doubleGEauxiliary(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💖 doubleGEauxiliary — LAPACK manual, section 3 💖

okay bestie, you found the LAPACK page for doubleGEauxiliary(3), and it's giving "double-precision auxiliary functions for GE matrices" energy — a whole little friend group of routines that do the gritty behind-the-scenes work 🦋

📛 NAME

doubleGEauxiliary — double

🎀 SYNOPSIS

🧚 Functions

subroutine dgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE) — DGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
subroutine dgetc2 (N, A, LDA, IPIV, JPIV, INFO) — DGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.
double precision function dlange (NORM, M, N, A, LDA, WORK) — DLANGE 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 dlaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED) — DLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
subroutine dtgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO) — DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

🌈 Detailed Description

This is the group of double auxiliary functions for GE matrices.

🔥 Function Documentation

🍓 subroutine dgesc2

subroutine dgesc2 (integer N, double precision, dimension( lda, * ) A,
    integer LDA, double precision, dimension( * ) RHS, integer, dimension( *
    ) IPIV, integer, dimension( * ) JPIV, double precision SCALE)

DGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.

🎯 Purpose:

DGESC2 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 DGETC2.

🧩 Parameters

N — INTEGER. The order of the matrix A.
A — DOUBLE PRECISION array, dimension (LDA,N). On entry, the LU part of the factorization of the n-by-n matrix A computed by DGETC2: A = P * L * U * Q.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1, N).
RHS — DOUBLE PRECISION array, dimension (N). On entry, the right hand side vector b. On exit, the solution vector X.
IPIV — INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).
JPIV — INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).
SCALE — 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.

🍓 subroutine dgetc2

subroutine dgetc2 (integer N, double precision, dimension( lda, * ) A,
    integer LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV,
    integer INFO)

DGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.

🎯 Purpose:

DGETC2 computes an LU factorization with 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 the Level 2 BLAS algorithm.

🧩 Parameters

N — INTEGER. The order of the matrix A. N >= 0.
A — DOUBLE PRECISION array, dimension (LDA, N). On entry, the n-by-n matrix A 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, i.e., giving a nonsingular perturbed system.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
IPIV — INTEGER array, dimension(N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).
JPIV — INTEGER array, dimension(N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).
INFO — INTEGER.
- = 0: successful exit
- > 0: if INFO = k, U(k, k) is likely to produce overflow if we try 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.

🍓 double precision function dlange

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

DLANGE 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:

DLANGE  returns the value of the one norm,  or the Frobenius norm, or
the  infinity norm,  or the  element of  largest absolute value  of a
real matrix A.

🎁 Returns

DLANGE = ( 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).

⚠️ heads up bestie: max(abs(A(i,j))) is not a consistent matrix norm. she's just built different, don't hold her to norm rules 💅

🧩 Parameters

NORM — CHARACTER*1. Specifies the value to be returned in DLANGE as described above.
M — INTEGER. The number of rows of the matrix A. M >= 0. When M = 0, DLANGE is set to zero.
N — INTEGER. The number of columns of the matrix A. N >= 0. When N = 0, DLANGE is set to zero.
A — DOUBLE PRECISION array, dimension (LDA,N). The m by n matrix A.
LDA — INTEGER. The leading dimension of the array A. LDA >= max(M,1).
WORK — 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.

🍓 subroutine dlaqge

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

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

🎯 Purpose:

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

🧩 Parameters

M — INTEGER. The number of rows of the matrix A. M >= 0.
N — INTEGER. The number of columns of the matrix A. N >= 0.
A — DOUBLE PRECISION 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 — INTEGER. The leading dimension of the array A. LDA >= max(M,1).
R — DOUBLE PRECISION array, dimension (M). The row scale factors for A.
C — DOUBLE PRECISION array, dimension (N). The column scale factors for A.
ROWCND — DOUBLE PRECISION. Ratio of the smallest R(i) to the largest R(i).
COLCND — DOUBLE PRECISION. Ratio of the smallest C(i) to the largest C(i).
AMAX — DOUBLE PRECISION. Absolute value of largest matrix entry.
EQUED — 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.

and that's part 1 bestie 💖 the dtgex2 deep-dive and the rest are coming in part 2 — stay tuned, don't go anywhere 🦋

`dtgex2` 💖

   subroutine dtgex2 (logical WANTQ, logical WANTZ, integer N, double
       precision, dimension( lda, * ) A, integer LDA, double precision,
       dimension( ldb, * ) B, integer LDB, double precision, dimension( ldq, *
       ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integer LDZ,
       integer J1, integer N1, integer N2, double precision, dimension( * )
       WORK, integer LWORK, integer INFO)

okay so bestie, DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation. no cap. ✨

Purpose 🦋

DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair (A, B) by an orthogonal equivalence transformation. it's giving linear algebra realness. 💅

(A, B) must be in generalized real Schur canonical form (as returned by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular. periodt.

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

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

Parameters 🎀

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 DOUBLE PRECISION array, dimensions (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 DOUBLE PRECISION array, dimensions (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 DOUBLE PRECISION array, dimension (LDQ,N). On entry, if WANTQ = .TRUE., the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ,N). On entry, if WANTZ =.TRUE., the orthogonal 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). 1 <= J1 <= N. 🎯
N1 — N1 is INTEGER. The order of the first block (A11, B11). N1 = 0, 1 or 2. 🌟
N2 — N2 is INTEGER. The order of the second block (A22, B22). N2 = 0, 1 or 2. 🌟
WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). 🛠️
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 ). 📏
INFO — INFO is INTEGER.
- =0: Successful exit
- >0: If INFO = 1, the transformed matrix (A, B) would be too far from generalized Schur form; the blocks are not swapped and (A, B) and (Q, Z) are unchanged. The problem of swapping is too ill-conditioned.
- <0: If INFO = -16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1).

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.

okay bestie, that's the whole dtgex2 tea. this page was generated automatically by Doxygen for LAPACK from the source code — version 3.11.0, dated Sun Nov 27 2022, for doubleGEauxiliary(3). bye! 💖✨

📦 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

doubleGEauxiliary(3)

💖 doubleGEauxiliary — LAPACK manual, section 3 💖

📛 NAME

🎀 SYNOPSIS

🧚 Functions

🌈 Detailed Description

🔥 Function Documentation

🍓 subroutine dgesc2

🎯 Purpose:

🧩 Parameters

👑 Author

💜 Contributors:

🍓 subroutine dgetc2

🎯 Purpose:

🧩 Parameters

👑 Author

💜 Contributors:

🍓 double precision function dlange

🎯 Purpose:

🎁 Returns

🧩 Parameters

👑 Author

🍓 subroutine dlaqge

🎯 Purpose:

🧩 Parameters

🫧 Internal Parameters:

👑 Author

dtgex2 💖

Purpose 🦋

Parameters 🎀

Author 👑

Further Details 💡

Contributors 🌈

References 📚

`dtgex2` 💖