dimension (LDA,N)

The N-by-N symmetric matrix whose scaling factors are to be computed.

LDA — LDA is INTEGER

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

S — S is REAL array, dimension (N)

If INFO = 0, S contains the scale factors for A.

SCOND — SCOND is REAL

If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S. (like, don't bother, she's fine 🙈)

AMAX — AMAX is REAL

Largest absolute value of any matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.

WORK — WORK is COMPLEX array, dimension (2*N)
INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, the i-th diagonal element is nonpositive. 💀

💜 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎀 References:

Livne, O.E. and Golub, G.H., 'Scaling by Binormalization', Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. DOI 10.1023/B:NUMA.0000016606.32820.69 Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

🌈 CSYRFS

subroutine csyrfs (character UPLO, integer N, integer NRHS, complex,
    dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, * ) AF,
    integer LDAF, integer, dimension( * ) IPIV, complex, dimension( ldb, * )
    B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real,
    dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * )
    WORK, real, dimension( * ) RWORK, integer INFO)

✨ Purpose:

CSYRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, and provides error bounds and backward error estimates for the solution. She's the glow-up your solution needed. 🔥

🌸 Parameters

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

The order of the matrix A. N >= 0.

NRHS — NRHS is INTEGER

The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

A — A is COMPLEX array, dimension (LDA,N)

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. (the other half is literally not invited 🫥)

LDA — LDA is INTEGER

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

AF — AF is COMPLEX array, dimension (LDAF,N)

The factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSYTRF.

LDAF — LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

IPIV — IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D as determined by CSYTRF.

B — B is COMPLEX array, dimension (LDB,NRHS)

The right hand side matrix B.

LDB — LDB is INTEGER

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

X — X is COMPLEX array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by CSYTRS. On exit, the improved solution matrix X. ⭐ (she got a makeover)

LDX — LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

FERR — FERR is REAL array, dimension (NRHS)

The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. (she's a lil dramatic, but in a cute way 💅)

BERR — BERR is REAL array, dimension (NRHS)

The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).

WORK — WORK is COMPLEX array, dimension (2*N)
RWORK — RWORK is REAL array, dimension (N)
INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

🧚 Internal Parameters:

ITMAX is the maximum number of steps of iterative refinement.

💜 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌟 CSYRFSX

subroutine csyrfsx (character UPLO, character EQUED, integer N, integer
    NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension(
    ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real,
    dimension( * ) S, complex, dimension( ldb, * ) B, integer LDB, complex,
    dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * )
    BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM,
    real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real,
    dimension( * ) PARAMS, complex, dimension( * ) WORK, real, dimension( *
    ) RWORK, integer INFO)

✨ Purpose:

CSYRFSX improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite, and provides error bounds and backward error estimates for the solution. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. See comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds. 👑

The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED and S below. In this case, the solution and error bounds returned are for the original unequilibrated system.

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument. (the "just trust me bro" option, but like, officially sanctioned ✨)

🌸 Parameters

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

Specifies the form of equilibration that was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.

= 'N': No equilibration
= 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). The right hand side B has been changed accordingly.

N — N is INTEGER

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 matrices B and X. NRHS >= 0.

A — okay bestie, hold on, her description is literally arriving in the next part, don't scroll away 💅

`A`

A is a COMPLEX array, dimension (LDA,N) — bestie, this is literally the symmetric matrix itself. And the way it's stored depends on UPLO, so listen up:

If UPLO = 'U': the leading N-by-N upper triangular part of A holds the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced (so like, don't bother filling it in, it's giving background extra 💅).
If UPLO = 'L': the leading N-by-N lower triangular part of A holds the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

`LDA`

LDA is INTEGER — the leading dimension of the array A, no cap. It has to satisfy LDA >= max(1,N).

`AF`

AF is a COMPLEX array, dimension (LDAF,N) — this is the factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSYTRF. Iconic, honestly.

`LDAF`

LDAF is INTEGER — the leading dimension of the array AF. Constraint: LDAF >= max(1,N). ✨

`IPIV`

IPIV is an INTEGER array, dimension (N) — it holds the deets on the interchanges and the block structure of D, as determined by CSYTRF. 🫶

`S`

S is a REAL array, dimension (N) — the scale factors for A.

Okay so here's the situation:

If EQUED = 'Y', A is multiplied on the left and right by diag(S).
S is an input argument if FACT = 'F'; otherwise, S is an output argument.
If FACT = 'F' and EQUED = 'Y', each element of S must be positive.
If S is output, each element of S is a power of the radix.
If S is input, each element of S should be a power of the radix to ensure a reliable solution and error estimates.

Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. 😭 So like, don't play with it.

`B`

B is a COMPLEX array, dimension (LDB,NRHS) — the right hand side matrix B. That's the whole job.

`LDB`

LDB is INTEGER — the leading dimension of the array B. You need LDB >= max(1,N).

`X`

X is a COMPLEX array, dimension (LDX,NRHS):

On entry: the solution matrix X, as computed by CGETRS.
On exit: the improved solution matrix X. Glow up, fr fr.

`LDX`

LDX is INTEGER — the leading dimension of the array X, and it needs LDX >= max(1,N).

`RCOND`

RCOND is REAL — the reciprocal scaled condition number. It's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if that happened).

If this is less than the machine precision (in particular, if it's zero), the matrix is singular to working precision. 💀

Note though: the error may still be small even if this number is very small and the matrix appears ill-conditioned. So like, don't panic-spiral immediately, bestie.

`BERR`

BERR is a REAL array, dimension (NRHS) — the componentwise relative backward error. Precisely: this is the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). She's precise. 💅

`N_ERR_BNDS`

N_ERR_BNDS is INTEGER — the number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below. 👇

`ERR_BNDS_NORM`

ERR_BNDS_NORM is a REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 — 'Guaranteed' error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 — Reciprocal condition number: estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*A, where S scales each row by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions. 📚

`ERR_BNDS_COMP`

ERR_BNDS_COMP is a REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))

If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

err = 1 — 'Trust/don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
err = 2 — 'Guaranteed' error bound: the estimated forward error, almost certainly within a factor of 10 of the true error so long as the next entry is greater than the threshold sqrt(n) * slamch('Epsilon'). This error bound should only be trusted if the previous boolean is true.
err = 3 — Reciprocal condition number: estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is 'guaranteed'. These reciprocal condition numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some appropriately scaled matrix Z. Let Z = S*(A*diag(x)), where x is the solution for the current right-hand side and S scales each row of A*diag(x) by a power of the radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions. 📚

`NPARAMS`

NPARAMS is INTEGER — it specifies the number of parameters set in PARAMS. If it's <= 0, the PARAMS array is never referenced and default values are used. Chill vibes only.

