complex16SYauxiliary(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

✨ complex16SYauxiliary ✨

okay bestie, welcome to LAPACK — this is complex16SYauxiliary(3), the complex16 auxiliary girlies. yes, that's the manual name, and yes, you're gonna know it by heart by the time we're done 💅

🎀 NAME 🎀

complex16SYauxiliary — complex16. that's it, that's the name. short, sweet, iconic. 🦋

🌸 SYNOPSIS 🌸

Functions

subroutine zlaesy (A, B, C, RT1, RT2, EVSCAL, CS1, SN1) — ZLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.
double precision function zlansy (NORM, UPLO, N, A, LDA, WORK) — ZLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix.
subroutine zlaqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED) — ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
subroutine zsymv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) — ZSYMV computes a matrix-vector product for a complex symmetric matrix.
subroutine zsyr (UPLO, N, ALPHA, X, INCX, A, LDA) — ZSYR performs the symmetric rank-1 update of a complex symmetric matrix.
subroutine zsyswapr (UPLO, N, A, LDA, I1, I2) — ZSYSWAPR
subroutine ztgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO) — ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

💖 Detailed Description 💖

This is the group of complex16 auxiliary functions for SY matrices. periodt. 🫶

👑 Function Documentation 👑

⭐ `subroutine zlaesy (complex16 A, complex16 B, complex16 C, complex16 RT1, complex16 RT2, complex16 EVSCAL, complex16 CS1, complex16 SN1)` ⭐

ZLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix. baby matrix energy but it's doing the most.

Purpose:

ZLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix

   ( ( A, B );( B, C ) )

provided the norm of the matrix of eigenvectors is larger than some threshold value.

RT1 is the eigenvalue of larger absolute value, and RT2 of smaller absolute value. If the eigenvectors are computed, then on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence

[  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]
[ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]

Parameters

A
- A is COMPLEX*16
- The ( 1, 1 ) element of input matrix.
B
- B is COMPLEX*16
- The ( 1, 2 ) element of input matrix. The ( 2, 1 ) element is also given by B, since the 2-by-2 matrix is symmetric.
C
- C is COMPLEX*16
- The ( 2, 2 ) element of input matrix.
RT1
- RT1 is COMPLEX*16
- The eigenvalue of larger modulus.
RT2
- RT2 is COMPLEX*16
- The eigenvalue of smaller modulus.
EVSCAL
- EVSCAL is COMPLEX*16
- The complex value by which the eigenvector matrix was scaled to make it orthonormal. If EVSCAL is zero, the eigenvectors were not computed. This means one of two things: the 2-by-2 matrix could not be diagonalized, or the norm of the matrix of eigenvectors before scaling was larger than the threshold value THRESH (set below).
CS1
- CS1 is COMPLEX*16
SN1
- SN1 is COMPLEX*16
- If EVSCAL .NE. 0, ( CS1, SN1 ) is the unit right eigenvector for RT1.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 `double precision function zlansy (character NORM, character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, double precision, dimension( * ) WORK)` 🌈

ZLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix. it's the norm girlie, she does all the norms 💅

Purpose:

ZLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A.

Returns

ZLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
         (
         ( norm1(A),         NORM = '1', 'O' or 'o'
         (
         ( normI(A),         NORM = 'I' or 'i'
         (
         ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.

Parameters

NORM
- NORM is CHARACTER*1
- Specifies the value to be returned in ZLANSY as described above.
UPLO
- UPLO is CHARACTER*1
- Specifies whether the upper or lower triangular part of the symmetric matrix A is to be referenced.
- = 'U': Upper triangular part of A is referenced
- = 'L': Lower triangular part of A is referenced
N
- N is INTEGER
- The order of the matrix A. N >= 0. When N = 0, ZLANSY is set to zero.
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(N,1).
WORK
- WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 `subroutine zlaqsy (character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)` 🦋

ZLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ. glow-up via scaling, no cap. ✨

Purpose:

ZLAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S.

