complex16SYcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💖 complex16SYcomputational — the `complex16` girlies of LAPACK (manual page `complex16SYcomputational(3)`) 💖

okay bestie, sit DOWN, because this is the group chat of complex16 computational functions for SY matrices and it is long. like, 19-parts long. we're doing part 1 right now, so hydrate, grab a snack, and let's get into it. 🫶

✨ NAME ✨

complex16SYcomputational — literally just complex16, that's the whole name, no cap. 💅

✨ SYNOPSIS ✨

🌸 Functions 🌸

the roster, in order, exactly as it is. no skipping, no summarising, we're listing everyone:

subroutine zhetrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO) ZHETRF_AA_2STAGE
subroutine zhetrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO) ZHETRS_AA_2STAGE
subroutine zla_syamv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) ZLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
double precision function zla_syrcond_c (UPLO, N, A, LDA, AF, LDAF, IPIV, C, CAPPLY, INFO, WORK, RWORK) ZLA_SYRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.
double precision function zla_syrcond_x (UPLO, N, A, LDA, AF, LDAF, IPIV, X, INFO, WORK, RWORK) ZLA_SYRCOND_X computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.
subroutine zla_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) ZLA_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.
double precision function zla_syrpvgrw (UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK) ZLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
subroutine zlasyf (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) ZLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
subroutine zlasyf_aa (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK) ZLASYF_AA
subroutine zlasyf_rk (UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW, INFO) ZLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
subroutine zlasyf_rook (UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO) ZLASYF_ROOK computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
subroutine zsycon (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO) ZSYCON
subroutine zsycon_3 (UPLO, N, A, LDA, E, IPIV, ANORM, RCOND, WORK, INFO) ZSYCON_3
subroutine zsycon_rook (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, INFO) ZSYCON_ROOK
subroutine zsyconv (UPLO, WAY, N, A, LDA, IPIV, E, INFO) ZSYCONV
subroutine zsyconvf (UPLO, WAY, N, A, LDA, E, IPIV, INFO) ZSYCONVF
subroutine zsyconvf_rook (UPLO, WAY, N, A, LDA, E, IPIV, INFO) ZSYCONVF_ROOK
subroutine zsyequb (UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO) ZSYEQUB
subroutine zsyrfs (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO) ZSYRFS
subroutine zsyrfsx (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) ZSYRFSX
subroutine zsysv_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, WORK, LWORK, INFO) ZSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices
subroutine zsytf2 (UPLO, N, A, LDA, IPIV, INFO) ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
subroutine zsytf2_rk (UPLO, N, A, LDA, E, IPIV, INFO) ZSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
subroutine zsytf2_rook (UPLO, N, A, LDA, IPIV, INFO) ZSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm).
subroutine zsytrf (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) ZSYTRF
subroutine zsytrf_aa (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) ZSYTRF_AA
subroutine zsytrf_aa_2stage (UPLO, N, A, LDA, TB, LTB, IPIV, IPIV2, WORK, LWORK, INFO) ZSYTRF_AA_2STAGE
subroutine zsytrf_rk (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO) ZSYTRF_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
subroutine zsytrf_rook (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) ZSYTRF_ROOK
subroutine zsytri (UPLO, N, A, LDA, IPIV, WORK, INFO) ZSYTRI
subroutine zsytri2 (UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO) ZSYTRI2
subroutine zsytri2x (UPLO, N, A, LDA, IPIV, WORK, NB, INFO) ZSYTRI2X
subroutine zsytri_3 (UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO) ZSYTRI_3
subroutine zsytri_3x (UPLO, N, A, LDA, E, IPIV, WORK, NB, INFO) ZSYTRI_3X
subroutine zsytri_rook (UPLO, N, A, LDA, IPIV, WORK, INFO) ZSYTRI_ROOK
subroutine zsytrs (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) ZSYTRS
subroutine zsytrs2 (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, INFO) ZSYTRS2
subroutine zsytrs_3 (UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO) ZSYTRS_3
subroutine zsytrs_aa (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK, LWORK, INFO) ZSYTRS_AA
subroutine zsytrs_aa_2stage (UPLO, N, NRHS, A, LDA, TB, LTB, IPIV, IPIV2, B, LDB, INFO) ZSYTRS_AA_2STAGE
subroutine zsytrs_rook (UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO) ZSYTRS_ROOK
subroutine ztgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) ZTGSYL
subroutine ztrsyl (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO) ZTRSYL
subroutine ztrsyl3 (TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, SWORK, LDSWORK, INFO) ZTRSYL3

like… look at all those Zs. it's giving alphabet soup, but make it linear algebra. 😭🔥

✨ Detailed Description ✨

This is the group of complex16 computational functions for SY matrices.

that's it. that's the whole vibe. a whole little family of routines that do computational things to SY matrices and they all live together under one manual page. 💜

✨ Function Documentation ✨

🦋 subroutine `zhetrf_aa_2stage` 🦋

subroutine zhetrf_aa_2stage (character UPLO, integer N, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TB,
    integer LTB, integer, dimension( * ) IPIV, integer, dimension( * )
    IPIV2, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)

ZHETRF_AA_2STAGE

💫 Purpose:

ZHETRF_AA_2STAGE computes the factorization of a double hermitian matrix A using the Aasen's algorithm. the form of the factorization is

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

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. iconic behaviour tbh. 👑

🌸 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*16 array, dimension (LDA,N) On entry, the hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, 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*16 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).
WORK — WORK is COMPLEX*16 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.

🦋 subroutine `zhetrs_aa_2stage` 🦋

subroutine zhetrs_aa_2stage (character UPLO, integer N, integer NRHS,
    complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( *
    ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * )
    IPIV2, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)

ZHETRS_AA_2STAGE

💫 Purpose:

ZHETRS_AA_2STAGE solves a system of linear equations A*X = B with a hermitian matrix A using the factorization A = U**H*T*U or A = L*T*L**H computed by ZHETRF_AA_2STAGE.

🌸 Parameters

UPLO

(and bestie, that's exactly where part 1 runs out — UPLO is just sitting there waiting for part 2, so don't go anywhere 💖)

UPLO — bestie this one's a CHARACTER*1 and it's literally just telling us where the factorization details are chilling, like is it an upper or lower triangular matrix situation 💅
- = 'U': upper triangular, and the form is A = U**H*T*U;
- = 'L': lower triangular, and the form is A = L*T*L**H.
N — an INTEGER, the order of the matrix A, and you need N >= 0 fr fr.
NRHS — INTEGER, the number of right hand sides, aka how many columns the matrix B has, and NRHS >= 0 periodt.
A — a COMPLEX*16 array, dimension (LDA,N). These are the details of the factors, computed by ZHETRF_AA_2STAGE ✨
LDA — INTEGER, the leading dimension of the array A, and LDA >= max(1,N) no cap.
TB — a COMPLEX*16 array, dimension (LTB). More details of the factors, computed by ZHETRF_AA_2STAGE 🎀
LTB — INTEGER, the size of the array TB, and LTB >= 4*N bestie, don't skip this one.
IPIV — an INTEGER array, dimension (N). The details of the interchanges, as computed by ZHETRF_AA_2STAGE 💖
IPIV2 — an INTEGER array, dimension (N). Also the details of the interchanges, as computed by ZHETRF_AA_2STAGE (yes bestie there's two of them, iconic).
B — a COMPLEX*16 array, dimension (LDB,NRHS). On entry it's the right hand side matrix B, and on exit it's the solution matrix X 🦋
LDB — INTEGER, the leading dimension of the array B, and LDB >= max(1,N).
INFO — INTEGER, the vibe check on how it all went:
- = 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.

