# tgsyl - Man Page

tgsyl: Sylvester equation

## Synopsis

### Functions

subroutine ctgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
CTGSYL
subroutine dtgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
DTGSYL
subroutine stgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
STGSYL
subroutine ztgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
ZTGSYL

## Function Documentation

### subroutine ctgsyl (character trans, integer ijob, integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldc, * ) c, integer ldc, complex, dimension( ldd, * ) d, integer ldd, complex, dimension( lde, * ) e, integer lde, complex, dimension( ldf, * ) f, integer ldf, real scale, real dif, complex, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

CTGSYL

Purpose:

``` CTGSYL solves the generalized Sylvester equation:

A * R - L * B = scale * C            (1)
D * R - L * E = scale * F

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 complex entries. A, B, D and E are upper
triangular (i.e., (A,D) and (B,E) in generalized Schur form).

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1
is an output scaling factor chosen to avoid overflow.

In matrix notation (1) is equivalent to solve Zx = scale*b, where Z
is defined as

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

Here Ix is the identity matrix of size x and X**H is the conjugate
transpose of X. Kron(X, Y) is the Kronecker product between the
matrices X and Y.

If TRANS = 'C', y in the conjugate transposed system Z**H *y = scale*b
is solved for, which is equivalent to solve for R and L in

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

This case (TRANS = 'C') is used to compute an one-norm-based estimate
of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
and (B,E), using CLACON.

If IJOB >= 1, CTGSYL computes a Frobenius norm-based estimate of
Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
reciprocal of the smallest singular value of Z.

This is a level-3 BLAS algorithm.```
Parameters

TRANS

```          TRANS is CHARACTER*1
= 'N': solve the generalized sylvester equation (1).
= 'C': solve the 'conjugate transposed' system (3).```

IJOB

```          IJOB is INTEGER
Specifies what kind of functionality to be performed.
=0: solve (1) only.
=1: The functionality of 0 and 3.
=2: The functionality of 0 and 4.
=3: Only an estimate of Dif[(A,D), (B,E)] is computed.
=4: Only an estimate of Dif[(A,D), (B,E)] is computed.
(CGECON on sub-systems is used).
Not referenced if TRANS = 'C'.```

M

```          M is INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L.```

N

```          N is INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L.```

A

```          A is COMPLEX array, dimension (LDA, M)
The upper triangular matrix A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1, M).```

B

```          B is COMPLEX array, dimension (LDB, N)
The upper triangular matrix B.```

LDB

```          LDB is INTEGER
The leading dimension of the array B. LDB >= max(1, N).```

C

```          C is COMPLEX array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, C has been overwritten by
the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
the solution achieved during the computation of the
Dif-estimate.```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1, M).```

D

```          D is COMPLEX array, dimension (LDD, M)
The upper triangular matrix D.```

LDD

```          LDD is INTEGER
The leading dimension of the array D. LDD >= max(1, M).```

E

```          E is COMPLEX array, dimension (LDE, N)
The upper triangular matrix E.```

LDE

```          LDE is INTEGER
The leading dimension of the array E. LDE >= max(1, N).```

F

```          F is COMPLEX array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, F has been overwritten by
the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
the solution achieved during the computation of the
Dif-estimate.```

LDF

```          LDF is INTEGER
The leading dimension of the array F. LDF >= max(1, M).```

DIF

```          DIF is REAL
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Dif-function, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2).
IF IJOB = 0 or TRANS = 'C', DIF is not referenced.```

SCALE

```          SCALE is REAL
On exit SCALE is the scaling factor in (1) or (3).
If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
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.```

WORK

```          WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK > = 1.
If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

IWORK

`          IWORK is INTEGER array, dimension (M+N+2)`

INFO

```          INFO is INTEGER
=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: (A, D) and (B, E) have common or very close
eigenvalues.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994.
[3] B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

Definition at line 292 of file ctgsyl.f.

### subroutine dtgsyl (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 dif, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DTGSYL

Purpose:

``` DTGSYL solves the generalized Sylvester equation:

A * R - L * B = scale * C                 (1)
D * R - L * E = scale * F

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 (real) 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 (1) is equivalent to solve  Zx = scale b, where
Z is defined as

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

Here 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.

If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b,
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 (TRANS = 'T') is used to compute an one-norm-based estimate
of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
and (B,E), using DLACON.

If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
reciprocal of the smallest singular value of Z. See [1-2] for more
information.