`PARAMS`

PARAMS is a REAL array, dimension NPARAMS — it specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.

PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative refinement or not.
- Default: 1.0
  - = 0.0: No refinement is performed, and no error bounds are computed.
  - = 1.0: Use the double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
  - (other values are reserved for future use)
PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual computations allowed for refinement.
- Default: 10
- Aggressive: Set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy. 😬
PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
- Default: 1.0 (attempt componentwise convergence)

`WORK`

WORK is a COMPLEX array, dimension (2*N). Just vibing in the background, doing the work. 💪

`RWORK`

RWORK is a REAL array, dimension (2*N). Same energy, real numbers this time.

`INFO`

okay bestie the tea on INFO is literally spilling over into the next part — hold that thought 🫖✨

= 0: Successful exit. The solution to every right-hand side is guaranteed. No notes, immaculate.
< 0: If INFO = -i, the i-th argument had an illegal value. bestie that argument is a crime scene.
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. singular era, honestly.
= N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP. tl;dr it only snitches on the first sus one, you gotta go look at the full arrays yourself.

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 subroutine `csysv_aa_2stage` 🦋

subroutine csysv_aa_2stage (character UPLO, integer N, integer NRHS,
    complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB,
    integer LTB, integer, dimension( * ) IPIV, integer, dimension( * )
    IPIV2, complex, dimension( ldb, * ) B, integer LDB, complex, dimension(
    * ) WORK, integer LWORK, integer INFO)

CSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices periodt 💅

✨ Purpose:

CSYSV_AA_2STAGE 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.

Aasen's 2-stage algorithm is used to factor A as

A = U**T * T * U,  if UPLO = 'U', or
A = L * T * L**T,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is symmetric and band. The matrix T is then LU-factored with partial pivoting. The factored form of A is then used to solve the system of equations A * X = B.

This is the blocked version of the algorithm, calling Level 3 BLAS. big girl algorithm energy, she does it in chunks.

✨ Parameters

UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER. The order of the matrix A. N >= 0.
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 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, L is stored below (or above) the subdiaonal blocks, when UPLO is 'L' (or 'U'). yes it said subdiaonal, we're keeping it, don't @ me.

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
TB — TB is COMPLEX array, dimension (LTB). On exit, details of the LU factorization of the band matrix. receipts, saved for later.
LTB — LTB is INTEGER. The size of the array TB. LTB >= 4*N, internally used to select NB such that LTB >= (3*NB+1)*N.

If LTB = -1, then a workspace query is assumed; the routine only calculates the optimal size of LTB, returns this value as the first entry of TB, and no error message related to LTB is issued by XERBLA.

IPIV — IPIV is INTEGER array, dimension (N). On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k).
IPIV2 — IPIV2 is INTEGER array, dimension (N). On exit, it contains the details of the interchanges, i.e., the row and column k of T were interchanged with the row and column IPIV(k).
B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. glow up, it becomes the answer 💖
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
WORK — WORK is COMPLEX workspace of size LWORK.
LWORK — LWORK is INTEGER. The size of WORK. LWORK >= N, internally used to select NB such that LWORK >= N*NB.

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 = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, band LU factorization failed on i-th column oof, column i said no.

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🎀 subroutine `csytf2` 🎀

subroutine csytf2 (character UPLO, integer N, complex, dimension( lda, * )
    A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

CSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). note: the blurb says real, the Purpose says complex — we report, we don't decide.

✨ Purpose:

CSYTF2 computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method:

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, U**T is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS. the chill, one-at-a-time sibling. no blocking, just vibes.

✨ 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 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). Details of the interchanges and the block structure of D.

If UPLO = 'U':

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 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':

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 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.

INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -k, the k-th argument had an illegal value
- > 0: if INFO = k, D(k,k) 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. 🚨 do NOT use it to solve anything, that's a division by zero, that's a whole disaster.

🌸 Author 🌸

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ Further Details:

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, 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). she's packing everything right back into A, efficient queen 💅

So when UPLO = 'L', bestie, we get A = L*D*L**T, where

   L = P(1)*L(1)* ... *P(k)*L(k)* ...,

like, L is a product of terms P(k)*L(k), and k walks from 1 to n in steps of 1 or 2, and D is that 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, then 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). Slay.

Contributors 💖

09-29-06 - patch from Bobby Cheng, MathWorks

Replace l.209 and l.377

IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN

IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN

1-96 - Based on modifications by J. Lewis, Boeing Computer Services Company

subroutine csytf2_rk 🦋

subroutine csytf2_rk (character UPLO, integer N, complex, dimension( lda, *
    ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * )
    IPIV, integer INFO)

CSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). no cap, that's the whole vibe of this one.

Purpose 🌸

CSYTF2_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 unblocked version of the algorithm, calling Level 2 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 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 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.

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 further details 🙈

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
01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

subroutine csytf2_rook ⭐

subroutine csytf2_rook (character UPLO, integer N, complex, dimension( lda,
    * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

CSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). 🎀

Purpose:

CSYTF2_ROOK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman ('rook') diagonal pivoting method:

   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, U**T is the transpose of U, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS. ✨

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 — a COMPLEX array, dimension (LDA,N). On entry, bestie, this is the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A holds the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced — like, it's giving background extra, don't even look at it. If UPLO = 'L', the leading n-by-n lower triangular part of A holds 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 — the receipts are all down there 💅).

LDA — LDA is INTEGER. It's the leading dimension of the array A. Constraint era: LDA >= max(1,N). ✨
IPIV — IPIV is an INTEGER array, dimension (N). It's got all the details of the interchanges and the block structure of D. 🎀

If UPLO = 'U':

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 IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were interchanged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = 'L':

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 IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were interchanged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

INFO — INFO is INTEGER, and it's the vibe check at the end 👑
- = 0: successful exit. slay.
- < 0: if INFO = -k, the k-th argument had an illegal value. rude.
- > 0: if INFO = k, D(k,k) 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. bestie. be so careful. 😭

🦋 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ Further Details:

If UPLO = 'U', then A = U*D*U**T, where

   U = P(n)*U(n)* ... *P(k)U(k)* ...,

           (   I    v    0   )   k-s
   U(k) =  (   0    I    0   )   s
           (   0    0    I   )   n-k
              k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**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, 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).