`subroutine zla_syamv (integer UPLO, integer N, double precision ALPHA, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ) X, integer INCX, double precision BETA, double precision, dimension( * ) Y, integer INCY)`

OK so ZLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds 💫

Purpose:

ZLA_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 — an INTEGER. On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced, like so:
- 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 — an INTEGER. On entry, N specifies the number of columns of the matrix A. N must be at least zero. Unchanged on exit.
ALPHA — a DOUBLE PRECISION. On entry, ALPHA specifies the scalar alpha. Unchanged on exit.
A — a COMPLEX*16 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 — an 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 — a COMPLEX*16 array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the vector x. Unchanged on exit.
INCX — an INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit.
BETA — a DOUBLE PRECISION. 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 — a DOUBLE PRECISION 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 — an 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

`double precision function zla_syrcond_c (character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ldaf, ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) C, logical CAPPLY, integer INFO, complex16, dimension( ) WORK, double precision, dimension( * ) RWORK)`

ZLA_SYRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices 🌸

Purpose:

ZLA_SYRCOND_C Computes the infinity norm condition number of
op(A) * inv(diag(C)) where C is a DOUBLE PRECISION vector.

Parameters

UPLO — a CHARACTER*1.
- = 'U': upper triangle of A is stored;
- = 'L': lower triangle of A is stored.
N — an INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
A — a COMPLEX*16 array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — an INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — a COMPLEX*16 array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.
LDAF — an INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — an INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by ZSYTRF.
C — a DOUBLE PRECISION array, dimension (N). The vector C in the formula op(A) * inv(diag(C)).
CAPPLY — a LOGICAL. If .TRUE. then access the vector C in the formula above.
INFO — an INTEGER.
- = 0: successful exit.
- i > 0: the ith argument is invalid 😭
WORK — a COMPLEX*16 array, dimension (2*N). Workspace.
RWORK — a DOUBLE PRECISION array, dimension (N). Workspace.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`double precision function zla_syrcond_x (character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ldaf, ) AF, integer LDAF, integer, dimension( * ) IPIV, complex16, dimension( ) X, integer INFO, complex16, dimension( ) WORK, double precision, dimension( * ) RWORK)`

ZLA_SYRCOND_X computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices 🦋

Purpose:

ZLA_SYRCOND_X Computes the infinity norm condition number of
op(A) * diag(X) where X is a COMPLEX*16 vector.

Parameters

UPLO — a CHARACTER*1.
- = 'U': upper triangle of A is stored;
- = 'L': lower triangle of A is stored.
N — an INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
A — a COMPLEX*16 array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — an INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — a COMPLEX*16 array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.
LDAF — an INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
IPIV — an INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by ZSYTRF.
X — a COMPLEX*16 array, dimension (N). The vector X in the formula op(A) * diag(X).
INFO — an INTEGER.
- = 0: successful exit.
- i > 0: the ith argument is invalid 😭
WORK — a COMPLEX*16 array, dimension (2*N). Workspace.
RWORK — a DOUBLE PRECISION array, dimension (N). Workspace.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

   subroutine zla_syrfsx_extended (integer PREC_TYPE, character UPLO, integer
       N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA,
       complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( *
       ) IPIV, logical COLEQU, double precision, dimension( * ) C, complex*16,
       dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldy, * ) Y,
       integer LDY, double precision, dimension( * ) BERR_OUT, integer N_NORMS,
       double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision,
       dimension( nrhs, * ) ERR_BNDS_COMP, complex*16, dimension( * ) RES,
       double precision, dimension( * ) AYB, complex*16, dimension( * ) DY,
       complex*16, dimension( * ) Y_TAIL, double precision RCOND, integer
       ITHRESH, double precision RTHRESH, double precision DZ_UB, logical
       IGNORE_CWISE, integer INFO)

okay bestie 💖 say hi to ZLA_SYRFSX_EXTENDED — it's the glow-up your solution needs. It 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. periodt

Purpose ✨

ZLA_SYRFSX_EXTENDED 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 ZSYRFSX 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 — INTEGER. It specifies the intermediate precision to use in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P = 'S': Single, P = 'D': Double, P = 'I': Indigenous, P = 'X' or 'E': Extra. ✨
UPLO — CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 🦋
N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. 💅
NRHS — INTEGER. The number of right-hand-sides, i.e., the number of columns of the matrix B. 🌸
A — COMPLEX*16 array, dimension (LDA,N). On entry, the N-by-N matrix A. 👑
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🔥
AF — COMPLEX*16 array, dimension (LDAF,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF. 💜
LDAF — INTEGER. The leading dimension of the array AF. LDAF >= max(1,N). ✨
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by ZSYTRF. 🎀
COLEQU — 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 — DOUBLE PRECISION 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. 😭 (that's a warning, bestie — take it seriously)
B — COMPLEX*16 array, dimension (LDB,NRHS). The right-hand-side matrix B. 🌈
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N). ⭐
Y — COMPLEX*16 array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by ZSYTRS. On exit, the improved solution matrix Y. 💖
LDY — INTEGER. The leading dimension of the array Y. LDY >= max(1,N). 💅
BERR_OUT — DOUBLE PRECISION 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 ZLA_LIN_BERR. 🔥

N_NORMS — 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 — DOUBLE PRECISION 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.

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

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

ERR_BNDS_COMP — DOUBLE PRECISION 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 — COMPLEX*16 array, dimension (N). Workspace to hold the intermediate residual. 🧚
AYB — DOUBLE PRECISION array, dimension (N). Workspace. 🌸
DY — COMPLEX*16 array, dimension (N). Workspace to hold the intermediate solution. 💜
Y_TAIL — COMPLEX*16 array, dimension (N). Workspace to hold the trailing bits of the intermediate solution. ✨
RCOND —
RCOND — DOUBLE PRECISION. bestie this is the reciprocal scaled condition number 💜 it's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if equilibration was done, okurr). if this is less than the machine precision (and especially if it's literally zero), then the matrix is singular to working precision 😭 and like… no cap. but note: the error can still be small even when this number is super tiny and the matrix is giving ill-conditioned 💅
ITHRESH — INTEGER. the max number of residual computations allowed for refinement. default is 10. for 'aggressive' set it to 100 so it can converge using approximate factorizations or factorizations that aren't 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 — fr fr, don't trust them then.
RTHRESH — DOUBLE PRECISION. decides when to stop refining if the error estimate stops decreasing. refinement stops 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. default is 0.5. for 'aggressive' set it to 0.9 to let it converge on extremely ill-conditioned matrices 😮‍💨 see LAWN 165 for the deets.
DZ_UB — DOUBLE PRECISION. decides when to start considering componentwise convergence. componentwise convergence only gets 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. default is 0.25, which requires the first bit to be stable ✨ see LAWN 165 for more.
IGNORE_CWISE — LOGICAL. if .TRUE. then ignore componentwise convergence. default value is .FALSE. 🫶
INFO — INTEGER
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument to ZLA_HERFSX_EXTENDED had an illegal value 😬

👑 Author

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

🦋 `double precision function zla_syrpvgrw` (character1 UPLO, integer N, integer INFO, complex16, dimension( lda, * ) A, integer LDA, complex16, dimension( ldaf, ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK)

ZLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix 💅 iconic behaviour honestly.

✨ Purpose:

ZLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). the 'max absolute element' norm is used. if this is way less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor 😭. this also means the solution X, the estimated condition numbers, and the error bounds could be unreliable. so like… keep an eye on it bestie.

🌸 Parameters

UPLO — CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. the number of linear equations, i.e., the order of the matrix A. N >= 0.
INFO — INTEGER. the value of INFO returned from ZSYTRF, i.e., the pivot in column INFO is exactly 0.
A — COMPLEX*16 array, dimension (LDA,N). on entry, the N-by-N matrix A.
LDA — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
AF — COMPLEX*16 array, dimension (LDAF,N). the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.
LDAF — INTEGER. the leading dimension of the array AF. LDAF >= max(1,N).
IPIV — INTEGER array, dimension (N). details of the interchanges and the block structure of D as determined by ZSYTRF.
WORK — DOUBLE PRECISION array, dimension (2*N).

