complex16GEauxiliary(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

okay so bestie, welcome to complex16GEauxiliary(3) from LAPACK — it's giving complex16 GE auxiliary squad energy, and the manual name is literally complex16GEauxiliary. The short-name vibe? complex16. We're about to go through the whole group chat of routines, and yes, every flag, type, constraint, author and contributor is staying exactly where it belongs. no cap. 💖✨

✨ NAME ✨

complex16GEauxiliary — complex16

🌸 SYNOPSIS 🌸

🫶 Functions 🫶

bestie, here's the lineup — signatures stay exactly as they are:

subroutine zgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE)
    ZGESC2 solves a system of linear equations using the LU
    factorization with complete pivoting computed by sgetc2.
subroutine zgetc2 (N, A, LDA, IPIV, JPIV, INFO)
    ZGETC2 computes the LU factorization with complete pivoting of the
    general n-by-n matrix.
double precision function zlange (NORM, M, N, A, LDA, WORK)
    ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-
    norm, or the largest absolute value of any element of a general
    rectangular matrix.
subroutine zlaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
    ZLAQGE scales a general rectangular matrix, using row and column
    scaling factors computed by sgeequ.
subroutine ztgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1,
    INFO)
    ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular
    matrix pair by an unitary equivalence transformation.

🌈 Detailed Description 🌈

This is the group of complex16 auxiliary functions for GE matrices. That's the whole description, and honestly it's iconic. 💅

💅 Function Documentation 💅

🦋 `zgesc2` 🦋

subroutine zgesc2 (integer N, complex*16, dimension( lda, * ) A, integer
    LDA, complex*16, dimension( * ) RHS, integer, dimension( * ) IPIV,
    integer, dimension( * ) JPIV, double precision SCALE)

ZGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2. yes, lowercase sgetc2 in that line, bestie — keeping it exact.

✨ Purpose ✨

ZGESC2 solves a system of linear equations

A * X = scale* RHS

with a general N-by-N matrix A using the LU factorization with complete pivoting computed by ZGETC2.

🎀 Parameters 🎀

N — N is INTEGER. The number of columns of the matrix A. (the page says columns, so we're saying columns, periodt.)
A — A is COMPLEX*16 array, dimension (LDA, N). On entry, the LU part of the factorization of the n-by-n matrix A computed by ZGETC2: A = P * L * U * Q.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1, N).
RHS — RHS is COMPLEX*16 array, dimension N. On entry, the right hand side vector b. On exit, the solution vector X.
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).
JPIV — JPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).
SCALE — SCALE is DOUBLE PRECISION. On exit, SCALE contains the scale factor. SCALE is chosen 0 <= SCALE <= 1 to prevent overflow in the solution.

👑 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.

🦋 `zgetc2` 🦋

subroutine zgetc2 (integer N, complex*16, dimension( lda, * ) A, integer
    LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer
    INFO)

ZGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix. it's giving factorization main character energy. 🔥

✨ Purpose ✨

ZGETC2 computes an LU factorization, using complete pivoting, of the n-by-n matrix A. The factorization has the form A = P * L * U * Q, where P and Q are permutation matrices, L is lower triangular with unit diagonal elements and U is upper triangular.

This is a level 1 BLAS version of the algorithm.

🎀 Parameters 🎀

N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is COMPLEX*16 array, dimension (LDA, N). On entry, the n-by-n matrix to be factored. On exit, the factors L and U from the factorization A = P*L*U*Q; the unit diagonal elements of L are not stored. If U(k, k) appears to be less than SMIN, U(k, k) is given the value of SMIN, giving a nonsingular perturbed system.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1, N).
IPIV — IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).
JPIV — JPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).
INFO — INFO is INTEGER.
- = 0: successful exit
- > 0: if INFO = k, U(k, k) is likely to produce overflow if one tries to solve for x in Ax = b. So U is perturbed to avoid the overflow.

👑 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.

🦋 `zlange` 🦋

double precision function zlange (character NORM, integer M, integer N,
    complex*16, dimension( lda, * ) A, integer LDA, double precision,
    dimension( * ) WORK)

ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix. normie? never heard of her. 💅

✨ Purpose ✨

ZLANGE  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 matrix A.

🌟 Returns 🌟

ZLANGE is set to:

            ZLANGE = ( 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 ZLANGE as described above.
M — M is INTEGER. The number of rows of the matrix A. M >= 0. When M = 0, ZLANGE is set to zero.
N — N is INTEGER. The number of columns of the matrix A. N >= 0. When N = 0, ZLANGE is set to zero.
A — A is COMPLEX*16 array, dimension (LDA,N). The m by n matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(M,1).
WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced.

👑 Author 👑

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

🦋 `zlaqge` 🦋

subroutine zlaqge (integer M, integer N, complex*16, dimension( lda, * ) A,
    integer LDA, double precision, dimension( * ) R, double precision,
    dimension( * ) C, double precision ROWCND, double precision COLCND,
    double precision AMAX, character EQUED)

ZLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ. balancing act, but make it complex16. ✨

✨ Purpose ✨

ZLAQGE equilibrates a general M by N matrix A using the row and column scaling factors in the vectors R and C.

