realSYauxiliary(3)

📦
version 3.11.0 · 2 versions
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✨ NAME ✨

Okay bestie, welcome to realSYauxiliary(3) — the manual page for the real auxiliary squad that handles SY matrices. It's giving "real" and it's giving iconic. 💅

🦋 SYNOPSIS 🦋

Functions

  • real function slansy (NORM, UPLO, N, A, LDA, WORK) — SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.
  • subroutine slaqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED) — SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
  • subroutine slasy2 (LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO) — SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
  • subroutine ssyswapr (UPLO, N, A, LDA, I1, I2) — SSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
  • subroutine stgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO) — STGSY2 solves the generalized Sylvester equation (unblocked algorithm).

🌸 DETAILED DESCRIPTION 🌸

This is the group of real auxiliary functions for SY matrices. That's the whole vibe check, bestie. 💖

💅 FUNCTION DOCUMENTATION 💅

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

SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix.

🎀 Purpose:

SLANSY 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 symmetric matrix A. Periodt. ✨

✨ Returns

SLANSY is set to one of these, depending on NORM:

SLANSY = ( 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. No cap. 💀

🫶 Parameters

  • NORM — NORM is CHARACTER*1. Specifies the value to be returned in SLANSY 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, SLANSY is set to zero.
  • A — A is REAL 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 slaqsy (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.

🎀 Purpose:

SLAQSY 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 REAL 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 slasy2 (logical LTRANL, logical LTRANR, integer ISGN, integer N1, integer N2, real, dimension( ldtl, * ) TL, integer LDTL, real, dimension( ldtr, * ) TR, integer LDTR, real, dimension( ldb, * ) B, integer LDB, real SCALE, real, dimension( ldx, * ) X, integer LDX, real XNORM, integer INFO)

SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.

🎀 Purpose:

SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in

op(TL)*X + ISGN*X*op(TR) = SCALE*B,

where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or -1. op(T) = T or T**T, where T**T denotes the transpose of T. ✨

🫶 Parameters

  • LTRANL — LTRANL is LOGICAL. On entry, LTRANL specifies the op(TL):
    • = .FALSE., op(TL) = TL,
    • = .TRUE., op(TL) = TL**T.
  • LTRANR — LTRANR is LOGICAL. On entry, LTRANR specifies the op(TR):
    • = .FALSE., op(TR) = TR,
    • = .TRUE., op(TR) = TR**T.
  • ISGN — ISGN is INTEGER. On entry, ISGN specifies the sign of the equation as described before. ISGN may only be 1 or -1.
  • N1 — N1 is INTEGER. On entry, N1 specifies the order of matrix TL. N1 may only be 0, 1 or 2.
  • N2 — N2 is INTEGER. On entry, N2 specifies the order of matrix TR. N2 may only be 0, 1 or 2.
  • TL — TL is REAL array, dimension (LDTL,2). On entry, TL contains an N1 by N1 matrix.
  • LDTL — LDTL is INTEGER. The leading dimension of the matrix TL. LDTL >= max(1,N1).
  • TR — TR is REAL array, dimension (LDTR,2). On entry, TR contains an N2 by N2 matrix.
  • LDTR — LDTR is INTEGER. The leading dimension of the matrix TR. LDTR >= max(1,N2).
  • B — B is REAL array, dimension (LDB,2). On entry, the N1 by N2 matrix B contains the right-hand side of the equation.
  • LDB — LDB is INTEGER. The leading dimension of the matrix B. LDB >= max(1,N1).
  • SCALE — SCALE is REAL. On exit, SCALE contains the scale factor. SCALE is chosen less than or equal to 1 to prevent the solution overflowing.
  • X — X is REAL array, dimension (LDX,2). On exit, X contains the N1 by N2 solution.
  • LDX — LDX is INTEGER. The leading dimension of the matrix X. LDX >= max(1,N1).
  • XNORM — XNORM is REAL. On exit, XNORM is the infinity-norm of the solution.
  • INFO — INFO is INTEGER. On exit, INFO is set to
    • 0: successful exit.
    • 1: TL and TR have too close eigenvalues, so TL or TR is perturbed to get a nonsingular equation.
NOTE: In the interests of speed, this routine does not check the inputs for errors. 🚨

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

💖 subroutine ssyswapr (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer I1, integer I2)

SSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.