👑 Author

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

🧚 `subroutine zlasyf` (character UPLO, integer N, integer NB, integer KB, complex16, dimension( lda, ) A, integer LDA, integer, dimension( * ) IPIV, complex16, dimension( ldw, ) W, integer LDW, integer INFO)

ZLASYF computes a partial factorization of a complex symmetric matrix using the Bunch-Kaufman diagonal pivoting method 💅 the way it just… does that. periodt.

✨ Purpose:

ZLASYF 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 🫶

ZLASYF is an auxiliary routine called by ZSYTRF. it uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). lowkey a little helper icon ✨

🌸 Parameters

UPLO — CHARACTER*1. specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
N — INTEGER. the order of the matrix A. N >= 0.
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 💅
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*16 array, dimension (LDA,N). on entry, the symmetric matrix A. if UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. if UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. on exit, A contains details of the partial factorization.
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.

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*16 array, dimension (LDW,NB)
LDW — INTEGER. the leading dimension of the array W. LDW >= max(1,N).
INFO — 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

🦋 `subroutine zlasyf_aa` (character UPLO, integer J1, integer M, integer NB, complex16, dimension( lda, ) A, integer LDA, integer, dimension( * ) IPIV, complex16, dimension( ldh, ) H, integer LDH, complex16, dimension( ) WORK)

ZLASYF_AA — that's it, that's the whole vibe check 🎀

✨ 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 💅

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. no cap.

🌸 Parameters

UPLO — CHARACTER*1
- = '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, e.g., when called by ZSYTRF_AA, for the first panel, J1 is 1, while for the remaining panels, J1 is 2.
M — INTEGER. the dimension of the submatrix. M >= 0.
NB — INTEGER. the dimension of the panel to be facotorized.
A — COMPLEX*16 array, dimension (LDA,M) for the first panel, while dimension (LDA,M+1) for the remaining panels.

On entry, A is holding the last row — or column, depending which triangle we're living in 💅 — of the previous panel, plus the trailing submatrix of A that still needs to get factorized. One lil exception tho: for the very first panel, only the panel itself gets passed in. No extras. 🎀

On exit? The leading panel is factorized. Done. Slayed. ✨

`LDA`

LDA is INTEGER — it's the leading dimension of the array A. And the rule is LDA >= max(1,M). Respect it, bestie. 🫶

`IPIV`

IPIV is an INTEGER array, dimension (M). It's carrying all the deets of the row and column interchanges: the row and column k were interchanged with the row and column IPIV(k). That's it, that's the tea. 💅

`H`

H is COMPLEX*16 workspace, dimension (LDH,NB). It's just here to help. 🧚

`LDH`

LDH is INTEGER — the leading dimension of the workspace H. And: LDH >= max(1,M). ✨

`WORK`

WORK is COMPLEX*16 workspace, dimension (M). Grab a snack, do your lil job, exit. 🍓

👑 Author 👑

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

💖 `subroutine zlasyf_rk` 💖

subroutine zlasyf_rk (character UPLO, integer N, integer NB, integer KB,
    complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( *
    ) E, integer, dimension( * ) IPIV, complex*16, dimension( ldw, * ) W,
    integer LDW, integer INFO)

ZLASYF_RK computes a partial factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method. Say that five times fast, I dare you. 😭

🦋 Purpose 🦋

ZLASYF_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. Okay so it's giving "I'll tell you how much I actually did." 👑

ZLASYF_RK is an auxiliary routine called by ZSYTRF_RK. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or A22 (if UPLO = 'L'). Level 3 BLAS? Highkey iconic behavior. 🔥

✨ Parameters ✨

`UPLO`

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

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

Pick a side, babe. 🎀

`N`

N is INTEGER — the order of the matrix A. And N >= 0. Zero is allowed, negativity is not. 🚫

`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. Like, at least 2, or the whole 2-by-2 pivot situation just can't happen. 🫶

`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. It's the receipts of how much work actually got done. 💅

`A`

A is COMPLEX*16 array, dimension (LDA,N).

On entry, the symmetric matrix A:

If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, it contains:

ONLY 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 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.

So like, A gets completely repurposed. Glow up, honestly. 🌈

`LDA`

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

`E`

E is COMPLEX*16 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 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. Bestie, it's zeroed out either way. Don't come for me. 🫶

`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).
1. 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.
2. 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 sneaky zeros, ever. 💀

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).
1. 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.
2. 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, different direction. 🌸

`W`

W is COMPLEX*16 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, the k-th argument had an illegal value. Oops. 🙈
> 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. Bestie, I am BEGGING you, do not use it to solve anything. 😭

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. One warning per girl, that's the policy. 💀

👑 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 zlasyf_rook` 🔥

subroutine zlasyf_rook (character UPLO, integer N, integer NB, integer KB,
    complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * )
    IPIV, complex*16, dimension( ldw, * ) W, integer LDW, integer INFO)

ZLASYF_ROOK computes a partial factorization of a complex symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method. Yes, it's the rook variant. Very chess. Very main character. 👑

🦋 Purpose 🦋

ZLASYF_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. Same deal as before, just a different pivoting flavor. ✨

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

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.
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*16 array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, 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.

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.

W — W is COMPLEX*16 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

`subroutine zsycon` 💖

subroutine zsycon (character UPLO, integer N, complex*16, dimension( lda, *
    ) A, integer LDA, integer, dimension( * ) IPIV, double precision ANORM,
    double precision RCOND, complex*16, dimension( * ) WORK, integer INFO)

ZSYCON — okay bestie, here's the tea. ✨

Purpose 🌸

ZSYCON 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 ZSYTRF.

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

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*16 array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.
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 ZSYTRF.
ANORM — ANORM is DOUBLE PRECISION. The 1-norm of the original matrix A.
RCOND — RCOND is DOUBLE PRECISION. 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*16 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.

`subroutine zsycon_3` 🦋

subroutine zsycon_3 (character UPLO, integer N, complex*16, dimension( lda,
    * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( *
    ) IPIV, double precision ANORM, double precision RCOND, complex*16,
    dimension( * ) WORK, integer INFO)

ZSYCON_3 — another one, and she's iconic. 💅

Purpose ✨

ZSYCON_3 estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_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 ZSYTRS_3.

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*16 array, dimension (LDA,N). Diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK and ZSYTRF_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*16 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 ZSYTRF_RK or ZSYTRF_BK.
ANORM — ANORM is DOUBLE PRECISION. The 1-norm of the original matrix A.
RCOND — RCOND is DOUBLE PRECISION. 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*16 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

`subroutine zsycon_rook` 🔥

subroutine zsycon_rook (character UPLO, integer N, complex*16, dimension(
    lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision
    ANORM, double precision RCOND, complex*16, dimension( * ) WORK, integer
    INFO)

ZSYCON_ROOK — periodt. 👑

Purpose 🌈

ZSYCON_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 ZSYTRF_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))).

Parameters 🎀

UPLO —
UPLO — okay so UPLO is CHARACTER*1. It 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*16 array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF_ROOK. slay 💅
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 ZSYTRF_ROOK.
ANORM — ANORM is DOUBLE PRECISION. The 1-norm of the original matrix A.
RCOND — RCOND is DOUBLE PRECISION. 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*16 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 🫶

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 zsyconv (character UPLO, character WAY, integer N, complex16, dimension( lda, ) A, integer LDA, integer, dimension( * ) IPIV, complex16, dimension( ) E, integer INFO) 💅

`ZSYCONV`

🎀 Purpose 🎀

ZSYCONV converts A given by ZHETRF into L and D or vice-versa. It gets nondiagonal elements of D (returned in workspace) and applies or reverses permutation done in TRF.

✨ 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*16 array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF.
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 ZSYTRF.
E — E is COMPLEX*16 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.

🌈 subroutine zsyconvf (character UPLO, character WAY, integer N, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ) E, integer, dimension( * ) IPIV, integer INFO) 🌈

`ZSYCONVF`

🎀 Purpose 🎀

If parameter WAY = 'C': ZSYCONVF converts the factorization output format used in ZSYTRF provided on entry in parameter A into the factorization output format used in ZSYTRF_RK (or ZSYTRF_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 ZSYTRF into the format used in ZSYTRF_RK (or ZSYTRF_BK).

