tgsy2 - Man Page

subroutine ctgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info)
CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
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).
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).
subroutine ztgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info)
ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).  

Purpose:

 CTGSY2 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. 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 solving equation (1) corresponds 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) ],

 Ik is the identity matrix of size k and X**H is the 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 is used to compute an estimate of Dif[(A, D), (B, E)] =
 = sigma_min(Z) using reverse communication with CLACON.

 CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL
 of an upper bound on the separation between to matrix pairs. Then
 the input (A, D), (B, E) are sub-pencils of two matrix pairs in
 CTGSYL.

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 COMPLEX array, dimension (LDA, M)
          On entry, A contains an upper triangular matrix.

LDA

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

B

          B is COMPLEX array, dimension (LDB, N)
          On entry, B contains an upper triangular matrix.

LDB

          LDB is INTEGER
          The leading dimension of the matrix 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).
          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 COMPLEX 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 COMPLEX 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 COMPLEX 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 CTGSYL, 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 CTGSY2 is called by
          CTGSYL.

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 CTGSY2 is called by
          CTGSYL.

INFO

          INFO is INTEGER
          On exit, if INFO is set to
            =0: Successful exit
            <0: If INFO = -i, input argument number i is illegal.
            >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 256 of file ctgsy2.f.

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.

Definition at line 271 of file dtgsy2.f.

STGSY2 solves the generalized Sylvester equation (unblocked algorithm).  

Purpose:

 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.

Definition at line 271 of file stgsy2.f.

ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).  

Purpose:

 ZTGSY2 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. 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 solving equation (1) corresponds 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) ],

 Ik is the identity matrix of size k 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 is used to compute an estimate of Dif[(A, D), (B, E)] =
 = sigma_min(Z) using reverse communication with ZLACON.

 ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL
 of an upper bound on the separation between to matrix pairs. Then
 the input (A, D), (B, E) are sub-pencils of two matrix pairs in
 ZTGSYL.

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 COMPLEX*16 array, dimension (LDA, M)
          On entry, A contains an upper triangular matrix.

LDA

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

B

          B is COMPLEX*16 array, dimension (LDB, N)
          On entry, B contains an upper triangular matrix.

LDB

          LDB is INTEGER
          The leading dimension of the matrix 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).
          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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 ZTGSYL, 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 ZTGSY2 is called by
          ZTGSYL.

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 ZTGSY2 is called by
          ZTGSYL.

INFO

          INFO is INTEGER
          On exit, if INFO is set to
            =0: Successful exit
            <0: If INFO = -i, input argument number i is illegal.
            >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 256 of file ztgsy2.f.