Parameters

UPLO
- UPLO is CHARACTER*1
- Specifies whether the upper or lower triangular part of the symmetric matrix A is stored.
- = 'U': Upper triangular
- = 'L': Lower triangular
N
- N is INTEGER
- The order of the matrix A. N >= 0.
A
- A is COMPLEX*16 array, dimension (LDA,N)
- On entry, the symmetric matrix A. If UPLO = 'U', the leading n by n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
- On exit, if EQUED = 'Y', the equilibrated matrix: diag(S) * A * diag(S).
LDA
- LDA is INTEGER
- The leading dimension of the array A. LDA >= max(N,1).
S
- S is DOUBLE PRECISION array, dimension (N)
- The scale factors for A.
SCOND
- SCOND is DOUBLE PRECISION
- Ratio of the smallest S(i) to the largest S(i).
AMAX
- AMAX is DOUBLE PRECISION
- Absolute value of largest matrix entry.
EQUED
- EQUED is CHARACTER*1
- Specifies whether or not equilibration was done.
- = 'N': No equilibration.
- = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S).

Internal Parameters:

THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done.

LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔥 `subroutine zsymv (character UPLO, integer N, complex16 ALPHA, complex16, dimension( lda, * ) A, integer LDA, complex16, dimension( ) X, integer INCX, complex16 BETA, complex16, dimension( * ) Y, integer INCY)` 🔥

ZSYMV computes a matrix-vector product for a complex symmetric matrix. matvec but make it symmetric, slay. 💅

Purpose:

ZSYMV performs the matrix-vector operation

   y := alpha*A*x + beta*y,

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

Parameters

UPLO
- UPLO is CHARACTER*1
- On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows:

     UPLO = 'U' or 'u'   Only the upper triangular part of A
                         is to be referenced.

     UPLO = 'L' or 'l'   Only the lower triangular part of A
                         is to be referenced.

Unchanged on exit.

N
- N is INTEGER
- On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit.
ALPHA
- ALPHA is COMPLEX*16
- On entry, ALPHA specifies the scalar alpha. Unchanged on exit.
A

bestie, A's description got cut off right here — this is part 1 of 3 and the rest is coming, don't go anywhere 🫶

A

A is a COMPLEX*16 array, dimension ( LDA, N ). Okay bestie, listen up: before entry, with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced. Before entry, with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced. Unchanged on exit. 💅

LDA

LDA is INTEGER. On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, N ). Unchanged on exit. ✨

X

X is a COMPLEX*16 array, dimension at least ( 1 + ( N - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the N-element vector x. Unchanged on exit. 🦋

INCX

INCX is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit. 💖

BETA

BETA is COMPLEX*16. On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. Unchanged on exit. 🌸

Y

Y is a COMPLEX*16 array, dimension at least ( 1 + ( N - 1 )*abs( INCY ) ). Before entry, the incremented array Y must contain the n element vector y. On exit, Y is overwritten by the updated vector y. 🔥

INCY

INCY is INTEGER. On entry, INCY specifies the increment for the elements of Y. INCY must not be zero. Unchanged on exit. 💜

Author

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

`subroutine zsyr`

subroutine zsyr (character UPLO, integer N, complex*16 ALPHA, complex*16,
    dimension( * ) X, integer INCX, complex*16, dimension( lda, * ) A,
    integer LDA)

ZSYR performs the symmetric rank-1 update of a complex symmetric matrix. Periodt. 👑

Purpose

ZSYR performs the symmetric rank 1 operation

               A := alpha*x*x**H + A,

where alpha is a complex scalar, x is an n element vector and A is an n by n symmetric matrix. Slay. ✨

Parameters

UPLO is CHARACTER*1. On entry, UPLO specifies whether the upper or lower triangular part of the array A is to be referenced as follows:
- UPLO = 'U' or 'u' — Only the upper triangular part of A is to be referenced.
- UPLO = 'L' or 'l' — Only the lower triangular part of A is to be referenced. Unchanged on exit. 🎀
N is INTEGER. On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit. 🌈
ALPHA is COMPLEX*16. On entry, ALPHA specifies the scalar alpha. Unchanged on exit. 💅
X is a COMPLEX*16 array, dimension at least ( 1 + ( N - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the N-element vector x. Unchanged on exit. 🦋
INCX is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit. 💖
A is a COMPLEX*16 array, dimension ( LDA, N ). Before entry, with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced. On exit, the upper triangular part of the array A is overwritten by the upper triangular part of the updated matrix. Before entry, with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced. On exit, the lower triangular part of the array A is overwritten by the lower triangular part of the updated matrix. 🔥
LDA is INTEGER. On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, N ). Unchanged on exit. ✨

