complexSYauxiliary(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

complexSYauxiliary(3) 💅✨

okay bestie, welcome to the LAPACK corner of the universe where we handle complex SY matrices — auxiliary style. yes it's complexSYauxiliary(3), yes it's part of LAPACK, and yes we're gonna get through all of it together. grab a strawberry 🍓 and let's go.

✨ NAME ✨

complexSYauxiliary — the complex auxiliary crew 🫶

🌸 SYNOPSIS 🌸

Functions

subroutine claesy (A, B, C, RT1, RT2, EVSCAL, CS1, SN1) — CLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix. it's giving eigen-decomposition energy 💅
real function clansy (NORM, UPLO, N, A, LDA, WORK) — CLANSY 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 claqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED) — CLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
subroutine csymv (UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY) — CSYMV computes a matrix-vector product for a complex symmetric matrix.
subroutine csyr (UPLO, N, ALPHA, X, INCX, A, LDA) — CSYR performs the symmetric rank-1 update of a complex symmetric matrix.
subroutine csyswapr (UPLO, N, A, LDA, I1, I2) — CSYSWAPR 👑 (that's literally all it says, bestie, we're not inventing lore)
subroutine ctgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO) — CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

💖 Detailed Description 💖

This is the group of complex auxiliary functions for SY matrices. periodt. that's the whole vibe. 🌈

🎀 Function Documentation 🎀

subroutine claesy (complex A, complex B, complex C, complex RT1, complex RT2, complex EVSCAL, complex CS1, complex SN1)

CLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix. iconic behavior, honestly. ✨

Purpose

CLAESY 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. The ( 1, 1 ) element of input matrix.
B — B is COMPLEX. 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. The ( 2, 2 ) element of input matrix.
RT1 — RT1 is COMPLEX. The eigenvalue of larger modulus.
RT2 — RT2 is COMPLEX. The eigenvalue of smaller modulus.
EVSCAL — EVSCAL is COMPLEX. 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). 😭 lowkey stressful but okay
CS1 — CS1 is COMPLEX
SN1 — SN1 is COMPLEX. 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.

real function clansy (character NORM, character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)

CLANSY 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. norm girlie behavior, we love her 🎀

Purpose

CLANSY 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

CLANSY

CLANSY = ( 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. 💀 noted.

Parameters

NORM — NORM is CHARACTER*1. Specifies the value to be returned in CLANSY 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, CLANSY is set to zero.
A — A is COMPLEX array, dimension (LDA,N). The symmetric matrix A. If UPLO = 'U', the leading n by n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(N,1).
WORK — WORK is REAL 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 claqsy (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

CLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ. it's the equilibrate-and-slay pipeline 🧚

Purpose

CLAQSY 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 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 REAL array, dimension (N). The scale factors for A.
SCOND — SCOND is REAL. Ratio of the smallest S(i) to the largest S(i).
AMAX — AMAX is REAL. 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 csymv (character UPLO, integer N, complex ALPHA, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, integer INCX, complex BETA, complex, dimension( * ) Y, integer INCY)

CSYMV computes a matrix-vector product for a complex symmetric matrix. the way it just multiplies... main character energy fr fr 🔥

Purpose

CSYMV 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.

On entry, ALPHA specifies the scalar alpha. Unchanged on exit.

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

Unchanged on exit.

LDA — 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 — COMPLEX 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 — INTEGER

On entry, INCX specifies the increment for the elements of X. INCX must not be zero.

Unchanged on exit.

BETA — COMPLEX

On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. ✨ (bestie, that's a free pass, use it)

Unchanged on exit.

Y — COMPLEX 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 — 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 csyr (character UPLO, integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex, dimension( lda, * ) A, integer LDA)`

CSYR performs the symmetric rank-1 update of a complex symmetric matrix. okay so slay, one vector, one rank, one glow-up of your matrix 💖

🌸 Purpose

CSYR 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.

🎀 Parameters

UPLO — 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 — INTEGER

On entry, N specifies the order of the matrix A. N must be at least zero.

Unchanged on exit.

ALPHA — COMPLEX

On entry, ALPHA specifies the scalar alpha.

Unchanged on exit.

X — COMPLEX 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 — INTEGER

On entry, INCX specifies the increment for the elements of X. INCX must not be zero.

Unchanged on exit.

A — COMPLEX 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 — 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 csyswapr (character UPLO, integer N, complex, dimension( lda, n ) A, integer LDA, integer I1, integer I2)`

CSYSWAPR — no cap, the name is the whole plot twist, but here's the official tea anyway ☕

🦋 Purpose

CSYSWAPR applies an elementary permutation on the rows and the columns of a symmetric matrix. swapping rows and columns, keeping the vibe symmetric, iconic behavior honestly ✨

💖 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.