🌸 Contributors:

November 2013, Igor Kozachenko, Computer Science Division, University of California, Berkeley

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

01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd, USA

💖 subroutine `csytrf`

subroutine csytrf (character UPLO, integer N, complex, dimension( lda, * )
    A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * )
    WORK, integer LWORK, integer INFO)
    CSYTRF

🎀 Purpose:

CSYTRF 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*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.

This is the blocked version of the algorithm, calling Level 3 BLAS. it's giving optimized, no cap. 🔥

🌈 Parameters

UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER. The order of the matrix A. Constraint: N >= 0.
A — A is a 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, 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. Constraint: LDA >= max(1,N).
IPIV — IPIV is an INTEGER array, dimension (N). Details of the interchanges and the block structure of D. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
WORK — WORK is a 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. Constraint: LWORK >= 1. For best performance, LWORK >= N*NB, where NB is the block size returned by ILAENV.

INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💜 Further Details:

If UPLO = 'U', then A = U*D*U**T, where

   U = P(n)*U(n)* ... *P(k)U(k)* ...,

           (   I    v    0   )   k-s
   U(k) =  (   0    I    0   )   s
           (   0    0    I   )   n-k
              k-s   s   n-k

If UPLO = 'L', then A = L*D*L**T, where

   L = P(1)*L(1)* ... *P(k)*L(k)* ...,

           (   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

🦋 subroutine `csytrf_aa`

subroutine csytrf_aa (character UPLO, integer N, complex, dimension( lda, *
) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * )
WORK, integer LWORK, integer INFO)
CSYTRF_AA

✨ Purpose:

CSYTRF_AA computes the factorization of a complex symmetric matrix A using the Aasen's algorithm. The form of the factorization is

   A = U**T*T*U  or  A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a complex symmetric tridiagonal matrix.

This is the blocked version of the algorithm, calling Level 3 BLAS. 🔥

🍓 Parameters

UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER. The order of the matrix A. Constraint: N >= 0.
A

`A`

Okay so</p> A is a COMPLEX array, dimension (LDA,N).

On entry, this is the symmetric matrix A ✨. If UPLO = 'U', the leading N-by-N upper triangular part of A holds the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced (like, we literally do not look at it, bestie). If UPLO = 'L', the leading N-by-N lower triangular part of A holds the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, the tridiagonal matrix is stored in the diagonals and the subdiagonals of A just below (or above) the diagonals, and L is stored below (or above) the subdiagonals, when UPLO is 'L' (or 'U'). 💅

`LDA`

LDA is INTEGER.

The leading dimension of the array A. So like, LDA >= max(1,N). Respect the math, queen.

`IPIV`

IPIV is INTEGER array, dimension (N).

On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k). 🔄

`WORK`

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

On exit, if INFO = 0, WORK(1) returns the optimal LWORK. It's giving "here's how much room I actually wanted" energy.

`LWORK`

LWORK is INTEGER.

The length of WORK. LWORK >= MAX(1,2*N). For optimum performance LWORK >= N*(1+NB), where NB is the optimal blocksize.

`INFO`

INFO is INTEGER.

= 0: successful exit ✅
< 0: if INFO = -i, the i-th argument had an illegal value. 🙈

Author

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

CSYTRF_AA_2STAGE 👑

subroutine csytrf_aa_2stage (character UPLO, integer N, complex, dimension(
    lda, * ) A, integer LDA, complex, dimension( * ) TB, integer LTB,
    integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex,
    dimension( * ) WORK, integer LWORK, integer INFO)

Purpose

Not me about to explain the Aasen's algorithm to you, bestie, because it's lowkey iconic. CSYTRF_AA_2STAGE computes the factorization of a complex symmetric matrix A using the Aasen's algorithm. The form of the factorization is

   A = U**T*T*U  or  A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is a complex symmetric band matrix with the bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is LU factorized with partial pivoting). ✨

This is the blocked version of the algorithm, calling Level 3 BLAS. Level 3 BLAS?? We love a fast bestie. 🔥

Parameters

`UPLO`

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

`N`

N is INTEGER

The order of the matrix A. N >= 0.

`A`

A is COMPLEX 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, L is stored below (or above) the subdiagonal blocks, when UPLO is 'L' (or 'U'). 💅

`LDA`

LDA is INTEGER

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

`TB`

TB is COMPLEX array, dimension (LTB)

On exit, details of the LU factorization of the band matrix. 🎀

`LTB`

LTB is INTEGER

The size of the array TB. LTB >= 4*N, internally used to select NB such that LTB >= (3*NB+1)*N.

`IPIV`

IPIV is INTEGER array, dimension (N)

On exit, it contains the details of the interchanges, i.e., the row and column k of A were interchanged with the row and column IPIV(k). 🔄

`IPIV2`

IPIV2 is INTEGER array, dimension (N)

On exit, it contains the details of the interchanges, i.e., the row and column k of T were interchanged with the row and column IPIV(k).

`WORK`

WORK is COMPLEX workspace of size LWORK

`LWORK`

LWORK is INTEGER

The size of WORK. LWORK >= N, internally used to select NB such that LWORK >= N*NB.

`INFO`

INFO is INTEGER

= 0: successful exit ✅
< 0: if INFO = -i, the i-th argument had an illegal value. 🙈
> 0: if INFO = i, band LU factorization failed on i-th column 💀

Author

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

CSYTRF_RK 🦋

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 (BLAS3 blocked algorithm). 👑

Purpose

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),

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 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:

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.