🎀 Parameters 🎀

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the M by N matrix A. On exit, the equilibrated matrix. See EQUED for the form of the equilibrated matrix.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(M,1).
R — R is DOUBLE PRECISION array, dimension (M). The row scale factors for A.
C — C is DOUBLE PRECISION array, dimension (N). The column scale factors for A.
ROWCND — ROWCND is DOUBLE PRECISION. Ratio of the smallest R(i) to the largest R(i).
COLCND — COLCND is DOUBLE PRECISION. Ratio of the smallest C(i) to the largest C(i).
AMAX — AMAX is DOUBLE PRECISION. Absolute value of largest matrix entry.
EQUED — EQUED is CHARACTER*1. Specifies the form of equilibration that was done.
- = 'N': No equilibration
- = 'R': Row equilibration, i.e., A has been premultiplied by diag(R).
- = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C).
- = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).

🔍 Internal Parameters: 🔍

THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done.

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

👑 Author 👑

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

🦋 `ztgex2` 🦋

subroutine ztgex2 (logical WANTQ, logical WANTZ, integer N, complex*16,
    dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B,
    integer LDB, complex*16, dimension( ldq, * ) Q, integer LDQ, complex*16,
    dimension( ldz, * ) Z, integer LDZ, integer J1, integer INFO)

ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation. (yes, "an unitary" — keeping the original grammar exactly, bestie. 💀)

✨ Purpose ✨

bestie, this is where part 1 pulls up — the ztgex2 Purpose section continues in part 2, and we are not signing off, we're just pausing on a cliffhanger. 💅

ZTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22) in an upper triangular matrix pair (A, B) by an unitary equivalence transformation. 💅

And fr fr, (A, B) must be in generalized Schur canonical form — that is, A and B are both upper triangular. No cap.

Optionally, the matrices Q and Z of generalized Schur vectors are updated. The vibe is:

Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H
Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H

Parameters 🎀

WANTQ — WANTQ is LOGICAL. .TRUE. means update the left transformation matrix Q; .FALSE. means do not update Q. Periodt. 💖
WANTZ — WANTZ is LOGICAL. .TRUE. means update the right transformation matrix Z; .FALSE. means do not update Z. Slay. ✨
N — N is INTEGER. The order of the matrices A and B. N >= 0. 👑
A — A is a COMPLEX*16 array, dimensions (LDA,N). On entry, it's the matrix A in the pair (A, B). On exit, it's the updated matrix A. Iconic. 💅
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). You got this. 🌸
B — B is a COMPLEX*16 array, dimensions (LDB,N). On entry, it's the matrix B in the pair (A, B). On exit, it's the updated matrix B. 🔥
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N). Fr fr. 🦋
Q — Q is a COMPLEX*16 array, dimension (LDQ,N). If WANTQ = .TRUE., on entry it's the unitary matrix Q; on exit, it's the updated matrix Q. Not referenced if WANTQ = .FALSE.. Iykyk. 💜
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1; if WANTQ = .TRUE., LDQ >= N. 🎀
Z — Z is a COMPLEX*16 array, dimension (LDZ,N). If WANTZ = .TRUE., on entry it's the unitary matrix Z; on exit, it's the updated matrix Z. Not referenced if WANTZ = .FALSE.. 🌈
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1; if WANTZ = .TRUE., LDZ >= N. ⭐
J1 — J1 is INTEGER. The index to the first block (A11, B11). 🍓
INFO — INFO is INTEGER. =0: successful exit, yay! =1: the transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill-conditioned. 😭

Author 💖

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

Further Details 🧚

In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details. 👀

Contributors 🫶

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

References 📚

B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF-94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

Author 🤖

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

This page is from LAPACK version 3.11.0, dated Sun Nov 27 2022, and it's all about complex16GEauxiliary(3). Bye bestie! 💅✨

📦 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

complex16GEauxiliary(3)

✨ NAME ✨

🌸 SYNOPSIS 🌸

🫶 Functions 🫶

🌈 Detailed Description 🌈

💅 Function Documentation 💅

🦋 zgesc2 🦋

✨ Purpose ✨

🎀 Parameters 🎀

👑 Author 👑

🫶 Contributors: 🫶

🦋 zgetc2 🦋

✨ Purpose ✨

🎀 Parameters 🎀

👑 Author 👑

🫶 Contributors: 🫶

🦋 zlange 🦋

✨ Purpose ✨

🌟 Returns 🌟

🎀 Parameters 🎀

👑 Author 👑

🦋 zlaqge 🦋

✨ Purpose ✨

🎀 Parameters 🎀

🔍 Internal Parameters: 🔍

👑 Author 👑

🦋 ztgex2 🦋

✨ Purpose ✨

Parameters 🎀

Author 💖

Further Details 🧚

Contributors 🫶

References 📚

Author 🤖

🦋 `zgesc2` 🦋

🦋 `zgetc2` 🦋

🦋 `zlange` 🦋

🦋 `zlaqge` 🦋

🦋 `ztgex2` 🦋