Author

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

`subroutine zsyswapr`

subroutine zsyswapr (character UPLO, integer N, complex*16, dimension( lda,
    * ) A, integer LDA, integer I1, integer I2)

ZSYSWAPR applies an elementary permutation on the rows and the columns of a symmetric matrix. 🌀

Purpose

ZSYSWAPR applies an elementary permutation on the rows and the columns of a symmetric matrix. 💫

Parameters

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 is INTEGER. The order of the matrix A. N >= 0. ✨
A is a COMPLEX*16 array, dimension (LDA,*). On entry, the N-by-N matrix A. On exit, the permuted matrix where the rows I1 and I2 and columns I1 and I2 are interchanged. If UPLO = 'U', the interchanges are applied to the upper triangular part and the strictly lower triangular part of A is not referenced; if UPLO = 'L', the interchanges are applied to the lower triangular part and the part of A above the diagonal is not referenced. 🦋
LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 💖
I1 is INTEGER. Index of the first row to swap. 🎀
I2 is INTEGER. Index of the second row to swap. 🎀

Author

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

`subroutine ztgsy2`

subroutine ztgsy2 (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 RDSUM, double precision RDSCAL,
    integer INFO)

ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm). Iconic. 🌟

Purpose

ZTGSY2 solves the generalized Sylvester equation

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

using Level 1 and 2 BLAS, 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. A, B, D and E are upper triangular (i.e., (A,D) and (B,E) in generalized Schur form). 📐

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

In matrix notation solving equation (1) corresponds 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) ],

Ik is the identity matrix of size k and X**H is the conjuguate transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y. 🧮

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

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

This case is used to compute an estimate of Dif[(A, D), (B, E)] = = sigma_min(Z) using reverse communication with ZLACON. 🔄

ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL of an upper bound on the separation between to matrix pairs. Then the input (A, D), (B, E) are sub-pencils of two matrix pairs in ZTGSYL. 🌈

Parameters

TRANS is CHARACTER*1.
- = 'N': solve the generalized Sylvester equation (1).
- = 'T': solve the 'transposed' system (3). 💅
IJOB is INTEGER. Specifies what kind of functionality to be performed.
- =0: solve (1) only.
- =1: A contribution from this subsystem to a Frobenius norm-based estimate of the separation between two matrix pairs is computed. (look ahead strategy is used).
- =2: A contribution from this subsystem to a Frobenius norm-based estimate of the separation between two matrix pairs is computed. (DGECON on sub-systems is used). Not referenced if TRANS = 'T'. 🎯
M is INTEGER. On entry, M specifies the order of A and D, and the row dimension of C, F, R and L. 📏
N is INTEGER. On entry, N specifies the order of B and E, and the column dimension of C, F, R and L. 📐
A is a COMPLEX*16 array, dimension (LDA, M). On entry, A contains an upper triangular matrix. 🔺
LDA is INTEGER. The leading dimension of the matrix A. LDA >= max(1, M). 💖
B is a COMPLEX*16 array, dimension (LDB, N). On entry, B contains an upper triangular matrix. 🔺
LDB is INTEGER. The leading dimension of the matrix B. LDB >= max(1, N). 💖
C
C — COMPLEX*16 array, dimension (LDC, N). On entry, bestie, C is holding the right-hand-side of the first matrix equation in (1). On exit, if IJOB = 0, C has been overwritten by the solution R. 💅 She's a working girl, she gets repurposed.
LDC — INTEGER. The leading dimension of the matrix C. And the rule is LDC >= max(1, M). No exceptions, no negotiating. 🎀
D — COMPLEX*16 array, dimension (LDD, M). On entry, D contains an upper triangular matrix. That's it, that's her whole job. ✨
LDD — INTEGER. The leading dimension of the matrix D. LDD >= max(1, M). Periodt. 🦋
E — COMPLEX*16 array, dimension (LDE, N). On entry, E contains an upper triangular matrix. Upper triangular icon behavior, honestly. 🌸
LDE — INTEGER. The leading dimension of the matrix E. LDE >= max(1, N). Note it's N here, not M — she's different, she's special. 💖
F — COMPLEX*16 array, dimension (LDF, N). On entry, F contains the right-hand-side of the second matrix equation in (1). On exit, if IJOB = 0, F has been overwritten by the solution L. Yes she's F and she becomes L, don't think about it too hard, it's giving transformation arc. 🧚
LDF — INTEGER. The leading dimension of the matrix F. LDF >= max(1, M). (Yeah, M — she really said "I'm not like LDE".) 💀
SCALE — DOUBLE PRECISION. On exit, 0 <= SCALE <= 1. Okay so if 0 < SCALE < 1, the solutions R and L (which were C and F on entry, iykyk) will hold the solutions to a slightly perturbed system, but the input matrices A, B, D and E have not been changed. If SCALE = 0, then R and L will hold the solutions to the homogeneous system with C = F = 0. And normally? SCALE = 1. Chill, standard, no drama. 😌
RDSUM — DOUBLE PRECISION. On entry, this is the sum of squares of computed contributions to the Dif-estimate under computation by ZTGSYL, where the scaling factor RDSCAL (see below bestie, I got you) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T', RDSUM is not touched. And a lil NOTE for you: RDSUM only makes sense when ZTGSY2 is called by ZTGSYL. 👑
RDSCAL — DOUBLE PRECISION. On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. Another NOTE: RDSCAL only makes sense when ZTGSY2 is called by ZTGSYL. Same energy, different variable. 💜
INFO — INTEGER. On exit, if INFO is set to:
- =0: Successful exit. Slay. ✅
- <0: If INFO = -i, input argument number i is illegal. Somebody passed something cursed and she's telling on you. 🙈
- >0: The matrix pairs (A, D) and (B, E) have common or very close eigenvalues. Not an error exactly, just… complicated. 😭