`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 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 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, like IPIV(k) < 0 and IPIV(k-1) < 0, means one thing and one thing only bestie: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs, no solo negativity here 💅)
1. If -IPIV(k) != k, then rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred, she stayed put 🫶
2. If -IPIV(k-1) != k-1, then 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), it's always ABS( IPIV(k) ) <= k. periodt.
d) NOTE: any entry IPIV(k) is always NONZERO on output. no zeros allowed, zero is not invited to this party 🚫

If UPLO = 'L', (and in factorization order, k increases from 1 to N, so she's climbing 📈):

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, we've been over this 💀)
1. If -IPIV(k) != k, then 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, then 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), it's always ABS( IPIV(k) ) >= k.
d) NOTE: any entry IPIV(k) is always NONZERO on output. yes we're saying it again, it's that important 💖

🧺 WORK

WORK is COMPLEX array, dimension ( MAX(1,LWORK) ). On exit, if INFO = 0, then WORK(1) returns the optimal LWORK. she's self-reporting her ideal size, iconic behavior 👑

📏 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. (give her room to breathe bestie, she performs better 💅)

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 don't worry, nobody's yelling at you here 🫶

🚦 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.

bestie listen to me. INFO > 0 means the matrix is singular and dividing by zero WILL happen if you try to solve a system with it. that's not a vibe, that's a crash. respect the INFO 💀

👑 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

🌈 subroutine csytrf_rook

subroutine csytrf_rook (character UPLO, integer N, complex, dimension( lda,
    * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( *
    ) WORK, integer LWORK, integer INFO)

✨ CSYTRF_ROOK

🎯 Purpose:

CSYTRF_ROOK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. The form of the factorization is:

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.

This is the blocked version of the algorithm, calling Level 3 BLAS. (level 3? she's highkey eating 🔥)

🧩 Parameters

UPLO — UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX 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). Details of the interchanges and the block structure of D.

If UPLO = 'U':

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 IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = 'L':

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 IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

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.

INFO — INFO is INTEGER:

= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
      has been completed, but the block diagonal matrix D is
      exactly singular, and division by zero will occur if it
      is used to solve a system of equations.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔍 Further Details:

If UPLO = 'U', then A = U*D*U**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, 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 UPLO = 'L', 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, 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

🫶 Contributors:

 June 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

🦋 subroutine csytri

subroutine csytri (character UPLO, integer N, complex, dimension( lda, * )
    A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * )
    WORK, integer INFO)

✨ CSYTRI

🎯 Purpose:

okay so CSYTRI is literally the inverse-machine 💫 it computes the inverse of a complex symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF. bestie, that's the whole plot: you hand it the factorization, it hands you back the inverse. periodt.

UPLO — CHARACTER*1. tells you whether the details of the factorization are stored as an upper or lower triangular matrix. no third option, we're not doing that today:
- = 'U': upper triangular, form is A = U*D*U**T;
- = 'L': lower triangular, form is A = L*D*L**T.
N — INTEGER. the order of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N).
- on entry: the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF. 💾
- on exit, if INFO = 0: the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced. so mind which half you're reading, ok? 🙈
LDA — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). the details of the interchanges and the block structure of D as determined by CSYTRF.
WORK — COMPLEX array, dimension (2*N).
INFO — INTEGER. the exit-status tea ☕:
- = 0: successful exit. slay.
- < 0: if INFO = -i, the i-th argument had an illegal value. rude.
- > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed. 💀

Author — Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`CSYTRI2` 🎀

subroutine csytri2 (character UPLO, integer N, complex, dimension( lda, * )
A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( * )
WORK, integer LWORK, integer INFO)

CSYTRI2 ✨

Purpose:

CSYTRI2 computes the inverse of a complex symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF. and then — the lil glow-up — CSYTRI2 sets the LEADING DIMENSION of the workspace before calling CSYTRI2X that actually computes the inverse. so it's giving backstage-manager energy, setting things up for the main character. 👑

UPLO — CHARACTER*1. specifies whether the details of the factorization are stored as an upper or lower triangular matrix:
- = 'U': upper triangular, form is A = U*D*U**T;
- = 'L': lower triangular, form is A = L*D*L**T.
N — INTEGER. the order of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N).
- on entry: the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
- on exit, if INFO = 0: the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). the details of the interchanges and the block structure of D as determined by CSYTRF.
WORK — COMPLEX array, dimension (N+NB+1)*(NB+3).
LWORK — INTEGER. the dimension of the array WORK.
- WORK is size >= (N+NB+1)*(NB+3)
- if LWORK = -1, then a workspace query is assumed; the routine:
  - 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 — INTEGER.
- = 0: successful exit. 🫶
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.

Author — Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`CSYTRI2X` 🦋

subroutine csytri2x (character UPLO, integer N, complex, dimension( lda, * )
A, integer LDA, integer, dimension( * ) IPIV, complex, dimension(
n+nb+1,* ) WORK, integer NB, integer INFO)

CSYTRI2X ✨

Purpose:

CSYTRI2X computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF.

UPLO — CHARACTER*1. specifies whether the details of the factorization are stored as an upper or lower triangular matrix:
- = 'U': upper triangular, form is A = U*D*U**T;
- = 'L': lower triangular, form is A = L*D*L**T.
N — INTEGER. the order of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N).
- on entry: the NNB diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
- on exit, if INFO = 0: the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension (N). the details of the interchanges and the NNB structure of D as determined by CSYTRF.
WORK — COMPLEX array, dimension (N+NB+1,NB+3).
NB — INTEGER. block size 📦
INFO — INTEGER.
- = 0: successful exit. ✨
- < 0: if INFO = -i, the i-th argument had an illegal value.
- > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.

Author — Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`CSYTRI_3` 💜

subroutine csytri_3 (character UPLO, integer N, complex, dimension( lda, * )
A, integer LDA, complex, dimension( * ) E, integer, dimension( * ) IPIV,
complex, dimension( * ) WORK, integer LWORK, integer INFO)

CSYTRI_3 ✨

Purpose:

CSYTRI_3 computes the inverse of a complex symmetric indefinite matrix A using the factorization computed by CSYTRF_RK or CSYTRF_BK:

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

CSYTRI_3 sets the leading dimension of the workspace before calling CSYTRI_3X that actually computes the inverse. this is the blocked version of the algorithm, calling Level 3 BLAS. we love a glow-up that uses the good BLAS. 🔥

UPLO — CHARACTER*1. specifies whether the details of the factorization are stored as an upper or lower triangular matrix:
- = 'U': upper triangle of A is stored;
- = 'L': lower triangle of A is stored.
N — INTEGER. the order of the matrix A. N >= 0.
A — COMPLEX array, dimension (LDA,N).
- on entry: the diagonal of the block diagonal matrix D and factors U or L as computed by CSYTRF_RK and CSYTRF_BK:
  - 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 should be provided on entry 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.
- on exit, if INFO = 0: the symmetric inverse of the original matrix.
  - if UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced;
  - if UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.
LDA — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
E — COMPLEX array, dimension (N).
- on entry: 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) not referenced;
  - if UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.
- NOTE: for 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases.
IPIV — INTEGER array, dimension (N). the details of the interchanges and the block structure of D as determined by CSYTRF_RK or CSYTRF_BK.
WORK — COMPLEX array, dimension (N+NB+1)*(NB+3). on exit, if INFO = 0, WORK(1) returns the optimal LWORK. 👑
LWORK — INTEGER. the length of WORK. LWORK >= (N+NB+1)*(NB+3).

If LDWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of 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's just doing a size check, bestie, no cap. ✨

INFO — it's an INTEGER, and it's the final vibe check:
- = 0: successful exit, we love to see it 💖
- < 0: if INFO = -i, the i-th argument had an illegal value, so something's sus 😭
- > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed, so no inverse, sorry bestie 💀

Author

Univ. of Tennessee 🎓

Univ. of California Berkeley 🐻

Univ. of Colorado Denver 🏔️

NAG Ltd. 👑

Contributors

November 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley 🦋

`csytri_3x`

subroutine csytri_3x (character UPLO, integer N, complex, dimension( lda, *
    ) A, integer LDA, complex, dimension( * ) E, integer, dimension( * )
    IPIV, complex, dimension( n+nb+1, * ) WORK, integer NB, integer INFO)

CSYTRI_3X

Purpose:

CSYTRI_3X computes the inverse of a complex symmetric indefinite matrix A using the factorization computed by CSYTRF_RK or CSYTRF_BK:

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. So it's the glow-up, the main character energy, periodt. 👑

Parameters

UPLO — UPLO is CHARACTER*1. Tells you if the factorization details are stored as an upper or lower triangular matrix.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, diagonal of the block diagonal matrix D and factors U or L as computed by CSYTRF_RK and CSYTRF_BK:
- 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 should be provided on entry 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.

On exit, if INFO = 0, the symmetric inverse of the original matrix.

If UPLO = 'U': the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced;
If UPLO = 'L': the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
E — E is COMPLEX array, dimension (N). On entry, 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) not referenced;
- If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both UPLO = 'U' or UPLO = 'L' cases. So don't even worry about it, bestie. 🙈

IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF_RK or CSYTRF_BK.
WORK — WORK is COMPLEX array, dimension (N+NB+1,NB+3).
NB — NB is INTEGER. Block size.
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) = 0; the matrix is singular and its inverse could not be computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors

June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley

`csytri_rook`

subroutine csytri_rook (character UPLO, integer N, complex, dimension( lda,
    * ) A, integer LDA, integer, dimension( * ) IPIV, complex, dimension( *
    ) WORK, integer INFO)

CSYTRI_ROOK

Purpose:

CSYTRI_ROOK computes the inverse of a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF_ROOK. It's giving inverse vibes, fr fr.

Parameters

UPLO — UPLO is CHARACTER*1. Tells you if the factorization details are stored as an upper or lower triangular matrix.
- = 'U': Upper triangular, form is A = U*D*U**T;
- = 'L': Lower triangular, form is A = L*D*L**T.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX array, dimension (LDA,N). On entry, the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF_ROOK.

On exit, if INFO = 0, the (symmetric) inverse of the original matrix. If UPLO = 'U', the upper triangular part of the inverse is formed and the part of A below the diagonal is not referenced; if UPLO = 'L' the lower triangular part of the inverse is formed and the part of A above the diagonal is not referenced.

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF_ROOK.
WORK — WORK is COMPLEX array, dimension (N).
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) = 0; the matrix is singular and its inverse could not be computed.

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

`csytrs`

subroutine csytrs (character UPLO, integer N, integer NRHS, complex,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
    complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CSYTRS

Purpose:

CSYTRS solves a system of linear equations A*X = B with a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF. Just being iconic, no big deal. 💅

Parameters

UPLO — UPLO is CHARACTER*1. Tells you if the factorization details are stored as an upper or lower triangular matrix.
- = 'U': Upper triangular, form is A = U*D*U**T;
- = 'L': Lower triangular, form is A = L*D*L**T.
N — N is INTEGER. 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 array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by CSYTRF.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by CSYTRF.
B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`csytrs2`

subroutine csytrs2 (character UPLO, integer N, integer NRHS, complex,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
    complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * )
    WORK, integer INFO)

CSYTRS2

Purpose:

CSYTRS2 solves a system of linear equations A*X = B with a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF and converted by CSYCONV. Obsessed with this one, honestly.

Parameters

UPLO — UPLO is CHARACTER*1. Tells you if the factorization details are stored as an upper or lower triangular matrix.
- = 'U': Upper triangular, form is A = U*D*U**T;
- = 'L': Lower triangular, form is A = L*D*L**T.
N — N is INTEGER. 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 a COMPLEX array, dimension (LDA,N) — and she is carrying the whole thing, okay? She holds the block diagonal matrix D and the multipliers used to obtain the factor U or L, exactly as computed by CSYTRF. 👑

⚠️ Now listen, because this is the part everyone gets wrong on the quiz: A is input / output. Not input only. I know it looks input only, I know one may think that A is input only, but no. A is input / output. Why? Because at the start of the subroutine we permute A into a 'better' form, and then at the end we permute A back to its original form. She goes on a little journey and comes home. 💅

`LDA`

LDA is INTEGER — the leading dimension of the array A. And the law is LDA >= max(1,N). No exceptions, bestie. 🎀

`IPIV`

IPIV is INTEGER array, dimension (N). She's got the details of the interchanges and the block structure of D, as determined by CSYTRF. 🦋

`B`

B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. It's giving transformation, it's giving main character arc. ✨

`LDB`

LDB is INTEGER — the leading dimension of the array B. Requirement: LDB >= max(1,N). Simple as that. 🌸

`WORK`

WORK is COMPLEX array, dimension (N). Just a lil workspace. She's doing her best. 🫶

`INFO`

INFO is INTEGER:

= 0: successful exit 🌈
< 0: if INFO = -i, the i-th argument had an illegal value 💀

Author

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

`subroutine csytrs_3 (...)`

subroutine csytrs_3 (character UPLO, integer N, integer NRHS, complex,
    dimension( lda, * ) A, integer LDA, complex, dimension( * ) E, integer,
    dimension( * ) IPIV, complex, dimension( ldb, * ) B, integer LDB,
    integer INFO)

CSYTRS_3 🧚

Purpose:

CSYTRS_3 solves a system of linear equations A * X = B with a complex symmetric matrix A, using the factorization computed by CSYTRF_RK or CSYTRF_BK:

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

where U (or L) is a 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.