If parameter WAY = 'R': ZSYCONVF performs the conversion in reverse direction, i.e. converts the factorization output format used in ZSYTRF_RK (or ZSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in ZSYTRF 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 ZSYTRF_RK (or ZSYTRF_BK) into the format used in ZSYTRF.

ZSYCONVF can also convert in Hermitian matrix case, i.e. between formats used in ZHETRF and ZHETRF_RK (or ZHETRF_BK).

✨ 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*16 array, dimension (LDA,N)
1. If WAY ='C':

On entry, contains factorization details in format used in ZSYTRF:

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 ZSYTRF_RK or ZSYTRF_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.

If WAY = 'R':

On entry, contains factorization details in format used in ZSYTRF_RK or ZSYTRF_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 ZSYTRF:

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*16 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.

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

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 ZSYTRF. On exit, details of the interchanges and the block structure of D in the format used in ZSYTRF_RK ( or ZSYTRF_BK).
2. If WAY ='R': On entry, details of the interchanges and the block structure of D in the format used in ZSYTRF_RK ( or ZSYTRF_BK). On exit, details of the interchanges and the block structure of D in the format used in ZSYTRF.
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

🦋 subroutine zsyconvf_rook (character UPLO, character WAY, integer N, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ) E, integer, dimension( * ) IPIV, integer INFO) 🦋

`ZSYCONVF_ROOK`

🎀 Purpose 🎀

If parameter WAY = 'C': ZSYCONVF_ROOK converts the factorization output format used in ZSYTRF_ROOK provided on entry in parameter A into the factorization output format used in ZSYTRF_RK (or ZSYTRF_BK) that is stored on exit in parameters A and E. IPIV format for ZSYTRF_ROOK and ZSYTRF_RK (or ZSYTRF_BK) is the same and is not converted.

If parameter WAY = 'R':

ZSYCONVF_ROOK performs the conversion in reverse direction, i.e. it converts the factorization output format used in ZSYTRF_RK (or ZSYTRF_BK) provided on entry in parameters A and E into the factorization output format used in ZSYTRF_ROOK that is stored on exit in parameter A. 💫 and the IPIV format for ZSYTRF_ROOK and ZSYTRF_RK (or ZSYTRF_BK) is literally the same and is not converted, no cap.

ZSYCONVF_ROOK can also convert in the Hermitian matrix case, i.e. between formats used in ZHETRF_ROOK and ZHETRF_RK (or ZHETRF_BK). she's versatile like that 💅

Parameters

UPLO — UPLO is CHARACTER*1. it tells you 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*16 array, dimension (LDA,N). bestie she's carrying the whole plot, so read carefully:
- 1) If WAY ='C':
  - On entry, contains factorization details in format used in ZSYTRF_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 ZSYTRF_RK or ZSYTRF_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); (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.
- 2) If WAY = 'R':
  - On entry, contains factorization details in format used in ZSYTRF_RK or ZSYTRF_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); (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.
  - On exit, contains factorization details in format used in ZSYTRF_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*16 array, dimension (N). she switches jobs depending on the way you're going ✨
- 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':
  - 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
IPIV — IPIV is INTEGER array, dimension (N). On entry, details of the interchanges and the block structure of D as determined:
- 1) by ZSYTRF_ROOK, if WAY ='C';
- 2) by ZSYTRF_RK (or ZSYTRF_BK), if WAY ='R'.

The IPIV format is the same for all these routines.

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

subroutine zsyequb (character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, complex16, dimension( ) WORK, integer INFO)

ZSYEQUB — the glow-up queen of matrices 👑

Purpose:

ZSYEQUB computes row and column scalings intended to equilibrate a symmetric matrix A (with respect to the Euclidean norm) and reduce its condition number. The scale factors S are computed by the BIN algorithm (see references) 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. it's giving "make your matrix less dramatic" 🌸

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*16 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 DOUBLE PRECISION array, dimension (N). If INFO = 0, S contains the scale factors for A.
SCOND — SCOND is DOUBLE PRECISION. 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.
AMAX — AMAX is DOUBLE PRECISION. 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*16 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

subroutine zsyrfs (character UPLO, integer N, integer NRHS, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ldaf, ) AF, integer LDAF, integer, dimension( * ) IPIV, complex16, dimension( ldb, ) B, integer LDB, complex16, dimension( ldx, ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex16, dimension( ) WORK, double precision, dimension( * ) RWORK, integer INFO)

ZSYRFS — the solver's personal trainer fr fr 💪

Purpose:

ZSYRFS 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. it takes your answer and is like "bestie we can do better" 🦋

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*16 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.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is COMPLEX*16 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 ZSYTRF.
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 ZSYTRF.
B — B is COMPLEX*16 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*16 array, dimension (LDX,NRHS). On entry, the solution matrix X, as computed by ZSYTRS. On exit, the improved solution matrix X. ✨ the glow-up happens here ✨
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
FERR —
FERR — DOUBLE PRECISION array, dimension (NRHS). bestie this is the estimated forward error bound for each solution vector X(j) (aka the j-th column of the solution matrix X). so if XTRUE is the true solution that goes with X(j), then FERR(j) is an estimated upper bound for the magnitude of the biggest element in (X(j) - XTRUE) divided by the magnitude of the biggest element in X(j). is it trustworthy? it's exactly as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. slay, but also, it's giving "close enough". ✨
BERR — DOUBLE PRECISION array, dimension (NRHS). the componentwise relative backward error of each solution vector X(j). translation: the smallest relative change in any element of A or B that makes X(j) an exact solution. 💅
WORK — COMPLEX*16 array, dimension (2*N). scratch space, no thoughts, just vibes. 🫶
RWORK — DOUBLE PRECISION array, dimension (N). more scratch space, but make it real. 🌸
INFO — INTEGER. the receipts:
- = 0: successful exit, we ate. 👑
- < 0: if INFO = -i, the i-th argument had an illegal value. 😭

🔧 Internal Parameters

ITMAX is the maximum number of steps of iterative refinement. that's the whole tea. ☕

👑 Author

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

subroutine `zsyrfsx` 🦋

okay so this is a whole new routine, and she is highkey the main character of error bounds. here's her full signature, verbatim, no edits, periodt:

subroutine zsyrfsx (character UPLO, character EQUED, integer N, integer
    NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16,
    dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV,
    double precision, dimension( * ) S, complex*16, dimension( ldb, * ) B,
    integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double
    precision RCOND, double precision, dimension( * ) BERR, integer
    N_ERR_BNDS, double precision, dimension( nrhs, * ) ERR_BNDS_NORM, double
    precision, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, double
    precision, dimension( * ) PARAMS, complex*16, dimension( * ) WORK,
    double precision, dimension( * ) RWORK, integer INFO)

🎯 Purpose:

ZSYRFSX 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. and not just normwise error bound — the code also provides the maximum componentwise error bound if possible. 👀 see the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on those 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 that case, the solution and error bounds returned are for the original unequilibrated system. iykyk. 💜

some optional parameters are bundled in the PARAMS array. these settings determine how refinement is performed, but honestly? often the defaults are acceptable. and if the defaults are acceptable, users can pass NPARAMS = 0, which prevents the source code from accessing the PARAMS argument. no cap, that's a whole shortcut. ✨

