doubleSYauxiliary(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

, in debian bookworm

💖 NAME 💖

doubleSYauxiliary — double. This is doubleSYauxiliary(3) from LAPACK, bestie. 💅

✨ SYNOPSIS ✨

Okay so here are the functions, bestie. Read the signatures exactly — no vibes-based renaming. 🦋

double precision function dlansy (NORM, UPLO, N, A, LDA, WORK) — DLANSY 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 dlaqsy (UPLO, N, A, LDA, S, SCOND, AMAX, EQUED) — DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
subroutine dlasy2 (LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR, LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO) — DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
subroutine dsyswapr (UPLO, N, A, LDA, I1, I2) — DSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
subroutine dtgsy2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO) — DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

🌸 Detailed Description 🌸

This is the group of double auxiliary functions for SY matrices. That's the whole vibe. ✨

🦋 Function Documentation 🦋

💫 `dlansy` 💫

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

DLANSY 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

DLANSY 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.

Returns

DLANSY

               DLANSY = ( 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 DLANSY as described above. You already saw the menu, bestie. ✨
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, DLANSY is set to zero.
A — A is DOUBLE PRECISION 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 DOUBLE PRECISION 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.

💫 `dlaqsy` 💫

   subroutine dlaqsy (character UPLO, integer N, double precision, dimension(
       lda, * ) A, integer LDA, double precision, dimension( * ) S, double
       precision SCOND, double precision AMAX, character EQUED)

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

Purpose

DLAQSY 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N). The scale factors for A.
SCOND — SCOND is DOUBLE PRECISION. Ratio of the smallest S(i) to the largest S(i).
AMAX — AMAX is DOUBLE PRECISION. 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.

💫 `dlasy2` 💫

   subroutine dlasy2 (logical LTRANL, logical LTRANR, integer ISGN, integer N1,
       integer N2, double precision, dimension( ldtl, * ) TL, integer LDTL,
       double precision, dimension( ldtr, * ) TR, integer LDTR, double
       precision, dimension( ldb, * ) B, integer LDB, double precision SCALE,
       double precision, dimension( ldx, * ) X, integer LDX, double precision
       XNORM, integer INFO)

DLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.

Purpose

DLASY2 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION. On exit, SCALE contains the scale factor. SCALE is chosen less than or equal to 1 to prevent the solution overflowing.
X — X is DOUBLE PRECISION 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 DOUBLE PRECISION. 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.

💫 `dsyswapr` 💫

   subroutine dsyswapr (character UPLO, integer N, double precision, dimension(
       lda, * ) A, integer LDA, integer I1, integer I2)

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

Purpose

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

Parameters

UPLO
UPLO — UPLO is CHARACTER*1. It 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 is DOUBLE PRECISION array, dimension (LDA,*). On entry, the N-by-N matrix A. On exit, 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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
I1 — I1 is INTEGER. Index of the first row to swap
I2 — I2 is INTEGER. Index of the second row to swap

👑 Author 👑

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

✨ `subroutine dtgsy2` ✨

subroutine dtgsy2 (character TRANS, integer IJOB, integer M, integer N,
    double precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldb, * ) B, integer LDB, double precision, dimension( ldc, *
    ) C, integer LDC, double precision, dimension( ldd, * ) D, integer LDD,
    double precision, dimension( lde, * ) E, integer LDE, double precision,
    dimension( ldf, * ) F, integer LDF, double precision SCALE, double
    precision RDSUM, double precision RDSCAL, integer, dimension( * ) IWORK,
    integer PQ, integer INFO)

DTGSY2 solves the generalized Sylvester equation (unblocked algorithm). ✨

🎯 Purpose 🎯

DTGSY2 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 DLACON.

DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL 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 DTGSYL. See DTGSYL 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. (DGECON 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION. 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 DOUBLE PRECISION. On entry, the sum of squares of computed contributions to the Dif-estimate under computation by DTGSYL, 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 DTGSY2 is called by DTGSYL.
RDSCAL — RDSCAL is DOUBLE PRECISION. 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 DTGSY2 is called by DTGSYL.
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 📜

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That's the whole vibe, bestie! 💖 This page was auto-cooked by Doxygen for LAPACK from the source code — Version 3.11.0, dated Sun Nov 27 2022 — and it's the doubleSYauxiliary(3) manual page. Slay, read, and go solve those matrix equations. 💅✨

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doubleSYauxiliary(3)

💖 NAME 💖

✨ SYNOPSIS ✨

🌸 Detailed Description 🌸

🦋 Function Documentation 🦋

💫 dlansy 💫

Purpose

Returns

Parameters

Author

💫 dlaqsy 💫

Purpose

Parameters

Internal Parameters

Author

💫 dlasy2 💫

Purpose

Parameters

Author

💫 dsyswapr 💫

Purpose

Parameters

👑 Author 👑

✨ subroutine dtgsy2 ✨

🎯 Purpose 🎯

🔧 Parameters 🔧

👑 Author 👑

🌟 Contributors 🌟

📜 Author 📜

💫 `dlansy` 💫

💫 `dlaqsy` 💫

💫 `dlasy2` 💫

💫 `dsyswapr` 💫

✨ `subroutine dtgsy2` ✨