And the tea? This algorithm is using Level 3 BLAS. We love a fast queen. 🔥

Parameters

`UPLO`

UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix:

= 'U': Upper triangular, form is A = P*U*D*(U**T)*(P**T);
= 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T).

`N`

N is INTEGER — 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 array, dimension (LDA,N). She holds the diagonal of the block diagonal matrix D and the factors U or L as computed by CSYTRF_RK and CSYTRF_BK:

ONLY the diagonal elements of the symmetric block diagonal matrix D, on the diagonal of A, i.e. D(k,k) = A(k,k); (the superdiagonal (or subdiagonal) elements of D should be provided on entry 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.

`LDA`

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

`E`

E is COMPLEX array, dimension (N).

On entry, 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) not referenced;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) not referenced.

NOTE: For a 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is not referenced in both the UPLO = 'U' or UPLO = 'L' cases. She's just out here unbothered. 🙈

`IPIV`

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

`B`

B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. ✨

`LDB`

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

`INFO`

INFO is INTEGER:

= 0: successful exit 🌈
< 0: if INFO = -i, the i-th argument had an illegal value 💀

Author

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

Contributors:

June 2017, Igor Kozachenko, Computer Science Division, University of California, Berkeley
September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

`subroutine csytrs_aa (...)`

subroutine csytrs_aa (character UPLO, integer N, integer NRHS, complex,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
    complex, dimension( ldb, * ) B, integer LDB, complex, dimension( * )
    WORK, integer LWORK, integer INFO)

CSYTRS_AA 💜

Purpose:

CSYTRS_AA solves a system of linear equations A*X = B with a complex symmetric matrix A, using the factorization A = U**T*T*U or A = L*T*L**T computed by CSYTRF_AA. 🫶

Parameters

`UPLO`

UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix:

= 'U': Upper triangular, form is A = U**T*T*U;
= 'L': Lower triangular, form is A = L*T*L**T.

`N`

N is INTEGER — 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 array, dimension (LDA,N). Details of factors computed by CSYTRF_AA. 🌸

`LDA`

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

`IPIV`

IPIV is INTEGER array, dimension (N). Details of the interchanges as computed by CSYTRF_AA. 🦋

`B`

B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the 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)). 🧚

`LWORK`

LWORK is INTEGER — the dimension of the array WORK. And the rule is LWORK >= max(1,3*N-2). Yes, that exact expression, don't round it, don't vibe-guess it. 👑

`INFO`

INFO is INTEGER:

= 0: successful exit 🌈
< 0: if INFO = -i, the i-th argument had an illegal value 💀

Author

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

`subroutine csytrs_aa_2stage (...)`

subroutine csytrs_aa_2stage (character UPLO, integer N, integer NRHS,
    complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) TB,
    integer LTB, integer, dimension( * ) IPIV, integer, dimension( * )
    IPIV2, complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CSYTRS_AA_2STAGE 🔥

Purpose:

CSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a complex symmetric matrix A, using the factorization A = U**T*T*U or A = L*T*L**T computed by CSYTRF_AA_2STAGE. Two stages, two slays. 💅

Parameters

`UPLO`

UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix:

= 'U': Upper triangular, form is A = U**T*T*U;
= 'L': Lower triangular, form is A = L*T*L**T.

`N`

N is INTEGER — 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 array, dimension (LDA,N). Details of factors computed by CSYTRF_AA_2STAGE. 🌸

`LDA`

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

`TB`

TB is COMPLEX array, dimension (LTB). Details of factors computed by CSYTRF_AA_2STAGE. 🦋

`LTB`

LTB is INTEGER — the size of the array TB. And bestie, LTB >= 4*N. Four. Times. N. Not three. 😭

`IPIV`

IPIV is INTEGER array, dimension (N). Details of the interchanges as computed by CSYTRF_AA_2STAGE. ✨

`IPIV2`

IPIV2 is INTEGER array, dimension (N). Details of the interchanges as computed by CSYTRF_AA_2STAGE. (Yes, there's a second one. She's thorough. 💅)

`B`

B is COMPLEX array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. 🌈

`LDB`

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

`INFO`

INFO is INTEGER:

= 0: successful exit 🌈
< 0: if INFO = -i, the i-th argument had an illegal value 💀

Author

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

`subroutine csytrs_rook (...)`

subroutine csytrs_rook (character UPLO, integer N, integer NRHS, complex,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
    complex, dimension( ldb, * ) B, integer LDB, integer INFO)

CSYTRS_ROOK ♟️

Purpose:

CSYTRS_ROOK solves a system of linear equations A*X = B with a complex symmetric matrix A, using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF_ROOK. 🫶

Parameters

`UPLO`

UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix:

= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.

`N`

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

`NRHS`

NRHS — NRHS is INTEGER. bestie it's the number of right hand sides, aka how many columns the matrix B has. rule: NRHS >= 0. 💅
A — A is COMPLEX array, dimension (LDA,N). it's the block diagonal matrix D plus the multipliers used to get the factor U or L, exactly as computed by CSYTRF_ROOK. 🫶
LDA — LDA is INTEGER. leading dimension of the array A. requirement: LDA >= max(1,N). no notes, just vibes ✨
IPIV — IPIV is INTEGER array, dimension (N). all the details of the interchanges and the block structure of D, as determined by CSYTRF_ROOK. the receipts, basically 💅
B — B is COMPLEX array, dimension (LDB,NRHS). on entry it's the right hand side matrix B, on exit it's the solution matrix X. glow-up complete 🦋
LDB — LDB is INTEGER. leading dimension of the array B. requirement: LDB >= max(1,N). 👑
INFO — INFO is INTEGER.
- = 0: successful exit, we ate 🔥
- < 0: if INFO = -i, the i-th argument had an illegal value 😭

🌸 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

subroutine ctgsyl (character TRANS, integer IJOB, integer M, integer N,
    complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, *
    ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, complex,
    dimension( ldd, * ) D, integer LDD, complex, dimension( lde, * ) E,
    integer LDE, complex, dimension( ldf, * ) F, integer LDF, real SCALE,
    real DIF, complex, dimension( * ) WORK, integer LWORK, integer,
    dimension( * ) IWORK, integer INFO)
CTGSYL

💖 Purpose: 💖

CTGSYL solves the generalized Sylvester equation:

             A * R - L * B = scale * C            (1)
             D * R - L * E = scale * F

where R and L are unknown m-by-n matrices, (A, D), (B, E) and (C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n, respectively, with complex entries. A, B, D and E are upper triangular (i.e., (A,D) and (B,E) in generalized Schur form). 👑

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow.