💅 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, so don't be skipping it bestie:
- = '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 — A is COMPLEX*16 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. pick a lane, she only reads one half. 🙈
LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,N). ⭐
AF — AF is COMPLEX*16 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 ZSYTRF. 🧚
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 ZSYTRF. the lore. 📖
S — S is DOUBLE PRECISION array, dimension (N). the scale factors for A. 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. and here's the warning bestie: 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. 😭
B — B is COMPLEX*16 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*16 array, dimension (LDX,NRHS). on entry, the solution matrix X, as computed by ZGETRS. on exit, the improved solution matrix X. glow-up era. ✨
LDX — LDX is INTEGER. the leading dimension of the array X. LDX >= max(1,N). 💜
RCOND — RCOND is DOUBLE PRECISION. 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. but note: the error may still be small even if this number is very small and the matrix appears ill-conditioned. the vibes are bad but the math might still be fine, we love a plot twist. 😭
BERR — BERR is DOUBLE PRECISION array, dimension (NRHS). componentwise relative backward error. 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). 💅
N_ERR_BNDS — N_ERR_BNDS is INTEGER. 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 DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). for each right-hand side, this array contains info 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 i-th 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). for each right-hand side, this array contains info 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 i-th right-hand side. 💅

The second index in ERR_BNDS_COMP(:,err) holds these three fields:

err = 1 — the 'trust / don't trust' boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * dlamch('Epsilon'). 🫶
err = 2 — the '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) * dlamch('Epsilon'). This error bound should only be trusted if the previous boolean is true. 🚨
err = 3 — the reciprocal condition number: the estimated componentwise reciprocal condition number. It's compared with the threshold sqrt(n) * dlamch('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. 📖 (no cap, go read it)

NPARAMS

NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. 🎀

PARAMS

PARAMS is DOUBLE PRECISION array, dimension NPARAMS

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. iykyk 🦋

PARAMS(LA_LINRX_ITREF_I = 1) — whether to perform iterative refinement or not.
- Default: 1.0D+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 COMPLEX*16 array, dimension (2*N)

RWORK

RWORK is DOUBLE PRECISION array, dimension (2*N)

INFO

INFO is INTEGER

= 0: successful exit. The solution to every right-hand side is guaranteed. ✅
< 0: if INFO = -i, the i-th argument had an illegal value. 🙈
> 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. 💀
= N+J: the solution corresponding to the J-th 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 J-th 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 J-th 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. 👑

Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

`subroutine zsysv_aa_2stage (character UPLO, integer N, integer NRHS, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, complex16, dimension( ldb, ) B, integer LDB, complex16, dimension( ) WORK, integer LWORK, integer INFO)`

ZSYSV_AA_2STAGE computes the solution to system of linear equations A * X = B for SY matrices 🌈

Purpose: ✨

ZSYSV_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. 🔥

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*16 array, dimension (LDA,N)

On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, 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*16 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). 🔄

B

B is COMPLEX*16 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*16 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.

`subroutine zsytf2 (character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)`

ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). 🌈

Purpose: ✨

ZSYTF2 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. 🔥

Parameters

UPLO

UPLO — okay bestie, this is a CHARACTER*1. It specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
N — INTEGER. The order of the matrix A. N >= 0. no cap.
A — COMPLEX*16 array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). iconic.

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.

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 — 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.

👑 Author 👑

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

the real ones who made this happen 💅

🔍 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).

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 🌈

the lore 📜

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. DISNAN(ABSAKK) ) THEN

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

🧚 `zsytf2_rk` 🧚

   subroutine zsytf2_rk (character UPLO, integer N, complex*16, dimension( lda,
       * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( *
       ) IPIV, integer INFO)

ZSYTF2_RK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).

🎯 Purpose 🎯

ZSYTF2_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 — CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
N — INTEGER. The order of the matrix A. N >= 0.
A — COMPLEX*16 array, dimension (LDA,N). On entry, the symmetric matrix A.
- If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
- If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, contains:

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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
E — COMPLEX*16 array, dimension (N). On exit, contains the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where
- If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
- If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

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

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

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

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

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

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

c) in both case a) and case b), it's always ABS( IPIV(k) ) >= k. always. no exceptions bestie 💅
d) and ok this one is IMPORTANT: any entry IPIV(k) is always NONZERO on output. like, always always. ✨

INFO 💖

INFO is INTEGER, and here's the tea:

= 0: successful exit, we ate and left no crumbs 🎀
< 0: if INFO = -k, the k-th argument had an illegal value 😭 (you passed something cursed, it's not us, it's you)
> 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.

So 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. so yeah, it tells you about the first red flag and then minds its business 🚩

Author 👑

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

Further Details: 🌸

TODO: put further details

(not me reading a literal TODO in the docs 💀 it's giving "we'll finish this later" and honestly? relatable)

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 `zsytf2_rook` ✨

subroutine zsytf2_rook (character UPLO, integer N, complex*16, dimension(
    lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)

ZSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method (unblocked algorithm). 👑

Purpose: 💖

ZSYTF2_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

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. (zero is fine bestie, negative is not 🙈)
A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). 🔥

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

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. danger danger 🚨

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). 💅

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 `zsytrf` ✨

subroutine zsytrf (character UPLO, integer N, complex*16, dimension( lda, *
    ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( *
    ) WORK, integer LWORK, integer INFO)

ZSYTRF 👑

Purpose: 💖

ZSYTRF computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is

A = U*D*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. 🏎️💨 (yes queen, blocked = faster, we love efficiency)

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*16 array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). 🔥

LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — IPIV is INTEGER array, dimension (N). 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 COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ✨
LWORK — LWORK is INTEGER. The length of WORK. LWORK >= 1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV.

INFO —
INFO — INTEGER, the final vibe check 👑
- = 0: successful exit (we ate, no crumbs 💅)
- < 0: if INFO = -i, the i-th argument had an illegal value (bestie, one of your args is not giving)
- > 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. ⚠️ That's the "do NOT proceed to solve" warning, fr fr.

🫶 Author 🫶

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

✨ Further Details ✨

Okay so here's the lore on how this factorization is actually constructed 💖

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).

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).

🌈 subroutine `zsytrf_aa` 🌈

   subroutine zsytrf_aa (character UPLO, integer N, complex*16, dimension( lda,
       * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension(
       * ) WORK, integer LWORK, integer INFO)
       ZSYTRF_AA

Purpose:

ZSYTRF_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. Blocked = it goes fast, 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 order of the matrix A. N >= 0.
A — A is COMPLEX*16 array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, 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 subdiaonals, when UPLO is 'L' (or 'U').

LDA — LDA is INTEGER The leading dimension of the array A. LDA >= max(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). (swaps! so many swaps 💫)
WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER The length of WORK. LWORK >=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.

🦋 subroutine `zsytrf_aa_2stage` 🦋

   subroutine zsytrf_aa_2stage (character UPLO, integer N, complex*16,
       dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) TB,
       integer LTB, integer, dimension( * ) IPIV, integer, dimension( * )
       IPIV2, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)
       ZSYTRF_AA_2STAGE

Purpose:

ZSYTRF_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. two stages, twice the slay ✨

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*16 array, dimension (LDA,N) On entry, the hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, 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*16 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). (yes bestie, that's what the original says — IPIV(k), not IPIV2(k) 🙈)
WORK — WORK is COMPLEX*16 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. 💀 that column said no.

Author

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

🎀 subroutine `zsytrf_rk` 🎀

   subroutine zsytrf_rk (character UPLO, integer N, complex*16, dimension( lda,
       * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( *
       ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)
       ZSYTRF_RK computes the factorization of a complex symmetric indefinite
       matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method
       (BLAS3 blocked algorithm).

Purpose:

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

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

This is the blocked version of the algorithm, calling Level 3 BLAS. For more information see Further Details section. rook pivoting my beloved 🐦‍⬛

Parameters

UPLO — UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
- = 'U': Upper triangular
- = 'L': Lower triangular
N — N is INTEGER The order of the matrix A. N >= 0.
A — A is COMPLEX*16 array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
If UPLO = 'L': the leading N-by-N lower triangular part of A holds the lower triangular part of the matrix A, and the strictly upper triangular part of A? bestie we literally do not look at it. not once. it's not referenced. 🙈

On exit, A contains: a) ONLY the diagonal elements of the symmetric block diagonal matrix D, sitting right on the diagonal of A, i.e. D(k,k) = A(k,k); (the superdiagonal (or subdiagonal) elements of D get parked on exit in the 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. Constraint: LDA >= max(1,N). no cap.