🎀 Purpose:

SSYSWAPR applies an elementary permutation on the rows and the columns of a symmetric matrix. ✨

🫶 Parameters

  • UPLO — UPLO is 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 — N is INTEGER. The order of the matrix A. N >= 0.
  • A —
  • A — okay bestie, this is a REAL array, dimension (LDA,*). On entry, it's the N-by-N matrix A. On exit, it's 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 stgsy2

   subroutine stgsy2 (character TRANS, integer IJOB, integer M, integer N,
       real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B,
       integer LDB, real, dimension( ldc, * ) C, integer LDC, real, dimension(
       ldd, * ) D, integer LDD, real, dimension( lde, * ) E, integer LDE, real,
       dimension( ldf, * ) F, integer LDF, real SCALE, real RDSUM, real RDSCAL,
       integer, dimension( * ) IWORK, integer PQ, integer INFO)

STGSY2 solves the generalized Sylvester equation (unblocked algorithm). ✨

Purpose

Okay so here's the tea ☕ — STGSY2 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, with real entries. (A, D) and (B, E) must be in generalized Schur canonical form, i.e. A, B are upper quasi triangular and D, E are upper triangular. The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output scaling factor chosen to avoid overflow.

In matrix notation solving equation (1) corresponds to solve Z*x = scale*b, where Z is defined as:

Z = [ kron(In, A)  -kron(B**T, Im) ]             (2)
    [ kron(In, D)  -kron(E**T, Im) ],

Ik is the identity matrix of size k and X**T is the transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y. In the process of solving (1), we solve a number of such systems where Dim(In), Dim(In) = 1 or 2.

If TRANS = 'T', solve the transposed system Z**T*y = scale*b for y, which is equivalent to solve for R and L in:

A**T * R  + D**T * L   = scale * C           (3)
R  * B**T + L  * E**T  = scale * -F

This case is used to compute an estimate of Dif[(A, D), (B, E)] = sigma_min(Z) using reverse communication with SLACON.

STGSY2 also (IJOB >= 1) contributes to the computation in STGSYL of an upper bound on the separation between to matrix pairs. Then the input (A, D), (B, E) are sub-pencils of the matrix pair in STGSYL. See STGSYL for details.

Parameters

  • TRANS — TRANS is CHARACTER*1
    • = 'N': solve the generalized Sylvester equation (1).
    • = 'T': solve the 'transposed' system (3).
  • IJOB — IJOB is 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 — M is INTEGER. On entry, M specifies the order of A and D, and the row dimension of C, F, R and L.
  • N — N is INTEGER. On entry, N specifies the order of B and E, and the column dimension of C, F, R and L.
  • A — A is REAL array, dimension (LDA, M). On entry, A contains an upper quasi triangular matrix.
  • LDA — LDA is INTEGER. The leading dimension of the matrix A. LDA >= max(1, M).
  • B — B is REAL array, dimension (LDB, N). On entry, B contains an upper quasi triangular matrix.
  • LDB — LDB is INTEGER. The leading dimension of the matrix B. LDB >= max(1, N).
  • C — C is REAL 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 — LDC is INTEGER. The leading dimension of the matrix C. LDC >= max(1, M).
  • D — D is REAL array, dimension (LDD, M). On entry, D contains an upper triangular matrix.
  • LDD — LDD is INTEGER. The leading dimension of the matrix D. LDD >= max(1, M).
  • E — E is REAL array, dimension (LDE, N). On entry, E contains an upper triangular matrix.
  • LDE — LDE is INTEGER. The leading dimension of the matrix E. LDE >= max(1, N).
  • F — F is REAL 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 — LDF is INTEGER. The leading dimension of the matrix F. LDF >= max(1, M).
  • SCALE — 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 — RDSUM is REAL. On entry, the sum of squares of computed contributions to the Dif-estimate under computation by STGSYL, 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 STGSY2 is called by STGSYL.
  • RDSCAL — 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 STGSY2 is called by STGSYL.
  • IWORK — IWORK is INTEGER array, dimension (M+N+2).
  • PQ — PQ is INTEGER. On exit, the number of subsystems (of size 2-by-2, 4-by-4 and 8-by-8) solved by this routine.
  • INFO — INFO is INTEGER. On exit, if INFO is set to
    • =0: Successful exit
    • <0: If INFO = -i, the i-th argument had an illegal value.
    • >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.

And that's the whole vibe, bestie! 💖 This page is from LAPACK version 3.11.0, dated Sun Nov 27 2022, and it's the realSYauxiliary(3) man page. Stay slay, keep your matrices in Schur form, and don't let SCALE hit zero unless you're ready for the homogeneous tea. 🫶✨

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