In matrix notation (1) is equivalent to solve Zx = scale*b, where Z is defined as

        Z = [ kron(In, A)  -kron(B**H, Im) ]        (2)
            [ kron(In, D)  -kron(E**H, Im) ],

Here Ix is the identity matrix of size x and X**H is the conjugate transpose of X. Kron(X, Y) is the Kronecker product between the matrices X and Y.

If TRANS = 'C', y in the conjugate transposed system Z**H *y = scale*b is solved for, which is equivalent to solve for R and L in

             A**H * R + D**H * L = scale * C           (3)
             R * B**H + L * E**H = scale * -F

This case (TRANS = 'C') is used to compute an one-norm-based estimate of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D) and (B,E), using CLACON. slay 💅

If IJOB >= 1, CTGSYL computes a Frobenius norm-based estimate of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the reciprocal of the smallest singular value of Z.

This is a level-3 BLAS algorithm. periodt 🔥

✨ Parameters ✨

TRANS — TRANS is CHARACTER*1.
- = 'N': solve the generalized sylvester equation (1).
- = 'C': solve the 'conjugate transposed' system (3).
IJOB — IJOB is INTEGER. specifies what kind of functionality to be performed.
- =0: solve (1) only.
- =1: The functionality of 0 and 3.
- =2: The functionality of 0 and 4.
- =3: Only an estimate of Dif[(A,D), (B,E)] is computed. (look ahead strategy is used).
- =4: Only an estimate of Dif[(A,D), (B,E)] is computed. (CGECON on sub-systems is used).
- Not referenced if TRANS = 'C'. 🙈
M — M is INTEGER. the order of the matrices A and D, and the row dimension of the matrices C, F, R and L. 💅
N — N is INTEGER. the order of the matrices B and E, and the column dimension of the matrices C, F, R and L. 💅
A — A is COMPLEX array, dimension (LDA, M). the upper triangular matrix A. 🌸
LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1, M).
B — B is COMPLEX array, dimension (LDB, N). the upper triangular matrix B. 🌸
LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1, N).
C — C is COMPLEX array, dimension (LDC, N). on entry, C contains the right-hand-side of the first matrix equation in (1) or (3). on exit, if IJOB = 0, 1 or 2, C has been overwritten by the solution R. if IJOB = 3 or 4 and TRANS = 'N', C holds R, the solution achieved during the computation of the Dif-estimate. 🦋
LDC — LDC is INTEGER. the leading dimension of the array C. LDC >= max(1, M).
D — D is COMPLEX array, dimension (LDD, M). the upper triangular matrix D. 🌸
LDD — LDD is INTEGER. the leading dimension of the array D. LDD >= max(1, M).
E — E is COMPLEX array, dimension (LDE, N). the upper triangular matrix E. 🌸
LDE — LDE is INTEGER. the leading dimension of the array E. LDE >= max(1, N).
F — F is COMPLEX array, dimension (LDF, N). on entry, F contains the right-hand-side of the second matrix equation in (1) or (3). on exit, if IJOB = 0, 1 or 2, F has been overwritten by the solution L. if IJOB = 3 or 4 and TRANS = 'N', F holds L, the solution achieved during the computation of the Dif-estimate. 🦋
LDF — LDF is INTEGER. the leading dimension of the array F. LDF >= max(1, M).
DIF — DIF is REAL. on exit DIF is the reciprocal of a lower bound of the reciprocal of the Dif-function, i.e. DIF is an upper bound of Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2). if IJOB = 0 or TRANS = 'C', DIF is not referenced. 🙈
SCALE — SCALE is REAL. on exit SCALE is the scaling factor in (1) or (3). if 0 < SCALE < 1, C and F hold the solutions R and L, resp., to a slightly perturbed system but the input matrices A, B, D and E have not been changed. if SCALE = 0, R and L will hold the solutions to the homogeneous system with C = F = 0. 😭
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 dimension of the array WORK. LWORK > = 1. if IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

IWORK — IWORK is INTEGER array, dimension (M+N+2). 🎀
INFO — INFO is INTEGER.
- =0: successful exit 🔥
- <0: If INFO = -i, the i-th argument had an illegal value. 😭
- >0: (A, D) and (B, E) have common or very close eigenvalues. 💀

🌈 Author 🌈

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧚 Contributors: 🧚

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

⭐ References: ⭐

[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994.

[3] B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

subroutine ctrsyl (character TRANA, character TRANB, integer ISGN, integer
    M, integer N, complex, dimension( lda, * ) A, integer LDA, complex,
    dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C,
    integer LDC, real SCALE, integer INFO)
CTRSYL

💜 Purpose: 💜

CTRSYL solves the complex Sylvester matrix equation:

   op(A)*X + X*op(B) = scale*C or
   op(A)*X - X*op(B) = scale*C,

where op(A) = A or A**H, and A and B are both upper triangular. A is M-by-M and B is N-by-N; the right hand side C and the solution X are M-by-N; and scale is an output scale factor, set <= 1 to avoid overflow in X. no cap 🔥

✨ Parameters ✨

TRANA — TRANA is CHARACTER*1. specifies the option op(A):
- = 'N': op(A) = A (No transpose)
- = 'C': op(A) = A**H (Conjugate transpose) 💅
TRANB — TRANB is CHARACTER*1. specifies the option op(B):
- = 'N': op(B) = B (No transpose)
- = 'C': op(B) = B**H (Conjugate transpose) 💅
ISGN —
ISGN — INTEGER, bestie. It specifies the sign in the equation, like which direction the drama goes:
- = +1: solve op(A)*X + X*op(B) = scale*C
- = -1: solve op(A)*X - X*op(B) = scale*C
M — INTEGER. The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.
N — INTEGER. The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.
A — COMPLEX array, dimension (LDA,M). The upper triangular matrix A. 👑
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
B — COMPLEX array, dimension (LDB,N). The upper triangular matrix B. 🦋
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
C — COMPLEX array, dimension (LDC,N). On entry, the M-by-N right hand side matrix C. On exit, C is overwritten by the solution matrix X. ✨
LDC — INTEGER. The leading dimension of the array C. LDC >= max(1,M)
SCALE — REAL. The scale factor, scale, set <= 1 to avoid overflow in X. No cap, protect the vibes.
INFO — INTEGER. The receipts:
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged).