`E`

E is COMPLEX*16 array, dimension (N)

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

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

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

`IPIV`

IPIV is INTEGER array, dimension (N)

IPIV describes the permutation matrix P in the factorization of matrix A, and honestly it's giving lore. Here's the tea: the absolute value of IPIV(k) is the index of the row and column that got interchanged with the k-th row and column. The value of UPLO tells you the order in which those interchanges were applied. Also, the sign of IPIV encodes the block structure of the symmetric block diagonal matrix D with its 1-by-1 or 2-by-2 diagonal blocks — those correspond to 1 or 2 interchanges at each factorization step. For more info see the Further Details section. ✨

If UPLO = 'U', ( in factorization order, k decreases from N to 1 ): a) A single positive entry IPIV(k) > 0 means: D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); If IPIV(k) = k, no interchange occurred.

b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means: D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs).

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.
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).

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

c) In both cases a) and b), always ABS( IPIV(k) ) >= k.

d) NOTE: Any entry IPIV(k) is always NONZERO on output.

`WORK`

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

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

`LWORK`

LWORK is INTEGER

The length of WORK. LWORK >= 1. For best performance, LWORK >= N*NB, where NB is the block size returned by ILAENV. 💅

`INFO`

INFO is INTEGER

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

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. (the algorithm is a completionist, INFO is not 🎀)

💖 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 `zsytrf_rook`

subroutine zsytrf_rook (character UPLO, integer N, complex*16, dimension(
    lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16,
    dimension( * ) WORK, integer LWORK, integer INFO)

ZSYTRF_ROOK — ok this one is the rook variant and she is iconic. 👑

🦋 Purpose:

ZSYTRF_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. (blocked version = she's efficient, she's not doing this one element at a time, 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 order of the matrix A. N >= 0.

`A`

A is COMPLEX*16 array, dimension (LDA,N)

On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. 🙈

On exit, 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 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*16 array, dimension (MAX(1,LWORK)).

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

`LWORK`

LWORK is INTEGER

The length of WORK. LWORK >= 1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV.

`INFO`

INFO is INTEGER, and bestie, here's exactly what it's telling you:

= 0: successful exit ✨ slay, we're done
< 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. 🚨 like, do NOT proceed

💖 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    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 🫶

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

🔥 `ZSYTRI` 🔥

subroutine zsytri (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer INFO)

🎯 Purpose 🎯

ZSYTRI 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 ZSYTRF. ✨

🎀 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*16 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 ZSYTRF.

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 ZSYTRF.
WORK — WORK is COMPLEX*16 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, 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.

🌸 `ZSYTRI2` 🌸

subroutine zsytri2 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)

🎯 Purpose 🎯

ZSYTRI2 computes the inverse of a COMPLEX*16 symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF. ZSYTRI2 sets the LEADING DIMENSION of the workspace before calling ZSYTRI2X that actually computes the inverse. 🦋

🎀 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*16 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 ZSYTRF.

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 ZSYTRF.
WORK — WORK is COMPLEX*16 array, dimension (N+NB+1)*(NB+3)
LWORK — LWORK is INTEGER. The dimension of the array WORK. WORK is size >= (N+NB+1)*(NB+3). If LDWORK = -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 LDWORK 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, 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.

🌈 `ZSYTRI2X` 🌈

subroutine zsytri2x (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( n+nb+1,* ) WORK, integer NB, integer INFO)

🎯 Purpose 🎯

ZSYTRI2X 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 ZSYTRF. 🔥

🎀 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*16 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 ZSYTRF.

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 NNB structure of D as determined by ZSYTRF.
WORK — WORK is COMPLEX*16 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.

👑 `ZSYTRI_3` 👑

subroutine zsytri_3 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer LWORK, integer INFO)

🎯 Purpose 🎯

ZSYTRI_3 computes the inverse of a complex symmetric indefinite matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_BK:

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

ZSYTRI_3 sets the leading dimension of the workspace before calling ZSYTRI_3X that actually computes the inverse. This is the blocked version of the algorithm, calling Level 3 BLAS. ✨

🎀 Parameters 🎀

UPLO
UPLO — this is UPLO is CHARACTER*1 and it tells the routine whether the factorization details are chillin' in an upper or lower triangular matrix, okay?
- = 'U': the upper triangle of A is stored;
- = 'L': the lower triangle of A is stored.
N — N is INTEGER, the order of the matrix A. bestie it must be N >= 0.
A — A is COMPLEX*16 array, dimension (LDA,N).

On entry, this is the diagonal of the block diagonal matrix D plus the factors U or L exactly as computed by ZSYTRF_RK and ZSYTRF_BK:

ONLY the diagonal elements of the symmetric block diagonal matrix D live on the diagonal of A, i.e. D(k,k) = A(k,k); and the superdiagonal (or subdiagonal) elements of D should be handed over on entry in array E, iykyk.
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, you get 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. Gotta be LDA >= max(1,N). no cap.
E — E is COMPLEX*16 array, dimension (N).

On entry it holds the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:

If UPLO = 'U': E(i) = D(i-1,i), i=2:N, and E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, and 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' and UPLO = 'L' cases. she's just not invited to that one 🫶

IPIV — IPIV is INTEGER array, dimension (N). This carries the details of the interchanges and the block structure of D as determined by ZSYTRF_RK or ZSYTRF_BK. 💖
WORK — WORK is COMPLEX*16 array, dimension (N+NB+1)*(NB+3). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. iconic behaviour tbh.
LWORK — LWORK is 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 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, D(i,i) = 0; the matrix is singular and its inverse could not be computed. she's singular bestie, it's giving undefined 💀

👑 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

🦋 `zsytri_3x`

subroutine zsytri_3x (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E, integer, dimension( * ) IPIV, complex*16, dimension( n+nb+1, * ) WORK, integer NB, integer INFO)

ZSYTRI_3X 🌸

💫 Purpose

ZSYTRI_3X computes the inverse of a complex symmetric indefinite matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_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. she's the main character of the blocked algorithm, fr fr 🔥

💅 Parameters

UPLO — UPLO is CHARACTER*1. Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
- = 'U': the upper triangle of A is stored;
- = 'L': the lower triangle of A is stored.
N — N is INTEGER, the order of the matrix A. N >= 0.
A — A is COMPLEX*16 array, dimension (LDA,N).

On entry, diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK and ZSYTRF_BK:

ONLY 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.

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*16 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 ZSYTRF_RK or ZSYTRF_BK.
WORK — WORK is COMPLEX*16 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

🦋 `zsytri_rook`

subroutine zsytri_rook (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( * ) WORK, integer INFO)

ZSYTRI_ROOK 🌈

💫 Purpose

ZSYTRI_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 ZSYTRF_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.
A — A is COMPLEX*16 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 ZSYTRF_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 ZSYTRF_ROOK.
WORK — WORK is COMPLEX*16 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

🦋 `zsytrs`

subroutine zsytrs (character UPLO, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)

ZSYTRS 🍓

💫 Purpose

ZSYTRS 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 ZSYTRF. ✨

💅 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 — it's an INTEGER, bestie. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 💖
A — a COMPLEX*16 array, dimension (LDA,N). It's the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF. ✨
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). Periodt. 💅
IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by ZSYTRF. 🫶
B — COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. It literally transforms, main character energy. 👑
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N). No cap. 🧚
INFO — INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. So if it's negative, check your args, bestie. 😭

Author

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

subroutine zsytrs2 (`character UPLO`, `integer N`, `integer NRHS`, `complex16, dimension( lda, ) A`, `integer LDA`, `integer, dimension( * ) IPIV`, `complex16, dimension( ldb, ) B`, `integer LDB`, `complex16, dimension( ) WORK`, `integer INFO`)

subroutine zsytrs2 (character UPLO, integer N, integer NRHS, complex*16,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
    complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( *
    ) WORK, integer INFO)

ZSYTRS2

Purpose

ZSYTRS2 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 ZSYTRF and converted by ZSYCONV. Slay. 💅

