SRC/dtgsy2.f(3)

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💖 SRC/dtgsy2.f 💖

Okay so this is the Library Functions Manual page for SRC/dtgsy2.f, and we are about to make the generalized Sylvester equation make sense, bestie. No cap.

🎀 NAME 🎀

SRC/dtgsy2.f — that's the name, that's the file, that's the whole vibe of this section. Slay.

✨ SYNOPSIS ✨

🌟 Functions/Subroutines 🌟

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). Periodt.

💅 Function/Subroutine Documentation 💅

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). Yes bestie, again, because this is the documentation section and we keep the receipts. 💖

💫 Purpose 💫

Okay so 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.

Definition at line 271 of file dtgsy2.f.

🪄 Author 🪄

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Cute sign-off: LAPACK Version 3.12.0, source SRC/dtgsy2.f(3), and the footer date slot is not printed here — mwah 💋🫶

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