This is a level 3 BLAS algorithm.```
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: The functionality of 0 and 3.
= 2: The functionality of 0 and 4.
= 3: Only an estimate of Dif[(A,D), (B,E)] is computed.
(look ahead strategy IJOB  = 1 is used).
= 4: Only an estimate of Dif[(A,D), (B,E)] is computed.
( DGECON on sub-systems is used ).
Not referenced if TRANS = 'T'.```

M

```          M is INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L.```

N

```          N is INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L.```

A

```          A is DOUBLE PRECISION array, dimension (LDA, M)
The upper quasi triangular matrix A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1, M).```

B

```          B is DOUBLE PRECISION array, dimension (LDB, N)
The upper quasi triangular matrix B.```

LDB

```          LDB is INTEGER
The leading dimension of the array 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) or (3).
On exit, if IJOB = 0, 1 or 2, C has been overwritten by
the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
the solution achieved during the computation of the
Dif-estimate.```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1, M).```

D

```          D is DOUBLE PRECISION array, dimension (LDD, M)
The upper triangular matrix D.```

LDD

```          LDD is INTEGER
The leading dimension of the array D. LDD >= max(1, M).```

E

```          E is DOUBLE PRECISION array, dimension (LDE, N)
The upper triangular matrix E.```

LDE

```          LDE is INTEGER
The leading dimension of the array 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) or (3).
On exit, if IJOB = 0, 1 or 2, F has been overwritten by
the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
the solution achieved during the computation of the
Dif-estimate.```

LDF

```          LDF is INTEGER
The leading dimension of the array F. LDF >= max(1, M).```

DIF

```          DIF is DOUBLE PRECISION
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Dif-function, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
IF IJOB = 0 or TRANS = 'T', DIF is not touched.```

SCALE

```          SCALE is DOUBLE PRECISION
On exit SCALE is the scaling factor in (1) or (3).
If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
to a slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, C and F hold the
solutions R and L, respectively, to the homogeneous system
with C = F = 0. Normally, SCALE = 1.```

WORK

```          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK > = 1.
If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

IWORK

`          IWORK is INTEGER array, dimension (M+N+6)`

INFO

```          INFO is INTEGER
=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: (A, D) and (B, E) have common or close eigenvalues.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

```  [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
No 1, 1996.

[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
Appl., 15(4):1045-1060, 1994

[3] B. Kagstrom and L. Westin, Generalized Schur Methods with
Condition Estimators for Solving the Generalized Sylvester
Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
July 1989, pp 745-751.```

Definition at line 296 of file dtgsyl.f.

### subroutine stgsyl (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 dif, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

STGSYL

Purpose:

``` STGSYL solves the generalized Sylvester equation:

A * R - L * B = scale * C                 (1)
D * R - L * E = scale * F

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 (real) 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 (1) is equivalent to solve  Zx = scale b, where
Z is defined as

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

Here 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.

If TRANS = 'T', STGSYL solves the transposed system Z**T*y = scale*b,
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 (TRANS = 'T') is used to compute an one-norm-based estimate
of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
and (B,E), using SLACON.

If IJOB >= 1, STGSYL computes a Frobenius norm-based estimate
of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
reciprocal of the smallest singular value of Z. See [1-2] for more
information.

This is a level 3 BLAS algorithm.```
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: The functionality of 0 and 3.
= 2: The functionality of 0 and 4.
= 3: Only an estimate of Dif[(A,D), (B,E)] is computed.
(look ahead strategy IJOB  = 1 is used).
= 4: Only an estimate of Dif[(A,D), (B,E)] is computed.
( SGECON on sub-systems is used ).
Not referenced if TRANS = 'T'.```

M

```          M is INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L.```

N

```          N is INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L.```

A

```          A is REAL array, dimension (LDA, M)
The upper quasi triangular matrix A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1, M).```

B

```          B is REAL array, dimension (LDB, N)
The upper quasi triangular matrix B.```

LDB

```          LDB is INTEGER
The leading dimension of the array 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) or (3).
On exit, if IJOB = 0, 1 or 2, C has been overwritten by
the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
the solution achieved during the computation of the
Dif-estimate.```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1, M).```

D

```          D is REAL array, dimension (LDD, M)
The upper triangular matrix D.```

LDD

```          LDD is INTEGER
The leading dimension of the array D. LDD >= max(1, M).```

E

```          E is REAL array, dimension (LDE, N)
The upper triangular matrix E.```

LDE

```          LDE is INTEGER
The leading dimension of the array 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) or (3).
On exit, if IJOB = 0, 1 or 2, F has been overwritten by
the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
the solution achieved during the computation of the
Dif-estimate.```

LDF

```          LDF is INTEGER
The leading dimension of the array F. LDF >= max(1, M).```

DIF

```          DIF is REAL
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Dif-function, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
IF IJOB = 0 or TRANS = 'T', DIF is not touched.```

SCALE

```          SCALE is REAL
On exit SCALE is the scaling factor in (1) or (3).
If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
to a slightly perturbed system but the input matrices A, B, D
and E have not been changed. If SCALE = 0, C and F hold the
solutions R and L, respectively, to the homogeneous system
with C = F = 0. Normally, SCALE = 1.```

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK > = 1.
If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

IWORK

`          IWORK is INTEGER array, dimension (M+N+6)`

INFO

```          INFO is INTEGER
=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: (A, D) and (B, E) have common or close eigenvalues.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

```  [1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
No 1, 1996.