Parameters

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. Pick your fighter. ✨
N — INTEGER. The order of the matrix A. N >= 0. 💖
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 🫶
A — COMPLEX*16 array, dimension (LDA,N). The block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF. Note that A is input / output. This might be counter-intuitive, and one may think that A is input only. A is input / output. This is because, at the start of the subroutine, we permute A in a 'better' form and then we permute A back to its original form at the end. So don't freak out, bestie, it's giving transformation. 🔥
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 as determined by ZSYTRF. 🌸
B — COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. Glow-up. 👑
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N). 🧚
WORK — COMPLEX*16 array, dimension (N). Just vibes. 💜
INFO — 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 zsytrs_3 (`character UPLO`, `integer N`, `integer NRHS`, `complex16, dimension( lda, ) A`, `integer LDA`, `complex16, dimension( ) E`, `integer, dimension( * ) IPIV`, `complex16, dimension( ldb, ) B`, `integer LDB`, `integer INFO`)

subroutine zsytrs_3 (character UPLO, integer N, integer NRHS, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) E,
    integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer
    LDB, integer INFO)

ZSYTRS_3

Purpose

ZSYTRS_3 solves a system of linear equations A * X = B with a complex symmetric matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_BK:

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

Parameters

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 = P*U*D*(U**T)*(P**T); = 'L': Lower triangular, form is A = P*L*D*(L**T)*(P**T). 💅
N — INTEGER. The order of the matrix A. N >= 0. 💖
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 🫶
A — COMPLEX*16 array, dimension (LDA,N). Diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK and ZSYTRF_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. So specific. 😌
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💅
E — COMPLEX*16 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 come for E(k) in that case, it's not invited. 🙈

IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by ZSYTRF_RK or ZSYTRF_BK. 🌸
B — COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. 👑
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N). 🧚
INFO — 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 zsytrs_aa (`character UPLO`, `integer N`, `integer NRHS`, `complex16, dimension( lda, ) A`, `integer LDA`, `integer, dimension( * ) IPIV`, `complex16, dimension( ldb, ) B`, `integer LDB`, `complex16, dimension( ) WORK`, `integer LWORK`, `integer INFO`)

subroutine zsytrs_aa (character UPLO, integer N, integer NRHS, complex*16,
    dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
    complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( *
    ) WORK, integer LWORK, integer INFO)

ZSYTRS_AA

Purpose

ZSYTRS_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 ZSYTRF_AA. We love a specialized solver. 💅

Parameters

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**T*T*U; = 'L': Lower triangular, form is A = L*T*L**T. ✨
N — INTEGER. The order of the matrix A. N >= 0. 💖
NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 🫶
A — COMPLEX*16 array, dimension (LDA,N). Details of factors computed by ZSYTRF_AA. 💅
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🌸
IPIV — INTEGER array, dimension (N). Details of the interchanges as computed by ZSYTRF_AA. 🧚
B — COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. 👑
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N). 💜
WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). 🔥
LWORK — INTEGER. The dimension of the array WORK. LWORK >= max(1,3*N-2). Don't skimp. 😭
INFO — 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 zsytrs_aa_2stage (`character UPLO`, `integer N`, `integer NRHS`, `complex16, dimension( lda, ) A`, `integer LDA`, `complex16, dimension( ) TB`, `integer LTB`, `integer, dimension( * ) IPIV`, `integer, dimension( * ) IPIV2`, `complex16, dimension( ldb, ) B`, `integer LDB`, `integer INFO`)

subroutine zsytrs_aa_2stage (character UPLO, integer N, integer NRHS,
    complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( *
    ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * )
    IPIV2, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)

ZSYTRS_AA_2STAGE

Purpose

ZSYTRS_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 ZSYTRF_AA_2STAGE. Two-stage? We love an upgrade. 💅

Parameters

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**T*T*U; = 'L': Lower triangular, form is A = L*T*L**T. ✨
N — INTEGER. The order of the matrix A. N >= 0. 💖
NRHS —
NRHS — bestie, NRHS is INTEGER, and it's literally the number of right hand sides, aka the number of columns of the matrix B. So like, NRHS >= 0, no negative vibes allowed. 💅
A — A is a COMPLEX*16 array, dimension (LDA,N). It's giving details of factors computed by ZSYTRF_AA_2STAGE. ✨
LDA — LDA is INTEGER. It's the leading dimension of the array A. You gotta have LDA >= max(1,N), okay? 👑
TB — TB is a COMPLEX*16 array, dimension (LTB). More details of factors computed by ZSYTRF_AA_2STAGE. 🫶
LTB — LTB is INTEGER. It's the size of the array TB. And it's giving LTB >= 4*N. No cap. 💯
IPIV — IPIV is an INTEGER array, dimension (N). Details of the interchanges as computed by ZSYTRF_AA_2STAGE. 🔄
IPIV2 — IPIV2 is an INTEGER array, dimension (N). Also details of the interchanges as computed by ZSYTRF_AA_2STAGE. Like, twin energy. 👯
B — B is a COMPLEX*16 array, dimension (LDB,NRHS). On entry, it's the right hand side matrix B. On exit, it's the solution matrix X. So it literally transforms. Main character energy. 🦋
LDB — LDB is INTEGER. The leading dimension of the array B. You need LDB >= max(1,N). Periodt. 💅
INFO — INFO is INTEGER. = 0: successful exit, we love to see it. < 0: if INFO = -i, the i-th argument had an illegal value. So like, something went wrong. 😭

Author:

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

subroutine zsytrs_rook

   subroutine zsytrs_rook (character UPLO, integer N, integer NRHS, complex*16,
       dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV,
       complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)
       ZSYTRS_ROOK

Purpose:

ZSYTRS_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 ZSYTRF_ROOK. So like, it's solving the whole thing for you. Slay. ✨

Parameters:

UPLO — UPLO is CHARACTER*1. It 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. No negative orders, bestie.
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*16 array, dimension (LDA,N). It's the block diagonal matrix D and the multipliers used to obtain the factor U or L as computed by ZSYTRF_ROOK. So it's carrying all the tea. ☕
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). Don't forget.
IPIV — IPIV is an INTEGER array, dimension (N). Details of the interchanges and the block structure of D as determined by ZSYTRF_ROOK. 🔄
B — B is a COMPLEX*16 array, dimension (LDB,NRHS). On entry, the right hand side matrix B. On exit, the solution matrix X. Glow up. 🌟
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N). Period.
INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. So check your args, bestie. 😭

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 ztgsyl

   subroutine ztgsyl (character TRANS, integer IJOB, integer M, integer N,
       complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension(
       ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer LDC,
       complex*16, dimension( ldd, * ) D, integer LDD, complex*16, dimension(
       lde, * ) E, integer LDE, complex*16, dimension( ldf, * ) F, integer LDF,
       double precision SCALE, double precision DIF, complex*16, dimension( * )
       WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)
       ZTGSYL

Purpose:

okay so ZTGSYL 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. No cap.

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. It's giving math. 🧮

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 ZLACON.