N — INTEGER

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

A — COMPLEX array, dimension (LDA,N)

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 — INTEGER

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

I1 — INTEGER

Index of the first row to swap

I2 — INTEGER

Index of the second row to swap

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💜 `subroutine ctgsy2 (character TRANS, integer IJOB, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( ldd, * ) D, integer LDD, complex, dimension( lde, * ) E, integer LDE, complex, dimension( ldf, * ) F, integer LDF, real SCALE, real RDSUM, real RDSCAL, integer INFO)`

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm). the math is mathing and the matrices are triangular, bestie, we love structure 😭💖

🌟 Purpose

CTGSY2 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. 🚨 (overflow is NOT the vibe, hence the scale)

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 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 CLACON.

CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL 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 CTGSYL. 🫶

🌸 Parameters

TRANS — CHARACTER*1
- = 'N': solve the generalized Sylvester equation (1).
- = 'T': solve the 'transposed' system (3).
IJOB — 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. (SGECON on sub-systems is used.)

Not referenced if TRANS = 'T'. 🙈

M — INTEGER

On entry, M specifies the order of A and D, and the row dimension of C, F, R and L.

N — INTEGER

On entry, N specifies the order of B and E, and the column dimension of C, F, R and L.

A — COMPLEX array, dimension (LDA, M)

On entry, A contains an upper triangular matrix.

LDA — INTEGER

The leading dimension of the matrix A. LDA >= max(1, M).

B — COMPLEX array, dimension (LDB, N)

On entry, B contains an upper triangular matrix.

LDB — INTEGER

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

C
C — bestie, this one is a COMPLEX array, dimension (LDC, N). On entry, C contains the right-hand-side of the first matrix equation in (1). On exit, if IJOB = 0, C has been overwritten by the solution R. 💖
LDC — is INTEGER. The leading dimension of the matrix C. LDC >= max(1, M). 📏
D — is a COMPLEX array, dimension (LDD, M). On entry, D contains an upper triangular matrix. 🔺
LDD — is INTEGER. The leading dimension of the matrix D. LDD >= max(1, M). 📐
E — is a COMPLEX array, dimension (LDE, N). On entry, E contains an upper triangular matrix. 🔺
LDE — is INTEGER. The leading dimension of the matrix E. LDE >= max(1, N). 📏
F — is a COMPLEX 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. 💖
LDF — is INTEGER. The leading dimension of the matrix F. LDF >= max(1, M). 📐
SCALE — is REAL. On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions R and L (C and F on entry) 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, R and L will hold the solutions to the homogeneous system with C = F = 0. Normally, SCALE = 1. ⚖️
RDSUM — is REAL. On entry, the sum of squares of computed contributions to the Dif-estimate under computation by CTGSYL, where the scaling factor RDSCAL (see below) 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. NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL. 🧮
RDSCAL — is REAL. 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. NOTE: RDSCAL only makes sense when CTGSY2 is called by CTGSYL. 🧮
INFO — is INTEGER. On exit, if INFO is set to:
- =0: Successful exit 🎉
- <0: If INFO = -i, input argument number i is illegal. 🚨
- >0: The matrix pairs (A, D) and (B, E) have common or very close eigenvalues. 👀

✨ Author ✨

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

🫶 Contributors: 🫶

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

🌸 Author 🌸

Generated automatically by Doxygen for LAPACK from the source code.

That's the whole page, bestie! Version 3.11.0, dated Sun Nov 27 2022, and it's literally the end of complexSYauxiliary(3). Periodt. 💅✨

📦 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

complexSYauxiliary(3)

complexSYauxiliary(3) 💅✨

✨ NAME ✨

🌸 SYNOPSIS 🌸

Functions

💖 Detailed Description 💖

🎀 Function Documentation 🎀

subroutine claesy (complex A, complex B, complex C, complex RT1, complex RT2, complex EVSCAL, complex CS1, complex SN1)

Purpose

Parameters

Author

real function clansy (character NORM, character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)

Purpose

Returns

Parameters

Author

subroutine claqsy (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

Purpose

Parameters

Internal Parameters

Author

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

Purpose

Parameters

👑 Author

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

🌸 Purpose

🎀 Parameters

👑 Author

🌈 subroutine csyswapr (character UPLO, integer N, complex, dimension( lda, n ) A, integer LDA, integer I1, integer I2)

🦋 Purpose

💖 Parameters

👑 Author

🌟 Purpose

🌸 Parameters

✨ Author ✨

🫶 Contributors: 🫶

🌸 Author 🌸

🦋 `subroutine csyr (character UPLO, integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex, dimension( lda, * ) A, integer LDA)`

🌈 `subroutine csyswapr (character UPLO, integer N, complex, dimension( lda, n ) A, integer LDA, integer I1, integer I2)`