realGEauxiliary(3)

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3.10.0 · ubuntu jammy

💅 realGEauxiliary(3) 💅

okay bestie, this is realGEauxiliary — LAPACK, section 3. it's a whole group of real auxiliary functions for GE matrices and ngl it's a lot, so this comes in two parts and you're reading part one. no cap. let's go ✨

🎀 NAME 🎀

realGEauxiliary — real

that's it. that's the whole name section. the actual tea is down in Detailed Description, keep scrolling, i promise it gets good 💖

✨ SYNOPSIS ✨

🫶 Functions 🫶

subroutine sgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE) — SGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
subroutine sgetc2 (N, A, LDA, IPIV, JPIV, INFO) — SGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.
real function slange (NORM, M, N, A, LDA, WORK) — SLANGE 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 slaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED) — SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
subroutine stgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO) — STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

🌸 Detailed Description 🌸

This is the group of real auxiliary functions for GE matrices. that's the whole thing. it's giving "the utility drawer of the GE matrix universe" and honestly? iconic. 💜

👑 Function Documentation 👑

💫 subroutine sgesc2 (integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) RHS, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, real SCALE) 💫

SGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.

🍓 Purpose 🍓

SGESC2 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 SGETC2.

🧚 Parameters 🧚

N — N is INTEGER. the order of the matrix A. that's the size of your whole situation, bestie.
A — A is REAL array, dimension (LDA,N). on entry, the LU part of the factorization of the n-by-n matrix A computed by SGETC2: A = P * L * U * Q. already factored, don't redo it.
LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1, N).
RHS — RHS is REAL array, dimension (N). on entry, the right hand side vector b. on exit, the solution vector X. it literally becomes the answer, that's the glow-up. ✨
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 sgetc2 (integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer INFO) 🦋

SGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.

🍓 Purpose 🍓

SGETC2 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.

okay so real talk: this is the Level 2 BLAS algorithm. just so you know what you're working with, bestie.

🧚 Parameters 🧚

N — N is INTEGER. the order of the matrix A. N >= 0.
A — A is REAL 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 — 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 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.

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

SLANGE 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 🍓

SLANGE 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 🧚

SLANGE —

SLANGE = ( 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.

heads up bestie: that last one is not a consistent matrix norm. fr fr. the vibes are off on that one, but it is what it is. 💀

🧚 Parameters 🧚

NORM — NORM is CHARACTER*1. specifies the value to be returned in SLANGE as described above. pick your fighter. 💅
M — M is INTEGER. the number of rows of the matrix A. M >= 0. When M = 0, SLANGE is set to zero.
N — N is INTEGER. the number of columns of the matrix A. N >= 0. When N = 0, SLANGE is set to zero.
A — A is REAL 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. so it only gets invited to the party when you ask for the infinity norm. 🙈

💖 Author 💖

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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

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

🍓 Purpose 🍓

SLAQGE 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 REAL 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 stgex2 (logical WANTQ, logical WANTZ, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( ldz, * ) Z, integer LDZ, integer J1, integer N1, integer N2, real, dimension( * ) WORK, integer LWORK, integer INFO) 🧚

STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

🍓 Purpose 🍓

STGEX2 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.

okay bestie, real talk: (A, B) has to be in generalized real Schur canonical form (the exact vibe SGGES hands back), meaning A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks, and B is upper triangular. periodt. 💅 no cap.

Optionally, the matrices Q and Z of generalized Schur vectors get updated too. the equations are literally giving main character energy:

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. iykyk 💅
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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (MAX(1,LWORK)). 🫶
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= MAX( 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: 📚

[1] 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.
[2] 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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