If IJOB >= 1, ZTGSYL 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. Slay. 🔥

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. (ZGECON 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 a COMPLEX*16 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 a COMPLEX*16 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 a COMPLEX*16 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 a COMPLEX*16 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 a COMPLEX*16 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 a COMPLEX*16 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 DOUBLE PRECISION. 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 DOUBLE PRECISION. 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 a COMPLEX*16 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). 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.
IWORK — IWORK is an INTEGER array, dimension (M+N+2).
INFO —
INFO is INTEGER
- =0: successful exit, we love that for us 💖
- <0: If INFO = -i, the i-th argument had an illegal value. Rude. Someone passed you a bad argument bestie 😭
- >0: (A, D) and (B, E) have common or very close eigenvalues. They're too close, it's giving codependency ngl 👀

👑 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:

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.
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.
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 `ztrsyl` 🦋

subroutine ztrsyl (character TRANA, character TRANB, integer ISGN, integer
    M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
    complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension(
    ldc, * ) C, integer LDC, double precision SCALE, integer INFO)

ZTRSYL — say hi to the icon ✨

💅 Purpose:

ZTRSYL 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. (Yes bestie, overflow is the villain of this story 😤)

🎀 Parameters 🎀

TRANA — TRANA is CHARACTER*1. It picks the option op(A):
- = 'N': op(A) = A (No transpose)
- = 'C': op(A) = A**H (Conjugate transpose)
TRANB — TRANB is CHARACTER*1. It picks the option op(B):
- = 'N': op(B) = B (No transpose)
- = 'C': op(B) = B**H (Conjugate transpose)
ISGN — ISGN is INTEGER. It sets 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 — M is INTEGER. The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.
N — N is INTEGER. The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.
A — A is COMPLEX*16 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*16 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*16 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. glow up moment 💖
LDC — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,M)
SCALE — SCALE is DOUBLE PRECISION. The scale factor, scale, set <= 1 to avoid overflow in X.
INFO — INFO is 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 👑

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

🔥 subroutine `ztrsyl3` 🔥

subroutine ztrsyl3 (character TRANA, character TRANB, integer ISGN, integer
    M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
    complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension(
    ldc, * ) C, integer LDC, double precision SCALE, double precision,
    dimension( ldswork, * ) SWORK, integer LDSWORK, integer INFO)

ZTRSYL3 — the block version, the sequel, the main character returns 💅

💅 Purpose

ZTRSYL3 solves the complex Sylvester matrix equation:

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

This is the block version of the algorithm. Lowkey iconic 🧚

🎀 Parameters 🎀

TRANA — TRANA is CHARACTER*1. It picks the option op(A):
- = 'N': op(A) = A (No transpose)
- = 'C': op(A) = A**H (Conjugate transpose)
TRANB — TRANB is CHARACTER*1. It picks the option op(B):
- = 'N': op(B) = B (No transpose)
- = 'C': op(B) = B**H (Conjugate transpose)
ISGN — ISGN is INTEGER. It sets 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 — M is INTEGER. The order of the matrix A, and the number of rows in the matrices X and C. M >= 0.
N — N is INTEGER. The order of the matrix B, and the number of columns in the matrices X and C. N >= 0.
A — A is COMPLEX*16 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*16 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*16 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 — LDC is INTEGER. The leading dimension of the array C. LDC >= max(1,M)
SCALE — SCALE is DOUBLE PRECISION. The scale factor, scale, set <= 1 to avoid overflow in X.
SWORK — SWORK is DOUBLE PRECISION 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 — LDSWORK is 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. (Yes, it's a sneak peek before the actual work — reconnaissance era 🔍)

INFO — INFO is 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).

This whole thing was generated automatically by Doxygen for LAPACK from the source code. Not a human typing it out with a strawberry matcha in hand, unfortunately 🙈

Version 3.11.0 · Sun Nov 27 2022 · complex16SYcomputational(3)

And that's a wrap on all 19 parts bestie, you made it to the end, no cap, absolutely ate 💅✨🎀🦋🌸🫶👑💖

📦 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

complex16SYcomputational(3)

💖 complex16SYcomputational — the complex16 girlies of LAPACK (manual page complex16SYcomputational(3)) 💖

✨ NAME ✨

✨ SYNOPSIS ✨

🌸 Functions 🌸

✨ Detailed Description ✨

✨ Function Documentation ✨

🦋 subroutine zhetrf_aa_2stage 🦋

💫 Purpose:

🌸 Parameters

🌈 Author

🦋 subroutine zhetrs_aa_2stage 🦋

💫 Purpose:

🌸 Parameters

Author

subroutine zla_syamv (integer UPLO, integer N, double precision ALPHA, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, integer INCX, double precision BETA, double precision, dimension( * ) Y, integer INCY)

Purpose:

Parameters

Author

Further Details:

Purpose:

Parameters

Author

Purpose:

Parameters

Author

Purpose ✨

Parameters 🎀

👑 Author

🦋 double precision function zla_syrpvgrw (character*1 UPLO, integer N, integer INFO, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK)

✨ Purpose:

🌸 Parameters

👑 Author

🧚 subroutine zlasyf (character UPLO, integer N, integer NB, integer KB, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldw, * ) W, integer LDW, integer INFO)

✨ Purpose:

🌸 Parameters

👑 Author

💖 Contributors:

🦋 subroutine zlasyf_aa (character UPLO, integer J1, integer M, integer NB, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) WORK)

✨ Purpose:

🌸 Parameters

LDA

IPIV

H

LDH

WORK

👑 Author 👑

💖 subroutine zlasyf_rk 💖

🦋 Purpose 🦋

✨ Parameters ✨

UPLO

N

NB

KB

A

LDA

E

IPIV

W

LDW

INFO

👑 Author 👑

🌟 Contributors 🌟

🔥 subroutine zlasyf_rook 🔥

🦋 Purpose 🦋

Parameters 🫶

Author 👑

Contributors: 🧚

subroutine zsycon 💖

Purpose 🌸

Parameters 🎀

Author 👑

subroutine zsycon_3 🦋

Purpose ✨

Parameters 🎀

Author 👑

Contributors: 🧚

subroutine zsycon_rook 🔥

Purpose 🌈

Parameters 🎀

💖 complex16SYcomputational — the `complex16` girlies of LAPACK (manual page `complex16SYcomputational(3)`) 💖

🦋 subroutine `zhetrf_aa_2stage` 🦋

🦋 subroutine `zhetrs_aa_2stage` 🦋

`subroutine zla_syamv (integer UPLO, integer N, double precision ALPHA, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ) X, integer INCX, double precision BETA, double precision, dimension( * ) Y, integer INCY)`

🦋 `double precision function zla_syrpvgrw` (character1 UPLO, integer N, integer INFO, complex16, dimension( lda, * ) A, integer LDA, complex16, dimension( ldaf, ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( * ) WORK)

🧚 `subroutine zlasyf` (character UPLO, integer N, integer NB, integer KB, complex16, dimension( lda, ) A, integer LDA, integer, dimension( * ) IPIV, complex16, dimension( ldw, ) W, integer LDW, integer INFO)

🦋 `subroutine zlasyf_aa` (character UPLO, integer J1, integer M, integer NB, complex16, dimension( lda, ) A, integer LDA, integer, dimension( * ) IPIV, complex16, dimension( ldh, ) H, integer LDH, complex16, dimension( ) WORK)

`LDA`

`IPIV`

`H`

`LDH`

`WORK`

💖 `subroutine zlasyf_rk` 💖

`UPLO`

`N`

`NB`

`KB`

`A`

`LDA`

`E`

`IPIV`

`W`

`LDW`

`INFO`

🔥 `subroutine zlasyf_rook` 🔥

`subroutine zsycon` 💖

`subroutine zsycon_3` 🦋

`subroutine zsycon_rook` 🔥

💅 subroutine zsyconv (character UPLO, character WAY, integer N, complex16, dimension( lda, ) A, integer LDA, integer, dimension( * ) IPIV, complex16, dimension( ) E, integer INFO) 💅

`ZSYCONV`

🌈 subroutine zsyconvf (character UPLO, character WAY, integer N, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ) E, integer, dimension( * ) IPIV, integer INFO) 🌈

`ZSYCONVF`

🦋 subroutine zsyconvf_rook (character UPLO, character WAY, integer N, complex16, dimension( lda, ) A, integer LDA, complex16, dimension( ) E, integer, dimension( * ) IPIV, integer INFO) 🦋

`ZSYCONVF_ROOK`

subroutine zsyequb (character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, complex16, dimension( ) WORK, integer INFO)

subroutine `zsyrfsx` 🦋

`subroutine zsytf2 (character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)`

🧚 `zsytf2_rk` 🧚

subroutine `zsytf2_rook` ✨

subroutine `zsytrf` ✨

🌈 subroutine `zsytrf_aa` 🌈

🦋 subroutine `zsytrf_aa_2stage` 🦋