[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
Appl., 15(4):1045-1060, 1994

[3] B. Kagstrom and L. Westin, Generalized Schur Methods with
Condition Estimators for Solving the Generalized Sylvester
Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
July 1989, pp 745-751.```

Definition at line 296 of file stgsyl.f.

### subroutine ztgsyl (character trans, integer ijob, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldc, * ) c, integer ldc, complex*16, dimension( ldd, * ) d, integer ldd, complex*16, dimension( lde, * ) e, integer lde, complex*16, dimension( ldf, * ) f, integer ldf, double precision scale, double precision dif, complex*16, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

ZTGSYL

Purpose:

``` ZTGSYL solves the generalized Sylvester equation:

A * R - L * B = scale * C            (1)
D * R - L * E = scale * F

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 complex entries. A, B, D and E are upper
triangular (i.e., (A,D) and (B,E) in generalized Schur form).

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1
is an output scaling factor chosen to avoid overflow.

In matrix notation (1) is equivalent to solve Zx = scale*b, where Z
is defined as

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

Here Ix is the identity matrix of size x and X**H is the conjugate
transpose of X. Kron(X, Y) is the Kronecker product between the
matrices X and Y.

If TRANS = 'C', y in the conjugate transposed system Z**H *y = scale*b
is solved for, which is equivalent to solve for R and L in

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

This case (TRANS = 'C') is used to compute an one-norm-based estimate
of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
and (B,E), using ZLACON.

If IJOB >= 1, ZTGSYL computes a Frobenius norm-based estimate of
Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
reciprocal of the smallest singular value of Z.

This is a level-3 BLAS algorithm.```
Parameters

TRANS

```          TRANS is CHARACTER*1
= 'N': solve the generalized sylvester equation (1).
= 'C': solve the 'conjugate transposed' system (3).```

IJOB

```          IJOB is INTEGER
Specifies what kind of functionality to be performed.
=0: solve (1) only.
=1: The functionality of 0 and 3.
=2: The functionality of 0 and 4.
=3: Only an estimate of Dif[(A,D), (B,E)] is computed.
=4: Only an estimate of Dif[(A,D), (B,E)] is computed.
(ZGECON on sub-systems is used).
Not referenced if TRANS = 'C'.```

M

```          M is INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L.```

N

```          N is INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L.```

A

```          A is COMPLEX*16 array, dimension (LDA, M)
The upper triangular matrix A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A. LDA >= max(1, M).```

B

```          B is COMPLEX*16 array, dimension (LDB, N)
The upper triangular matrix B.```

LDB

```          LDB is INTEGER
The leading dimension of the array B. LDB >= max(1, N).```

C

```          C is COMPLEX*16 array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, C has been overwritten by
the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
the solution achieved during the computation of the
Dif-estimate.```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1, M).```

D

```          D is COMPLEX*16 array, dimension (LDD, M)
The upper triangular matrix D.```

LDD

```          LDD is INTEGER
The leading dimension of the array D. LDD >= max(1, M).```

E

```          E is COMPLEX*16 array, dimension (LDE, N)
The upper triangular matrix E.```

LDE

```          LDE is INTEGER
The leading dimension of the array E. LDE >= max(1, N).```

F

```          F is COMPLEX*16 array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, F has been overwritten by
the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
the solution achieved during the computation of the
Dif-estimate.```

LDF

```          LDF is INTEGER
The leading dimension of the array F. LDF >= max(1, M).```

DIF

```          DIF is DOUBLE PRECISION
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Dif-function, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigma-min(Z), where Z as in (2).
IF IJOB = 0 or TRANS = 'C', DIF is not referenced.```

SCALE

```          SCALE is DOUBLE PRECISION
On exit SCALE is the scaling factor in (1) or (3).
If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
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.```

WORK

```          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK > = 1.
If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

IWORK

`          IWORK is INTEGER array, dimension (M+N+2)`

INFO

```          INFO is INTEGER
=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: (A, D) and (B, E) have common or very close
eigenvalues.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal. Appl., 15(4):1045-1060, 1994.
[3] B. Kagstrom and L. Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

Definition at line 292 of file ztgsyl.f.

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