complexGEauxiliary(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

complexGEauxiliary 💖 (that's the manual name bestie, `complexGEauxiliary(3)`)

okay so this is the complex auxiliary functions for GE matrices page — the little support squad that shows up when the main character (your GE matrix) needs help. it's LONG so it comes in two parts, and this is part one. buckle up 🎀

✨ SYNOPSIS ✨

🦋 Functions

subroutine cgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE) — CGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
subroutine cgetc2 (N, A, LDA, IPIV, JPIV, INFO) — CGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.
real function clange (NORM, M, N, A, LDA, WORK) — CLANGE 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 claqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED) — CLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
subroutine ctgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, INFO) — CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

🌈 Detailed Description 🌈

This is the group of complex auxiliary functions for GE matrices. no cap, that's literally the whole vibe of this page. ✨

💅 Function Documentation 💅

🌸 subroutine cgesc2

subroutine cgesc2 (integer N, complex, dimension( lda, * ) A, integer LDA,
    complex, dimension( * ) RHS, integer, dimension( * ) IPIV, integer,
    dimension( * ) JPIV, real SCALE)

CGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.

🎀 Purpose:

CGESC2 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 CGETC2.

💜 Parameters

N
- N is INTEGER
- The number of columns of the matrix A.
A
- A is COMPLEX array, dimension (LDA, N)
- On entry, the LU part of the factorization of the n-by-n matrix A computed by CGETC2: A = P * L * U * Q
LDA
- LDA is INTEGER
- The leading dimension of the array A. LDA >= max(1, N).
RHS
- RHS is COMPLEX 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 REAL
- 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.

🦋 subroutine cgetc2

subroutine cgetc2 (integer N, complex, dimension( lda, * ) A, integer LDA,
    integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer
    INFO)

CGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.

🎀 Purpose:

CGETC2 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 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.

🧚 real function clange

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

CLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.

🎀 Purpose:

CLANGE  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

CLANGE

CLANGE = ( 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 CLANGE as described above.
M
- M is INTEGER
- The number of rows of the matrix A. M >= 0. When M = 0, CLANGE is set to zero.
N
- N is INTEGER
- The number of columns of the matrix A. N >= 0. When N = 0, CLANGE is set to zero.
A
- A is COMPLEX 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 REAL 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.

🌸 subroutine claqge

subroutine claqge (integer M, integer N, complex, dimension( lda, * ) A,
    integer LDA, real, dimension( * ) R, real, dimension( * ) C, real
    ROWCND, real COLCND, real AMAX, character EQUED)

CLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.

🎀 Purpose:

CLAQGE 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 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 REAL array, dimension (M)
- The row scale factors for A.
C
- C is REAL array, dimension (N)
- The column scale factors for A.
ROWCND
- ROWCND is REAL
- Ratio of the smallest R(i) to the largest R(i).
COLCND
- COLCND is REAL
- Ratio of the smallest C(i) to the largest C(i).
AMAX
- AMAX is REAL
- 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.

👑 subroutine ctgex2

subroutine ctgex2 (logical WANTQ, logical WANTZ, integer N, complex,
    dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B,
    integer LDB, complex, dimension( ldq, * ) Q, integer LDQ, complex,
    dimension( ldz, * ) Z, integer LDZ, integer J1, integer INFO)

CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

🎀 Purpose:

CTGEX2 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.

(A, B) must be in generalized Schur canonical form, that is, A and
B are both upper triangular.

okay bestie, that's the end of part 1 💖 keep scrolling, there's more coming and we are NOT done yet 🎀

Optionally, the matrices Q and Z of generalized Schur vectors are updated.

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 ✨

okay bestie, here's the tea on every single argument — don't skip a single one, iykyk. 🫶

WANTQ — WANTQ is LOGICAL. .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q. 💅
WANTZ — WANTZ is LOGICAL. .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z. 💅
N — N is INTEGER. The order of the matrices A and B. N >= 0. ✨
A — A is COMPLEX array, dimension (LDA,N). On entry, the matrix A in the pair (A, B). On exit, the updated matrix A. 🦋
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🌿
B — B is COMPLEX array, dimension (LDB,N). On entry, the matrix B in the pair (A, B). On exit, the updated matrix B. 🦋
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N). 🌿
Q — Q is COMPLEX array, dimension (LDQ,N). If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE.. 👑
LDQ — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= 1; If WANTQ = .TRUE., LDQ >= N. 🌿
Z — Z is COMPLEX array, dimension (LDZ,N). If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit, 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. =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

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And that's a wrap, bestie! This page was brought to you by LAPACK version 3.11.0, dated Sun Nov 27 2022, for complexGEauxiliary(3). Stay iconic. 💅✨

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complexGEauxiliary(3)

complexGEauxiliary 💖 (that's the manual name bestie, complexGEauxiliary(3))

✨ SYNOPSIS ✨

🦋 Functions

🌈 Detailed Description 🌈

💅 Function Documentation 💅

🌸 subroutine cgesc2

🎀 Purpose:

💜 Parameters

⭐ Author

🫶 Contributors:

🦋 subroutine cgetc2

🎀 Purpose:

💜 Parameters

⭐ Author

🫶 Contributors:

🧚 real function clange

🎀 Purpose:

💜 Returns

💜 Parameters

⭐ Author

🌸 subroutine claqge

🎀 Purpose:

💜 Parameters

🌟 Internal Parameters:

⭐ Author

👑 subroutine ctgex2

🎀 Purpose:

✨ Parameters ✨

📝 Author

🔍 Further Details:

🌟 Contributors:

📚 References:

🤖 Author

complexGEauxiliary 💖 (that's the manual name bestie, `complexGEauxiliary(3)`)