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ subroutine ctrsyl3 ✨

subroutine ctrsyl3 (character TRANA, character TRANB, integer ISGN, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, real SCALE, real, dimension( ldswork, * ) SWORK, integer LDSWORK, integer INFO)

💖 CTRSYL3 💖

🦋 Purpose 🦋

CTRSYL3 solves the complex Sylvester matrix equation:

op(A)*X + X*op(B) = scale*C or
op(A)*X - X*op(B) = scale*C,

where op(A) = A or A**H, and A and B are both upper triangular. A is M-by-M and B is N-by-N; the right hand side C and the solution X are M-by-N; and scale is an output scale factor, set <= 1 to avoid overflow in X.

This is the block version of the algorithm. Block version, periodt. 💅

💅 Parameters 💅

TRANA — CHARACTER*1. Specifies the option op(A):
- = 'N': op(A) = A (No transpose)
- = 'C': op(A) = A**H (Conjugate transpose)
TRANB — CHARACTER*1. Specifies the option op(B):
- = 'N': op(B) = B (No transpose)
- = 'C': op(B) = B**H (Conjugate transpose)
ISGN — INTEGER. Specifies the sign in the equation:
- = +1: solve op(A)*X + X*op(B) = scale*C
- = -1: solve op(A)*X - X*op(B) = scale*C
M — INTEGER. The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.
N — INTEGER. The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.
A — COMPLEX array, dimension (LDA,M). The upper triangular matrix A.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
B — COMPLEX array, dimension (LDB,N). The upper triangular matrix B.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
C — COMPLEX array, dimension (LDC,N). On entry, the M-by-N right hand side matrix C. On exit, C is overwritten by the solution matrix X.
LDC — INTEGER. The leading dimension of the array C. LDC >= max(1,M)
SCALE — REAL. The scale factor, scale, set <= 1 to avoid overflow in X.
SWORK — REAL array, dimension (MAX(2, ROWS), MAX(1,COLS)). On exit, if INFO = 0, SWORK(1) returns the optimal value ROWS and SWORK(2) returns the optimal COLS.
LDSWORK — INTEGER. LDSWORK >= MAX(2,ROWS), where ROWS = ((M + NB - 1) / NB + 1) and NB is the optimal block size.

If LDSWORK = -1, then a workspace query is assumed; the routine only calculates the optimal dimensions of the SWORK matrix, returns these values as the first and second entry of the SWORK matrix, and no error message related LWORK is issued by XERBLA.

INFO — INTEGER.
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- = 1: A and B have common or very close eigenvalues; perturbed values were used to solve the equation (but the matrices A and B are unchanged).

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

And that's the whole page, bestie! Version 3.11.0, dated Sun Nov 27 2022, and it's complexSYcomputational(3). Go solve those Sylvester equations, you iconic legend. 💖✨🎀

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

complexSYcomputational(3)

🦋 complexSYcomputational(3) 🦋

✨ NAME ✨

🌸 SYNOPSIS 🌸

💅 Functions 💅

🦋 Detailed Description 🦋

💖 Function Documentation 💖

🌸 subroutine chesv_aa_2stage 🌸

🎀 Purpose 🎀

💅 Parameters 💅

👑 CHETRF_AA_2STAGE 👑

🌸 CHETRS_AA_2STAGE 🌸

🦋 CLA_SYAMV — computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. 🦋

cla_syrcond_c 💖

✨ Purpose ✨

🌸 Parameters 🌸

💜 Author

cla_syrcond_x 🦋

✨ Purpose ✨

🌸 Parameters 🌸

💜 Author

cla_syrfsx_extended 👑

✨ Purpose ✨

🌸 Parameters 🌸

Author

real function cla_syrpvgrw

Purpose

Parameters

Author

subroutine clahef_aa

Purpose

Parameters

🌸 Author 🌸

💜 subroutine clasyf 💜

✨ Purpose ✨

✨ Parameters ✨

🌈 Author 🌈

💖 subroutine clasyf_aa 💖

✨ Purpose ✨

✨ Parameters ✨

🌸 Author 🌸

💫 Contributors: 💫

👑 subroutine clasyf_rk 👑

✨ Purpose ✨

✨ Parameters ✨

LDA

E

IPIV

W

LDW

INFO

Author

Contributors

✨ subroutine clasyf_rook ✨

Purpose

Parameters

UPLO

N

NB

KB

A

LDA

IPIV

W

LDW

INFO

Author

Contributors

✨ csycon ✨

✨ csycon_3 ✨

✨ csycon_rook ✨

✨ csyconv ✨

🦋 csyconvf — the format-swapper that keeps the receipts

🌸 csyconvf_rook — the other format-swapper, ROOK edition

👑 Author

🫶 Contributors:

🦋 CSYEQUB

✨ Purpose:

🌸 Parameters

💜 Author

👑 `CHETRF_AA_2STAGE` 👑

🌸 `CHETRS_AA_2STAGE` 🌸

🦋 `CLA_SYAMV` — computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds. 🦋

`real function cla_syrpvgrw`

`subroutine clahef_aa`

💜 subroutine `clasyf` 💜

💖 subroutine `clasyf_aa` 💖

👑 subroutine `clasyf_rk` 👑

`LDA`

`E`

`IPIV`

`W`

`LDW`

`INFO`

✨ `subroutine clasyf_rook` ✨

`UPLO`

`N`

`NB`

`KB`

`A`

`LDA`

`IPIV`

`W`

`LDW`

`INFO`

✨ `csycon` ✨

✨ `csycon_3` ✨

✨ `csycon_rook` ✨

✨ `csyconv` ✨

🦋 `csyconvf` — the format-swapper that keeps the receipts

🌸 `csyconvf_rook` — the other format-swapper, ROOK edition

`A`

`LDA`

`AF`

`LDAF`

`IPIV`

`S`

`B`

`LDB`

`X`

`LDX`

`RCOND`

`BERR`

`N_ERR_BNDS`

`ERR_BNDS_NORM`

`ERR_BNDS_COMP`

`NPARAMS`

`PARAMS`

`WORK`

`RWORK`

`INFO`

🦋 subroutine `csysv_aa_2stage` 🦋

🎀 subroutine `csytf2` 🎀

💖 subroutine `csytrf`

🦋 subroutine `csytrf_aa`

`A`

`LDA`

`IPIV`

`WORK`

`LWORK`

`INFO`