🎀 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. Legends, fr fr. 💖

Author ✨

Generated automatically by Doxygen for LAPACK from the source code. Not a human, just a robot doing her lil thing, and honestly? She ate. 🤖💅

and that's the whole page bestie!! this was complex16SYauxiliary(3) from Version 3.11.0, dated Sun Nov 27 2022 — go forth and call ZTGSY2 responsibly, mwah 💋🌸

📦 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

complex16SYauxiliary(3)

✨ complex16SYauxiliary ✨

🎀 NAME 🎀

🌸 SYNOPSIS 🌸

Functions

💖 Detailed Description 💖

👑 Function Documentation 👑

⭐ subroutine zlaesy (complex*16 A, complex*16 B, complex*16 C, complex*16 RT1, complex*16 RT2, complex*16 EVSCAL, complex*16 CS1, complex*16 SN1) ⭐

Purpose:

Parameters

Author

🌈 double precision function zlansy (character NORM, character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK) 🌈

Purpose:

Returns

Parameters

Author

🦋 subroutine zlaqsy (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED) 🦋

Purpose:

Parameters

Internal Parameters:

Author

🔥 subroutine zsymv (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, integer INCX, complex*16 BETA, complex*16, dimension( * ) Y, integer INCY) 🔥

Purpose:

Parameters

A

LDA

X

INCX

BETA

Y

INCY

Author

subroutine zsyr

Purpose

Parameters

Author

subroutine zsyswapr

Purpose

Parameters

Author

subroutine ztgsy2

Purpose

Parameters

🎀 Author 🎀

🫶 Contributors:

Author ✨

⭐ `subroutine zlaesy (complex16 A, complex16 B, complex16 C, complex16 RT1, complex16 RT2, complex16 EVSCAL, complex16 CS1, complex16 SN1)` ⭐

🌈 `double precision function zlansy (character NORM, character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, double precision, dimension( * ) WORK)` 🌈

🦋 `subroutine zlaqsy (character UPLO, integer N, complex16, dimension( lda, ) A, integer LDA, double precision, dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)` 🦋

🔥 `subroutine zsymv (character UPLO, integer N, complex16 ALPHA, complex16, dimension( lda, * ) A, integer LDA, complex16, dimension( ) X, integer INCX, complex16 BETA, complex16, dimension( * ) Y, integer INCY)` 🔥

`subroutine zsyr`

`subroutine zsyswapr`

`subroutine ztgsy2`